The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups

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EMS Tracts in Mathematics 2

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EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions

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Karl H. Hofmann Sidney A. Morris

The Lie Theory of Connected Pro-Lie Groups A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups

M

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S E M E S

S E M E S

European Mathematical Society

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Authors: Karl Heinrich Hofmann Fachbereich Mathematik, AG 5 Technische Universität Darmstadt Schloßgartenstraße 7 64289 Darmstadt Germany E-Mail: hofmann @mathematik.tu-darmstadt.de

Sidney A. Morris School of Information Technology and Mathematical Sciences University of Ballarat P.O. Box 663 Ballarat, Victoria 3353 Australia E-mail: S.Morris @ballarat.edu.au

2000 Mathematics Subject Classification: Primary 22-02; secondary 17B65, 22D05, 22E20, 22E65, 44A13, 44M40, 58B25.

Key words: pro-Lie groups, pro-Lie algebras, Lie theory of connected pro-Lie groups, exponential function, structure theory of locally compact groups, completeness, quotient groups, open mapping theorem, Levi–Malcev splitting, local Iwasawa splitting.

ISBN 978-3-03719-032-6 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Preface

Sophus Marius Lie (1842–1899) laid the foundation of the theory named Lie theory in honor of its creator. Several mathematicians, likewise prominent in the history of modern mathematics, contributed to its inception in the decades following 1873, which was the year in which Lie started to occupy himself intensively in the study of what he called continuous groups, notably: Friedrich Engel, Wilhelm Killing, Élie Cartan, Henri Poincaré, and Hermann Weyl. From the beginning, however, the advance of Lie theory bifurcated into two separate major highways, which is the reason why the words Lie theory mean different things to different people. Lie himself aimed at accomplishing for the solution of differential equations (in the widest sense) what Évariste Galois and Lie’s countryman Niels Henrik Abel achieved for the solution of algebraic equations: A profound understanding and, to the best extent possible, a classification in terms of groups. Even though Lie considered himself a “geometer,” he created a territory of analysis that is called “Lie theory” by those working in it, and that is represented by the well-known text by Peter J. Olver entitled “Applications of Lie Groups to Differential Equations” [Springer-Verlag, Berlin, New York, etc., 1986]. We should say in the beginning, that the project of Lie theory which we shall discuss in this book, in philosophy and thrust, does not belong to this line. A second highway was taken by Killing and Cartan. It led to a study of what soon became known as Lie algebras, of the group and structure theory of Lie groups, and to the geometry of homogeneous spaces. The latter notably yielded the classification of symmetric spaces by Élie Cartan. At long last it merged into the encyclopedic attempt by Nicolas Bourbaki of the nineteen hundred sixties and seventies, to summarize what had been achieved, and to the emergence of an immense collection of textbooks at all levels. In 1973 Jean Dieudonné quipped “Les groupes de Lie sont devenus le centre des Mathématiques; on ne peut rien faire de sérieux sans eux.” (Lie groups have moved to the center of mathematics. One cannot seriously undertake anything without them. Gazette des Mathématiciens, Société Mathématique de France, Octobre 1974, p. 77.) By and large, in this line of “Lie theory” the words meant the structure theory of Lie algebras and Lie groups, and in particular how the latter is based on the former. The term ‘Lie group’ originally meant ‘finite-dimensional Lie group’ and most people understand the words in this sense today. However even Sophus Lie spoke of “unendliche Gruppen” by which he meant something like infinite-dimensional Lie groups. But reasonable concepts of dimension were not yet available in the 19th century before topology was on its way. And indeed Lie’s attempts in this direction did not appear to have gotten off the ground. The significance of Lie’s discoveries was emphasized by David Hilbert by raising the question in 1900 whether (in later terminology) a locally euclidean topological group is in fact an analytic group in the sense of Lie. This was the fifth of his famous 23 problems which foreshadowed so much of the mathematical creativity

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of the 20th century. It required half a century of effort on the part of several generations of eminent mathematicians until it was settled in the affirmative. Partial solutions came along as the structure of topological groups was understood better and better: Hermann Weyl and his student F. Peter in 1923 laid the foundations of the representation and structure theory of compact groups, and a positive answer to Hilbert’s Fifth Problem for compact groups was a consequence, drawn by John von Neumann in 1932. Lev Semyonovich Pontryagin and Egbert Rudolf van Kampen developed in 1932, respectively, 1936, the duality theory of locally compact abelian groups laying the foundations for an abstract harmonic analysis flourishing throughout the second half of the 20th century and providing the central method for attacking the structure theory of compact abelian groups via duality. Again a positive response to Hilbert’s question for locally euclidean abelian groups followed in the wash. One of the most significant and seminal papers in topological group theory was published in 1949 by Kenkichi Iwasawa, some three years before Hilbert’s Problem was finally settled by the concerted contribution of Andrew Mattei Gleason, Dean Montgomery, Leon Zippin, and Hidehiko Yamabe. It was Iwasawa who clearly recognized for the first time that the structure theory of locally compact groups reduced to that of compact groups and finite-dimensional Lie groups provided one knew that they happen to be approximated by finite-dimensional Lie groups in the sense of projective limits, in other words, if they were pro-Lie groups in our parlance. And this is what Yamabe established in 1953 for all locally compact groups which have a compact factor group modulo their identity component – almost connected locally compact groups as we shall say. The most influential monograph collecting these results was the book by Montgomery and Zippin of 1955 with the title “Topological Transformation Groups”. The theories of compact groups and of abelian locally compact groups had introduced in the first half of the century classes of groups with an explicit structure theory without the restriction of finite-dimensionality, and in the middle of the century these results opened up an explicit development for numerous results on the structure theory of locally compact groups. What are the coordinates of our book in this historical thread? It was recognized in 1957 by Richard Kenneth Lashof that any locally compact group G has a Lie algebra g. If g is appropriately defined, then the exponential function exp : g → G is supplied along with the definition. Yet the fact that these observations are the nucleus of a complete and rich, although infinite-dimensional Lie theory was never exploited. The present book is devoted to the foundations, and the exploitation of such a Lie theory. At a point in the overall historical development where infinite-dimensional Lie theories gain increasing acceptance and attract much interest, this appears to be timely. The Lie theory we unfold is based on projective limits, both on the group level and on the Lie algebra level. We shall find it very helpful that category theory, as a tool for the “working mathematician” as Saunders Mac Lane formulated it, is so well developed that we see immediately what we need, and we shall exploit it. In our case, we need the theory of limits in a complete category, that is, in a category in which all limits exist, and we need the theory of pairs of adjoint functors, which is closely linked with limits.

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The machinery of projective limits is familiar to mathematicians dealing with profinite groups in their work on Galois theory and arithmetics, quite generally. But the apparatus of projective limits is also familiar to mathematicians dealing with compact groups, their representation theory and abstract harmonic analysis. Indeed all group theoreticians working on the structure theory of locally compact groups encounter projective limits sooner or later. In this book we shall call projective limits of projective systems (or, as some authors say, inverse systems) of finite-dimensional Lie groups pro-Lie groups. That is, pro-Lie groups relate to finite-dimensional Lie groups exactly as profinite groups relate to finite groups. However, in the theory of locally compact groups, one encounters a special kind of projective limit, namely, limit situations where limit maps and bonding maps are proper, that is, are closed continuous homomorphisms between locally compact groups having compact kernels. Some authors call such maps perfect. This type of projective limit has a significant element of compactness already built into its definition, and it is this type of limit that has shaped the intuitions of group theoreticians for fifty years or more. From the vantage point of category theory, however, such a restriction is entirely unnatural, as is indeed the entire focus on locally compact pro-Lie groups: The class of locally compact groups is not even closed under the formation of products – as the example of the groups RN or ZN shows immediately. Mathematicians will be naturally attracted to the problem of eliminating the focus on locally compact groups. As one proceeds in the direction of pro-Lie groups in general, however, one comes to realize that the restriction to locally compact groups is unnatural also for reasons that are entirely interior to the mathematics of topological groups and Lie groups. For several years we have been engaged in the laying of the foundations of a general theory of the category of pro-Lie groups. The results are presented in this book. On the first 60 pages, the reader will find a panoramic overview of what is contained in its 14 chapters, and the user of the book should get a more compact overview by perusing its table of contents. The Lie theory of finite-dimensional Lie groups works because for a connected Lie group G, its Lie algebra g and its exponential function exp : g → G largely determine the structure of G. We hasten to add that, except for the case that G is simply connected, they do not do so completely. As the title of our book indicates, we focus on a Lie theory for connected pro-Lie groups. As a consequence, our structure theory is one that is mainly concerned with connected pro-Lie groups, sometimes going a bit further, but rarely much beyond almost connected groups. In view of Yamabe’s Theorem, the structure theory of connected or almost connected pro-Lie groups applies at once to connected or even almost connected locally compact groups. There are several key elements to the structure theory of pro-Lie groups. Firstly, a thorough understanding of the working of projective limits is needed without the crutch of thinking in terms of proper maps all the time. Chapter 1 deals with many facets of this issue. But only after Chapter 3 will we have understood all aspects of what this means for the very definition of pro-Lie groups itself.

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Secondly, the entire theory depends on our accepting that pro-Lie groups, even though not being Lie groups, nevertheless have a working Lie theory, complete with the appropriate Lie algebras which we shall call pro-Lie algebras and working exponential functions that mediate between pro-Lie groups and their Lie algebras. Indeed we must become aware at an early stage that there is a good Lie algebra functor from the category of pro-Lie groups to the category of pro-Lie algebras. One of the very positive side effects of facing wider categories than the conventional ones in developing a Lie theory is that this enlargement of scope forces us to realize in great clarity that the Lie algebra functor is opposed by a Lie group functor that encapsulates lucidly the contents of Lie’s Third Fundamental Theorem. This applies to the classical situation as well, but it is not recognized there because the theory of universal covering Lie groups, while providing topologically satisfying results in general, tends to obscure the precise functorial set-up. Since for pro-Lie groups a classical covering theory is impossible as one knows from the theory of compact connected abelian groups, it is mandatory that one understands the functorial background of a more general universal covering theory. We shall discuss this in Chapters 2, 4, 6 and 8. Thirdly, the success of the structure theory of pro-Lie groups depends in a large measure on our success in dealing with the structure theory of pro-Lie algebras. This pervades the whole book, but most of this is done in our rather long Chapter 7. The point is that the topological vector spaces underlying pro-Lie algebras are what we call weakly complete topological vector spaces, because they are exactly the duals of real vector spaces given the weak ∗-topology, that is, the topology of pointwise convergence of linear functionals. Since the vector space duality is crucial for this class of topological vector spaces and hence for the structure theory of pro-Lie algebras we present the essential features of the linear algebra of weakly complete topological vector spaces in an appendix, namely, Appendix 2. The relevance of weakly complete topological vector spaces in the structure theory of pro-Lie groups themselves is evidenced in that chapter in which we discuss the structure of commutative pro-Lie groups, and that is Chapter 5. With all of these foundations done, the Lie and structure theory of pro-Lie groups can proceed, as it does in Chapters 9 through 13. This preface is not the place to go into the details, but we shall present to our readers in the beginning of the book, in our panoramic overview, the results which we obtain. One of the lead motives of our structure theory is to reduce the structure of connected pro-Lie groups in the optimal extent possible to the structure theory of compact connected groups, weakly complete topological vector spaces, and finite-dimensional Lie groups. We will prove some major structure theorems which expose that we, in essence, achieve this goal. Acknowledgements. Our mathematical collaboration in the area of compact and locally compact groups originated in the late seventies. We record with deep gratitude the steady support of our families that we enjoyed throughout this time, particularly on the part of our wives Isolde Hofmann and Elizabeth Morris. Our work on the structure of pro-Lie groups began in the fall of the year 2000; we are grateful to our academic home institutions, notably to the University of Ballarat in Victoria, Australia, the Darmstadt

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University of Technology in Germany, the University of South Australia in Adelaide, Australia, and Tulane University in New Orleans, USA. We thank Dr. Manfred Karbe who accompanied the project from its beginning with great interest and finally published it at the European Mathematical Society Publishing House. Our special thanks go to Dr. Irene Zimmermann who acted as our copy editor and typsetter. Only a mathematician such as Dr. Zimmermann could have identified errors and inconsistencies as she did. On the top of this she polished rough spots in the text and asked questions which resulted in improving the presentation. In short we have been blessed to have such a professional handling of our TEX script. Darmstadt and Ballarat, March 2007

Karl. H. Hofmann Sidney A. Morris

Contents

Preface Panoramic Overview Part 1. The Base Theory of Pro-Lie Groups . . . . . . . . . . Part 2. The Algebra of Pro-Lie Algebras . . . . . . . . . . . . Part 3. The Fine Lie Theory of Pro-Lie Groups . . . . . . . . Part 4. Global Structure Theory of Connected Pro-Lie Groups Part 5. The Role of Compactness on the Pro-Lie Algebra Level Part 6. The Role of Compact Subgroups of Pro-Lie Groups . . Part 7. Local Splitting According to Iwasawa . . . . . . . . . 1

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Limits of Topological Groups Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The External Approach to Projective Limits . . . . . . . . . . Projective Limits and Local Compactness . . . . . . . . . . . The Fundamental Theorem on Projective Limits . . . . . . . . The Internal Approach to Projective Limits . . . . . . . . . . . Projective Limits and Completeness . . . . . . . . . . . . . . The Closed Subgroup Theorem . . . . . . . . . . . . . . . . . The Role of Local Compactness . . . . . . . . . . . . . . . . The Role of Closed Full Subcategories in Complete Categories Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Lie Groups and the Lie Theory of Topological Groups The General Definition of a Lie Group . . . . . . . . . . . . . . . . . The Exponential Function of Topological Groups . . . . . . . . . . . The Lie Algebra of a Topological Group . . . . . . . . . . . . . . . . The Category of Topological Groups with Lie Algebras . . . . . . . . The Lie Algebra Functor Has a Left Adjoint Functor . . . . . . . . . . Sophus Lie’s Third Fundamental Theorem . . . . . . . . . . . . . . . The Adjoint Representation of a Topological Group with a Lie Algebra Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Pro-Lie Groups Projective Limits of Lie Groups . . . . . . . . . . . . . . . . . The Lie Algebras of Projective Limits of Lie Groups . . . . . . Pro-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . Weakly Complete Topological Vector Spaces and Lie Algebras Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . .

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Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 An Overview of the Definitions of a Pro-Lie Group . . . . . . . . . . . . . 160 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4

Quotients of Pro-Lie Groups Quotient Objects in Categories . . . . . . . . . . . . . Quotient Groups of Pro-Lie Groups . . . . . . . . . . The Exponential Function of Compact Abelian Groups phisms . . . . . . . . . . . . . . . . . . . . . . The One Parameter Subgroup Lifting Theorem . . . . Sufficient Conditions for Quotients to be Complete . . Quotients and Quotient Maps between Pro-Lie Groups Postscript . . . . . . . . . . . . . . . . . . . . . . . .

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5 Abelian Pro-Lie Groups Examples of Abelian Pro-Lie Groups . . . . . . . . . . . . . . Weil’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Group Splitting Theorems . . . . . . . . . . . . . . . . Compactly Generated Abelian Pro-Lie Groups . . . . . . . . . Weakly Complete Topological Vector Spaces Revisited . . . . The Duality Theory of Abelian Pro-Lie Groups . . . . . . . . The Toral Homomorphic Images of an Abelian Pro-Lie Group Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 169 170 173 182 194 208 210

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Lie’s Third Fundamental Theorem Lie’s Third Fundamental Theorem for Pro-Lie Groups . . . . . . . . . . . . Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 264 266

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Profinite-Dimensional Modules and Lie Algebras Modules over a Lie Algebra . . . . . . . . . . . . . Duality of Modules . . . . . . . . . . . . . . . . . Semisimple and Reductive Modules . . . . . . . . Reductive Pro-Lie Algebras . . . . . . . . . . . . . Transfinitely Solvable Lie Algebras . . . . . . . . . The Radical and Levi–Mal’cev: Existence . . . . . Transfinitely Nilpotent Lie Algebras . . . . . . . . The Nilpotent Radicals . . . . . . . . . . . . . . . Special Endomorphisms of Pro-Lie Algebras . . . . Levi–Mal’cev: Uniqueness . . . . . . . . . . . . . Direct and Semidirect Sums Revisited . . . . . . . Cartan Subalgebras of Pro-Lie Algebras . . . . . . Theorem of Ado . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . .

269 269 272 277 281 284 291 296 300 305 309 313 315 330 332

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8 The Structure of Simply Connected Pro-Lie Groups The Adjoint Action . . . . . . . . . . . . . . . . . . . . . Simply Connected Pronilpotent Pro-Lie Groups . . . . . . The Topological Splitting Technique . . . . . . . . . . . . Simple Connectivity . . . . . . . . . . . . . . . . . . . . . Universal Morphism versus Universal Covering Morphism Postscript . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups The Exponential Function on the Inner Derivation Algebra . . Analytic Subgroups . . . . . . . . . . . . . . . . . . . . . . . Automorphisms and Invariant Analytic Subgroups . . . . . . . Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . Subalgebras and Subgroups . . . . . . . . . . . . . . . . . . . The Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Commutator Subgroup . . . . . . . . . . . . . . . . . . . Finite-Dimensional Connected Pro-Lie Groups . . . . . . . . Compact Central Subgroups . . . . . . . . . . . . . . . . . . Divisibility of Groups and Connected Pro-Lie Groups . . . . . The Open Mapping Theorem . . . . . . . . . . . . . . . . . . Completing Proto-Lie Groups . . . . . . . . . . . . . . . . . Unitary Representations . . . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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356 356 360 369 370 372 374 376 376 385 402 404 409 413 415 416

10 The Global Structure of Connected Pro-Lie Groups Solvability of Pro-Lie Groups . . . . . . . . . . . . . . The Radical . . . . . . . . . . . . . . . . . . . . . . . Semisimple and Reductive Groups . . . . . . . . . . . Nilpotency of Pro-Lie Groups . . . . . . . . . . . . . The Nilradical and the Coreductive Radical . . . . . . The Structure of Reductive Pro-Lie Groups . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . .

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11 Splitting Theorems for Pro-Lie Groups Splitting Reductive Groups Semidirectly . . . . . . . Vector Group Splitting in Noncommutative Groups . The Structure of Pronilpotent and Prosolvable Groups Conjugacy Theorems . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . .

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12 Compact Subgroups of Pro-Lie Groups Procompact Modules and Lie Algebras . . . . . . . . . . . . . . . . . . . . Procompact Lie Algebras and Compactly Embedded Lie Subalgebras of ProLie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximal Compactly Embedded Subalgebras of Pro-Lie Algebras . . . . . . Conjugacy of Maximal Compactly Embedded Subalgebras . . . . . . . . . Compact Connected Groups . . . . . . . . . . . . . . . . . . . . . . . . . Compact Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potentially Compact Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . The Conjugacy of Maximal Compact Connected Subgroups . . . . . . . . . The Analytic Subgroups Having a Full Lie Algebra . . . . . . . . . . . . . Maximal Compact Subgroups of Connected Pro-Lie Groups . . . . . . . . An Alternative Open Mapping Theorem . . . . . . . . . . . . . . . . . . . On the Center of a Connected Pro-Lie Group . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500 504 507 516 519 521 524 532 544 556 558 561

13 Iwasawa’s Local Splitting Theorem Locally Splitting Lie Group Quotients of Pro-Lie Groups The Lie Algebra Theory of the Local Splitting . . . . . . Splitting on the Group Level . . . . . . . . . . . . . . . Some Comments on Connectedness . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Catalog of Examples Classification of the Examples in the Catalog . . . . . . . . . . Abelian Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . . A Simple Construction . . . . . . . . . . . . . . . . . . . . . . Pronilpotent Pro-Lie Groups . . . . . . . . . . . . . . . . . . . Prosolvable Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . Semisimple and Reductive Pro-Lie Groups . . . . . . . . . . . . Mixed Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples Concerning the Definition of Lie and Pro-Lie Groups . Analytic Subgroups of Pro-Lie Groups . . . . . . . . . . . . . . Examples Concerning Simple Connectivity . . . . . . . . . . . Example Concerning g-Module Theory . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 Appendix 1 The Campbell–Hausdorff Formalism

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2 Appendix 2 Weakly Complete Topological Vector Spaces

629

3 Appendix 3 Various Pieces of Information on Semisimple Lie Algebras 651 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

Contents

xv

Bibliography

657

List of Symbols

667

Index

669

Panoramic Overview

Compactness is a core concept in general topology, because it introduces finiteness in otherwise infinite geometric objects. When we combine compactness with group theory and its enormous background we can expect a theory rich in results, varied in direction, and fertile in applications. And we get it as is evidenced through a sizeable body of monographs and texts having come about in the second half of last century. Naturally, we like to cite our book on compact groups [102] that appeared in 1998 and that experienced a second revised and augmented edition in 2006. The standard examples are linear groups such as the orthogonal and unitary groups, or the additive groups of p-adic integers, and this confirms that the concept of a compact group is natural. The class of compact groups is closed in the class of all topological groups under the formation of arbitrary products and the passage to closed subgroups. This makes it a closed category in its own right, and that in itself is a fact from which many desirable properties of this category follow. But there are before our eyes just as natural examples of groups that illustrate that there are many topological groups basic to analysis, geometry and algebra which are not compact; easily perceived examples are the additive groups of Rn or, more generally, finite-dimensional vector spaces over locally compact fields, and linear groups like the full linear groups Gl(n, R) and their closed subgroups. All of these groups, however, are locally compact. The most important locally compact groups are real Lie groups which are connected or have, at most, finitely many connected components. One definition of a Lie group is that it is a real analytic manifold with a group structure such that multiplication and inversion are analytic functions. A topological group which is isomorphic to a closed subgroup of the topological group Gl(n, R) is a Lie group, and we shall call such a Lie group a linear Lie group. Let us emphasize at this point that here, and in the following, when we speak about two topological groups as being isomorphic, we mean them to be isomorphic as topological groups; some writers like to stress this by saying that they are “isomorphic algebraically and topologically”. We give a definition of a general Lie group in Appendix 1 to this book which allows a quicker access to the group theoretical aspects of Lie group theory than one involving analytical manifolds. In our book [102] on compact groups we devote a whole chapter to an introduction to linear Lie groups. It is shown in that book that all compact Lie groups are matrix groups, that is, linear Lie groups. All these groups possess identity neighborhoods which are homeomorphic to Rn for a suitable dimension n: they are locally euclidean. In 1900 David Hilbert raised the question whether every locally euclidean group is a Lie group. It took half a century until this question was answered in the affirmative by the concerted joint efforts of Gleason [63], Montgomery and Zippin [144] published back to back in the Annals of Mathematics. The monograph [145] by Montgomery and Zippin appeared three years later and summarized the entire development including

2

Panoramic Overview

the important complements by Yamabe [206], [207] which followed one year later and to which we shall return presently. Montgomery and Zippin’s book became a classic which has not been replaced to this day, in spite of an excellent secondary source authored by Kaplansky [129] sixteen years later. Hilbert’s Problems numbered 23 in all; they were formulated in order to indicate the directions which research in mathematics was to take in the 20th century. The problem concerning Lie groups is number 5, and its difficulty as well as the sheer quantity of research that it fertilized was very indicative of Hilbert’s vision. So it is only natural that something even more influential came along with the affirmative solution of Hilbert’s Fifth Problem, namely, Yamabe’s Theorem. Yamabe’s Theorem tells us that every connected locally compact group G is approximated by a connected Lie group in the sense that G contains arbitrarily small normal subgroups N such that G/N is a Lie group ([206], [207]). In fact Yamabe’s Theorem applies to more than connected groups: it says that every locally compact group for which the group of connected components G/G0 is compact is approximated by Lie groups in the sense just explained. The concept of being approximated by Lie groups is so important that it certainly deserves a definition of its own. For this purpose let us first recall that a topological group is complete if every Cauchy filter (or every Cauchy net) converges; this aptly generalizes the concept of completeness of a metric space which is complete if every Cauchy sequence converges. Every locally compact group is complete and so no mention of completeness need be made when one deals with locally compact groups. Definition 1. (i) A topological group G is called a pro-Lie group if it is complete and if every identity neighborhood of G contains a normal subgroup N such that G/N is a Lie group. The category of all pro-Lie groups with continuous group morphisms between them is written proLieGr. (3.39) (ii) A topological group G is called almost connected if the factor group G/G0 of G modulo the connected component G0 of the identity is compact. Let us then reformulate Yamabe’s Theorem in this terminology: Every almost connected locally compact group is a pro-Lie group. It is a generally adopted notation that for a category A and objects A and B in A, the set of all morphisms A → B is denoted by A(A, B). For instance, if TopGr denotes the category of all topological groups and continuous group homomorphisms between them, then TopGr(G, H ) denotes the set of all continuous homomorphisms from the topological group G to the topological group H ; if G and H are pro-Lie groups, then we have proLieGr(G, H ) = TopGr(G, H ) by definition. This means that the category proLieGr is a full subcategory of the category TopGr of all topological groups. Algebraists, in particular ring theorists, are rather familiar with a concept similar to that of pro-Lie groups, namely, profinite groups. A group G is profinite if it is a complete topological group such that every identity neighborhood of G contains a normal open subgroup N such that G/N is finite. Profinite groups are compact, and

Panoramic Overview

3

they are pro-Lie groups. Profinite groups generalize finite groups in the exact same way as pro-Lie groups generalize Lie groups. Only three years before the solution of Hilbert’s Fifth Problem was found by Gleason, Montgomery and Zippin, a seminal paper by Iwasawa had appeared in the Annals of Mathematics [120]. In that paper he exposed fundamental properties of locally compact pro-Lie groups. So Yamabe’s result made all of this available for the study of the structure and the representation theory of almost connected locally compact groups. This was the culmination of half a century of research on topological groups following Hilbert’s vision in 1900. But at the same time, and certainly not less significant from the present vantage point, the work by Iwasawa, Gleason, Montgomery, Zippin and Yamabe provided motivation and incentive for another half a century’s worth of research on locally compact groups during the second half of the twentieth century. What went into this entire century of research naturally was the full body of highly developed structure and representation theory of finite-dimensional Lie groups and finite-dimensional Lie algebras. Let us briefly say what we mean by the Lie theory of a topological group on a very general level; after all, the words Lie theory appear in the title of this book. To each topological group G one can easily associate a topological space L(G), namely, the space Hom(R, G) of all continuous group homomorphisms from the additive topological group R of real numbers to the topological group G, endowed with the topology of uniform convergence on compact sets. We also have a continuous function exp : L(G) → G given by exp X = X(1) and a “scalar multiplication” (r, X) → r · X : R × L(G) → L(G) given by (r · X)(s) = X(sr). Whether these concepts are useful depends in large measure on the degree to which additional properties are satisfied. In Chapter 2 we shall elaborate on the following definition. Definition 2 (2.6ff.). A topological group G is said to have a Lie algebra, if L(G) has a continuous addition and bracket multiplication making it into a topological Lie algebra in such a fashion that   r   r  n (X + Y )(r) = lim X Y n→∞ n n and

  r   r   r −1  r −1 n2 X Y . Y X n→∞ n n n n If G has a Lie algebra, then L(G) is called the Lie algebra of G and exp : L(G) → G is called its exponential function. [X, Y ](r 2 ) = lim

Clearly a topological group G has a Lie algebra if and only if the connected component G0 of the identity has a Lie algebra G0 and L(G) = Hom(R, G) = Hom(R, G0 ) = L(G0 ). The image of the exponential function is contained in G0 . If we believe that L(G) and the exponential function encapsulate the Lie theory of G, then it is true that the identity component G0 already captures the Lie theory of G.

4

Panoramic Overview

We show in this book that every pro-Lie group G has a Lie algebra L(G) and the image exp L(G) of the exponential function algebraically generates a subgroup which is dense in the connected component G0 of the identity. We shall have much more to say about the topological Lie algebra L(G) that can arise in this fashion. But for the moment we observe this: In every totally disconnected locally compact group, the open (hence closed) subgroups form a basis of the neighborhood filter of the identity element. If G is a locally compact group, then G/G0 is a locally compact totally disconnected group, and so there is an open subgroup U of G containing G0 such that U/G0 is compact. Then U is almost connected and thus, by Yamabe’s Theorem, is a pro-Lie group. Therefore every locally compact group has an open subgroup which is a pro-Lie group and which captures the Lie theory of G. Apart from individual studies such as [134], [64], [103], [104], [106], the Lie theory of locally compact groups has never been systematically considered or exploited, although a start was made in [102] for the purpose of a structure theory of compact groups. One of the thrusts of this book is to change this situation with determination. In addition to the successful resolution of Hilbert’s Fifth Problem there is yet a second prime reason for the success of the structure and representation theory of locally compact groups: The 1932 proof by A. Haar of the existence and uniqueness of left invariant integration on a locally compact group G. Its full power for abstract harmonic analysis was recognized by A. Weil in his influential monograph [198] of 1941. Haar measure is the key to the representation theory of compact and locally compact groups on Hilbert space, and the wide field of abstract harmonic analysis with ever so many ramifications (including e.g. abstract probability theory on locally compact groups). A theorem due to A. Weil shows that, conversely, a complete topological group with a left- (or right-) invariant σ -finite measure is locally compact (see e.g. [76], [198]). Thus the category of locally compact groups is that which is exactly suited for real analysis resting on the existence of an invariant integral based on σ -additive measures. One cannot expect to extend this aspect of locally compact groups to larger classes without abandoning σ -additivity. (Bourbaki indicates in Chapter 9 of his “Intégration” [23], pp. 50–55, 70ff., how such an extension may be handled; however we shall not consider this aspect in this book.) In quiet moments of introspection one might even admire the small miracle inherent in the fact that measure theory carries as far as locally compact groups go. The proper domain for an invariant measure theory again appears to be the category of compact groups, where one has a unique invariant two sided invariant measure P with respect to which G is measurable with measure P (G) = 1. That is, P is a veritable probability measure that allows averaging over G as a remarkably simple but effective device ([102]). Yet there it is, Haar measure of locally compact groups, infinite but eminently useful making locally compact groups the analysts’ delight. However, from each of a group theoretical, of a Lie theoretical, and of a category theoretical point of view, the class of a locally compact groups has serious defects which go rather deep.

Panoramic Overview

5

Indeed, if we consider a family of Lie groups Gj , j ∈ J for an index set J , then its product j ∈J Gj is a perfectly good Hausdorff topological group with a lucid structure, but it fails to be locally compact whenever infinitely many of the Gj fail to be compact. Furthermore, while every locally compact group G does have a Lie algebra L(G), the additive group of the Lie algebra is never locally compact unless it is finitedimensional. Indeed even the additive topological group of the Lie algebra of a compact def

abelian group need not be locally compact; for example the product G = TJ of circle groups T = R/Z has a Lie algebra L(G) isomorphic to RJ and thus fails to be locally compact as soon as J is infinite, while the group TJ is comfortably within the realm of compact groups. Each Lie group G has a tangent bundle which is again a Lie group, namely, the semidirect product L(G) Ad G with G acting on its Lie algebra by adjoint action induced by inner automorphisms. Does a locally compact group have a tangent bundle? The answer is yes, it does, in fact every pro-Lie group has one (as we shall show in this book), but it is almost never a locally compact group except when the group itself is finite-dimensional. Thus the category of locally compact groups appears to have two major drawbacks: – The topological abelian group underlying the Lie algebra L(G) and the tangent bundle of a locally compact group fail to be locally compact unless L(G) is finitedimensional. In other words, the very Lie theory that makes the structure theory of locally compact groups interesting leads us outside the class. – The category of locally compact groups is not closed under the forming of products, even of copies of R; it is not closed under projective limits of projective systems of finite-dimensional Lie groups, let alone under arbitrary limits. In other words, the category of locally compact groups is badly incomplete. This book presents an argument for a shift in the vantage point of looking at locally compact groups. We plead for a structure theory of topological groups that places the focus squarely and systematically on pro-Lie groups. Recall that we denote the category of all (Hausdorff) topological groups and continuous group homomorphisms by TopGr. It turns out that the full subcategory proLieGr of TopGr consisting of all projective limits of finite-dimensional Lie groups avoids both of these difficulties. This would perhaps not yet be a sufficient reason for advocating this category if it were not for two facts: – Firstly, while not every locally compact group is a projective limit of Lie groups, every locally compact group has an open subgroup which is a projective limit of Lie groups, so that, in particular, every connected locally compact group is a pro-Lie group; also all compact groups and all locally compact abelian groups are pro-Lie groups. – Secondly, the category proLieGr is astonishingly well-behaved. Not only is it a complete category, it is closed under passing to closed subgroups and to those quotients which are complete, and it has a demonstrably good Lie theory.

6

Panoramic Overview

It is therefore indeed surprising that this class of groups has been little investigated in a systematic fashion. A serious attempt at such an investigation is made in this book where it is submitted that not only a general structure theory of locally compact groups can be based on a good understanding of the category proLieGr of pro-Lie groups, but that the category of pro-Lie groups is well worth a thorough study on its own account. In this book we will prove general structure theorems on pro-Lie groups which will include the best known general structure theorems on locally compact groups. Since the main strategy of the book is to provide a structure theory via Lie theory, en route we shall have to develop a full grown structure theory of those topological Lie algebras which occur as Lie algebras of pro-Lie groups. We shall call these pro-Lie algebras, because each of them is a complete topological Lie algebra such that every 0-neighborhood contains a closed ideal modulo which it is finite-dimensional (3.6).

Part 1. The Base Theory of Pro-Lie Groups For a description of some basic results on the theory of projective limits of Lie groups some technical background information appears inevitable even for an overview, long before we delve into the actual study of our topic.

Core Definitions and Facts on Pro-Lie Groups Definition 3. A projective system D of topological groups is a family of topological groups (Cj )j ∈J indexed by a directed set J and a family of morphisms {fj k : Ck → Cj | (j, k) ∈ J × J, j ≤ k}, such that fjj is always the identity morphism and i ≤ j ≤ k in J implies fik= fij  fj k . Then the projective limit of the system limj ∈J Cj is the subgroup of j ∈J Cj consisting of all J -tuples (xj )j ∈J for which the equation xj = fj k (xk ) holds for all j, k ∈ J such that j ≤ k. Every cartesian product of topological groups may be considered as a projective limit. Indeed, if (Gα )α∈A is an arbitrary family of topological groups indexed by an infinite set A, one obtains a projective system by considering J to be the set of finite  subsets of A directed by inclusion, by setting Cj = a∈j Ga for j ∈ J , and by letting fj k : Ck → Cj for j ≤ k in J be the projection onto the partial product. The projective  limit of this system is isomorphic to a∈A Ga . There are a few sample facts one should recall about the basic properties of projective limits (see e.g. [25], [64], [107], or this book 1.27 and 1.33): Let G = limj ∈J Gj be a projective limit of a projective system P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of topological groups with limit morphisms fj : G → Gj , and let Uj denote the filter of identity neighborhoods of Gj , U the filter of identity neighborhoods of G,

Panoramic Overview

7

and N the set {ker fj | j ∈ J }. Then U has a basis of identity neighborhoods {fk−1 (U ) | k ∈ J, U ∈ Uk } and N is a filter basis of closed normal subgroups converging to 1. If all bonding maps fj k : Gk → Gj are quotient morphisms and all limit maps fj are surjective, then the limit maps fj : G → Gj are quotient morphisms. The limit G is complete if all Gj are complete. Definition 4 (3.25). For a topological group G let N (G) denote the set of closed normal subgroups N such that all quotient groups G/N are finite-dimensional real Lie groups. Then G ∈ N (G) and G is said to be a proto-Lie group if every identity neighborhood contains a member of N (G). By our earlier Definition 1, if in addition, G is a complete topological group, then G is a pro-Lie group. While not every topological group can be embedded as a subgroup into a complete topological group, this is the case for proto-Lie groups, indeed every proto-Lie group has a completion which is a pro-Lie group. (See 4.1.)  Every product of a family of finite-dimensional Lie groups j ∈J Gj is a pro-Lie group. In particular, RJ is a pro-Lie group for any set J which is locally compact if and only if the set J is finite. The product ZN , accordingly, is a pro-Lie group. It is well known that the space ZN is homeomorphic to the space of irrational real numbers in the natural topology. We may formulate this by saying that the space of irrational numbers supports the structure of a pro-Lie group. It is a remarkable fact (which we discuss in Chapter 4) that the free abelian group Z(N) in countably many generators carries the structure of a nondiscrete pro-Lie group. The underlying topological space cannot be a Baire space and so certainly cannot be Polish (second countable completely metrizable), nor locally compact; indeed a countable homogeneous Baire space is necessarily discrete. These examples help us to realize from the beginning, that our general intuition of the topology of pro-Lie groups cannot be based on experience gathered from locally compact groups. If {Gj : j ∈ J } is a family of finite-dimensional real Lie groups then the subgroup    (gj )j ∈J ∈ Gj : {j ∈ J : gj = 1} is finite j ∈J

 of the direct product j ∈J Gj is a proto-Lie group which is not a pro-Lie group if J is infinite and the Gj are nonsingleton. We reiterate that a topological group G is called almost connected if the factor group G/G0 modulo the connected component G0 of the identity is compact. Everything that is proved for almost connected topological groups therefore is true for all connected groups and for all compact groups. One of the very weighty reasons why this concept is relevant for the theory of topological groups is the existence of Yamabe’s crucial result:

8

Panoramic Overview

Every almost connected locally compact group is a pro-Lie group. The group PSl(2, Qp ) of projective transformations of the p-adic projective line is locally compact, but has no nontrivial normal subgroups and is therefore a locally compact group which is not a pro-Lie group in our sense, while it is, of course, a p-adic Lie group. Every pro-Lie group G gives rise to a projective system {pN M : G/M → G/N : M ⊇ N in N (G)} whose projective limit it is (up to isomorphism). The converse is a difficult issue, but it is true. Theorem 5 (3.34, 3.35 (The Closed Subgroup Theorem)). Every projective limit of pro-Lie groups is a pro-Lie group. Every closed subgroup of a pro-Lie group is a pro-Lie group. A topological group is a pro-Lie group if and only if it is isomorphic to a closed subgroup of a product of Lie groups. In fact in simple category theoretical parlance the following theorem holds. Theorem 6 (3.3, 3.36). The category proLieGr of pro-Lie groups is closed in TopGr under the formation of all limits and is therefore complete. It is the smallest full subcategory of TopGr that contains all finite-dimensional Lie groups and is closed under the formation of all limits. This shows that the category proLieGr does not have some of the shortcomings of the category of locally compact groups which is obviously incomplete. It remains yet to be seen how good the Lie theory of the category proLieGr is and we shall say good things about it shortly. However, one must, at an early stage, admit that the category of pro-Lie groups has certain problems which are invisible as long as one stays inside the subcategory of locally compact pro-Lie groups. Indeed, every quotient group of a locally compact group is locally compact (which is a consequence of the fairly elementary observation that in any topological group, the product H K of a closed subset H and a compact subset K is closed, and the application of this fact to the case that H is a closed (normal) subgroup and K a compact identity neighborhood of G). It is one of the less elementary facts of Lie group theory that a quotient of a Lie group is a Lie group. Indeed the quotient of a linear Lie group need not be a linear Lie group, but is a Lie group nevertheless. The simplest example is the group of upper triangular matrices ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 x z def ⎝ 0 1 y ⎠ : x, y, z ∈ R G = ⎭ ⎩ 0 0 1 and the discrete central subgroup ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 0 n def Z = ⎝0 1 0 ⎠ : n ∈ Z ; ⎭ ⎩ 0 0 1

Panoramic Overview

9

here G is clearly a linear Lie group but the factor group G/Z, which is even locally isomorphic to G is not a linear Lie group. This was proved by Garret Birkhoff in 1936 [9] by astute but elementary linear algebra. (See also [102], p. 169ff.) It is a much more debilitating fact for the study of pro-Lie groups that quotient groups of pro-Lie groups need not be pro-Lie groups. (Corollary 4.11) Still, every quotient group of a pro-Lie group is a proto-Lie group and has a completion which is a pro-Lie group. (4.1) So the defect here arises from a phenomenon that is well observed and studied, that quotients of complete topological groups may fail to be complete. (See [176].) We shall explain in Chapter 4 that the additive group of the topological vector space R[0,1] has a nondiscrete closed subgroup K algebraically isomorphic to the free abelian group Z(N) in countably many generators such that R[0,1] /K is an abelian proto-Lie group which is dense in a compact connected and locally connected group (Corollary 4.11). We use this example in various places in the book to construct counterexamples. In this sense, this example is very helpful to build up our intuition on certain aspects of pro-Lie group theory that are invisible as long as we stay in the locally compact domain. Curiously, the counterexample itself arises from the theory of compact abelian groups, and it was discovered not so long ago ([106]). The defect of proLieGr of not being closed under the passing to quotients is, as we have said, debilitating, because passing to quotient groups is an extremely helpful device of reduction to simple situations in many proofs; therefore it is a handicap not having this tool available at all times inside proLieGr. Fortunately, we shall see that, even regarding quotients, the category proLieGr has its redeeming features. Theorem 7 (The Quotient Theorem; 4.28). Let G be an almost connected pro-Lie group and N a closed normal subgroup. Then G/N is a pro-Lie group provided at least one of the following conditions are satisfied by N : (i) N is almost connected. (ii) N is the kernel of a morphism from G onto some pro-Lie group. (iii) N is locally compact or Polish. Part (iii) of this theorem arises from general topological group theory, and we refer to sources like the book [176] of Dierolf and Roelke for such pieces of information. Parts (i) and (ii) belong to the proper substance of this book, and neither of the two is a trivial matter (See Theorem 4.28 and Corollary 9.58.) In fact, Part (ii) is a consequence of another core result concerning pro-Lie groups, namely, the Open Mapping Theorem that is well known to functional analysts as applying to a variety of operators between suitable topological vector spaces, and that is equally well known to people working with locally compact or Polish topological groups. If conditions are right, then the surjectivity of a continuous group homomorphism f : G → H from a topological

10

Panoramic Overview

group onto another implies already that f is an open function, or, in equivalent terms that the canonical decomposition G ⏐ ⏐ quot G/ ker f

f

−−−−−→ H  ⏐id ⏐ H −−−−−→ H F

produces an isomorphism of topological groups F : G/ ker f → H . If we let G be the additive group of real numbers Rd with the discrete topology, H the same group R but considered with its natural topology, then the identity function f : G → H is a bijective morphism between locally compact metric groups that is not open. We mentioned earlier that we shall expose a nondiscrete pro-Lie structure on the countable free abelian group H with infinitely many generators. So the identity morphism f from the discrete countable (hence locally compact Polish) group G = Z(N) to H is a continuous morphism between pro-Lie groups which is not open. These examples show that the following theorem is not likely to be either obvious or trivial: Theorem 8 (Open Mapping Theorem for Pro-Lie Groups; 9.60). Let G be an almost connected pro-Lie group and f : G → H a continuous group homomorphism onto a pro-Lie group. Then f is an open mapping. That is, under these circumstances, f is equivalent to a quotient homomorphism. One of the major impediments in the group theory of topological groups is the unavailability of the Second Isomorphism Theorem. The so called First Isomorphism Theorem says that if G is a topological group and M ⊆ N are normal subgroups of G then the morphism gM  → gN : G/M → G/N factors through an isomorphism of topological groups (G/M)/(N/M) → G/N. This is a very robust theorem belonging to universal algebra. The environment of the so-called Second Isomorphism Theorem is as follows: Assume that G is a topological group with a closed normal subgroup N and a closed subgroup H such that G = H N = N H . Then the surjective morphism h  → hN : H → G/N factors through a bijective continuous group homomorphism H /(H ∩ N) → H N/N . This may fail to be open even if H ∩ N = {1}. In [108] there is an example of a topological abelian group G and two (isomorphic) closed subgroups H and N such that G is algebraically the direct sum of H and N and G/H and G/N are (isomorphic) compact groups, while G blatantly fails to be compact. However, if G is a pro-Lie group, then a closed subgroup H is a pro-Lie group by the Closed Subgroup Theorem. If N is an almost connected closed normal subgroup of G and G is almost connected, then G/N is a pro-Lie group by the Quotient Theorem. Therefore, from the Open Mapping Theorem we get the next theorem. Theorem 9 (The Second Isomorphism Theorem for Pro-Lie Groups; 9.62). Let N be an almost connected normal subgroup and H an almost connected subgroup of a topological group G and assume that H , N , and H N are pro-Lie groups. Then N/(H ∩ N) and H N/N are naturally isomorphic.

Panoramic Overview

11

The Coarse Lie Theory of Pro-Lie Groups Let us consider a topological Lie algebra g and on it the filter basis of closed ideals j such that dim g/j < ∞; we shall denote it by  (g). Definition 10 (3.6). A topological Lie algebra g is called a pro-Lie algebra (short for profinite-dimensional Lie algebra) if  (g) converges to 0 and if g is a complete topological vector space. Under these circumstances, g ∼ = limj∈ (g) g/j, and the underlying vector space is a weakly complete topological vector space, that is, it is the algebraic dual of a real vector space with the weak ∗-topology. We give a systematic treatment of the duality of vector spaces and weakly complete topological vector spaces in Appendix 2 of this book. The category of pro-Lie algebras and continuous Lie algebra morphisms is denoted proLieAlg. Proposition 11 (3.3, 3.36). The category proLieAlg of pro-Lie algebras is closed in the category of topological Lie algebras and continuous Lie algebra morphisms under the formation of all limits and is therefore complete. It is the smallest category that contains all finite-dimensional Lie algebras and is closed under the formation of all limits. See also [104]. Our demonstration that Lie theory is applicable to pro-Lie groups begins with our showing results like the following: Theorem 12 (3.12, 2.25). Every pro-Lie group G has a pro-Lie algebra g as Liealgebra, and the assignment L which associates with a pro-Lie group G its pro-Lie algebra is a limit preserving functor. These matters will be shown in Chapters 2 and 3. In fact, a portion of this set-up allows a considerable improvement which we summarize in the next section.

The Category Theoretical Version of Lie’s Third Theorem Theorem 13 (Lie’s Third Theorem for Pro-Lie groups; 6.5, 6.6, 8.15). The Lie algebra functor L : proLieGr → proLieAlg has a left adjoint . It associates with every pro-Lie algebra g a unique simply connected pro-Lie group (g) and a natural isomorphism ηg : g → L((g)) such that for every morphism ϕ : g → L(G) there is a unique morphism ϕ  : (g) → G such that ϕ = L(ϕ  )  ηg . A good portion of this theorem we shall prove in Chapter 6, but we find it necessary to introduce a preliminary concept of simple connectivity. Indeed we shall call a proLie group prosimply connected if every member of N (G) contains a member N of N (G) such that G/N is a simply connected Lie group. This turns out to be, for a while, a very useful concept of simple connectivity for pro-Lie groups in all respects, and it

12

Panoramic Overview

reduces correctly to simple connectivity in the case of finite-dimensional Lie groups. Once we have developed enough structure theory we will be able in Chapter 8 to show that a pro-Lie group is prosimply connected if and only if it is simply connected. (See Theorem 8.15.) Indeed, for each pro-Lie algebra g, the group (g) is a projective limit of a projective system of simply connected Lie groups. The fact that L is a right adjoint confirms its property of preserving all limits. There is more to Theorem 13 than meets the eye, and we should alert the reader to these circumstances because they shed new light on the situation even when everything is restricted to the classical situation of finite-dimensional Lie groups. The adjointness of the two functors L and  may be expressed in terms of universal properties as follows. There is a natural isomorphism ηg : g → L((g)) such that for any morphism f : g → L(H ) of topological Lie algebras there is a unique morphism f  : (g) → H such that f = L(f  )  ηg . In diagrams: proLieAlg

proLieGr

ηg

g ⏐ ⏐ ∀f 

−−−−−→ L((g)) ⏐ ⏐  L(f )

L(H ) −−−−−→ idL(H )

(g) ⏐ ⏐  ∃!f

L(H )

H.

In fact, the natural isomorphism really allows us to identify g with the Lie algebra of (g). Sophus Lie’s Third Fundamental Theorem (in his own terminology) says that for every finite-dimensional Lie algebra there is a Lie group having as Lie algebra the given one. So this theorem persists for pro-Lie groups. The natural morphism η is what category theoreticians call the front adjunction or the unit of the adjunction. But any adjoint situation between two functors also has a back adjunction or counit with an appropriate version of the universal property. In the case of the present adjoint situation between L and , the back adjunction set-up is as follows. There is a natural morphism πG : (L(G)) → G of pro-Lie groups with the following universal property: Given a pro-Lie group G and any morphism f : (h) → G for some pro-Lie algebra h, there is a unique morphism f  : h → L(G) of pro-Lie algebras such that f = πG  (f  ). proLieAlg

L(G)  ⏐ ∃!f  ⏐ h

proLieGr πG

(L(G)) −−−−−→  ⏐  ⏐(f ) (h)

G  ⏐∀f ⏐

−−−−−→ (h) id(h)

Panoramic Overview

13

 and call the morphism πG : G  → G the We shall abbreviate (L(G)) by G, universal morphism of G. If G happens to be a pro-Lie group which has a universal covering group in the topological sense (in particular, if G is a finite dimensional Lie  → G is the universal covering morphism (8.21). In general the group), then πG : G universal morphism is neither surjective nor a local isomorphism. This is best realized at an early stage by considering any connected compact abelian group G together with  R), its exponential function expG : L(G) → G, L(G) = Hom(R, G) ∼ = Hom(G,  where G = Hom(G, T), T = R/Z is the discrete character group of G. These things  may be equated are explained in great detail in [102], Chapters 7 and 8. In this case G with the additive group of L(G) and πG with expG : L(G) → G. The world of compact abelian groups of course is full of examples for which the exponential function fails to be surjective, beginning with the one-dimensional examples that are different from the circle group, that is, the solenoids, the character groups of which are the noncyclic infinite subgroups of Q. Let us consider within the complete category proLieGr the full subcategory proSimpConLieGr of all simply connected pro-Lie groups. Then we have the following corollary. Corollary 14 (6.6(vi)). The restrictions and corestrictions of the functors L and  implement an equivalence of categories proLieAlg

L

−−−−−→ ←−−−−−

proSimpConLieGr.



Therefore, the category of pro-Lie algebras has a faithful copy inside the category of all pro-Lie groups, namely, the full subcategory of all simply connected pro-Lie groups.  → G is a group theoretical substitute for In this light, the universal morphism πG : G the exponential function expG : g → G; indeed for abelian pro-Lie groups the two functions agree for all practical intents and purposes. These matters are discussed in Chapter 6 but thereafter will pervade the whole book. Considering the problems we have encountered with quotients in the category of pro-Lie groups, it is remarkable that the functor L behaves well with regard to quotient morphisms. Indeed we see next that L not only preserves all limits, but some colimits as well.

Conservation Laws for L and  Theorem 15 (4.20). The functor L preserves quotients. Specifically, assume that G is a pro-Lie group and N a closed normal subgroup and denote by q : G → G/N the quotient morphism. Then G/N is a proto-Lie group whose Lie algebra L(G/N ) is a pro-Lie algebra and the induced morphism of pro-Lie algebras L(q) : L(G) → L(G/N) is a quotient morphism. The exact sequence 0 → L(N ) → L(G) → L(G/N ) → 0

14

Panoramic Overview

induces an isomorphism X + L(N ) → L(f )(X) : L(G)/L(N ) → L(G/N ). The core of Theorem 15 is proved by showing that for every quotient morphism f : G → H of topological groups, where G is a pro-Lie group, every one parameter subgroup Y : R → H lifts to one of G, that is, there is a one parameter subgroup σ of G such that Y = f  σ . (See 4.19, 4.20.) This requires the Axiom of Choice. It should be emphasized that, according to Theorem 15, a quotient group of a pro-Lie group always has a complete Lie algebra even if it is itself incomplete. Therefore, a proto-Lie group with an incomplete Lie algebra such as R(N) cannot be a quotient of a pro-Lie group. Corollary 16 (4.21). Let G be a pro-Lie group. Then {L(N ) | N ∈ N (G)} converges to zero and every closed ideal i of L(G) such that L(G)/i is finite-dimensional contains an L(N) for some N ∈ N (G). Furthermore, L(G) is the projective limit limN ∈N (G) L(G)/L(N ) of a projective system of bonding morphisms and limit maps all of which are quotient morphisms, and there is a commutative diagram L(γG ) G ∼ L(G) −−−−−→ L(limN ∈N ⏐ ⏐ (G) N ) = limn∈N (G) ⏐ ⏐ expG  L(limN∈N (G) expG/N ) G −−−−−→ limN ∈N (G) G/N.

L(G) L(N )

γG

Theorem 15 expresses a version of exactness of L. But there is also an exactness theorem for , the left adjoint of L. Theorem 17 (6.7, 6.8, 6.9). If h is a closed ideal of a pro-Lie algebra g, then the exact sequence q i 0 → h −−−→ g −−−→ g/h → 0 induces an exact sequence (j )

(q)

1 → (h) −−−→ (g) −−−→ (g/h) → 1, in which (j ) is an algebraic and topological embedding and (q) is a quotient morphism. There are some other noteworthy consequences of Theorem 15. Proposition 18 (4.22 (iv)). Any quotient morphism f : G → H of pro-Lie groups onto a finite-dimensional Lie group is a locally trivial fibration. Proposition 19 (4.22 (i)). For a pro-Lie group G, the subgroup exp g generated by the image of the exponential function, is dense in G0 , that is, exp g = G0 . In particular, a connected nonsingleton pro-Lie group has nontrivial one parameter subgroups. This may be viewed as an Existence Theorem for one parameter subgroups in proLie groups, indeed of an abundance of them – unless of course, the group in question is totally disconnected. So, as an illuminating consequence we get the following characterisation of a pro-Lie group to be totally disconnected.

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15

Corollary 20 (4.23). For a pro-Lie group G the following statements are equivalent: (a) G is totally disconnected. (b) L(G) = {0}. (c) The filter basis of open normal subgroups of G converges to 1. In this book we shall call topological groups satisfying these equivalent conditions prodiscrete groups. So every prodiscrete group is, in particular, a pro-Lie group. As we already mentioned and will observe again later, there are locally compact totally disconnected groups which are not prodiscrete. The group ZN in the product topology is prodiscrete but not locally compact. It is, as we remarked earlier, homeomorphic to the space of irrational numbers. Semidirect products of two topological groups (and semidirect sums of topological Lie algebras) permeate the whole book, beginning from Chapter 1 where we remind the reader of its definition in Exercise E1.5 through Chapter 11, that is specifically devoted to the splitting of pro-Lie groups, that is to results that assert that, under suitable circumstances, a given pro-Lie group may be represented as a semidirect product. If π : H → Aut(N ) is a representation of a topological group in the group def

of automorphisms of a topological group N such that the function (h, n)  → h · n = π(n)(h) : H × N → N is continuous, then the semidirect product N π H of N by H is the topological product N × H endowed with the multiplication (m, h)(n, k) = (m(h · n), hk). That N π H is a topological group is straightforwardly verified. Very simple examples show that semidirect products of pro-Lie groups need not be pro-Lie groups (see Examples 4.29). We shall demonstrate in this book that every proLie group acts under what will be called the adjoint action or adjoint representation Ad : G → Gl(L(G)) on L(G) (see 2.27ff.). So we can form the semidirect product L(G) Ad G,

(X, g)(Y, h) = (X + Ad(g)Y, gh),

and obtain this result. def

Proposition 21 (4.29 (iii)). For each pro-Lie group G, the semidirect product T (G) = L(G) Ad G is a well-defined pro-Lie group.

We call T (G) the tangent bundle of G. Thus pro-Lie groups have tangent bundles that are pro-Lie groups. In particular, all (almost) connected locally compact groups have tangent bundles within the category of pro-Lie groups. However, for a locally compact group G its tangent bundle T (G) is locally compact only if G is finite-dimensional. We have seen that the category of pro-Lie groups – contains all finite-dimensional real Lie groups, – is closed in the category of topological groups under the formation of all limits and the passing to closed subgroups, – has a substantial Lie algebra functor that possesses a very reasonable left adjoint,

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– is closed under the passing from a group to the additive group of its Lie algebra and under the passing from a group to its tangent bundle. In other words, we have seen that the category of pro-Lie groups has none of the defects which plague the category of locally compact groups while it still contains all almost connected locally compact groups. That is, it still houses comfortably all those locally compact groups that support all the Lie theory there is for locally compact groups. But can we exhibit, one might ask, enough fine structure theory of pro-Lie algebras and pro-Lie groups so that at least the known structure theory of locally compact groups is recovered? Like with all categories of groups, the first test that a group theory has to face is how well it elucidates the structure of its abelian representatives.

Abelian Pro-Lie groups Apart from a territory far removed from the domain of connected or even almost connected commutative pro-Lie groups, the situation is very satisfactory and is, as a first coarse approximation to the general structure theory of almost connected proLie groups, rather representative and a good guide for one’s intuition. A weakly complete vector space is a real topological vector space V for which there is a real vector space E such that V ∼ = E ∗ , where E ∗ is the algebraic dual E ∗ HomR (E, R) ⊆ R endowed with the weak -topology, that is, the topology of pointwise convergence induced from RE given the product topology. (See Appendix 2, notably Theorem A2.8.) If the cardinal dim E is the linear dimension of E, that is, the cardinality of one, hence every basis of E, then E ∼ = R(dim E) and thus V ∼ = Rdim E . Therefore, an equivalent definition of a weakly complete topological vector space is the postulate that there be a set J such that V ∼ = RJ (see Corollary A2.9). If N S(V ) denotes the filter basis of all closed vector subspaces F of a locally convex Hausdorff topological vector space V such that dim V /F < ∞, then V is a weakly complete vector space if and only if the natural morphism λV : V → limF ∈N S(V ) V /F , λV (v) = (v + F )F ∈N S(V ) is an isomorphism of topological vector spaces. If an abelian topological group is isomorphic to the additive group of a weakly complete topological vector space, that is, to RJ for some set J , then we shall call it a weakly complete vector group. The abelian pro-Lie groups we know best are the compact abelian groups and the weakly complete vector groups. So it is very pleasing that we can state the following result. Lemma 22 (Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups; 5.12). Any abelian almost connected pro-Lie group is isomorphic to the direct product of a weakly complete vector group and a compact abelian group.

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17

This result is succinct and very lucid. It illustrates that abelian pro-Lie groups, at least if they are almost connected are built up from weakly complete vector groups and compact abelian groups in a certainly simple fashion. In reality, we have better and more accurate information. For the more accurate information we have to pay a price: the formulations get more complicated. First we have a clean cut intermediate result showing that weakly complete topological vector spaces and tori are injectives in the category of abelian pro-Lie groups. Theorem 23 (5.19). Assume that G is an abelian pro-Lie group with a closed subgroup G1 and assume that there are sets I and J such that G1 ∼ = RI × TJ . Then G1 is a homomorphic retract of G, that is, G1 is a direct summand algebraically and topologically. So G ∼ = G1 × G/G1 . This allows us to argue that every abelian pro-Lie G group has a weakly complete vector subgroup V such that G is isomorphic to the direct product V × (G/V ) where the factor G/V has no nontrivial vector subgroup. We call any such subgroup V a vector group complement. For a topological group G we let comp(G) denote the set of all elements which are contained in a compact subgroup. Theorem 24 (Vector Group Splitting Theorem for Abelian Pro-Lie Groups; 5.20). Let G be an abelian pro-Lie group and V a vector group complement. Then there is a closed subgroup H such that (i) (v, h) → v + h : V × H → G is an isomorphism of topological groups. (ii) H0 is compact and equals comp G0 and comp(H ) = comp(G); in particular, comp(G) ⊆ H . (iii) H /H0 ∼ = G/G0 , and this group is prodiscrete. (iv) G/ comp(G) ∼ = V × S for some prodiscrete abelian group S without nontrivial compact subgroups. (v) G has a characteristic closed subgroup G1 = G0 + comp(G) which is isomorphic to V × comp(H ) such that G/G1 is prodiscrete without nontrivial compact subgroups. (vi) The exponential function expG of G = V ⊕ H decomposes as expG = expV ⊕ expH where expV : L(V ) → V is an isomorphism of weakly complete vector groups and expH = expcomp(G0 ) : L((comp(G0 )) → comp(G0 ) is the exponential function of the unique largest compact connected subgroup; here L(comp(G0 )) = comp(L)(G) is the set of relatively compact one parameter subgroups of G. (vii) The arc component Ga of G is V ⊕Ha = V ⊕comp(G0 )a = im L(G). Moreover, if h is a closed vector subspace of L(G) such that exp h = Ga , then h = L(G). This theorem actually is the basis of a rather explicit structure theory of abelian pro-Lie groups. We recall that all locally compact abelian groups belong to this class. There are still some portions of an abelian pro-Lie group G which we do not fully control:

18

Panoramic Overview

– The factor group G/G1 ∼ = H / comp G is prodiscrete and has no compact subgroups but is otherwise uncharted. – comp(G) = comp(H ) is a pro-Lie group that is a directed union of compact subgroups. We do not know much more in the absence of local compactness. If we impose certain natural additional hypothesis that are traditionally invoked in topological group theory, the situation is at once much better. A topological group is called compactly generated if it is algebraically generated by a compact subset. Theorem 25 (The Compact Generation Theorem for Abelian Pro-Lie Groups; 5.32). (i) For a compactly generated abelian pro-Lie group G the characteristic closed subgroup comp(G) is compact and the characteristic closed subgroup G1 is locally compact. (ii) In particular, every vector group complement V is isomorphic to a euclidean group Rm for some m ∈ N0 = {0, 1, 2, . . . }. (iii) The factor group G/G1 is a compactly generated prodiscrete group without compact subgroups. If G/G1 is Polish, then G is locally compact and G∼ = Rm × comp(G) × Zn . (iv) If G is a pro-Lie group containing a finitely generated abelian dense subgroup, then comp(G) is compact and G ∼ = comp(G)× Zm . In particular, G is locally compact. (v) A finitely generated abelian pro-Lie group is discrete. The full subcategory of locally compact abelian groups in the category of abelian pro-Lie groups has a celebrated structure theory that is primarily due to the highly effective and elegant duality theory going back to L. S. Pontryagin [169] and E. R. van Kampen [125] in the early thirties of the 20th century. For any topological abelian group  = Hom(G, T) denote its dual with the compact open topology. (See e.g. G we let G   given [102, Chapter 7].) There is a natural morphism of abelian groups ηG : G → G by ηG (g)(χ) = χ (g) which may or may not be continuous; information regarding this issue is to be found for instance in [102, pp. 298ff], notably in Theorem 7.7 on p. 300.   is bijective and We shall call a topological abelian group semireflexive if ηG : G → G reflexive if ηG is an isomorphism of topological groups; in the latter case G is also said to have duality (see [102, p. 305]). In the direction of a duality theory of abelian pro-Lie groups we offer the following results. Proposition 26 (5.35). Let G be an abelian pro-Lie group and let V be a vector group complement. Then G is reflexive, respectively, semireflexive iff G/V is reflexive,  is isomorphic to a product E × A respectively, semireflexive. The character group G where E is the additive group of a real vector space with its finest locally convex topology and A is the character group of an abelian pro-Lie group whose identity component is compact. Theorem 27 (5.36). Every almost connected abelian pro-Lie group is reflexive, and its character group is a direct sum of the additive topological group of a real vector space

Panoramic Overview

19

endowed with the finest locally convex topology and a discrete abelian group. Pontryagin duality establishes a contravariant functorial bijection between the categories of almost connected abelian pro-Lie groups and the full subcategory of the category of topological abelian groups containing all direct sums of vector groups with the finest locally convex topology and discrete abelian groups.

Part 2. The Algebra of Pro-Lie Algebras The success of the Lie theory of classical Lie groups as well as in our case the Lie theory of pro-Lie groups depends on the effectiveness of the mechanism that allows us to translate problems of the topological group structure on the group level to algebraic problems on the Lie algebra level and back. Experience demonstrates that problems are more easily attacked in a purely algebraic environment. In the present case we know, however, that the Lie algebra of a pro-Lie group is a topological algebra itself. So we hope to repeat the classical success story only to the extent to which the topological algebra and the representation theory of pro-Lie groups themselves reduce to pure algebra – more or less. We shall see that this is largely the case for pro-Lie algebras due to the fact that the underlying topological vector spaces are weakly complete vector spaces and that these have a perfect duality theory that allows us to translate their topological linear algebra to pure linear algebra upon passing to the vector space duals. (See Appendix 2.)

The Module Theory of Pro-Lie Algebras We saw that for every pro-Lie group G there exists a simply connected pro-Lie group  → G. Thus the structure of  and a natural morphism with dense image πG : G G simply connected pro-Lie groups has no small influence on the structure of pro-Lie groups in general. We further saw that the structure of simply connected pro-Lie groups, in a well-understood sense, is completely determined by the structure of their Lie algebra. The lesson learned from Lie Theory of finite-dimensional Lie groups is that one must first study the structure of Lie algebras carefully and then apply the information gathered in this fashion to the group theory of Lie groups. It is no different with pro-Lie groups even though the connection between pro-Lie algebras and pro-Lie groups is more tenuous than in the finite-dimensional case. We develop the representation theory and structure theory of pro-Lie groups simultaneously. Elementary module theory is usually preceded by a rush of simple definitions which still turn out to be very effective. We record some to the extent they are necessary for the reader to follow this overview. Let L be a Lie algebra and E a vector space. Then E is an L-module if there is a bilinear map (x, v)  → x · v : L × E → E

satisfying

[x, y] · v = x · (y · v) − y · (x · v)

20

Panoramic Overview

for all x, y ∈ L and v ∈ E. A function f : E1 → E2 between L-modules is said to be a morphism of L-modules if it is linear and satisfies (∀x ∈ L, v ∈ E1 )

f (x · v) = x · f (v).

A submodule F of an L-module E is a vector subspace such that L · F ⊆ F . An L-module E is said to be simple if {0} and E = {0} are its only submodules. An L-module E is called semisimple if every submodule is a direct module summand. If L is a topological Lie algebra, then a topological vector space V is said to be a topological L-module if (x, v)  → x · v : L × V → V is continuous in each variable separately. If the topological vector space V is weakly complete, and if the filter basis of closed submodules W such that dim V /W < ∞ converges to 0, then V is said to be a profinite-dimensional L-module. The profinite-dimensional modules have a perfect duality; indeed if E is the topological dual of a profinite-dimensional L- module, then E is an L-module with respect to the module operation defined by x · ω, v = −ω, x · v for x ∈ L, v ∈ V and ω ∈ E. Duality permits us to transfer concepts from algebraic module theory to topological module theory. For instance, let V be a profinite-dimensional topological vector space and an L-module. Then the module is said to be reductive if its dual module is semisimple. Duality then permits us to prove theorems like the following: Theorem 28 (7.18). (a) Let V be a profinite-dimensional L-module for a Lie algebra L. Then the following statements are equivalent: (i) (ii) (iii) (iv)

V is reductive. Every finite-dimensional quotient module of V is reductive. V is the projective limit of finite-dimensional reductive module quotients. V is isomorphic to a product of finite-dimensional simple modules.

(b) Every profinite-dimensional L-module has a unique smallest submodule V ss such that V /V ss is reductive. The theory and duality of L-modules are discussed in great detail, among many other things, in Chapter 7. Now these module theoretical concepts apply to the structure theory of pro-Lie algebras. The key is the following remark. If g is a pro-Lie algebra, then the underlying weakly complete topological vector space |g| is a topological L-module with respect to the module operation defined by x · v = [x, v] for x ∈ g and v ∈ |g|. This module is called the adjoint module gad . A pro-Lie algebra g is called reductive if its adjoint module gad is a reductive g-module. It is called semisimple if it is reductive and its center z(g) is zero.

Panoramic Overview

21

While the duality theory of profinite-dimensional L-modules works perfectly, the theory of pro-Lie algebras has no duality theory in the sense that a pro-Lie algebra g could have a Lie algebra as a dual object. However, its adjoint g-module gad has a dual g-module, also called its coadjoint module gcoad . This module duality, however, attaches to each pro-Lie algebra g an almost purely algebraic object, the coadjoint module gcoad , and that is extremely helpful for the structure theory of pro-Lie algebras as the following results will show. Theorem 29 (The Structure Theorem of Reductive and Semisimple Pro-Lie Algebras). (a) For a pro-Lie algebra g the following conditions are equivalent. (i) g is reductive. (ii) g is the product of a family of finite-dimensional simple or one-dimensional ideals of g. (b) Let g be a reductive pro-Lie algebra. Then the commutator algebra [g, g] is closed and is a product of finite simple real Lie algebras. Further g ∼ = z(g) ⊕ [g, g] I ∼ algebraically and topologically, and z(g) = R for some set I . (c) A pro-Lie algebra is semisimple iff it is a product of finite simple real Lie algebras. (d) Every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. Consequently, for a pro-Lie algebra, the following statements are equivalent. (I) g is semisimple. (II) g is the product of a family of finite-dimensional simple ideals of g. (7.27, 7.29) In the light of the fact that pro-Lie algebras g arise as the Lie algebras of pro-Lie groups G, the very appealing duality theory of profinite-dimensional g-modules is a surprisingly effective tool for making the structure theory of pro-Lie groups algebraic.

Pro-Lie Algebras and Solvability Recalling in the structure theory of finite-dimensional Lie algebras that there is always a unique largest solvable ideal, called the radical, we cannot hope to be able to bypass the question of solvability in any structure theory of pro-Lie algebras that is deserving of this name. The fact that the underlying vector spaces of pro-Lie algebras are infinitedimensional as soon as the theory begins to be new and interesting is an ominous warning that solvability is going to be a delicate matter likely to involve set theory including well-ordering and ordinals. Firstly, on a purely algebraic basis, in any Lie algebra we must define a transfinite commutator series and use this transfinite series to define a general concept of solvability. This proceeds as follows. Let g be a Lie algebra. Set g(0) = g and define sequences of ideals g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g(α) is defined for α < β.

22

Panoramic Overview

 (i) If β is a limit ordinal, set g(β) = α<β g(α) . (ii) If β = α + 1, set g(β) = [g(α) , g(α) ]. For cardinality reasons, there is a smallest ordinal γ such that g(γ +1) = g(γ ) . Set = g(γ ) . Let ω denote the first infinite ordinal. Then g is said to be transfinitely solvable if g(∞) = {0}. If g is transfinitely solvable and γ ≤ ω, then g is called countably solvable. If γ is finite and g(γ ) = {0}, then g is called solvable. Thus we have implications solvable ⇒ countably solvable ⇒ transfinitely solvable. Any simple Lie algebra such as sl(2, R) (that is, the Lie algebra of 2 × 2-matrices with trace 0) yields an example with γ = 0 and g(∞) = g = {0}. But we are dealing with topological Lie algebras. The natural objects here are the members of the closed commutator series, giving another three reasonable concepts of solvability right away. Indeed, let g be a topological Lie subalgebra of a topological Lie algebra h. (For instance, h = g.) Set g((0)) = g and define sequences of ideals g((α)) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g((α)) is defined for α < β.  (i) If β is a limit ordinal, set g((β)) = α<β g((α)) . g(∞)

(ii) If β = α + 1, set g((β)) = [g((α)) , g((α)) ]. For cardinality reasons, there is a smallest ordinal γ such that g((γ +1)) = g((γ )) . Set g((∞)) = g((γ )) . Let ω denote the first infinite ordinal. Then g is said to be transfinitely topologically solvable, if g(∞) = {0}. If g is transfinitely topologically solvable and γ ≤ ω, then g is called countably topologically solvable. If γ is finite and g((γ )) = {0}, then g is called topologically solvable. However, the Lie algebras we have to consider here are not only topological Lie algebras, they are in fact pro-Lie algebras, that is projective limits of finite-dimensional ones. That suggests yet another concept of solvability, namely, a pro-Lie algebra g is called prosolvable if every finite-dimensional quotient algebra of g is solvable. It is known and proved just as in the case of topological groups in general that a topological Lie algebra is solvable if and only if it is topologically solvable. Thus there is a glimmer of hope that some of these seven reasonable concepts of solvability coincide. Theorem 30 (The Equivalence Theorem for Solvability of Pro-Lie Algebras; 7.53). Let g be a pro-Lie algebra. Then the following assertions are equivalent: (i) g is transfinitely solvable. (ii) g is transfinitely topologically solvable. (iii) g is countably solvable.

Panoramic Overview

23

(iv) g is countably topologically solvable. (v) g is prosolvable. (vi) g does not contain a finite-dimensional simple Lie algebra. What a relief! We have to deal with only two concepts: The classical concept of solvability (which does not play a great theoretical role in our context) and one concept of “infinite” solvability which, in the later parts of the book, will be called prosolvability. Actually this theorem is astonishing. Simple examples show that there are prosolvable algebras that are not solvable such as an infinite product of a family of solvable algebras with an unbounded family of solvable lengths. But it is not a priori clear that there cannot exist a prosolvable pro-Lie algebra with transfinite commutator series of arbitrary length in terms of ordinals. We shall show that a pro-Lie algebra g has a unique largest prosolvable ideal which is called its radical or solvable radical and is denoted by r(g).

Pro-Lie Algebras and Nilpotency It is of course no surprise, that we play a similar game with the nilpotency of pro-Lie algebras arriving, somewhere down the line, at the following result. Theorem 31 (The Equivalence Theorem for Nilpotency of Pro-Lie Algebras; 7.57). Let g be a pro-Lie algebra. Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

g is transfinitely nilpotent. g is transfinitely topologically nilpotent. g is countably nilpotent. g is countably topologically nilpotent. g is pronilpotent. For every pair x, y of elements in g the vector space endomorphism ad x satisfies limn (ad x)n y = 0.

A comparison of the preceding two results produces a certain difference in the structure of condition (vi) in the two cases. This in indicative of the fact, that the treatment of the two cases is not entirely parallel. In the case of nilpotency we shall show that a pro-Lie algebra g has a unique largest pronilpotent ideal which is called its nilradical and is denoted by n(g). We remarked earlier that every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. We shall see (7.66 and 7.67) that ncored (g) is pronilpotent and that ncored (g) = [g, g] ∩ r(g) = [g, r(g)].

24

Panoramic Overview def

Clearly, we have a chain of radicals to which we might add the center z(g) = {x ∈ g : (∀y ∈ g) [x, y] = 0}: z(g) ⊆ ncored (g) ⊆ n(g) ⊆ r(g) and they may very well all be different. While some results of the classical Lie algebra theory are rather difficult to recover for the case of pro-Lie algebras such as for instance the theory of Cartan subalgebras, there are others that are obtainable with comparatively little effort by some well chosen basic definitions and the classical results. An example of these is the Theorem of Ado. Theorem 32 (Theorem of Ado for Pro-Lie Algebras; 7.105). Every pro-Lie algebra g has a faithful profinite-dimensional module M. In addition, this module has the property that for every cofinite-dimensional submodule N and the associated finite-dimensional representation πN : g → gl(M/N ) for every element x from the nilradical, the endomorphism πN (x) of the finite-dimensional vector space M/N is nilpotent.

The Levi–Mal’cev Theorem for Pro-Lie Algebras One of the core results of the theory of finite-dimensional Lie algebras over fields of characteristic 0 is the semidirect splitting of the radical, a result that is usually labelled the Levi–Mal’cev Theorem. It is one of the remarkable facts of the theory of pro-Lie algebras that this theorem generalizes intact even though this does not happen by a simple generalization of the finite-dimensional proof. Nor is the proof based on a simple passage to the limit. We recall that in order merely to define the concepts of the various radicals, one had to understand first what solvability and nilpotency meant in the infinite-dimensional case. The Levi–Mal’cev Theorem for a finite-dimensional Lie algebra g has two essential parts: The first is an existence statement saying that there is a subalgebra s such that the vector space g is the direct sum of the radical r(g) and s. The second part tells us in which sense two subalgebras s1 and s2 satisfying these conditions agree: There is an element x of the coreductive radical for which, due to its provenance, the vector space endomorphism ad x, defined by (ad x)(y) = [x, y], is nilpotent and yields ead x s2 = s1 . Since ad x is nilpotent, ead x is a polynomial and is therefore defined over any field of characteristic 0 and without any recourse to convergence and topology. In the case of pro-Lie algebras, there is topology involved in the semidirect sum decomposition and extra effort has to go into the question whether indeed s is closed and whether the vector space direct sum is also a topological one. If x ∈ ncored (g), then ead x = idg + ad x + 2!1 · (ad x)2 + · · · is a well-defined automorphism of g called the special automorphism (implemented by x); in order to show that it is well defined, issues of convergence with respect to the topology of g have to be clarified, and this topology is rarely first countable.

Panoramic Overview

25

Let us now summarize what we shall show in the line of Levi–Mal’cev type results on pro-Lie algebras. Theorem 33 (Levi–Mal’cev Theorem for Pro-Lie Algebras: Existence and Conjugacy; 7.52, 7.77). (i) A pro-Lie algebra g is the semidirect sum r(g) ⊕ s of the radical and a closed semisimple subalgebra s. (ii) The radical r(g) is prosolvable, and any Levi summand is the cartesian product of a family of finite-dimensional simple Lie algebras. (iii) If h is a closed subalgebra of g such that g = r(g) + h then h contains a Levi summand s of g. (iv) Two Levi summands of a pro-Lie algebra are conjugate under a special automorphism. (v) A pro-Lie algebra g has only one Levi summand s, if and only if g is the direct sum, algebraically and topologically, r(g) ⊕ s. (vi) If m is a semisimple closed subalgebra of a pro-Lie algebra g and s is a Levi summand of g, then a conjugate of m under an inner automorphism of g is contained in s, and m is contained in some Levi summand. (vii) A semisimple closed ideal is contained in every Levi summand.

Simply Connected Pro-Lie Groups Revisited Of course we wish to apply the Levi–Mal’cev Theorem for pro-Lie algebras to finding out information about the structure theory of pro-Lie groups. This is really where the well-known structure theory of compact connected groups arises (see [102, Chapter 9]), and in that special case, the group structure rather well reflects the algebra structure. The noncompact situation is much more complicated. However, in the class of simply connected pro-Lie groups, the group structure perfectly reflects the Lie algebra structure. This was anticipated in our statement 14 above. Recalling that, on the level of pro-Lie algebras we have an algebraic and topological semidirect sum g = r(g)⊕s with a semisimple Levi–Mal’cev summand s and recalling that the structure of semisimple pro-Lie algebras is very lucid by Theorem 29 above, we have to clarify first the structure of simply connected prosolvable groups. Let us first remark, that we shall be able to show that on a pronilpotent pro-Lie algebra n the Campbell–Hausdorff multiplication (x, y) → x ∗ y = x + y + 21 · [x, y] + · · · is well defined, since the infinite series that defines it in the ring of formal power series in two noncommuting variables can be shown to be summable for all pairs (x, y) ∈ n × n. With respect to this multiplication, (n, ∗) is a pro-Lie group. Theorem 34 (Theorem on the Topological Structure of Simply Connected Pro-Lie Groups with Prosolvable Lie Algebras; 8.13). Let G be a prosimply connected pro-Lie group whose Lie algebra g = L(G) is prosolvable, that is, which is its own radical.

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Let n denote its Lie radical or its reductive radical, as the case may be. Then the following statements hold: (i) (n) ∼ = (n, ∗) may be considered as a closed normal subgroup N of G such that G/N is an abelian pro-Lie group whose exponential function is a homeomorphism, and L(G/N ) is naturally isomorphic to g/n. Indeed, (ii) expG/N : g/n → G/N is an isomorphism of weakly complete vector groups. (iii) The quotient morphism q : G → G/N admits a continuous cross section σ : G/N → G such that σ (N) = 1 (iv) There is an N-equivariant homeomorphism ϕ : G → N × (G/N ) such that ϕ(n) = (n, N ) for all n ∈ N, and pr G/N  ϕ = q. (v) G is homeomorphic to RJ for some set J . (vi) G is simply connected in any sense for which the additive group of a weakly complete topological vector space is simply connected. This is indeed, for simply connected pro-Lie groups with a prosolvable pro-Lie algebra, a fairly satisfactory state of affairs. The class of examples that illustrates how prosolvable pro-Lie groups arise by extending the nilradical by an abelian group is as follows. Lemma 35 (The Center-Free Embedding Lemma; 9.41). Let K be any pro-Lie group possessing enough finite-dimensional fixed point-free representations to separate the points. This is the case for all compact groups K and all locally compact abelian groups and all almost connected abelian pro-Lie groups. Then there is a center-free pro-Lie group G with a normal subgroup V such that G/V ∼ = K. The construction is surprisingly straightforward: Let {Vj : j ∈ J } be a family of fixed point free finite-dimensional K-modules providing enough representations  πj : K → Gl(Vj ) of K to separate the points. Then we set V = j ∈J Vj and define π : K → Gl(V ) by π(k)(vj )j ∈J = (πj (k)(vj ))j ∈J . Then we define G to be the semidirect product V π K. If K is itself a weakly complete vector group, then G is a simply connected metabelian pro-Lie group (that is, solvable with commutative commutator subgroup) such that n = V × {0} is the nilradical and coreductive radical. Another instructive example arises when we let K be any compact connected abelian  determines an irreducible representation πχ : K → group. Then each character χ ∈ K 2 ∼ C = R . The construction of Lemma 35 provides us with a center-free metabelian pro  R), and the Lie group G = CK π K. The Lie algebra k of K is isomorphic to Hom(K, image exp k of the exponential function is the dense proper analytic subgroup A(k, K),  which is exactly the arc component Ka of K and Ga = CK π |Ka Ka is the unique dense subgroup A(g, G) of G with Lie algebra g. We discuss “analytic” subgroups of pro-Lie groups extensively in Chapter 9 of this book.  Z). Thus The abstract quotient group π0 (K) = K/Ka is isomorphic to Ext(K, whenever this group is nonzero, the analytic subgroup A(k, K) is proper and dense. It then follows that A(g, G) is a proper subgroup as well. All nontrivial Lie group

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homomorphic images G/N of G are of the form R2p × Tq with q > 0 and thus have an infinite Poincaré group. d of the discrete additive group A good special case is the character group K = Q  of K may be of rational numbers. Then by Pontryagin Duality the character group K identified with Qd . Here for each N ∈ N (G) the factor group G/N is a circle group and so P (G/N) ∼ = Z while G/N0 is of the form R2p × K and thus has a nontrivial center. Within any category of locally compact groups the construction of such groups would be impossible as G is rarely locally compact in these examples. The structure of simply connected pro-Lie groups is now rather completely described in the next theorem. Theorem 36 (Structure Theorem for Simply Connected Pro-Lie Groups; 8.14). Let G be a simply connected pro-Lie group with Lie algebra g. Then (i) G is the semidirect product R I S of a closed normal subgroup R whose Lie algebra L(R) is the radical r(g) and a closed subgroup S whose Lie algebra s is a Levi summand of g. (ii) Thereis a family of simply connected simple Lie groups Sj , j ∈ J such that S∼ = j ∈J Sj . (iii) There is a closed normal subgroup N of G contained in R such that the pro-Lie algebra L(N ) = ncored (g) is the coreductive radical of g and that there is an N -equivariant isomorphism ϕ : R → N × (R/N ), where N ∼ = (ncored (g), ∗) and where R/N ∼ = r(g)/ncored (g) is a vector group. (iv) R is homeomorphic to RJ for some set J . (v) G is homeomorphic to a product of copies of R and of a family of simply connected real finite-dimensional simple Lie groups. (vi) If C denotes the identity component of the center of G, then G is C-equivariantly homeomorphic to C × G/C. The actual course of events will be this: we shall arrive at this structure theorem with the hypothesis of G being prosimply connected. This result will then allow us to demonstrate finally that the following are equivalent statements for a pro-Lie group G: (i) (ii) (iii) (iv)

G is prosimply connected. G is simply connected.  → G is an isomorphism. πG : G  → G is bijective. πG : G

The equivalence of (iii) and (iv) is a consequence of the Open Mapping Theorem 8. In the course of the development of the theory, however, the equivalence of (iii) and (iv) is proved directly at this point in Chapter 8.

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Part 3. The Fine Lie Theory of Pro-Lie Groups We have seen that pro-Lie algebras have a remarkably good structure theory and that, in a first step, the structure of pro-Lie algebra translates in an almost one-to-one fashion to simply connected pro-Lie algebras. Let us reiterate again that in the category of locally compact groups this very satisfactory phenomenon cannot be seen because the  = (L(G)) of a locally compact group is rarely simply connected manifestation G locally compact; indeed it is locally compact if and only if the radical r(L(G)) is finitedimensional and all simple factors of the semisimple pro-Lie algebra L(G)/r(L(G)) are compact simple Lie algebras (that is, simple real Lie algebras with a negative definite Killing form) with the possible exception of finitely many simple factors. Yet as soon as one renounces simple connectivity the problems of a global structure theory start and are, as a rule, more serious than in the theory of Lie groups. First we have to deal with the issue of what, in the situation of a pro-Lie group, constitutes an analytic subgroup and what the relation between (closed) subalgebras of the Lie algebra and connected subgroups might be.

The Lie Theory of Analytic Subgroups In the theory of topological groups in general one gets accustomed to thinking of subgroups as being closed. If one has a closed normal subgroup N of a topological group, then G/N is a Hausdorff topological group, and in the absence of normality, the quotient space is still a good Hausdorff homogeneous space on which G acts transitively. However, as soon as Lie groups emerge in the picture, certain nonclosed subgroups simply have to be taken into account, namely, the so-called analytic subgroups. A subgroup H of a Lie group G is called analytic if it is an immersed submanifold. The classical example upon which people base their intuition is the torus G = R2 /Z2 and def

the subgroups H = (R · (1, r) + Z2 )/Z2 which are compact and therefore closed if and only if r is rational and are nonclosed and dense in G if and only if r is irrational. These, together with the subgroup (R ·(0, 1)+ Z2 )/Z2 are all analytic subgroups other than the singleton one and G itself. Each one corresponds uniquely to a vector subspace (and subalgebra) of the Lie algebra R2 of G and all of them have to be taken into account to make this correspondence work well. The example illustrates well the fact that analytic subgroups depend in a chaotic fashion on the subalgebras. As one progresses more deeply into the theory of finite-dimensional Lie groups, one learns that the example is perhaps more typical than meets the eye at a first encounter. Of course, the closed connected subgroups of a finite-dimensional Lie group are indeed analytic, these are the “good” analytic subgroups, and the nonclosed ones which we illustrated above are the “bad” analytic subgroups – but they are needed. Another thing one learns in finite-dimensional Lie group theory is that analytic subgroups can be characterized group theoretically in one of a variety of ways. We wish to generalize the concept of “analytic subgroup” to the environment of pro-Lie

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groups, and an elaborate analysis and manifold theory is not immediately available here – nor is it needed in the context of a structure theory of pro-Lie groups. Therefore we should select a flexible and useful version of the various characterisations of analytic subgroups and use it for a definition of an analytic subgroup of a pro-Lie group in the general case. The one we opted for says that a subgroup H of a Lie group G is analytic if and only there is some connected Lie group C and a morphism of topological groups f : C → G such that H = f (C). In Chapter 9 we select this definition for pro-Lie groups: Definition 37 (9.5). (i) Let G be a pro-Lie group and H a subgroup. Then H is said to be an analytic subgroup of G if there is a morphism f : C → G of topological groups from a connected pro-Lie group C into G such that H = f (C) and L(f )(L(C)) is closed in L(G). (ii) A subgroup H of a pro-Lie group G is said to be exponentially generated if def

h = L(H ) is a closed Lie subalgebra of L(G) and H = exp h. In any infinite-dimensional Lie theory there is the added complication that subalgebras of the Lie algebra may or may not be closed, and on the Lie algebra level we insist that the Lie subalgebras we consider are closed. The definition above does not demand that im L(f ) = L(H ), however we shall show (9.6 (ii)) that this is the case. This means that any analytic subgroup H of a pro-Lie group uniquely determines its own Lie algebra L(H ). Every connected closed subgroup of a pro-Lie group is analytic (9.7). For each closed subalgebra h of the Lie algebra g = L(G) of a pro-Lie group the  and in view of the universal inclusion h → g induces a morphism (h) → (g) = G,  → G, the composition yields a morphism ih : (h) → G. If we morphism πG : G write L((h)) = h, then ih : (h) → G is the unique morphism inducing the inclusion L(ih ) : h → g. It is this morphism that has the special analytic subgroup A(h) or, more accurately, A(h, G) of G as its image. Let us retain this notation in the formulation of the following proposition: Proposition 38 (9.10, 9.11). For each closed subalgebra h of the Lie algebra L(G) of a pro-Lie group G, there is at least one analytic subgroup H such that L(H ) = h, namely, H = A(h, G). If h is an ideal, then (h) may and will be identified with a closed normal subgroup A(h, (g)) of (g). The analytic subgroup A(h) is exponentially generated and satisfies L(A(h)) = h. In particular, the analytic subgroup A(h) is arcwise connected. Among all analytic subgroups H satisfying L(H ) = h, the subgroup A(h) is the smallest; it is contained in each H satisfying L(H ) = h. As a consequence of this proposition we obtain the following assertion that is one possible generalisation of the classical correspondence between subalgebras of the Lie algebra of a Lie group and its analytic subgroups.

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Scholium (Scholium following 9.12). Let G be a pro-Lie group and g its Lie algebra. Denote the set of all analytic subgroups of G by A(G) and the set of all minimal analytic subgroups of G by A0 (G). Then the assignment H  → L(H ) : A0 (G) → C(g) is a bijection with inverse function h  → A(h) and the function h  → A(h) : C(g) → A(G) is a surjection. However, from most compact connected abelian non-Lie groups we can learn what happens with analytic subgroups. Indeed let G be a compact connected abelian group and Ga the arc component of the identity (see for instance [102, Chapter 8]). Then Ga = expG g = A(g, G) is the minimal analytic subgroup with Lie algebra g = L(G), and G is the largest analytic subgroup with Lie algebra g. All subgroups H with Ga ⊆ H ⊆ G satisfy L(H ) = g and some of these may very well be analytic: For instance (9.8 (iv)) the compact connected metric abelian group whose character group is the discrete group Q(N) has a continuum cardinality of different analytic subgroups all of whose Lie algebras agree with g. In formulating a nomenclature for analytic subgroups of a pro-Lie group G, one is in a quandary. On the one hand, it is pretty clear that a closed connected subgroup of a pro-Lie group should be called analytic. These have always, classically or otherwise, been the “good” analytic subgroups. The relevant representatives of the nonclosed analytic subgroups are the minimal ones of the type A(h), and they are in a clean bijective correspondence with the closed Lie subalgebras of the Lie algebra L(G). They are exactly the subgroups expG h for the closed subalgebras; they are the ones that are amenable to all deeper developments of a Lie theory of pro-Lie groups. They are the “good bad” analytic subgroups. They are certainly needed in the Lie theory of pro-Lie groups. One might be tempted to reserve the term “analytic subgroup” for these subgroups exclusively – but then one would have excluded the “good analytic subgroups”, the closed connected ones. So we think that our terminology is a good compromise, still extending the classical concept, including the “good analytic subgroups” and the “good bad analytic subgroups” under one common roof. But we do have to allow for the presence of the myriad other analytic subgroups that have little, if any theoretical significance we can perceive. But there they are, and as long as they do not upset our scheme of things, they may stay.

Centralizers and Normalizers Like in classical Lie theory, centralizers pose no problems. Let G be a pro-Lie group and H any subset. Then the centralizer or commutant Z(H, G) = {g ∈ G : (∀h ∈ H ) gh = hg} is closed in G and is therefore a pro-Lie group.

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Let H be a subgroup of a connected pro-Lie group G and assume that H ⊆ expG h, where h = L(H ). (This assumption is automatically satisfied if H is exponentially generated or analytic.) Then the following conclusions hold: (i) An automorphism α of G satisfies α(h) = h for all h ∈ H iff L(α)X = X for all X ∈ h.  (i ) An element g ∈ G is in Z(H, G) iff Ad(g)X = X for all X ∈ h. (ii) L(Z(H, G)) = z(h, g). (iii) Z(H, G)0 = expG z(h, g). These results satisfy most demands when it comes to centralizers, and we provide a few additional pieces of information in the book, but we do not need to go into them here. If H = G then Z(H, G) is the center Z(G) of G; likewise z(h, g) is the center of g on the Lie algebra level. So L(Z(G)) = z(g) and Z(G)0 = expG z(g). In particular here is a proof that a connected pro-Lie group is abelian iff its Lie algebra is abelian. The normalizer story is a bit more delicate. Let H be a subgroup of a group G. The normalizer of H in G is the set N(H, G) = {g ∈ G : gH g −1 = H }. If h is a subalgebra of a Lie algebra g, then the normalizer of h in g is the set n(h, g) = {X ∈ g : [X, h] ⊆ h}. Sometimes n(h, g) is said to be the idealizer of the subalgebra h in g. We shall prove the following facts (9.20), that illustrate well the significance of the maximal and the minimal analytic subgroups having a fixed Lie algebra. Let H be a subgroup of a pro-Lie group G and assume that H satisfies at least one of the following conditions: (a) H is a minimal analytic subgroup of G. (b) H is a closed connected subgroup. Then the following conclusions hold: (i) (i ) (ii) (iii)

An automorphism α of G satisfies α(H ) = H iff L(α)(h) = h. Let g be an element of G. Then gH g −1 = H iff Ad(g)h = h. The normalizer N(H, G) is closed in G. L(N(H, G)) = n(h, g).

Item (ii) may be surprising, since H may be a nonclosed analytic subgroup. The closedness of the normalizer arises from the possibility of transporting the issue to the Lie algebra level where H has a closed Lie algebra. This is not very much different from the way it is in classical Lie theory for finite-dimensional Lie groups (see for instance [102], Proposition 5.54), but some extra care is required.

Commutator Subgroups The Lie theory of commutator subgroups of analytic groups is hard already in finite dimensions, and many fairly elementary examples in the domain of pro-Lie groups

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show that certain difficulties we encounter here cannot be circumnavigated. We give a pronilpotent simply connected pro-Lie group whose commutator subgroup is not analytic. What we can prove is sampled by the following theorems, in which we denote the closed commutator subgroup (respectively, closed commutator subalgebra) by dots. Theorem 39 (9.26). For a connected pro-Lie group G, the closed commutator subgroup ˙ is a closed analytic subgroup which agrees with the closure A(˙g) of the unique G smallest analytic subgroup whose Lie algebra is the closed commutator subalgebra g˙ of the Lie algebra g = L(G) of G. Theorem 40 (Theorem on Commutator Subgroups of DenseAnalytic Subgroups; 9.32). Let G be a connected pro-Lie group and h a closed subalgebra of g = L(G). Then (i) L(A(h)) ˙ ⊆ h˙ ⊆ h. (ii) In particular, L(A(h))/h is abelian. (iii) The abstract group A(L(A(h)))/A(h) is abelian. Corollary 41 (9.34). Let G be a pro-Lie group and H a dense analytic subgroup with Lie algebra h, then for any N ∈ N (G), the algebraic commutator subgroup G of G ˙ is contained in A(h)N ⊆ H N. As a consequence  ˙ ˙ G ⊆ A(h)N ⊆ A(h) N ∈N (G)

and ˙ ⊆ A(˙g) = G. ˙ A(˙g) = A(h)

Finite Dimensional Connected Pro-Lie Groups We shall say, provisionally, but in perfect accord with a finer theory of topological dimension, that a pro-Lie group is finite-dimensional, if dim L(G) < ∞. Armed with the arsenal of analytic subgroups we are able to deal with finite-dimensional pro–Lie groups and show, that they are rather close to finite-dimensional Lie groups in most respects. The class of almost connected finite-dimensional pro-Lie groups is seen to coincide with the class of almost connected locally compact groups. (For the compact case see for instance [102], Theorem 9.52.)  → G by P (G) and We shall denote the kernel of the universal morphism πG : G call it the Poincaré group of G. It is natural that in a theory of pro-Lie groups, finitedimensional pro-Lie groups play a significant role. This is primarily due to the fact that for every normal subgroup N ∈ N (G) and its identity component N0 the pro-Lie group G/N0 is finite-dimensional, having the same Lie algebra as the Lie group G/N . If G itself is finite-dimensional, then N0 = {1} for every sufficiently small N ∈ N (G). In our discussion of finite-dimensional pro-Lie groups, we pass through the following theorem, which in itself does not refer to the hypothesis of finite-dimensionality but

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is, despite its technical character at the root of a number of significant results. It deals with a connected pro-Lie group G, a normal subgroup N such that G/N is a Lie group, and with an intermediate member N1 of N (G), N0 ⊆ N1 ⊆ N and the Lie group def

L = G/N1 : P (G/N −−−−−→ P (G/N ⏐ 0 ) −−−−−→ P (L) ⏐ ⏐ ) ⏐ ⏐ ⏐   

N1 /N0

−−−−−→

 L ⏐ ⏐ ϕ G/N0

=

 L ⏐ ⏐ πL  ρ −−−−−→ L −−−−−→

=

−−−−−→ −−−−−→

 L ⏐ ⏐πG/N  G/N.

Theorem 42 (9.39). Let G be a connected pro-Lie group and N an arbitrary member of the filter basis N (G). Then there is a characteristic subgroup N1 ∈ N (G) that is def open in N such that the connected Lie group L = G/N1 and its universal covering  → L, together with the quotient map ρ : G/N0 → L satisfy the following πL : L conditions:  → G/N0 such that πL = ρ  ϕ, (∗) There is a lifting morphism ϕ : L (∗∗) N1 /N0 = ker(ρ1 )N = ϕ(P (L)) = comp(G/N0 ) is a compact metric totally disconnected central subgroup of G/N0 . The group P (L)/(P (L) ∩ ker ϕ) is finite iff N0 has finite index in N1 iff N0 ∈ N (G). This is the case if P (L) is finite and this is the case if P (G/N) = ker πG/N is finite. def  : g ∈ P (L)} ∼ (∗∗∗) Let D = {(ϕ(g)−1 , g) ∈ N1 /N0 × L = P (L); then (nN0 , g)D → nN0 · ϕ(g) :

 N1 /N0 × L → G/N0 D

is a well-defined isomorphism of locally compact metric groups, and  N/N0 × L  and G/N0 are all locally isomorphic. N/N1 × L,  )/D is isomorphic to the minimal (†) The subgroup (ϕ(P (L)) × G/N analytic subgroup A(L(G/N0 ), G/N0 ) of G/N0 with Lie algebra L(G/N0 ) ∼ = L(G/N ). While this statement is fairly technical, it proves its value in the subsequent conclusions: Proposition 43 (9.40). Let G be a connected pro-Lie group with the following property: (•) In the filter basis N (G) every member contains a member N such that N/N0 is finite. Then G = A(g, G).

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Theorem 44 (9.44). Let G be a finite-dimensional connected pro-Lie group with Lie algebra g. Then G is locally compact metric, and there is a compact metric totally def disconnected member  ∈ N (G) such that the Lie group F = G/ and the quotient morphism ρ : G → F satisfy the following conditions:  → G such that πF = ρ  ϕ. (∗) There is a morphism ϕ : F (∗∗)  = ϕ(P (F )). def (∗∗∗) Let D = {(ϕ(g)−1 , g) : g ∈ P (F )} ∼ = P (F ); then  ×F (c, g)D  → cϕ(g) : →G D is a well-defined isomorphism of locally compact metric groups, and  × F ,  and G are all locally isomorphic. ×F  (†) The subgroup ϕ(P (FD))×F is isomorphic to the minimal analytic subgroup A(L(G), G) of G with Lie algebra g ∼ = L(F ). We keep in mind: Connected finite-dimensional pro-Lie groups are locally compact metric. Consequently, almost connected finite-dimensional pro-Lie groups are locally compact. For any connected finite-dimensional pro-Lie group G there are a totally disconnected compact abelian group  and a simply connected Lie group L such that we have a quotient morphism  × L → G whose kernel is a discrete central subgroup of  × L projecting onto a dense subgroup of . We shall conclude a number of useful pieces of information from these developments. It is sometimes useful to know that the limit representation G ∼ = limN∈N (G) G/N of a pro-Lie group in terms of its Lie group quotients yields a representation G∼ = lim G/N0 N ∈N (G)

of G in terms of finite-dimensional metric quotients of G modulo connected normal subgroups N0 . There is a fairly significant conclusion coming out of this context: Theorem 45 (Existence of the Largest Compact Normal Abelian Subgroup; 9.50). Let G be a connected pro-Lie group. (i) Then G has a unique largest compact central subgroup KZ(G). The factor group G/ KZ(G) does not have nondegenerate compact central subgroups. (ii) The center Z(G) is a direct product of a weakly complete vector group V and a subgroup A of Z(G) containing the characteristic subgroup KZ(G); moreover, the factor group Z(G)/V KZ(G) ∼ = A/ KZ(G) is prodiscrete and free of nonsingleton compact subgroups. The characteristic closed subgroup Z(G)0 comp(Z(G)) is the direct product of V and KZ(G). By way of illustration, this theorem says that a connected pro-Lie group cannot contain a central subgroup isomorphic to the additive group Qp of a p-adic rational field or a discrete group isomorphic to a discrete Prüfer group Z(p∞ ) = p1∞ Z/Z.

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However, on the other hand, we give a construction that shows the following (9.51): Given an abelian pro-Lie group A such that the union comp(A) of its compact subgroups is compact then there is a connected (metabelian) pro-Lie group G such that A is (isomorphic to) the center of G. A second major result using conclusions from our discussion of finite-dimensional pro-Lie groups is the Open Mapping Theorem which we have already recorded in Theorem 8. But a lot of information that has accrued at this stage in the book enters its proof.

Part 4. Global Structure Theory of Connected Pro-Lie Groups Since we have established a reasonable correspondence between subalgebras and analytic subgroups and handled the Lie theory of commutator subgroups with some success, we may hope to embark upon a global structure theory and deal with issues like solvability, nilpotency, reductivity, semisimplicity in the absence of simple connectivity.

Solvability and Nilpotency of Pro-Lie Groups The question of solvability of infinite-dimensional Lie algebras that we discussed earlier is paralleled by the question of solvability of arbitrary groups. Definition 46 (10.1). Let G be a group. Set G(0) = G and define sequences of subgroups G(α) indexed by the ordinals α, card α ≤ card G via transfinite induction. Assume that G(α) is defined for α < β.  (i) If β is a limit ordinal, set G(β) = α<β G(α) . (ii) If β = α + 1, set G(β) = [G(α) , G(α) ]. For cardinality reasons, there is a smallest ordinal γ such that G(γ +1) = G(γ ) . Set = G(γ ) . Let ω denote the first infinite ordinal. Then G is said to be transfinitely solvable, if G(∞) = {0}. If G is transfinitely solvable and γ ≤ ω, then G is called countably solvable. If γ is finite and G(γ ) = {0}, then G is called solvable. G(∞)

We proceed to make a parallel definition for the infinite version of nilpotency. Definition 47 (10.5). Let G be a group. Set G[0] = G and define sequences of normal subgroups G[α] indexed by the ordinals α, card α ≤ card G via transfinite induction. Assume that G[α] is defined for α < β.  (i) If β is a limit ordinal, set G[β] = α<β G[α] .

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(ii) If β = α + 1, set G[β] = [G, G[α] ]. For cardinality reasons, there is a smallest ordinal δ such that G[δ+1] = G[δ] . Set = G[δ] . Then G is said to be transfinitely nilpotent if G[∞] = {0}. If G is transfinitely nilpotent and δ ≤ ω, then G is called countably nilpotent. If δ is finite and G[δ] = {0}, then G is called nilpotent. G[∞]

Since G(α) ⊆ G[α] , any transfinitely nilpotent Lie group is transfinitely solvable. As we are dealing with topological groups, we have topological versions of these concepts as well. Definition 48 (10.8). Let G be a subgroup of a topological group H . (For instance, H = G.) Set g = L(G) and G((0)) = G; define sequences of normal subgroups G((α)) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that G((α)) is defined for α < β.  (i) If β is a limit ordinal, set G((β)) = α<β G((α)) . (ii) If β = α + 1, set G((β)) = [G((α)) , G((α)) ]. For cardinality reasons, there is a smallest ordinal γ such that G((γ +1)) = G((γ )) . Set G((∞)) = G((γ )) . Let ω denote the first infinite ordinal. Then G is said to be transfinitely topologically solvable, if G((∞)) = {1}. If g is transfinitely topologically solvable and γ ≤ ω, then G is called countably topologically solvable. If γ is finite and G((γ )) = {0}, then G is called topologically solvable. And the nilpotent counterpart follows at once. Definition 49 (10.9). Let G be a subgroup of a topological group. We set G[[0]] = G and define sequences of closed normal subgroups G[[α]] indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that G[[α]] is defined for α < β.  (i) If β is a limit ordinal, set G[[β]] = α<β G[[α]] . (ii) If β = α + 1, set G[[β]] = [G, G[[α]] ]. For cardinality reasons, there is a smallest ordinal δ such that G[[δ+1]] = G[[δ]] . Set G[[∞]] = G[[δ]] . Then G is said to be transfinitely topologically nilpotent, if G[∞] = {0}. If G is transfinitely topologically nilpotent and δ ≤ ω, then G is called countably topologically nilpotent. If δ is finite and G[[δ]] = {0}, then G is called topologically nilpotent. All of this may look a bit tedious, but as we are dealing with infinite groups and with topological groups there does not appear any way to bypass these definitions. However, since we are dealing here with pro-Lie groups more definitions are to follow inevitably.

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Definition 50 (10.12). A pro-Lie group G is called prosolvable if every (finite-dimensional) quotient Lie group G/N, N ∈ N (G) is solvable. It is called pronilpotent if every (finite-dimensional) quotient Lie group G/N , N ∈ N (G), is nilpotent. We have transfinite theories of solvability and nilpotency for pro-Lie algebras on the one hand and for pro-Lie groups on the other. The theory of analytic subgroups and their correspondence to closed subalgebras is now launched on this side-by-side situation with good success (see 10.14ff) The results are sizeable. Theorem 51 (The Equivalence Theorem for Solvability of Connected Pro-Lie Groups; 10.18). Let G be a connected pro-Lie group and g its Lie algebra L(G). Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

G is transfinitely solvable. G is countably solvable. G is transfinitely topologically solvable. G is countably topologically solvable. G is prosolvable. G does not contain a finite-dimensional analytic simple subgroup. g is prosolvable. g does not contain a finite-dimensional simple Lie algebra.

Again it is a remarkable feature of connected pro-Lie groups that all reasonable concepts of infinite solvability coalesce and that a genuine transfinite solvability does in fact not occur. The situation with nilpotency is, alas, not equally perfect as far as our knowledge is concerned. Theorem 52 (The Equivalence Theorem for Nilpotency of Connected pro-Lie Groups; 10.36). Let G be a connected pro-Lie group and g its Lie algebra L(G). Then the following assertions are equivalent: (i) (ii) (iii) (iv)

G is transfinitely topologically nilpotent. G is countably topologically nilpotent. G is pronilpotent. g is pronilpotent.

These conditions imply the following ones: (v) G is transfinitely nilpotent. (vi) G is countably nilpotent. Is a transfinitely nilpotent connected pro-Lie group pronilpotent? We do not know. A transfinitely nilpotent group has to be prosolvable since it is transfinitely solvable and then Theorem 51 applies. The impediment for a proof is the failure of transfinite nilpotency to be preserved by passing to quotients. Free topological groups are free

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groups in the algebraic sense and thus are countably nilpotent; but every topological group is a quotient of a free topological group and thus of a transfinitely countably nilpotent topological group. In the meantime, we are content to have what Theorem 52 gives us. The relationship between the topological commutator series and the topological descending central series on the Lie algebra and on the group level are expressed in a somewhat delicate fashion involving the minimal analytic subgroups associated with a closed subalgebra as follows: Theorem 53 (Theorem on the Commutator Series of Pro-Lie Groups; 10.20). Let G be a connected pro-Lie group. Then G((α)) = A(g((α)) ) for all ordinals α. Theorem 54 (Theorem on the Descending Central Series of Pro-Lie Groups; 10.38). Let G be a connected pro-Lie group. Then G[[α]] = A(g[[α]] ) for all ordinals α. It is quite natural that we should introduce the counterparts of the radical r(g), the nilradical n(r), and the coreductive radical ncored (g) of a pro-Lie algebra g for a connected pro-Lie group G. The definitions are a bit delicate, because some obvious attempts at a definition are not feasible. Here is the way we proceed: Definition 55 (10.23, 10.40). Let G be a pro-Lie group and g = L(G) its Lie algebra. Then the closed subgroup expG r(g) will be denoted by R(G). This group is called the radical of G or, if more clarity is required, the solvable radical of the group G. The closed subgroups expG n(g) and expG ncored (g) will be denoted by N (G), respectively Ncored (G). These groups are called the nilradical, respectively, coreductive radical of G. Recall that Z(G) denotes the center of G. Since Z(G)0 = expG z(g) and z(g) ⊆ n(g) ⊆ r(g) we notice that Z(G)0 ⊆ N (G) ⊆ R(G). We say that a connected pro-Lie group G is semisimple, if R(G) = {1}, and reductive, if R(G) = Z(G)0 . The radical R(G) is a prosolvable connected closed characteristic subgroup; the nilradical and the coreductive radical are pronilpotent connected closed characteristic subgroups. More precisely (10.25, 10.28): Theorem 56 (10.25). If G is a pro-Lie group, then the radical R(G) is the largest connected transfinitely topologically solvable normal subgroup and is a closed connected characteristic subgroup of G such that L(R(G)) = r(g). The factor group G0 /R(G) is semisimple. If f : G → H is a quotient morphism of connected pro-Lie groups, then f (R(G)) = R(H ). Recall that for a morphism of almost connected pro-Lie groups to be a quotient morphism, by the Open Mapping Theorem it suffices to be surjective.

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Theorem 57 (10.42). If G is a pro-Lie group, then the nilradical N (G) is the largest connected transfinitely topologically nilpotent normal subgroup and is a closed connected characteristic subgroup of G such that L(N (G)) = n(g). Theorem 58 (10.43). If G is a connected pro-Lie group, then the coreductive radical Ncored (G) is the smallest connected closed normal subgroup N such that G/N is reductive. In particular, G/Ncored (G) and G/N (G) are reductive. It is a closed connected characteristic subgroup of G such that L(Ncored (G)) = ncored (g). If f : G → H is a quotient morphism of connected pro-Lie groups, then f (Ncored (G)) = Ncored (H ). In a pro-Lie group we have therefore a hierarchy of characteristic connected closed subgroups: G | G0 | R(G) | N (G) | Ncored (G) | {1}. The factor group G0 /R(G) is semisimple, the factor group G0 /Ncored (G) is reductive. We remember also Z(G)0 ⊆ N (G). So it is clearly time to say something about semisimple and reductive groups: Theorem 59 (Characterisation of Semisimple and Reductive Connected Pro-Lie Groups; 10.29). Let G be a connected pro-Lie group. (i) G is semisimple iff g is semisimple,  and G is reductive iff g is reductive.  is a product (ii) G is semisimple iff G j ∈J Sj of simply connected simple finite is a product dimensional Lie groups Sj , j ∈ J . Also G is reductive iff G  j ∈J Sj of pro-Lie groups Sj , j ∈ J which are either simply connected simple finite-dimensional Lie groups or copies of R. (iii) Assume that P is a connected proto-Lie group, embedded into its completion G according to Theorem 4.1 and assume that g = L(P ) = L(G) such that g is a semisimple pro-Lie algebra. This assumption is satisfied if P = G is a semisimple pro-Lie group by (i) above. Then we have the following conclusions: ∼ =P  = (g) = (a) G j ∈J Sj where all Sj are simply connected simple Lie groups, and  = Sj → P πP : G j ∈J

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is  a morphism with dense image whose kernel D is a closed subgroup of  j ∈J Z(Sj ) and thus is a totally disconnected  central subgroup of G. (b) There is a quotient morphism f : P → j ∈J Sj /Z(Sj ) with a totally disconnected kernel ker f = Z(P ). The quotient P /Z(P ) is a center-free semisimple pro-Lie group. The completion G of P is Z(G)P , a semisimple connected pro-Lie group satisfying P /Z(P ) ∼ = G/Z(G). (c) (Sandwich Theorem) The group P is ‘sandwiched’ between two products via two morphisms   f πG Sj −−−→ P −−−→ Sj /Z(Sj ) j ∈J

j ∈J

whose composition is just the quotient morphism obtained by passing to the quotient Sj → Sj /Z(Sj ) in each factor. (d) Let G be a semisimple pro-Lie group and let A(g) be the minimal analytical subgroup with Lie algebra  g = L(G). Then G = Z(G)A(g), and A(g)/Z(A(g)) ∼ = G/Z(g) ∼ = j ∈J Gj /Z(Gj ). If A(g) is center-free then A(g) is complete and therefore equal to G. Theorem 60 (Theorem on the Closure of Semisimple Analytic Subgroups; 10.32). def

Let G be a pro-Lie group and s a closed semisimple subalgebra of g. Set H = A(s). Then the following conclusions hold: (i) (ii) (iii) (iv)

H is topologically perfect, that is, [H, H ] = H . [H, H ] ⊆ A(s). H is reductive such that [L(H ), L(H )] = s. In particular,  sj L(H ) ∼ = = RI × s, s ∼ j ∈J

for some set I and J and a family  of simple finite-dimensional Lie algebras sj , I ∼  j ∈ J . Therefore, H = R × j ∈J Sj , Sj = (sj ). (v) H = Z(H )A(s) and H /Z(H ) is a center-free pro-Lie group.  then (vi) Let S be the image of j ∈J (sj ) in G; (a) A(s) = πG (S), and (b) Z(H ) = Z(A(s)). (c) If L(H ) = s, that is, if H is semisimple, then KZ(H )A(s, H ) = H . (vii) The minimal analytic subgroup A(s) is closed in G if and only if its center Z(A(s)) is closed in G. This is the case for instance if Z(A(s)) is compact. Much of what has been said, but not everything, is included in the following summary: Theorem 61 (Characterisation of Reductive Pro-Lie Groups; 10.48). Let G be a connected pro-Lie group. Then the following statements are equivalent:

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(i) G is reductive. (ii) g is reductive. (iii) g = z(g) ⊕ g˙ for a unique semisimple pro-Lie algebra g˙ , obtained as the closed commutator subalgebra of g. (iv) g = a ⊕ r for a central closed subalgebra a ⊆ z(g) and a closed reductive def subalgebra r = L(A(˙g)). (v) G = AS = AS for a closed connected central subgroup A and a semisimple minimal analytic subgroup S. (vi) G = Z(G)S for a semisimple minimal analytic subgroup S, that is, S = A(s) for some semisimple subalgebra (indeed ideal) s. (vii) G = Z(G)S for the minimal analytic subgroup S = A(˙g), and g˙ is semisimple. Essential ingredientsof theorems on semisimple and reductive groups G are the simple factors of g˙ ∼ = j ∈J sj . We say that sj is of bounded type if the simply connected Lie group (sj ) has a compact center. (See 10.50 for more details.) We say that a semisimple pro-Lie algebra s is of bounded type if all of its simple factors are of bounded type. This amounts to saying that the center of (s) is compact. This concept is very important in determining the structure of the minimal analytic subgroup whose Lie algebra is a Levi summand. Theorem 62 (10.52). Let G be a pro-Lie group and let s be a semisimple pro-Lie def subalgebra of g = L(G) defining a minimal analytic subgroup A(s). (a) If Z(A(s)) is compact, then A(s) is closed. (b) The following statements are equivalent: (i) s is of bounded type. (ii) s ⊆ L(G) for a pro-Lie group L(G) then A(s, G) is closed in G. In other words, s is of bounded type if and only if A(s, G)) is closed in all pro-Lie groups G.

Splitting Theorems for Pro-Lie Groups An important class of structure theorems for topological groups is formed by the so called splitting theorems. Assume that N is a normal subgroup of a topological group G. There is a representation ι : G → Aut(N ) defined by ι(g)(n) = gng −1 ; we do not worry here about a topological group structure on the group Aut(N ) of automorphisms of the topological group N or any continuity properties of ι; however what is relevant here is that the function (g, n)  → ι(g)(n) = gng −1 : G × N → N is continuous, allowing us to define a semidirect product G ι N , that is, the product space N × G with the multiplication (m, g)(n, h) = (mι(g)n, gh). The function μ : N ι G → G defined by μ(n, g) = ng is a morphism of topological groups. If H is any subgroup of G, then N ι H is a subgroup of N ι G, and the morphism μ restricts to a

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morphism μH : N ι H → G whose image is N H . We note that the kernel of μH is {(h−1 , h) : h ∈ N ∩ H } and that h  → (h−1 , h) : N ∩ H → ker μH is an isomorphism of topological groups. In particular μH : N ι H → G is bijective if and only if N ∩ H = {1} and NH = G. In the absence of any Open Mapping Theorem we cannot assert that μH is an isomorphism. Now assume that we are given G and N; then a splitting theorem provides sufficient conditions for the existence of a subgroup H of G such that μH : N ι H → G is an isomorphism. This is sometimes expressed by saying that N is a semidirect factor of G and that H is a semidirect cofactor. In fact, under these circumstances one also writes G = N  H which some readers may consider a mild abuse of notation since the semidirect product sign  is reserved for the “external” semidirect product. We have already encountered some typical splitting theorems, for instance Theorem 24 and Theorem 36. Those two theorems had quite different proofs. This variety of methods will also be typical for the splitting theorems that we will discuss now and prove in Chapter 11. The information we have accumulated on reductive pro-Lie groups allows us to establish a splitting theorem for reductive pro-Lie groups as follows: Theorem 63 (Splitting Theorem for Reductive Pro-Lie Groups; 11.8). Let s = g˙ be the Levi summand of the pro-Lie algebra g of a semisimple connected pro-Lie group G, and assume that s is of bounded type. Then expG g, the unique minimal analytic ˙ of G, and is a subgroup with Lie algebra s, is the closed commutator subgroup G semidirect factor. That is, there is a closed connected abelian subgroup A of G acting ˙ under inner automorphisms such that ι(a)(n) = ana −1 and such that on G ˙ ι A → G, μA : G

μA (n, a) = na

is an isomorphism of pro-Lie groups. ∼ G/G, ˙ so we could express this as saying that a reductive pro-Lie Of course, A = group G whose Lie algebra has no simple factor of unbounded type is a semidirect ˙ and its commutator factor group G/G. ˙ If product of its closed commutator subgroup G s = sl(2, R), then this assertion fails already for locally compact connected groups of dimension 3. On the other hand, every compact connected group satisfies the hypothesis of the theorem yielding the so-called Borel–Hofmann–Scheerer Splitting Theorem (see [102, Theorem 9.39]): Corollary 64. Every compact connected group G is the semidirect product of its commutator subgroup G and a closed abelian subgroup isomorphic to G/G . Here one also uses the fact that the algebraic commutator subgroup of a compact connected group is closed. (See [102, Theorem 9.2].) One should draw the reader’s attention to the fact that under the hypotheses of ˙ × Z0 → G for Theorem 63, by Theorems 61 and 62 there is a surjective morphism G the identity component Z0 of the center Z which, by the Open Mapping Theorem, is

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in fact a quotient morphism. But in general it has a prodiscrete kernel isomorphic to ˙ Therefore G ˙ is not a direct factor. Z0 ∩ G. ˙ is answered by the following The question how many cofactors we can have for G remark: ˙ denote the set Assume that the hypotheses of Theorem 63 are satisfied. Let C(G) ˙ of cofactors of G in G. Then the function ˙  ) → C(G), ˙  : HomD (Z(G)0 , G

(f ) = {f (z)−1 z | z ∈ Z},

is a bijection.(11.9) An entirely different splitting theorem with rather powerful consequences arises when the quotient group modulo a normal vector subgroup is compact. Theorem 65 (The Vector Group Splitting Theorem for Compact Quotients; 11.15, 11.31). Let G be a pro-Lie group with a normal weakly complete vector subgroup N such that G/N is compact. Then G has a compact subgroup K such that G = N  K. Moreover, two semidirect cofactors K1 and K2 for N are conjugate under an inner automorphism implemented by an element of N . For Lie groups and indeed for locally compact groups this result is fairly well known to mathematicians working in the area of locally compact groups. (See e.g. [108].) Yet we prove it here for the first time for pro-Lie groups in general. Theorem 65 does generalize to prosolvable normal subgroups as follows: Corollary 66 (Splitting Simply Connected Prosolvable Groups; 11.17, 11.32). Let G be a pro-Lie group with a normal subgroup N such that N is simply connected prosolvable and G/N is compact. Then G has a compact subgroup K such that G = N  K. Moreover, two semidirect cofactors K1 and K2 for N are conjugate under an inner automorphisms implemented by an element of N. These results allow us to prove strong structure theorems for prosolvable groups which give us a reasonably good insight into the structure of prosolvable connected pro-Lie groups. First we need to attend to some business concerning pronilpotent proLie groups; we recall that the closed commutator subgroup of a connected prosolvable group is pronilpotent. Lemma 67 (Simple Connectivity of Pronilpotent Pro-Lie Groups; 11.27). For a connected pronilpotent pro-Lie group G, the following statements are equivalent: (i) (ii) (iii) (iv) (v)

G has no compact subgroups. G has no compact normal subgroups. G has no compact normal connected subgroups. G is simply connected. expG : (g, ∗) → G is an isomorphism of topological groups.

We note that Conditions (i), (ii), (iii) are group theoretical conditions, Condition (iv) is a topological condition, and (v) is a Lie theoretical condition. The last one reconfirms

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what we said earlier: The Lie algebra here determines the group, and does so in a particularly explicit way. Now we have all the ingredients to prove the following structure theorem which belongs to the type of “almost” splitting theorems. Theorem 68 (Structure of Almost Connected Prosolvable Pro-Lie Groups; 11.28). Let G be an almost connected pro-Lie group whose identity component is prosolvable. Let C be the unique largest compact central connected subgroup, clearly contained in the nilradical N(G0 ). Then there is a maximal compact subgroup K which is abelian containing C, and there is a connected closed normal subgroup V containing N (G0 ) such that V /C is simply connected and G/C = V /C  K/C. There is a compact abelian subgroup M of G such that G = V M and V ∩ M is totally disconnected central. G    V   N(G)  0   C    {1}

 

compact, splits mod V ∩ M ∼ = (L(V /N (G0 )), +) splits topologically over N (G0 )



∼ = (L(N (G0 ))/C, ∗)  = maximal compact connected central compact central

Roughly speaking this theorem says that the compact subgroups come in two kinds: one is deep down in the center and is connected, while the other is at the top and “almost” splits as a semidirect cofactor for V which is simply connected modulo C. From these results we can get a group theoretical characterisation of simple connectivity for prosolvable groups in the spirit of Lemma 67 for pronilpotent groups. Corollary 69 (Simple Connectivity of Prosolvable Lie Groups; 11.29). Let G be a connected prosolvable pro-Lie group. Then the following statements are equivalent: (i) G does not contain any nontrivial compact subgroup. (ii) G is simply connected. (iii) G is homeomorphic to (L(N (G)), ∗) × (L(G/N (G)), +). If these conditions are satisfied, then G = expG g.

Part 5. The Role of Compactness on the Pro-Lie Algebra Level We have seen in various results on the structure of pro-Lie groups that compact subgroups play an important role. In Theorem 45 we say that each connected pro-Lie

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group contains a unique largest compact central subgroup. In describing the global structure of pronilpotent and prosolvable pro-Lie groups (see notably Theorem 68) we observed the crucial role of compact subgroups. In Chapter 12 of this book we investigate this role further in a systematic way. In particular, we are looking for maximal compact connected subgroups and aim to show their conjugacy, where possible. Our strategy, however, must be Lie theoretical. That is, we should be able to detect compact connected subgroups by identifying their Lie algebras. This requires that we look back at the module theory of pro-Lie algebras and try to transform the topological property of compactness on the group level into algebraic properties on the algebra level.

Lie Algebra Modules and Compactness Definition 70 (7.8). Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. (i) V is called a profinite-dimensional L-module if it is complete as a topological vector space and the filter basis M of closed submodules M ⊆ V such that dim V /M < ∞ converges to 0. (ii) E is called a locally finite-dimensional L-module if for each finite subset S of E there is a finite-dimensional submodule F of E containing S. Definition 71 (12.1). (i) Let L be a Lie algebra and let V be an L-module. Then V is called a pre-Hilbert L-module if V is a real vector space with an inner product (• | •), that is, a symmetric positive bilinear form, such that (∀x ∈ L, v, w ∈ V )

(x · v|w) = −(v|x · w).

It is called a Hilbert L-module if, √ in addition, V is a complete topological vector space with respect to the norm v2 = (v|v). (ii) V is called a compact L-module if V can be given an inner product relative to which it is a Hilbert L-module, and dim V < ∞. (iii) An L-module V is called procompact if V is profinite-dimensional and if all finite-dimensional quotient modules are compact L-modules. (iv) An L-module V is called an algebraically locally compact L-module if V is locally finite-dimensional and if all finite-dimensional submodules are compact L-modules. The terminology of a “compact” L-module formulated in (ii) derives from the adjoint module of a compact Lie group; for details see for instance [102, pp. 188ff., notably Proposition 6.2]. We define an algebraic property of a locally finite-dimensional module as “locally compact” even though it has nothing to do with the topological property of local compactness, but this is not any more deleterious than calling certain finite-dimensional Lie algebras compact, and this practice is well established.

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It is not hard to see that some of the concepts we introduced appear in dual pairs. Indeed for a profinite-dimensional L-module V over a Lie algebra L, the following conditions are equivalent: (i) V is a procompact L-module. (ii) The topological dual V  of V is an algebraically locally compact L-module. Therefore we understand the structure of procompact L- modules if we understand the largely algebraic concept of an algebraically locally compact L-module. In this regard we shall prove the following theorem with the aid of the Axiom of Choice: Theorem 72 (The Structure of Algebraically Locally Compact Modules; 12.4). (i) An algebraically locally compact L-module is a pre-Hilbert L-module which is an orthogonal direct sum of compact submodules. (ii) Any L-submodule of an algebraically locally compact L-module is algebraically locally compact and is an orthogonally direct summand. (iii) Any L-module homomorphic image of an algebraically locally compact L-module is algebraically locally compact. (iv) For each algebraically locally compact L-module E there is a Hilbert L-module  in which E is dense in the Hilbert space norm. E As a consequence, every algebraically locally compact L-module E is a semisimple L-module. By way of duality, we now can instantly formulate the following results on procompact L-modules. Theorem 73 (The Structure of Procompact Modules; 12.6). (i) A procompact L-module is a direct product of compact simple L-modules and is a semisimple L-module. (ii) A continuous homomorphic image of a procompact L-module is procompact. (iii) A closed submodule of a procompact L-module is procompact. The concept of an L-module for a Lie algebra L is parallelled by that of a G-module where G is a group. This is what one learns in the elementary linear algebra of group representations. If V is a topological vector space and G a topological group, then V is called a jointly continuous topological G-module if the module action (g, v) → g · v : G × V → V is continuous. We mentioned that the adjoint module of a compact Lie group G is a compact g-module, and indeed the adjoint module of a compact group G is a procompact g modules as we shall observe presently. It is both noteworthy in its own right and useful in various applications we shall make that, conversely, procompact modules give rise to compact groups. Theorem 74 (The Compact Group Associated with a Procompact Module; 12.8). Let V be a procompact L-module. Then there is a compact connected group GV ⊆ Aut(V ) such that V is a jointly topological GV -module and there is a homomorphism of Lie algebras λ : L → L(GV ) such that (i) GV = expGV λ(L), (ii) (∀x ∈ L, v ∈ V ) x · v = limh→0,h =0 h1 ((expGV λ(h · x)) · v − v), and

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(iii) (∀x ∈ g, w ∈ V ) (expGV λ(x))(w) = w + xV (w) + 2!1 · xV2 (w) + · · · ∈ V . (iv) A closed vector subspace W of V is an L-module if and only if it is a GV -module. One of the applications which we derive from this result says that if L = L1 + L2 is a Lie algebra with two subalgebras L1 and L2 satisfying [L1 , L2 ] ⊆ L2 , and if V is a profinite-dimensional L-module such that V is a procompact Lj -module for each of j = 1 and j = 2, then V is a procompact L-module.

Procompact Lie Algebras and Compactly Embedded Lie Subalgebras of Pro-Lie Algebras If g is a pro-Lie algebra, then the adjoint module gad is a profinite-dimensional g-module, and the coadjoint module gcoad is a locally finite-dimensional g-module. If k is any Lie subalgebra of g, then the restriction of the adjoint action of g on gad to k makes gad a profinite-dimensional k-module and the restriction of the coadjoint action of g on gcoad to k makes gcoad into a locally finite-dimensional k-module which is dual to the k-module gad . In Definition 71 (iii) we defined the notion of a procompact L-module for a Lie algebra L. This allows us now to say when a subalgebra of a pro-Lie algebra is “compactly embedded.” Definition 75. Let g be a pro-Lie algebra and k a Lie subalgebra. Then k is said to be compactly embedded into g if the adjoint module gad is a procompact k-module. If g is compactly embedded into itself, then g is said to be a procompact pro-Lie algebra. Obviously, a closed compactly embedded subalgebra is procompact in its own right. One notices that every commutative pro-Lie algebra, that is, every weakly complete vector space is a procompact Lie algebra, but a noncentral one-dimensional subalgebra of the three-dimensional Heisenberg algebra is a compact (hence procompact) Lie algebra which is not compactly embedded into the Heisenberg algebra. In the Heisenberg algebra, every 2-dimensional vector subspace containing the 1-dimensional center and commutator subalgebra is a maximal abelian subalgebra and is also an ideal. This shows that maximal abelian subalgebras and ideals are not unique. It also shows that the analytic subgroup belonging to a maximal abelian subalgebra may not be compact in a Lie group having the Heisenberg algebra as Lie algebra. Here is what it means to be a procompact pro-Lie algebra: Theorem 76 (The Structure Theorem of Procompact Lie Algebras; 12.12). (A) Let g be a pro-Lie algebra. Then the following statements are equivalent: (i) (ii) (iii) (iv)

g is procompact. The coadjoint g-module gcoad is an algebraically locally compact g-module. The coadjoint g-module gcoad is a direct sum of simple compact g-modules. g is a direct product of simple compact Lie algebras or copies of R.

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(v) g is a direct product of its center z(g) and its commutator algebra g , and g is a product of simple compact Lie algebras. In particular, for a procompact Lie-algebra g, the radical r(g) agrees with its center z(g). (B) A closed subalgebra of a procompact pro-Lie algebra is procompact. (C) The image of a procompact pro-Lie algebra under a continuous morphism of Lie algebras is procompact. (D) A product of any family of procompact pro-Lie algebras is procompact. (E) A closed procompact semisimple subalgebra k of a pro-Lie algebra g is compactly embedded in g. (F) g/r(g) is procompact iff g/ncored (g) is procompact. (G) If V is a profinite-dimensional k-module for a semisimple procompact pro-Lie algebra k, then V is a procompact k-module. (H) If k is a procompact semisimple closed subalgebra of a pro-Lie algebra g, then it is compactly embedded in g. (I) Assume that a subalgebra g of a pro-Lie algebra h is the sum of two subalgebras g1 and g2 such that, firstly, g1 and g2 are compactly embedded in h, and, secondly, [g1 , g2 ] ⊆ g2 . Then g is compactly embedded in h. Compactly embedded subalgebras are preserved under homomorphisms in the following sense: If ϕ : g1 → g2 is a surjective morphism of pro-Lie algebras and k is a compactly embedded subalgebra of g1 , then ϕ(k) is compactly embedded in g2 . Compactly embedded subalgebras behave well in many ways: (i) If k is a compactly embedded subalgebra of g, then z(g) + k is compactly embedded. (ii) The center z(g) of g is compactly embedded. The closure of a compactly embedded subalgebra is compactly embedded. (iii) A compactly embedded subalgebra k contained in a pronilpotent ideal n is contained in the center z(g). In particular, a pronilpotent compactly embedded ideal is central. (iv) Any procompact subalgebra k of a prosolvable pro-Lie algebra g is abelian. In particular, a compactly embedded subalgebra of a prosolvable pro-Lie algebra is abelian.

Maximal Compactly Embedded Subalgebras of Pro-Lie Algebras With the Axiom of Choice one establishes without major difficulty the existence of maximal compactly embedded subalgebras: Theorem 77 (Maximal Compactly Embedded Subalgebras: Existence; 12.15). Let g be a pro-Lie algebra. Then

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(i) every compactly embedded subalgebra of g is contained in a maximal compactly embedded subalgebra, and (ii) every compactly embedded abelian subalgebra of g is contained in a maximal compactly embedded abelian subalgebra. So existence is an easy matter, but a proof of the fact that maximal compactly embedded subalgebras are conjugate is quite a challenge. It was primarily for a proof of this fact that we developed the theory of Cartan subalgebras since they turn out to be the key here.

Cartan Subalgebras of Pro-Lie Algebras Even in defining Cartan subalgebras in the context of pro-Lie algebras we need some auxiliary concepts. The essence is this Definition 78 (7.84). Let g be a pro-Lie algebra and h a subalgebra. Then we set g0 (h) = {x ∈ g : (∀h ∈ h, j ∈  (g))(∃n = n(x, h, j) ∈ N) (ad h)n (x) ∈ j}   = ((ad h)n )−1 (j) h∈h,j∈ (g) n∈N

=



((ad h)dim g/j )−1 (j),

h∈h,j∈ (g)

and g0 (h) = {x ∈ g : (∀h ∈ h, j ∈  (g)) (ad h)(x) ∈ j} = {x ∈ g : (∀h ∈ h) [h, x] = 0} = z(h, g), the centralizer of h in g. These definitions still reflect their finite-dimensional counterpart, but the adjustment to the pro-Lie algebra environment causes things to be more complicated. Accordingly, the following theorem is not exactly easy to prove. Theorem 79 (7.87). For a closed pronilpotent subalgebra h of a pro-Lie algebra g, the following conditions are equivalent: (i) g0 (h) = h. (ii) h is its own normalizer. Once we have this theorem we can at least proceed with the definition of a Cartan subalgebra of a pro-Lie algebra. Definition 80 (7.88). A subalgebra h of a pro-Lie algebra g is said to be a Cartan subalgebra if it is a closed pronilpotent subalgebra satisfying the equivalent conditions of Theorem 79. That is, a Cartan subalgebra of a pro-Lie algebra is a closed pronilpotent subalgebra that agrees with its own normalizer.

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For finite-dimensional Lie algebras, the existence of Cartan subalgebras is proved by establishing that each regular element is contained in a unique Cartan subalgebra where the regular elements are determined by finite-dimensional linear algebra and form an open dense subset. We cannot follow this path in our environment. But with the aid of the Axiom of Choice we prove: Theorem 81 (Existence of Cartan Subalgebras; 7.93). Let g be a pro-Lie algebra and i a cofinite-dimensional closed ideal. Then for each subalgebra hi of g containing i such that hi /i is a Cartan subalgebra of g/i there is a Cartan subalgebra h of g such that h + i = hi . In fact we shall see that the union of all Cartan subalgebras in a pro-Lie algebra is dense. It is also a very useful fact to know that if f : g1 → g2 is a surjective morphism of pro-Lie algebras and if h2 is a Cartan subalgebra of g2 , then there exists a Cartan subalgebra h1 of g1 such that f (h1 ) = h2 . We shall finally show the following result, generalizing a well-known fact on finite-dimensional Lie algebras but rather delicate to prove in the context of pro-Lie algebras: Theorem 82 (Conjugacy of Cartan Subalgebras of Prosolvable Pro-Lie Algebras; 7.101). Let h1 and h2 be two Cartan subalgebras of a prosolvable pro-Lie algebra g. Then there is an x ∈ ncored (g) such that ead x h1 = h2 . Now we are able to exploit these facts for a proof of the conjugacy of maximal compactly embedded subalgebras. First one verifies that a lemma which is prominently proved in Bourbaki [19, Chapter 7], persists for pro-Lie algebras. Indeed: Lemma 83 (12.17). Let g be a pro-Lie algebra and a a subalgebra satisfying the following two conditions: (i) a is abelian. (ii) g is a semisimple a-module under the adjoint action. Then a is contained in a Cartan subalgebra of g; in fact the nonempty set C(a, g) of Cartan subalgebras of g containing a is the set of Cartan subalgebras of z(a, g), the centralizer of a in g. Definition 84. An inner automorphism of a pro-Lie algebra is a finite composition of automorphisms of the form ead x with x ∈ g. An element a ∈ g is said to be conjugate to b in g if there is an inner automorphism ϕ such that ϕ(a) = b. Now we proceed by proving that (i) if in a pro-Lie algebra g all Cartan subalgebras are conjugate, then all maximal compactly embedded abelian subalgebras are conjugate, and we conclude from this information and Theorem 82 that (ii) in a prosolvable pro-Lie algebra all maximal compactly embedded subalgebras are conjugate under inner automorphisms.

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This is the first step in a longer chain of arguments which end up in the following conjugacy theorem. Theorem 85 (Maximal Compactly Embedded Subalgebras: Conjugacy; 12.27). Let g be a pro-Lie algebra. (i) If k1 and k2 are two maximal compactly embedded subalgebras of g, then there is an inner automorphism ϕ of g such that ϕ(k1 ) = k2 . (ii) If t1 and t2 are two maximal compactly embedded abelian subalgebras of g, then there is an inner automorphism ϕ of g such that ϕ(t1 ) = t2 . (iii) If t is a maximal compactly embedded abelian subalgebra of g then there is a compactly embedded subalgebra k of g containing t and k is unique modulo the coreductive radical ncored (g), that is, if k1 is a maximal compactly embedded subalgebra containing t, then there is an x ∈ ncored (g) such that ead x k1 = k. In particular, if g is reductive, then k is unique. Moreover, there is an inner automorphism ϕ of g such that ϕ(k1 ) = k and ϕ(t) = t. From the conjugacy theorem we can derive some rather immediate consequences. Corollary 86 (12.31). Let  be the set of pairs (a, k) where k is a maximal compactly embedded subalgebra of a pro-Lie algebra g and a is a maximal compactly embedded abelian subalgebra of g contained in k. Let the group Inn(g) of all inner automorphisms act on  via ϕ · (a, k) = (ϕ(a), ϕ(k)). Then the action is transitive. So in principle, if Inn(g)(a,k) is the isotropy subgroup of this action fixing the pair (a, k), then the set  is bijectively equivalent to the quotient space Inn(g)/ Inn(g)(a,k) . By Corollary 8.18 in the book, which exploits Corollary 14 above, it then follows that  is bijectively equivalent to a quotient space of (g). Theorem 85 permits us to intersect all maximal compactly embedded subalgebras and to conclude, that in this fashion we obtain a unique maximal compactly embedded ideal: Corollary 87 (The Largest Compactly Embedded Ideal of a Pro-Lie Algebra; 12.34). Let g be a pro-Lie algebra and r = r(g) its radical. Then (i) there is a unique largest compactly embedded ideal m(g); (ii) there is a unique largest compactly embedded abelian ideal, namely, the center z(m(g)) = z(g); (iii) m(g) ∩ r(g) = z(g) = z(m(g)) and m(g) = m(g) ∩ s for each Levi summand s of g. Further m(g), z(g), and m(g) are invariant under all automorphisms of g. As far as the quotient g/m(g) is concerned, we have to be circumspect; indeed if g is the 3-dimensional Heisenberg algebra, then m(g) = z(g) = [g, g] and g/m(g) is abelian, hence procompact. Nevertheless, this appears to be the only point of caution, because we have:

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Corollary 88 (12.35). The quotient algebra g/m(g) has no nondegenerate compactly embedded semisimple ideal.

Part 6. The Role of Compact Subgroups of Pro-Lie Groups The entire purpose of studying the nature of compactly embedded subalgebras of proLie algebras was to investigate compact subgroups of pro-Lie groups via their Lie theory. A test of how good our chances are that this will work is the following characterisation theorem of procompact pro-Lie algebras: Theorem 89 (Group Theoretical Characterisation of Procompact Pro-Lie Algebras; 12.36). For a pro-Lie algebra g the following statements are equivalent. (i) g is a procompact Lie algebra. (ii) There is a simply connected topological group G of the form RI × S for some set I and a compact simply connected compact group S such that g ∼ = L(G). (iii) There is a compact connected group G such that L(G) ∼ g. = (iv) There is  a unique projective compact connected group G such that L(G) ∼ = g. (v) (g) = i∈I Si for a family of Lie groups Si each of which is either isomorphic to R or else is a compact simply connected Lie group. In these circumstances, if L((g)) is identified with g as is possible by Theorem 6.4, then expg) g = (g). If H is a pro-Lie group with Lie algebra h containing g, then expH g is an analytic subgroup of H , indeed the minimal analytic subgroup with Lie algebra g. For the concept of a projective compact group, which occurred in a unique fashion in (iv) we must refer the reader to [102, Definition 9.75ff.] and Theorem 8.78ff. As a consequence and further test for the effectiveness of pro-Lie theory we derive the core structure theorem of compact connected groups from this result (12.37), which of course is discussed in source books on compact groups: Corollary 90 (The Structure Theorem of Semisimple Compact Connected Groups and the Levi–Mal’cev Structure Theorem for Connected Compact Groups; 12.37). Each compact connected group is a quotient  modulo a central totally disconnected compact subgroup of the product Z0 (G) × j ∈J Sj , where the Sj are simply connected simple Lie groups and where Z0 (G) is the identity component of the center of G. We are now beginning to look for maximal compact subgroups of a pro-Lie group, if there are any. The additive group of p-adic rationals (see Example 1.20 (A)(i)) is a nondiscrete locally compact but noncompact abelian group which is a union of an ascending chain of compact (open) subgroups; thus there is no maximal compact subgroup in such a pro-Lie group. Ofcourse, there are simple discrete abelian examples:  ∞ ∞ 1 The groups Z(p∞ ) = n=1 pn Z /Z and n=1 Z/mn Z (for a family of positive integers mn ) are countably infinite torsion groups which are the union of ascending towers

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of finite groups. It is therefore not a priori clear whether, for instance, connected pro-Lie groups have maximal compact subgroups at all. We attack this question by linking the group theory of connected pro-Lie groups with the Lie algebra theory of pro-Lie algebras. In one direction we have: Proposition 91 (12.41). Let G be a pro-Lie group and H a compact subgroup with def Lie algebra h = L(H ). Then eadg h = Ad(H ) ⊆ Aut(g) is a compact subgroup and h is compactly embedded. In the other direction we must recall what we said in Theorem 74 about the compact group GV associated with a procompact module V and find: Proposition 92 (12.42). Let G be a pro-Lie group and h a compactly embedded subalgebra of g = L(G). Assume that H is an analytic subgroup of G with Lie algebra h, and denote by V the weakly complete vector space g considered as the procompact h-module under the adjoint action. Then Ad(H ) ⊆ GV ⊆ Ad(g) ⊆ Aut(g), and Ad(H ) = Ad(H ). In particular, Ad(H ) is a compact group and agrees with ead h . When we discussed analytic subgroups we saw that for a closed subalgebra h of the Lie algebra g of a pro-Lie group G we could have many analytic subgroups H with L(H ) = h. There was always a minimal one, there may fail to be a maximal one, let alone a closed one. Yet if h is compactly embedded, the situation is better in this regard: Corollary 93. Let h be a maximal compactly embedded subalgebra of the Lie algebra g of a pro-Lie group G. Then for any analytic subgroup H of G with L(H ) = h one has L(H ) = h. In other words, among the analytic subgroups with Lie algebra h there is a closed one which is the unique largest one. If nothing else is being said we consider on Aut(g) the topology of pointwise convergence, that is, the one induced from gg . If we now summarize the essential features of this discussion we notice that on the group side, compactness occurs within the adjoint group. Indeed the center of any pro-Lie algebra is always compactly embedded, but the identity component of the center in the group may easily fail to be compact: Theorem 94 (12.45). Let h be a closed subalgebra of the Lie algebra g of a pro-Lie group G. Then the following statements are equivalent: (i) h is compactly embedded into g. (ii) Ad(exp h) = ead h is a compact subgroup of Aut(g). (iii) Ad(exp h) = ead h  is a compact subset of Aut(g). If one were to create a name for those groups which have procompact Lie algebras or those whose adjoint groups are compact one might come up with the following nomenclature:

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Definition 95 (12.46). (i) A connected pro-Lie group G is called potentially compact, if its Lie algebra g is a procompact pro-Lie algebra. (ii) A subgroup H of a pro-Lie group G is called compactly embedded, if Ad(H ) is compact in Aut(g) (with respect to the topology of pointwise convergence). With this terminology we get the following results: Corollary 96 (12.47). Let H be an analytic subgroup of a pro-Lie group G and h the Lie algebra of H inside the Lie algebra g of G. Then the following conditions are equivalent: (i) H is compactly embedded in G. (ii) h is compactly embedded in g. Theorem 97 (Characterisation of Potentially Compact Connected Pro-Lie Groups; 12.48). Let G be a connected pro-Lie group. Then the following conditions are equivalent. (i) G is potentially compact. (ii) G contains a closed weakly complete central vector subgroup V and a maximal compact subgroup C which is characteristic, such that the function μ : V × C → G, μ(v, c) = vc is an isomorphism of topological groups. (iii) There is a morphism f : G → K into a compact group with dense image such that L(f ) : L(G) → L(K) is an isomorphism. If these conditions are satisfied, then G = Z0 (G)G , where Z0 (G), the identity component of the center is isomorphic to V × comp(Z0 (G)), and where the algebraic commutator group G is a semisimple compact connected characteristic subgroup. That terminology we chose also allows us to formulate very briefly what corresponds, on the group level, to maximal compact embedded subalgebras, respectively, maximal compactly embedded abelian subalgebras on the Lie algebra level. Proposition 98 (12.52). Let G be a connected pro-Lie group and h a closed subalgebra of g = L(G). Then h is maximal compactly embedded in g, respectively, maximal compactly embedded abelian in g if and only if there is a maximal compactly embedded connected subgroup H of G, respectively, a maximal compactly embedded connected abelian subgroup H of G such that L(H ) = h.

The Conjugacy of Maximal Compact Connected Subgroups On our way to maximal compact connected subgroups of a connected pro-Lie group we are now reaching the crucial step (12.53): Theorem 99 (Maximal Compactly Embedded Connected Subgroups: Existence and Conjugacy; 12.53). Each connected pro-Lie group G contains maximal compactly embedded connected subgroups, respectively, maximal potentially compact connected abelian subgroups, and these are conjugate in G under inner automorphisms.

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The desired result on maximal connected compact, respectively, maximal compact connected abelian subgroups is now a relatively easy corollary. Corollary 100 (Maximal Compact Connected Subgroups: Existence and Conjugacy; 12.54). Each connected pro-Lie group G contains maximal compact connected subgroups, respectively, maximal compact connected abelian subgroups. Maximal compact connected subgroups, respectively, maximal compact connected abelian subgroups, are conjugate in G under inner automorphisms. Intersecting a conjugacy class of closed algebras yields an ideal, and intersecting a conjugacy class of closed subgroups yields a normal subgroup. These simple steps provide us now with these results: Theorem 101 (The Largest Compactly Embedded Connected Normal Subgroup of a Pro-Lie Group; 12.56). Let G be a connected pro-Lie group and R(G) its radical. Let Z(G)0 = V C denote the direct product decomposition of the identity component of the center into a vector group factor V and the maximal compact subgroup C = comp(Z(G)0 ) according to the Vector Group Splitting Lemma 22 for Connected Abelian Pro-Lie Groups. Then the following assertions hold. (i) There is a unique maximal compactly embedded connected normal subgroup MaxCE(G). (ii) L(MaxCE(G)) = m(g), the algebraic commutator subgroup MaxCE(G) is compact and agrees with exp m(g) , and MaxCE(G) = Z(G)0 exp m(g) . There is a quotient morphism V ×C×MaxCE(G) → MaxCE(G), given by (v, z, c)  → zc whose kernel is isomorphic to C ∩ MaxCE(G) . (iii) (MaxCE(G) ∩ R(G))0 = Z(G)0 = Z(MaxCE(G))0 and MaxCE(G) = (MaxCE(g) ∩ S)0 for each Levi factor S of G. (iv) There is a unique maximal compact connected normal subgroup MaxC(G) and MaxC(G) = comp(MaxCE(G)) = C MaxCE(G) . (v) The factor group G/ MaxC(G) has no nontrivial compact connected normal subgroups. Corollary 102 (12.57). Let G be a connected pro-Lie group. (i) G contains a largest compactly embedded connected abelian normal subgroup which is central, namely, Z(MaxCE(G))0 = Z(G)0 , the identity component of the center of MaxCE(G) and of G itself. (ii) Similarly, G contains a unique largest compact connected abelian normal subgroup which is also central, namely, Z(MaxC(G)0 ) = comp(Z(MaxCE(G))0 ) = comp(Z(G)0 ). (iii) The factor group G/ comp(Z(G)0 ) has no nontrivial compact connected central subgroups.

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Recall the three-dimensional Heisenberg group N = (R · X + R · Y + R · Z, ∗), [X, Y ] = Z, x ∗ y = x + y + 21 · [x, y]. Let G = N/Z · Z. Then Z(G) = MaxC(G) = C(G) = (R · Z)/(Z · Z) ∼ = R/Z = T. But G/ MaxC(G) ∼ = R2 is abelian and therefore 2 procompact; that is C(G/C(G)) = G/C(G) = R = {0}. Thus, in general, G/C(G) and G/ MaxC(G) may have compactly embedded normal subgroups. Sometimes it facilitates notation if one introduces a new name. Definition 103 (12.58). A group G will be called compactly simple, if it is a topological group in which every compact normal subgroup is singleton. With this terminology we finally get: Theorem 104 (Largest Compact Normal Subgroup of a Pro-Lie Group; 12.59). Every connected pro-Lie group G has a unique largest compact normal subgroup MaxK(G) and G/ MaxK(G) is a compactly simple connected pro-Lie group. Yamabe’s Theorem ([206], [207]) states that every locally compact almost connected group is a pro-Lie group. In a locally compact pro-Lie group G, all sufficiently small members of the standard filter basis N (G) are compact. In view of these two facts we see that a locally compact almost connected Lie group is compactly simple only if it is a compactly simple Lie group. Naturally, we now wish to know as much as possible about compactly simple groups. The following is a step in this direction. Proposition 105 (12.60). Let G be a connected pro-Lie group without nontrivial compact central subgroups. Then the following conclusions hold. (i) Every compact connected normal subgroup is semisimple and center-free. (ii) The center Z(G) of G is an abelian pro-Lie group isomorphic to Z(G)0 × H , where the identity component Z(G)0 of the center Z(G) is a vector group isomorphic to RI for some set I and where H is a totally disconnected subgroup of Z(G) which contains no compact subgroups. (iii) The nilradical N (G), that is, the largest pronilpotent connected closed normal subgroup, is simply connected and thus is isomorphic to (n(g), ∗) where n(g) is the nilradical of g = L(G). (iv) The radical R(G) is a semidirect product V  K of a simply connected normal subgroup V containing N (G) and a compact connected group.

The Analytic Subgroups Whose Lie Algebras Equal That of the Full Group We have the ingredients for significant results on the unique minimal analytic subgroup A(g, G) = expG g with full Lie algebra g in any pro-Lie group G. We discover, that a certain characteristic, but not necessarily connected, supergroup of the radical R(G) plays an important role.

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Definition 106 (12.63). Let G be a connected pro-Lie group, let q : G → G/R(G) def

denote the quotient map and define Q(G) = q −1 (KZ(G/R(G))). Call Q(G) the extended radical of G. So the extended radical is an extension of the radical by a compact totally disconnected abelian group. Of course we would like to know some typical properties of this extended radical. Proposition 107 (12.64). For a connected pro-Lie group G, the extended radical Q(G) satisfies the following conditions: (i) Q(G) is a prosolvable characteristic subgroup of G whose identity component is the radical R(G). (ii) Q(G) contains a closed subgroup V that is normal in Q(G), contains the nilradical N (G) of G and is simply connected modulo KZ(G)0 . (iii) Q(G) contains a maximal compact abelian subgroup K such that Q(G) = V K and V ∩ K = KZ(G)0 . (iv) K contains KZ(G). (v) There is a totally disconnected compact subgroup D of K such that K = A(L(K), K)D and Q(R) = R(G)D. Now we are ready for a completely general result on the dense characteristic subgroup A(g, G) = expG g of a connected pro-Lie group G. Theorem 108 (Supplementing the Minimal Analytic Subgroup Generated by the Full Lie Algebra; 12.65). Let G be a connected pro-Lie group and K a maximal compact subgroup of the extended radical Q(R). Then G = K · A(g, G). There is a totally disconnected compact subgroup D of K such that G = D · A(g, G) and Q(G) = R(G)D. There is a connected compact abelian subgroup C of G containing K. In particular, G = C · A(g, G) is generated by divisible groups. This result has many consequences. One immediate outcome is the following: Corollary 109 (12.66). The abstract group G/A(g, G) is abelian. The algebraic commutator subgroup of G is contained in A(g, G). The next corollary, however, requires the full power of Open Mapping Theorem for Almost Connected Pro-Lie Groups (Theorem 8). In addition one has to ascertain  via that the group G acts automorphically on the simply connected universal group G  a homomorphism α : G → Aut(G).

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Corollary 110 (The Resolution Theorem for Connected Pro-Lie Groups; 12.68). Let G be a connected pro-Lie group. Then there is a totally disconnected compact subgroup D of the extended radical and a quotient morphism of pro-Lie groups  α D → G, δ: G The morphism

δ(x, k) = πG (x)k.

−1 x  → (x −1 , πG (x)) : πG (D) → ker δ

is an isomorphism of prodiscrete groups. The morphism δ induces an isomorphism L(δ) of pro-Lie algebras. There is an exact sequence   D → G → 1. 1 → π −1 (D) → G δ

The following consequence is a kind of Levi–Mal’cev decomposition theorem on the group level. We recall that it is rather frustrating in general to imitate the clean Levi–Mal’cev splitting of the Lie algebra on the group level. Corollary 111 (12.69). Let G be a connected pro-Lie group and let g = r(g) + s be a Levi–Mal’cev decomposition. Then G = Q(G) · A(s, G), A(s, G) = expG s. Under special conditions more elegant conclusions may be drawn, like for instance the following: Theorem 112 (12.70). Let G be a connected pro-Lie group and let g = r(g) + s be a Levi decomposition of its Lie algebra. Assume the following hypotheses: (i) [r(g), s] = {0}, and (ii) R(G)/ KZ(G)0 is simply connected. Then G = KZ(G) · A(g, G). The preceding results motivate a further definition: Definition 113 (12.72). A connected pro-Lie group G will be called centrally supplemented if  G = Z(G)A(g, G) = Z(G)πG (G). By Theorem 112, for example, all reductive connected pro-Lie groups and all prosolvable pro-Lie groups which are simply connected modulo the maximal connected central compact subgroup are centrally supplemented. The Center-Free Embedding Lemma 35 gives us plenty center-free metabelian connected pro-Lie groups G in which A(g, G) is a proper subgroup and which, therefore, are not centrally supplemented. For centrally supplemented connected pro-Lie groups, there exist more elegant versions of Theorem 108 and Corollary 110, namely: Theorem 114 (The Resolution Theorem for Centrally Supplemented Pro-Lie Groups; 12.74). Let G be a connected centrally supplemented pro-Lie group. Then there is a prodiscrete central subgroup D of G such that there is a quotient morphism  × D → G, δ: G

δ(x, d) = πG (x)d

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inducing an isomorphism L(δ) of Lie algebras, and the kernel ker δ is prodiscrete and isomorphic to π −1 (D) ∩ A(g, G). There is an exact sequence  × D → G → 1. 1 → π −1 (D) → G δ

 → G that already the We noted in discussing the universal morphism πG : G example of compact connected abelian groups shows that for non-Lie groups, πG fails rather significantly to be a universal covering morphism (which it is if G is a Lie group!). In general it is neither surjective, nor open, let alone a local isomorphism. The morphism δ in the Resolution Theorems is as close as one can get to adjust πG in such a fashion that something resembling a universal covering results: At least it is surjective, open, and has a prodiscrete kernel. Resolution Theorems for compact abelian groups and compact groups were described for the first time in our book [102] in Chapters 8 (Theorem 8.20) and 9 (Theorem 9.51). One of the applications we are making of (a special case of) the Resolution Theorems is that we can prove a fact on maximal compact subgroups of connected pro-Lie groups that is overdue. We know that maximal compact connected subgroups exist and are conjugate, and we know that a unique maximal compact normal subgroup exists. Yet up to this point we do not know whether maximal compact subgroups exist. Now in this direction we show that maximal compact subgroups of connected pro-Lie groups exist and are connected, and therefore are all conjugate. However, the situation is really better. We shall come up with a significant result, that illustrates quite well our motivation to invest so much energy into the investigation of compact and notably maximal compact subgroups. The core theorem is the following. Theorem 115 (Theorem on the Maximal Compact Subgroups; 12.81). Let G be an arbitrary connected pro-Lie group. Then: (i) (ii) (iii) (iv)

G has at least one maximal compact subgroup C. Every maximal compact subgroup is connected. All maximal compact subgroups are conjugate under inner automorphisms. There exists a set J and a homeomorphism ε : C × RJ → G such that ε(C × {0}) = C. Also Ga = ε(Ca × RJ ) = expG g.

None of these statements is obvious. Our proof gives many more details about Statement (iv). This last one, taken together with the Borel–Scheerer–Hofmann Splitting Theorem (see [102, Theorem 9.39], and see also Theorem 11.8 in this book) gives the following statement: Theorem 116 (Theorem on the Topological Splitting of Pro-Lie Groups; 12.87). Every connected pro-Lie group is homeomorphic to a direct product of a compact connected semisimple group, a compact connected abelian group, and a space RJ for a set J . Clearly, this gives us a characterisation of the local compactness of a connected pro-Lie group. The cardinal of J is invariantly attached to G, and we are justified (as we shall see in detail) to call this cardinal the dimension of the quotient space G/C.

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We shall call this cardinal the manifold rank of G. The group G is locally compact if and only if its manifold rank is finite. These theorems will be anticipated a few times in the course of the book, so for instance in the structure theorem for connected abelian pro-Lie groups (Theorem 23) and Lemma 22, or the Structure Theorem 36 for Simply Connected Pro-Lie Groups. An obvious but striking consequence is that all the information that algebraic topology (homotopy, homology, cohomology) gives us on compact connected groups yields this precise information on connected pro-Lie groups, since according to the preceding theorems each maximal compact subgroup C of a connected pro-Lie group G is a homotopy deformation retract and thus is homotopically equivalent (that is, isomorphic in the homotopy category) to G. Another consequence will be that a connected pro-Lie group G is locally compact if and only if the linear codimension of the Lie algebra L(C) of a maximal compact subgroup C of G is finite. This is equivalent to saying that the factor group G/ MaxK(G) of G modulo the unique largest compact normal subgroup is a Lie group. (The proof of this fact uses Yamabe’s Theorem.) It does perhaps not come as a surprise that with the powerful tools that these results provide one can successfully revisit the scenes of earlier investigations. One example is the Open Mapping Theorem which we presented early in this overview, namely, in Theorem 8, being aware that it requires tools that become available at a later stage. The setting of Open Mapping Theorems is always the same: We have a surjective morphism f : G → H of topological groups and we are hoping for additional sufficient conditions which will allow us to conclude that f is an open mapping and thus is equivalent to a quotient morphism. The perennial illustration is the identity map Rd → R from the discrete additive group of real numbers to the additive group of real numbers in its natural topology. This is a bijective morphism between abelian Lie groups that fails to be open. If G and H are pro-Lie groups we have seen in Theorem 8 that if G/G0 is compact then f is open. We will show the following result. Theorem 117 (Alternative Open Mapping Theorem; 12.85). Assume that f : G → H is a surjective morphism from a pro-Lie group G onto a connected pro-Lie group H and that the quotient group G/ ker f is complete. If the morphism L(f ) : L(G) → L(H ) of pro-Lie algebras is surjective, then f is open. In Theorem 4.20 and its corollaries which we previewed in Theorem 15 above we shall show that if f is open, then L(f ) is a quotient morphism. We can express the gist of the situation by saying A bijective morphism f : G → H of pro-Lie groups, of which at least one is connected, is an isomorphism if and only if it induces an isomorphism L(f ) : L(G) → L(H ) of Lie algebras. When formulated in this fashion it becomes evident that this circle of ideas belongs genuinely to a Lie theory of pro-Lie groups. This last theorem permits us to prove a structure theorem, well known but not trivial for Lie groups and a hard fact for pro-Lie groups:

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Theorem 118 (12.87). The center of a connected pro-Lie group is contained in some closed connected abelian subgroup.

Part 7. Local Splitting According to Iwasawa We already referred to Iwasawa’s paper [120] of 1949 as one of the most influential contributions to the structure theory of topological groups. One of the results in that paper turned out to be immensely practical and like no other result it illustrates how the structure theory of locally compact groups reduces largely (but not completely!) to the structure theory of compact groups and classical Lie theory. Theorem 119 (Iwasawa’s Local Splitting Theorem for Connected Locally Compact Groups). Any identity neighborhood of a connected locally compact group G contains a compact normal subgroup N such that G/N is a Lie group and that the groups G and N × G/N are locally isomorphic.  is  to G, where L This enables us to produce a quotient morphism from N × G/N the universal covering group of a connected Lie group L and where the kernel of this morphism is discrete. In general, N is not connected, but this quotient map is the next best thing to a covering morphism. Any general theory of pro-Lie groups will have to be measured by the elucidation that it offers of this result. Does it generalize to connected pro-Lie groups? The answer, alas, is no. In Chapter 13, which deals with this topic we illustrate this negative situation by producing a center-free pronilpotent pro-Lie group, a metabelian center-free proLie group and a class two nilpotent pro-Lie group, none of which permits locally splitting in the sense of the Iwasawa Local Splitting Theorem. The latter example is the Heisenberg group of the realm of pro-Lie groups. So what, if anything, is the obstruction that prevents us from proving a straightforward generalisation? Let G be an arbitrary pro-Lie group. Note right away that we do not insist that it be connected. Its identity component G0 has a largest connected normal pronilpotent subgroup N(G0 ) which we call the nilradical which we already mentioned several times. Naturally, it contains the identity component Z(G0 )0 of the center of G0 . We shall call the quotient N(G0 )/Z(G0 )0 the nilcore of G, and we shall also abbreviate it by N (G). We shall show (13.16) that the nilcore always is a simply connected pronilpotent pro-Lie group. We know from Theorem 34 and its context that its Lie algebra n has an everywhere defined Campbell–Hausdorff multiplication ∗ : n × n → n,

X ∗ Y = X + Y + 21 [X, Y ] + · · ·

such that the exponential function exp : (n, ∗) → N (G) defines an isomorphism of pro-Lie groups. The Lie algebra of the nilcore determines its structure completely. It is homeomorphic to the weakly complete topological vector space underlying n which is isomorphic to RJ for a set J with a uniquely defined cardinality which we also call the nildimension of G, written ν(G). That is a cardinal valued invariant of G.

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The counterexamples mentioned above all have infinite nildimension. We show in Chapter 13 the following: Theorem 120 (The Local Splitting Theorem for Pro-Lie Groups; 13.19). Let G be a pro-Lie group and assume its nildimension ν(G) to be finite. Then every identity neighborhood contains an almost connected normal pro-Lie subgroup N such that G/N is a Lie group and the groups G and N × G/N are locally isomorphic. No connectivity is required, but apart from the hypothesis that nildimension be finite nothing is required – except, of course, that we are talking about pro-Lie groups. We recognize, that the obstruction is the nilcore. On the other hand, we should also recognize that it is easy to visualize even nilpotent pro-Lie groups of arbitrary nildimension that do have local splitting, for instance, if H3 is the three-dimensional Heisenberg def

group, then G = H3J for an infinite set J is a class two nilpotent pro-Lie algebra of nildimension ν(G) = card J , and by its very construction, G has local splitting. The hypothesis ν(G) < ∞ implies global structural results: Theorem 121 (The Finite Nildimension Theorem;13.19, 13.22). Let G be a pro-Lie group. Then the following statements are equivalent: (i) The nildimension ν(G) is finite. (ii) G is locally isomorphic to the product of a closed normal almost connected subgroup with a reductive identity component, and a connected Lie group. If these conditions are satisfied, then there is a compact totally disconnected central  → G, μ(d, x) = dπG (x) subgroup D of G and an open surjective morphism μ : D×G with a prodiscrete kernel. However, in Theorems 120 and 121, the hypothesis ν(G) < ∞ is acceptable insofar as it is implied by the assumption that G be locally compact. Indeed: if G is locally compact so is its nilcore N (G), and since we saw that N (G) is homeomorphic to RJ with card J = ν(G) we see that N (G) is locally compact iff ν(G) < ∞. Therefore we deduce the Iwasawa Local Splitting Theorem for Locally Compact Groups in the following form which is more general than the classical version: Corollary 122 (Iwasawa’s Local Splitting Theorem Revisited). Let G be a locally compact pro-Lie group. Then there is an open subgroup G of G and there are arbitrarily small compact normal subgroups N of G such that G/N is a Lie group and G and N × G/N are locally isomorphic. Classically, the Iwasawa Local Splitting Theorem 119 required the hypothesis of connectivity. The presentation of the local splitting theory in the frame work of proLie group theory, culminating in Theorem 120 and Corollary 118 does not require this hypothesis, thanks to an effective pro-Lie theory. ——————— Pro-Lie-tarians of the world: unite! Karl Morrix, 2007

Chapter 1

Limits of Topological Groups

One of our primary motivations is to expose the structure of locally compact connected groups. A basic structural feature of locally compact groups is that they are in many ways close to Lie groups. Indeed they have a Lie algebra and an exponential function, and their identity components can be expressed as projective limits of Lie groups, or, more informatively, can be approximated by Lie groups. Lie theory essentially captures only identity components. The category of locally compact groups has serious defects, for instance even RN is not in the category. Our approach is to find a well-behaved category which has a good Lie theory and contains at least all connected locally compact groups. Several categories present themselves for study, as does the categorical concept of projective limit, or more generally, limit in a category. So we begin the book with that part of category theory which is crucial for our purpose. Category theory is introduced not because it offers greater generality, but rather because it provides greater clarity. The projective limit of topological groups provides a systematic device for approximating a topological group by other topological groups which may be better understood. There are two approaches to projective limits, firstly, an external one building up the limit from families of given topological groups, secondly, an internal one showing how a given group can, in suitable circumstances, be approximated by quotient groups. For the structure theory of locally compact groups we shall see that both of these aspects are relevant. This chapter begins with a study of limits in a category and then proceeds to describe projective limits of topological groups from an external, and then an internal, perspective. Prerequisites. We shall assume that the reader is familiar with basic category theory for which the self-contained Appendix 3 of [102] is an appropriate source. For the external theory of projective limits we need only an elementary knowledge of set theory and some familiarity with morphisms (that is, continuous homomorphisms) of topological groups; for the internal theory we use elementary information about topological groups including completeness (see e.g. [25, Chap. III, §3] and [176]).

Limits We begin by considering a category C (see e.g. [102, Definition A3.1ff.]). We denote the class of objects of C by ob C. If C1 , C2 ∈ ob C, then we denote the set of all morphisms C1 → C2 by C(C1 , C2 ) or Hom(C1 , C2 ) if the category is understood.

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One of the most basic concepts in the analysis of functors (see e.g. [102, Definition A3.17]) is that of a natural transformation; we begin by recalling that concept. Definition 1.1. Let A and B be categories and let S, T : A → B be two functors. A natural transformation α : S → T assigns to each object A ∈ ob A a B-morphism αA : SA → T A such that the following diagram commutes for all f ∈ C(A, A ). SA ⏐ ⏐ αA 

Sf

−−−−−→

T A −−−−−→ Tf

 SA ⏐ ⏐α   A

T A

If all αA are isomorphisms, then α is called a natural isomorphism. A category is called small if its class of objects is a set rather than a proper class; then also its class of morphisms is a set as a union of a set of sets. We shall say that a functor D : J → C from an “index category” J to a category C is a diagram if J is a small category. Consider a fixed index category J and a category C; let ob C J be the class of all functors D : J → C, and for two functors D1 , D2 ∈ ob C J let C J (D1 , D2 ) be the class of all natural transformations α : D1 → D2 from D1 to D2 . Then C J is a “generalized” category: All axioms of a category are satisfied with the possible exception of that which demands that hom-sets be sets rather than proper classes; if J is a small category, this condition is satisfied as well and C J is called the category of diagrams on J . For each object A of C we obtain the constant functor const(A) : J → C mapping all objects j of J to A and all morphisms of J to idA . A natural transformation α : const(A) → D is called a cone with vertex A. Any morphism f : A → B in C gives a natural transformation of functors const(f ) : const(A) → const(B). If J is a small category, the assignment const is a functor const : C → C J . The Definition of Limits and Limit Cones Definition 1.2. Let J and C be categories. A functor D : J → C is said to have a limit lim D ∈ ob C if there is a cone λ : const(lim D) → D, called the limit cone such that for each cone α : const A → D there is a unique morphism α  : A → lim D with α = λ  const(α  ). It may serve a useful purpose to visualize this in the following diagram const(α  )

const(A) −−−−−→ const(lim ⏐ ⏐ D) ⏐ ⏐ α λ D −−−−−→ D. idD

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Limits are unique in the following sense. If λ : const(L) → D and μ : const(M) → D are limit cones then there exists a unique isomorphism ε : L → M such that the following diagram commutes: const(ε)

const(L) −−−−−→ const(M) ⏐ ⏐ ⏐ ⏐μ λ  D −−−−−→ D. idD

Indeed we take ε = λ and note ε−1 = μ . Proposition 1.3. The assignment ν : C(A, lim D) → C J (const(A), D),

ν(f ) = λ  const(f ) : const(A) → D

is a bijection with inverse ν −1 (α) = α  . Proof. Exercise E1.1. Exercise E1.1. Verify the details of a proof of the uniqueness of the limit. Prove Proposition 1.3. We shall hasten to see that many familiar constructs are in fact limits. A category J is called discrete if it has no morphisms except the identity morphisms. We may say that a discrete category has only objects and no morphisms (with the exception of those which it has to have, namely, the identity morphisms). The Definition of Products, Equalizers, Pullbacks, and Projective Limits Definition 1.4. (i) If J is a discrete category, then a diagram D : J → C is nothing but a family of objects {D(j ) | j ∈ J }. The limit lim D of D is called the product of the family in C, written P = j ∈J Aj , Aj = D(j ), and the morphisms λj : P → Aj are called projections, usually written prj . If D = {A, B}, one writes lim D = A × B and prA , prB for the projections. If α : X → A and β : X → B are morphisms, one writes (α, β) : X → A × B for the unique fill-in morphism of the product. (ii) If f, g : A → B are two morphisms in C, then an object E is called an equalizer eq(f, g) of f and g if there is a morphism e : E → A such that f e = ge and for every morphism ϕ : X → A with f ϕ = gϕ there is a unique morphism ϕ  : X → E with eϕ  = ϕ. (iii) If f : A → C and g : B → C are two morphisms in C, then an object P is called a pullback or fibered product of f and g written P = A ×C B (somewhat incompletely, since P depends on f and g), if there are morphisms p : P → A and q : P → B such that fp = gq and that for each pair of morphisms α : X → A and β : X → B with f α = gβ there is a unique map ξ : X → P such that α = pξ and β = qξ .

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The entire diagram

p

P ⏐ ⏐ q

−−−−−→ A ⏐ ⏐f 

B

−−−−−→ C g

is called a pullback if it has the universal property described above. If both f and g are monics, then the pullback P is called an intersection. (iv) The limit of a diagram D : J → C is called finite if J is finite. (v) Let J be a partially ordered set, that is, a set with a reflexive, transitive and antisymmetric relation ≤. Then J can be considered as a category, whose objects are the elements of J and in which for two objects j, k ∈ J a unique morphism k → j exists if j ≤ k. A set is said to be directed if it is partially ordered and every finite nonempty subset has an upper bound. If J is a directed set, then a diagram D : J → C is called a projective system in C, and if D has a limit lim D then it is called a projective limit in C. The projective system D is often written as a family of morphisms {fj k : Ck → Cj | (j, k) ∈ J × J, j ≤ k}, where D(j ) is written Cj and D(k → j ) is written fj k . Exercise E1.2. Show that equalizers and pullbacks are limits. Prove that for a directed set J , a family of maps and objects P = {fj k : Ck → Cj | (j, k) ∈ J × J, j ≤ k} is a projective system if the following conditions are satisfied: (i) (∀j ∈ J ) fjj : Cj → Cj is the identity morphism of Cj . (ii) (∀i, j, k ∈ J ) (i ≤ j ≤ k) ⇒ fik = fij fj k . [Hint. In order to show that equalizers and pullbacks are limits exhibit appropriate small categories J and functors D : J → C. Set ⎫ ⎧ ⎪ • ⎪ ⎬ ⎨   1 ↓1 • and J = Je = • → pb → ⎪ ⎭ ⎩• → • ⎪ 2 2

and notice that a functor D : Je → C is just a pair of morphisms with common domain and common codomain (ignoring the identity morphisms), the limit being their equalizer, while a functor D : Jpb → C is just a pair of morphisms with the same codomain, its limit being their pullback. If P is given, make J into a category with one morphism k → j iff j ≤ k and define an assignment D : J → C on objects by D(i) = Cj and on morphisms by D(k → j ) = fj k and use conditions (i) and (ii) to show that D is a functor.] It is important to recognize that frequently a limit can be constructed by using other types of limits. The following three theorems are significant examples. Theorem 1.5. (a) (Pullbacks via Finite Products and Equalizers) Let f : A → C and g : B → C be morphisms such that the product A × B exists and the equalizer

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e : P = eq(f pr A , g pr B ) → A × B exists. Then P is the pullback of f and g with p = pr A e and q = pr A e. (b) (Products via Projective Limits and Finite Products) Let (Gα : a ∈ A) be a family of objects. Let J be the set of all finite  subsets j of A. Then J is a directed set with respect to ⊆ and we define Gj = α∈j Gα for j ∈ J . For j ⊆ k in J let pr j k : Gj → Gk be the projection defined via the universal property of the product Gk . Then P = {pr j k : Gk → Gj | (j, k) ∈ J × J, j ⊆ k} is a projective system. Let G = lim P = limj ∈J Gj be the projective limit of P and let pj : G → Gj be the limit maps.Note Gα = G{α} for α ∈ A and define pr α : G → Gα by pr α = p{α} . Then G = α∈A Gα with the projections pr α : G → Gα . Proof. Exercise E1.3 (i). We shall denote the identity morphism of an object Z in a category by idZ . A morphism f : X → Y in a category is called a retraction (respectively, coretraction) if there is a morphism g : Y → X such that f g = idY (respectively, gf = idX ). If f : X → Y is a retraction, then Y is called a retract of X. Exercise E1.3. (i) Prove Theorem 1.5. (ii) Prove the following statement on pullbacks: Assume that δ D ⏐ −−−−−→ A ⏐ ⏐ ⏐a d  B −−−−−→ C b

is a pullback diagram and that a is a retraction, that is, there is a c : C → A such that ac = idC . Then d is a retraction. [Hint. (i) For a proof of (a), verify the universal property of the pullback by “diagram chasing”. In order to prove (b), let fα : X → Gα , α ∈ A be a family of morphisms. For each j ∈ J , by the universal property of the finite product Gj we find a unique morphism fj : X → Gj such that pr α fj = fα for α ∈ j . We note that j ⊆ k implies fj = pr j k fk (diagram chasing and uniqueness in the product property). Hence the universal property of the projective limit yields a unique f : X → G such that fj = pj  f for all j ∈ J . In particular, f is a unique map such that fα = prα  f (ii) The composition b  idB = b agrees with acb = b. By the universal property of the pullback there is a unique β : B → D such that dβ = id B (and δβ = cb).] Definition 1.6. It is said that a category has pullbacks if for each pair of morphisms with the same codomain the pullback exists. Similarly it is said that a category has equalizers if each pair of morphisms f, g : A → B has an equalizer, that it has products if each family has a product, that it has projective limits if each projective system has a limit, that it has finite limits, if every finite diagram has a limit and that it has limits if every (small) diagram has a limit.

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We say that two coretractions r, s : A → B are coherent if pr = ps = idA for one and the same retraction p. It is said that a category has intersections of coherent retracts if every pair of coherent coretractions has a pullback. We should mention that the coherent retracts constitute a technical concept that occasionally allows us to minimize the number of tests we have to perform in order to show that a given category has limits. The following theorem will turn out to be very important for certain applications to categories with which we work. Let us assume that we are given two morphisms f, g : A → B and that the product A × B exists. Let pr A : A × B → A and pr B : A × B → B denote the projections. Then the universal property of the product (see 1.4) yields two unique morphisms r, s : A → A × B such that (α) idA = pr A r = pr A s and (β) f = pr B r, g = pr B s, that is, r = (idA , f ) and s = (idA , g). In view of (α) we note that r and s are coherent coretractions and thus, in particular, are monics. Therefore, if the pullback p E A ⏐ −−−−−→ ⏐ ⏐ ⏐r q (P)  A −−−−−→ A × B s

exists, then it is an intersection. Theorem 1.7. (a) (Equalizers via Finite Products) Assume that f, g : A → B are morphisms, and that the product A × B and the pullback (P) exists. Then p = q and E is the equalizer of f and g with e = p = q. (b) (Intersections of Coherent Retracts) A category has pullbacks if it has finite products and intersections of coherent retracts. (c) (Products via Projective Limits and Finite Products) A category has products if it has projective limits and finite products. Proof. (a) From rp = sq we get p = idA p = (pr A r)p = pr A (rp) = pr A (sq) = · · · = q. We set e = p = q and claim that f

e −−−−→ E −−−−−→ A − −−−−−→ B

(E)

g

is an equalizer diagram. Indeed let ϕ : X → A be a morphism such that f ϕ = gϕ. We claim that this implies rϕ = sϕ. Indeed by the uniqueness of the universal property of the product (see 1.4) this follows from pr A rϕ = ϕ = pr A sϕ and pr B rϕ = f ϕ = gϕ = pr B sϕ. But now the universal property of the pullback (P) yields a unique morphism ε : X → E such that ϕ = pε = qε = eε. Thus the universal property of the equalizer (E) is verified. (b) Assume that f, g : A → B are morphisms in a category which has finite products and intersections of retracts. Since the category has finite products, the product A × B exists, and since it has intersections of retracts, the pullback (P) exists because r and s are coherent coretractions. Thus the equalizer of f, g : A → B exists by (i) above. (c) This is immediate from Theorem 1.5(b).

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Exercise E1.4. Complement the proof of Theorem 1.7 by drawing the appropriate diagrams which we used in the proof and verify our arguments by “diagram chasing”. Theorem 1.8 (Arbitrary Limits via Equalizers and Arbitrary Products). Let D : J → C denote an arbitrary diagram. Hypotheses: Assume that the following products exist:  (i) P = j ∈ob J D(j ), (ii) Q = f ∈morph J D(ran f ), where ran f is the codomain (or range) of f . For each f ∈ morph J define two morphisms αf , βf : P → D(ran f ) by αf = (Df )  pr dom f and βf = pr ran f . By the universal property of the product Q there exist unique morphisms α  , β  : P → Q with αf = pr f  α  and βf = pr f  β  . Assume that the following equalizer exists: (iii) L = eq(α  , β  ), e : L → P . Conclusions: The prescription λj = pr j  e : L → D(j ) defines a cone λ : const(L) → D and L is the limit of D with respect to this cone. Proof. (a) For each f : j → k in J we must show that λk = (Df )λj , i.e. pr k e = (Df ) pr j e. This means αf e = βf e which is equivalent to pr f α  e = pr f β  e, and since e is the equalizer of α  and β  this condition is satisfied. (b) Let ξ : const X → D be a cone. By the universal property of P , there is a unique morphism ξ  : X → P such that ξj = pr j ξ  for all j . Let f : j → k in J . Then pr f α  ξ  = αf ξ  = (Df ) pr j ξ  = (Df )ξj = ξk = pr k ξ  = βf ξ  = pr f β  ξ  . By the uniqueness in the universal property of Q we conclude α  ξ  = β  ξ  . Since L is the equalizer of α  and β  we conclude the existence of a unique ξ  : X → L such that ξ  = eξ  . Then ξj = pr j ξ  = pr j eξ  = λj ξ  . The uniqueness of ξ  follows from its construction. Definition 1.9. (i) A category is said to be complete if every diagram has a limit. (ii) A subcategory A of a category C is called full if for each pair of objects A1 , A2 in ob A, the equation A(A1 , A2 ) = C(A1 , A2 ) holds. (iii) A subcategory A of a category C is said to be closed in C under the formation of limits, or under forming limits if for every diagram D : J → A that has a limit lim D in C, an isomorphic copy of lim D is in ob A. (iv) A subcategory A of a category C is said to be closed under passing to intersections of coherent retracts if for each pullback P ⏐ −−−−−→ A ⏐ ⏐ ⏐r   A −−−−−→ B s

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in C for which r, s is a pair of coherent coretractions, an isomorphic copy of P is in ob A. Assume that D : J → A is a diagram in a full subcategory A of a category C. (∗) If a limit cone λ : const(lim D) → D exists in C, if lim D ∈ ob A, and if A is full, the morphisms λj are all in A, by fullness, λ : const(lim D) → D is a limit cone in A. Accordingly, we record that (∗∗) any full subcategory A of a complete category C is complete if it is closed under the formation of limits. The results we have recorded so far yield the following important criteria. The Limit Existence Theorem Theorem 1.10. (i) If a category has finite products and equalizers, then it has finite limits. If a category has arbitrary products and equalizers, then it is complete. (ii) If a category has finite limits and projective limits, then it is complete. (iii) If a category has arbitrary products and has intersections of coherent retracts, then it is complete. (iv) If a full subcategory A of a complete category C is closed in C under the formation of products and passing to intersections of coherent retracts, then it is closed under the formation of all limits and is therefore complete. Proof. (i) This follows from 1.8. (ii) This follows from (i) and 1.7 (c). (iii) By 1.7, if a category has products and intersections of coherent retracts, it has equalizers, and thus by (i) it is complete. (iv) In view of the remark (∗∗) preceding the theorem and by 1.7 the subcategory A of C is complete iff it is closed under the formation of all limits in C. By 1.8 and (∗) it is closed under all limits if it is closed under the formation of equalizers and arbitrary products. By 1.7 and (∗) it is closed under the formation of equalizers if it is closed under the formation of finite products and finite intersections of coherent retracts. Therefore A is complete. This theorem is of great importance in demonstrating the existence of limits. It reduces the verification of the existence of limits to the existence of intersections and products. In general, these are readily verified or refuted. In our applications it will be important to know that a full subcategory of a complete category is itself complete, if it is closed under the formation of products and the passing to intersections. It suffices in fact to test the passage to intersections of coherent retracts. Of course, a category has equalizers if it has finite products and intersections, and it has intersections if it has pullbacks. Our emphasis on the existence of the intersection of coherent retracts is the result of an effort to point out that one needs to test very few

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intersections in order to verify the completeness of a category. Nevertheless, it will often be the case that one is going to test the existence of either equalizers directly, or of pullbacks, or of intersections. Let TopGr denote the category of topological groups and continuous group homomorphisms between them. It is important for us to understand retractions and retracts in TopGr. The following exercise elucidates this issue. Exercise E1.5. Prove the following assertions: (i) Assume that p : G → H is a retraction in the category of topological groups. Set N = ker p, and let j : H → G be a morphism such that pj = idH . For h ∈ H let Ih denote the automorphism of N given by Ih (n) = j (h)nj (h)−1 . Then the product N ×H , endowed with the multiplication (n1 , h1 )(n2 , h2 ) = (n1 Ih1 (n2 ), h1 h2 ) is a topological group and the function μ : N × H → G, μ(n, h) = nj (h) is an isomorphism of topological groups. The topological group on the product N × H is called the semidirect product N  H ; we shall return to this subject time and again, but note here that N will be called a semidirect factor and H a semidirect cofactor. (ii) Let r, s : H → G be two coherent coretractions with respect to one and the same retraction p : G → H . Then H acts on N via h · n = r(h)nr(h)−1 , and by (i) we may assume without losing generality that G = N × H with the multiplication (n1 , h1 )(n2 , h2 ) = (n1 (h1 · n2 ), h1 h2 ) and that p(n, h) = h, r(h) = (1, h), and s(h) = (γ (h), h) with a continuous function γ : H → N satisfying γ (h1 h2 ) = γ (h1 )(h1 · γ (h2 )).

(∗)

Then im r = {1} × H and im s = {(γ (h), h) : h ∈ H } are coherent retracts of G and their intersection is isomorphic to ker γ = {h ∈ H : γ (h) = 1}. A function γ : H → N between groups H and N such that H acts on N in such a way that each n  → h · n : N → N is an automorphism of N satisfying (∗) is called a cocycle or a crossed homomorphism; we shall call the set of elements in its domain which are mapped to the identity of the range its kernel, just as for homomorphisms between groups. (iii) If we define ϕ, ψ : G → G by ϕ(n, h) = (1, h) and ψ(n, h) = (γ (h), h), then ϕ and ψ are idempotent endomorphisms of topological groups and their equalizer {g ∈ G : ϕ(g) = ψ(g)} is isomorphic to N × ker γ . (iv) If H is a topological group acting continuously on a topological group N via automorphisms, and if γ : H → N is a continuous cocycle, then ker γ is the intersection of a suitable pair of coherent retracts.

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[Hint. (i) Show that μ is a homeomorphism, and that it satisfies μ((n1 , h1 )(n2 , h2 )) = μ(n1 , h1 )μ(n2 , h2 ) iff N × H is given the multiplication as specified. (ii) Use the multiplication formula to verify the functional equation of γ , and observe that (n, h) ∈ im r ∩ im s iff n = γ (h) = 1. (iii) is straightforward. def

(iv) We consider G = N × H with the multiplication specified in (ii) and define r, s : H → G by r(h) = (1, h), s(h) = (γ (h), h). These functions are the required coherent retractions.] We have observed the following facts. In the category TopGr of topological groups the intersection of two coherent retracts is the kernel of a continuous cocycle, and that the kernel of a continuous cocycle is the intersection of two coherent retracts. A subcategory of TopGr is certainly closed under passing to coherent retracts if it is closed under the passing to closed subgroups. We introduce names for frequently cited subcategories of the category TopGr of topological groups: Let CompGr be the category of all (Hausdorff) compact groups and continuous group homomorphisms as morphisms, and LCGr the category of all (Hausdorff) locally compact groups with continuous group homomorphisms. The category of all (Hausdorff) connected groups and continuous group homomorphisms as morphisms is denoted by ConnGr. In a topological group G let G0 denote the connected component of the identity. Completeness of TopGr and of its Subcategories Theorem 1.11. (i) The categories TopGr and CompGr are complete categories. In particular, all projective limits exist in each of them. (ii) Let G be any full subcategory of TopGr. If G is closed in TopGr under the formation of products and the passing to intersections of coherent retracts, that is, passing to kernels of continuous cocycles, then G is closed in TopGr under the formation of all limits and is therefore a complete category. The subcategory G satisfies these conditions certainly if it is closed under products and passing to closed subgroups. (iii) The category LCGr is not complete. Indeed, a countably infinite family of copies of R fails to have a product in LCGr. (iv) Any family {Dj | j ∈ J }, of discrete  torsion-free abelian groups has a product in LCGr, namely, the cartesian product j ∈J Dj with the discrete topology. This is the case for example if each Dj equals Z, the discrete additive group of integers. (v) The category ConnGr is not closed in TopGr under the formation of limits. Yet ConnGr is a complete category. If D : J → ConnGr is a diagram and limTopGr D

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denotes the limit of D in TopGr, then (limTopGr D)0 is the limit of D in ConnGr with the restrictions of the limit maps giving the limit cone. Thus limConnGr D = (limTopGr D)0 . Proof. (i) We verify that TopGr and CompGr have products and equalizers. (Once the completeness of TopGr is secured, the completeness of CompGr also follows from the subsequent assertion (ii).) (ii) In view of E1.5 above, this statement is an immediate consequence of assertion (ii) of the Limit Existence Theorem 1.10. (iii) In order to show that LCGr fails to have products, we consider the family {Gn | n ∈ N}, Gn = R, for all n ∈ N, where the topological group R is the additive group of real numbers with the euclidean topology and N is the set of natural numbers 1, 2, 3, . . . . We claim that this family does not have a product in the category LCGr. Indeed, suppose that P is the product of this family in LCGr, and let pr n : P → R be the projections. By the universal property of the product RN in TopGr, there is a morphism P → RN which, by the uniqueness in the universal property of P , is injective. Hence P is a locally compact abelian group without compact subgroups and thus is of the form Rn × D for a discrete group D (see [102, Theorem 7.57 (i)]). Let πk : Rn+1 → R, k = 1, . . . , n+1, be the projections. We define a family of morphisms αm : Rn+1 → R, m ∈ N by " πk if k = 1, . . . , n + 1, αk = 0 otherwise. Then by the universal property of P there is a morphism α : Rn+1 → P , injective by the uniqueness in the universal property of Rn+1 , such that pr k α = αk for all k ∈ N. By the universal property of the product Rn+1 there is a morphism β : P → Rn+1 such that pr k = πk  β, k = 1, . . . , n + 1. Therefore, by the uniqueness in the universal property, πk = πk βα for k = 1, . . . , n + 1, and thus βα is the identity of Rn+1 . Thus α is injective and therefore induces an injective morphism of topological groups from Rn+1 into the identity component Rn × {0} of P . Since a continuous group homomorphism between finite-dimensional real vector spaces is R-linear, this is a contradiction. (iv) We verify that the discrete cartesian product has the universal property by using [102, Theorem 7.57] in order to show that every morphism fj : G → Dj from a locally compact abelian group has in its kernel the open subgroup G0 · comp(G) where the group comp(G) is the union of all compact subgroups. (v) The two closed subgroups (R × Z)/Z2 and (R(1, 1/2) + Z2 )/Z2 of the compact torus R2 /Z2 are two connected subgroups whose intersection is disconnected. Thus ConnGr is not closed in TopGr under the formation of intersections. (This holds even in the category of compact abelian groups.) Let D : J → ConnGr be a diagram and limTopGr D the limit of D in TopGr. Let λ : const(limTopGr D) → D denote the limit cone in TopGr. Now let C be a connected topological group and α const(C) → D a cone. By the universal property of limTopGr D

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there is a unique morphism of topological groups α  : C → limTopGr D such that α = λ  const(α  ). Since C is connected, the morphism α  : C → limTopGr D maps C into the identity component (limTopGr D)0 and therefore defines a corestriction α  : C → (limTopGr D)0 such that α = λ  const(α  ). Since the limit maps λj separate the points of limTopGr D they certainly separate the points of (limTopGr D)0 , and therefore α  is readily seen to be unique. Thus (limTopGr D)0 has the universal property of the limit in ConnGr. In 1.11 (iv) we observed that in the incomplete category LCGr, we may nevertheless have products of certain infinite families and that the product of such a family may differ from its product in the complete supercategory TopGr. In Corollary 1.26 we shall see that certain projective limits exist in LCGr. Statement 1.11 (ii) is a rather important criterion which yields workable sufficient conditions for a full subcategory of TopGr to be complete. But ConnGr in (v) is a very natural example of a full subcategory of TopGr which is complete in its own right but is not closed in TopGr under the formation of limits. Since arbitrary products of connected spaces are connected, it is closed under the formation of products. We now discuss the important concept of cofinality for which we shall have many applications even in this chapter. Definition 1.12. A functor C : K → J is called cofinal if it satisfies the following two conditions: (i) (∀j ∈ ob J )(∃k ∈ ob K, f ) f ∈ J (Ck, j ). This is encapsulated in the following simple diagram: f

j ← Ck. (ii) (∀fα ∈ J (Ckα , j ), kα ∈ ob K, α = 1, 2)(∃k ∈ ob K, gα ∈ K(k, kα )) f1 (Cg1 ) = f2 (Cg2 ). This too is encapsulated in a diagram: Cg1

Ck ⏐1 ⏐ f1 

←−−−−

Ck ⏐ ⏐Cg  2

j

←−−−−

Ck2 .

f2

A special case makes this concept more intuitive to readers who have previously worked with cofinality in the context of directed sets. Let J and K be two partially ordered sets and make them into categories as in 1.4 (v). A function between partially ordered sets is a functor iff it is order preserving. Now assume that K is directed. Then a function C : K → J is said to be cofinal if for each j ∈ J there is a k ∈ K such that j ≤ Ck. Thus a cofinal order preserving function is a cofinal functor in the sense of Definition 1.12; condition (ii) is automatically satisfied since K is directed.

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The Cofinality Lemma Theorem 1.13. Let C : K → J be a cofinal functor and D : J → C a diagram. If lim DC exists with limit cone λ : lim DC → DC, then D has the limit lim DC with suitable limit cone  λ : lim DC → D such that  λCk = λk . Proof. First we describe the construction of  λ: If j ∈ ob J , then by 1.12 (i) there is an f : Ck → j and we set λj,f = (Df )λk : lim DC → D(j ). Now assume that fi : Cki → j , i = 1, 2 are given. Then by 1.12 (ii) we find gi : k → Cki with f1 (Cg1 ) = f2 (Cg2 ). Now λj,f1 = (Df1 )λk1 = (Df1 )(DCg1 )λk = D(f1 (Cg1 ))λk = D(f2 Cg2 )λk = · · · = λj,f2 . Thus λj,f does not depend on the particular choice of f and we may set  λj = λj,f unambiguously. We note that  λCk = λk,idCk = λk . With  1.12 (ii) again we verify that λ : lim DC → D is a cone. We claim it has the universal property. Let α : A → D be a cone. Then αCk : A → DC(k) is a cone, and by the limit property there is a unique morphism α  : A → lim DC such that αCk = λk α  . If now j ∈ ob J by 1.12 (i) we find an f : Ck → j , and by the definition of  λ we have  λj = (Df )λk . Now αj = (Df )αCk (since α is λj α  . We note the uniqueness of α  as a fill-in for the cone natural) = (Df )λk α  =  αj : A → Dj : Let α  : A → lim DC be a morphism such that λj α  = λj α  for all j ∈ ob J . Let k ∈ ob K. Then λk α  =  λCk α  =  λCk α  , and the uniqueness of α  as a  fill-in for the cone αCk : A → DCk shows α = α  . The proof is now complete. A careful reader will notice that the existence of lim DC implies lim DC = lim D and thus the existence of lim D; this observation may be relevant in general if one is not working in a complete category. If the existence of limits is not a problem in a category (such as e.g. TopGr and CompGr) then we can consider the limit as a functor in a sense expressed below. Proposition 1.14. Let C be a complete category and J1 and J2 small categories. Let Dj : Jj → C, j = 1, 2, be diagrams and assume that there is a cofinal functor F : J1 → J2 and a natural transformation α : D1 → D2 F . Let λ1 : lim D1 → D1 and λ2 : lim D2 F → D2 F denote the limit cones. Then there is a unique morphism lim α : lim D1 → lim D2 such that the following diagram involving the limit cones λj : lim Dj → Dj , j = 1, 2, commutes: lim⏐D1 ⏐ λ1  D1

lim α

−−−−−→ lim ⏐ D2 = lim D2 F ⏐λ  2 −−−−−→ D2 F. α

In particular this applies to the case J2 = J1 = J and F the identity functor of J , in which case α → lim α defines a functor lim : C J → C. Proof. Exercise E1.6.

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Exercise E1.6. Prove Proposition 1.14. [Hint. By the Cofinality Lemma 1.13 we know lim D2 = lim D2 F . Use the universal property in Definition 1.2 to obtain lim α and its uniqueness. Lastly assume that the domains of the diagrams agree and F is the identity functor; in order to prove lim αβ = (lim α)  (lim β), use the uniqueness property in 1.2.] In fact, in the language of adjoint functors (see e.g. [102, Definition A3.29ff.]), Proposition 1.3, in the case of a complete category C, says that lim : C J → C is right adjoint to const : C → C J . We consider some concrete examples of the functoriality of the limit in the category TopGr.

Example 1.15. Let P = {αj k : Gk → Gj | (j, k) ∈ J1 × J1 , j ≤ k} and Q = {βj k : Hk → Hj | (j, k) ∈ J2 × J2 , j ≤ k} be two projective systems of topological groups with limits G and H , respectively, and assume that there is a cofinal function θ : J1 → J2 and there is a family of morphisms γj : Gj → Hθ (j ) , j ∈ J1 such that for all j ≤ k in J1 the diagram G⏐j γj ⏐  Hθ (j )

αj k

←−−−−

G⏐k ⏐γ k ←−−−− Hθ (k) βθ (j )θ (k)

commutes. Then there is a unique morphism limj ∈J1 γj : G → H such that for the limit maps αj : G → Gj and βθ (j ) : H → Hθ (j ) the following diagrams commute for all j ∈ J1 . αj G⏐j ←−−−− G ⏐ ⏐lim γj ⏐   k∈J1 γk Hθ (j ) ←−−−− H. βθ (j )

Proof. Let C = TopGr be the category of topological groups, and define D : J → C and E : J → C to be the projective systems in C given by D(j ) = Gj , D(k → j ) = αj k , j ≤ k in J1 and E(j ) = Ej , E(k → j ) = βj k , j ≤ k in J2 . Let F = θ ; then F is a functor J1 → J2 when J1 and J2 are considered as categories. Then the γj provide a natural transformation γ : D → EF , and the assertion follows from 1.14.

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The Definition of Continuous and Procontinuous Functors Definition 1.16. A functor F : A → B is said to preserve limits or to be continuous if for every diagram D : J → A with a limit lim D, λ : const(lim D) → D the diagram F D : J → B has the limit F (lim D), F λ : F (const(lim D)) → F D. The functor F is said to preserve projective limits or to be procontinuous if the condition above is satisfied for each projective system D : J → A. In compact notation we can write: “F is continuous iff lim F D ∼ = F (lim D).” ˇ In [102] we noted in an appropriate place that for instance Cech cohomology (as a functor from the category of compact Hausdorff spaces to the opposite1 of the category of graded commutative rings) preserves projective limits of compact spaces (cf. [102], ˇ Step 1 in the proof of 8.83, proof of Corollary A3.91); but Cech cohomology does not preserve products. Thus it provides a concrete example of a functor which is procontinuous but not continuous. The Continuity Theorem for Functors Theorem 1.17. The following conditions are equivalent for a functor U from a complete category to an arbitrary category: (i) (ii) (ii ) (iii)

U U U U

preserves limits, i.e. U is continuous. preserves intersections and products. preserves intersections of coherent retracts and products. preserves equalizers and products.

Proof. Exercise E1.7. Exercise E1.7. Prove Theorem 1.17. [Hint. Use Theorem 1.8 and Theorems 1.5 and 1.7.]

The External Approach to Projective Limits We take a closer look at projective limits in the category TopGr of topological groups. From Definition 1.4 (v) we recall the definition of a projective system and a projective limit. Let J be a directed set. A projective system of topological groups over J is a family of morphisms {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k}, where Gj , j ∈ J are topological groups, satisfying the following conditions: (i) fjj = idGj for all j ∈ J (ii) fj k  fkl = fj l for all j, k, l ∈ J with j ≤ k ≤ l. C is a category, then the opposite category C op has the same objects but has all arrows reversed, cf. [102, paragraph preceding Definition A3.25]. 1 If

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If D : J → TopGr is the functor given by D(j ) = Gj and D(k → j ) = fj k , then the projective limit lim D exists in TopGr since TopGr is a complete category (see 1.11 (i)). In our present situation we shall express lim D also in the form limj ∈J Gj , realizing that the morphisms fj k remain somewhat in the background with this notation. The limit cone λ : lim D → D of Definition 1.2 is a family of morphisms {fk : limj ∈J Gj → Gk | k ∈ J } such that fj = fj k fk for all k ∈ J . Customarily in this context, the morphisms fj k are called the bonding maps of the system and the morphisms fk are called the limit maps. Sometimes the projective limit of a projective system of topological groups is called the inverse limit of the system. It is very useful in the case of projective systems to recognize the specialization of Theorem 1.8 to the present situation and thus obtain a concrete realization of the limit with which one can work effectively in concrete situations. Indeed,  for a projective system of topological groups, define the topological group P by P = j ∈J Gj . Set G = {(gj )j ∈J ∈ P | (∀j, k ∈ J ) j ≤ k ⇒ fj k (gk ) = gj }. Then G is a closed subgroup of P . If incl : G → P denotes the inclusion and pr j : P → Gj the projection, then the function fj = pr j  incl : G → Gj is a morphism of topological groups for all j ∈ J , and for j ≤ k in J the relation fj = fj k  fk is satisfied. Proposition 1.18. In the notation above, G is the projective limit of D and the fj are the limit maps. Proof. Exercise E1.8. Exercise E1.8. Prove Proposition 1.18. [Hint. Reread the proof of Theorem 1.8 and verify that the equalizer L of that proof is exactly the closed subgroup G of the product P in the present situation.] In the category of topological groups, projective limits often have special properties. A very simple case is as follows: Remark 1.19. If all groups Gj in the projective system are compact, then the projective limit G is compact. Exercise E1.9. E1.9. Prove Remark 1.19. The category theoretical background of this remark of course is the fact that any limit whatsoever of a diagram of compact groups taken in the category of topological groups is a compact group, because products of compact groups are compact groups and equalizers of morphisms between compact groups are compact groups. Let us look at some concrete examples of projective limits of topological groups. Examples 1.20. (A) Assume that we have a sequence ϕn : Gn+1 → Gn , n ∈ N of morphisms of topological groups: ϕ1

ϕ2

ϕ3

ϕ4

G1 ←−− G2 ←−− G3 ←−− G4 ←−− · · ·

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79

We obtain a projective system of topological groups by defining, for natural numbers j ≤ k, the morphisms fj k : Gk → Gj defined by " if j = k, idGj fj k = ϕj ◦ ϕj +1 ◦ · · · ◦ ϕk−1 if j < k. (i) Let Z denote the discrete additive group of integers and Q the additive group of rational numbers. Choose a natural number p, set Q(pn ) = Q/pn Z and Z(pn ) = Z/p n Z, and give these groups the discrete topology. The most interesting case is when p is a prime number; but the construction as such has nothing to do with such a restriction. Define ψn : Z(pn+1 ) → Z(pn ) by ψn (z + p n+1 Z) = z + p n Z and ϕn : Q(pn+1 ) → Q(p n ) by ϕn (q + p n+1 Z) = q + p n Z. We have an infinite commutative diagram Z(p) ⏐ ⏐ incl Q(p)

ψ1

←−−

2 Z(p ⏐ ) ⏐ incl

ψ2

←−−

ψ3

←−−

3 Z(p ⏐ ) ⏐ incl

4 Z(p ⏐ ) ⏐ incl

ψ4

←−−

←−− Q(p2 ) ←−− Q(p3 ) ←−− Q(p4 ) ←−− ϕ1

ϕ2

ϕ3

ϕ4

··· ··· .

The projective limit of the upper row in the category TopGr is called the group Zp of p-adic integers and the projective limit of the second row is called the group Qp of p-adic rationals. By the definition of the limits we see that Zp ⊆ Qp and thus there is an inclusion morphism making the completed diagram commutative: Z(p) ⏐ ⏐ incl Q(p)

ψ1

←−−

2 Z(p ⏐ ) ⏐ incl

ψ2

←−−

3 Z(p ⏐ ) ⏐ incl

ψ3

←−−

4 Z(p ⏐ ) ⏐ incl

ψ4

←−− · · ·

←−− Q(p2 ) ←−− Q(p3 ) ←−− Q(p4 ) ←−− · · · ϕ1

ϕ2

ϕ3

ϕ4

Z⏐p ⏐ incl Qp .

The group Zp is compact. It is not yet clear that the group Qp is locally compact; this defect in our understanding calls for additional developments. In Exercise E1.13 below we shall in fact show that Zp is an open subgroup of Qp , which implies that Qp is locally compact. (ii) For any additively written abelian group A and any natural number p we can define an endomorphism μp : A → A by μp (g) = p · g for g ∈ A. It is customary, however, to write p in place of μp . Now for each n ∈ N we can set An = A and def

define ϕn : An+1 → An to be μp . Now set Gn = R and Hn = T = R/Z and consider the quotient map qn : Gn → Hn . Since this morphism is equivalent to the exponential def

function t → e2π it : R → SS 1 = {z ∈ C : |z| = 1} (which induces an isomorphism T → SS 1 ) we shall call it exp. Then we have an infinite commuting diagram R ⏐ ⏐ exp T

p

p

p

p

p

p

p

p

←−− R ⏐ ←−− R ⏐ ←−− R ⏐ ←−− ⏐ ⏐ ⏐ exp exp exp ←−− T ←−− T ←−− T ←−−

··· ··· .

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The projective limit of the upper row is R, identified with the subgroup of RN of all (p−n r)n∈N and the projective limit of the lower row is called the p-adic solenoid Tp . There is a morphism expTp : R → Tp induced by the morphism expTN : RN → TN , and we have the enlarged commutative diagram R ⏐ ⏐ exp T

p

p

p

p

p

p

p

p

←−− R ⏐ ←−− R ⏐ ←−− R ⏐ ←−− · · · ⏐ ⏐ ⏐ exp exp exp ←−− T ←−− T ←−− T ←−− · · ·

R ⏐ ⏐expT  p Tp .

The p-adic solenoid is compact. (B) The product G of any family {Gi | i ∈ I } of topological groups can be represented as a projective limit. Indeed define J to be the set of all finite subsets of I  and set Pj = i∈j Gi for each j ∈ J . The set J is a directed set with respect to inclusion. For j ⊆ k in J let fj k : Pk → Pj denote the projection onto the partial product and for j ∈ J define fj : G → Pj to be the projection onto the partial proddef

uct. Now P = {fj k : Pk → Pj | (j, k) ∈ J × J, j ⊆ k} is a projective system  def of topological groups. Then L = lim  j ∈J Pj is a closed subgroup of j ∈J Pj . The morphism g  → (fj (g))j ∈J : G → j ∈J Pj has range L and thus induces a morphism f : G → L. In fact, f is an isomorphism of topological groups. For instance, let I be an arbitrary set. Then the locally convex topological vector space RI is a projective limit of a projective system of finite-dimensional real topological vector spaces. This example will be very relevant to us in the context of Lie algebras of locally compact groups and indeed pro-Lie groups, once they are defined. (C) Let U be a filter basis of closed subgroups of a topological group G. Then U is directed with respect to ⊇. For H ⊇ K in U we let fH K : K → H denote the inclusion def

map. Then P = {fH K : K → H | (H, K) ∈ U × U, H ⊇ K} is a projective system def def  of topological groups, and L = limH ∈U H is a subgroup of P = H ∈U H . Let D denote the intersection of U. The morphism d  → (dH )H ∈U : D → P given by dH = d maps D into L and in fact induces an isomorphism of topological groups D → L. A special example is the case that U is the filter basis of all open (and hence closed) subgroups of G. If G is locally compact, then the intersection D is the connected component of the identity: indeed if G is totally disconnected this is shown in [102, Theorem 1.34], so it holds for G/G0 and thus for G, where G0 is the connected component of the identity (cf. [102], Exercise 1.12 (iii)). Exercise E1.10. Prove the assertions made in the discussion of the examples in 1.20. The crucial role of projective limits is evident in the entire structure theory of compact groups as one observes e.g. in [102]. We now return, in the concrete situation of projective limits of topological groups, to the issue of cofinality of 1.12 and 1.13 above and express it in the language of projective systems of topological groups.

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81

Let P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} be a projective system of topological groups, let  be a directed set, and let θ :  → J be an order preserving cofinal function, that is, for each j ∈ J there is an α ∈  such that j ≤ θ(α). For α ∈  we set Hα = Gθ (α) , and if α ≤ β we define ϕαβ : Hβ → Hα by ϕαβ = fθ (α)θ (β) . Then def

Pθ = {ϕαβ : Hβ → Hα | (α, β) ∈  × , α ≤ β} is straightforwardly seen to be a projective system of topological groups, called cofinal in P . Let ϕα : H = limβ∈ Hβ → Hα be the limit maps. Notice that an order preserving surjective function θ :  → J is always cofinal. One prominent example of cofinality in projective limits is as follows: For i ∈ J we denote by ↑i the set of all k ∈ J with i ≤ k. Then P  = {fj k : Gk → Gj | (j, k) ∈ ↑i × ↑i, j ≤ k} def

is an example of a projective system of topological groups which is cofinal in P ; indeed let  = ↑i and θ : ↑i → J the inclusion map. Now we have the following useful result: The Cofinality Lemma Revisited Lemma 1.21. (i) If H = limα∈ Hα is the projective limit of a projective system Pθ that is cofinal in a projective system P , then H is the limit of P with appropriate limit maps  ϕj : H → Gj satisfying  ϕθ (α) = ϕα . (ii) If G = limj ∈J Gj is a projective limit of topological groups and i ∈ J , then G∼ = limj ∈↑i Gj . Proof. Since the inclusion map ↑i → J is cofinal, Part (ii) is a special case of Part (i) and Part (i) is a special case of the category theoretical Cofinality Lemma 1.13 which was already discussed in Example 1.15. Because of the significance of cofinality it may serve a useful purpose to reprove explicitly the isomorphism of the two limits in the concrete case of projective limits of topological groups. Exercise E1.11. First compute the limits G = limj ∈J Gj and H = limα∈ Hα and find an explicit isomorphism q : G → H .   [Hint. Let H = limα∈ Hα . There is a morphism p : j ∈J Gj → α∈ Hα defined by p((gj )j ∈J ) = (gθ (α) )α∈ . If g = (gj )j ∈J ∈ G = limj ∈J Gj and h = (hα )α∈ = p(g), then ϕαβ (hβ ) = fθ (α)θ (β) (gθ (β) ) = gθ (α) = hα . Hence h ∈ H . Thus p induces a morphism of topological groups q : G → H . Next define a morphism q  : H → G as follows. Let h = (hα )α∈ ∈ H and let j ∈ J . If α1 , α2 ∈  such that j ≤ θ (α1 ), θ (α2 ) there is an α3 ≥ α1 , α2 and we note

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that fj θ (α1 ) (hα1 ) = fj θ (α1 ) (ϕα1 α3 (hα3 )) = fj θ (α1 ) (fθ (α1 )θ (α3 ) (hα3 )) = fj θ (α3 ) (hα3 ); by the same calculation, replacing “1” by “2”, we get fj θ (α2 ) (hα2 ) = fj θ (α3 ) (hα3 ). The conclusion is fj θ (α1 ) (hα1 ) = fj θ (α2 ) (hα2 ). Thus we can unambiguously define gj = fj θ (α) (hα )

for any α ∈  such that j ≤ θ (α),

and such an α exists by the very definition of θ in cofinality. We set q  (h) = (gj )j ∈J . If j ≤ k, then we can choose α ∈  such that k ≤ θ (α), then fj k (gk ) = fj k (fkθ (α) (hα )) = fj θ (α) (hα ) = gj by the definition of the gi . Hence (gj )j ∈J ∈ G. Since for each j ∈ J the function h  → gj = fj θ (α) (hα ) is a morphism of topological groups, the function q  : H → G is a morphism of topological groups (since a function from a space into a product is continuous iff its composition with each projection is continuous). We claim that q  q = idG and qq  = idH . Indeed first let g = (gj )j ∈J ∈ G; then q  q(g) = q  ((gθ (α) )α∈ ) = (gj )j ∈J , where gj = fj θ (α) (gθ (α) ) = gj for some α such that j ≤ θ (α). This proves the first half of the claim. Secondly let h = (hα )α∈ ∈ H ; then q  (h) = (gj )j ∈J where for each j we have gj = fj θ (α) (hα ) for any α ∈  with j ≤ θ (α). Then qq  (h) = (gθ (α) )α∈ . But gθ (α) = fθ (α)θ (α) (hα ) = hα by the definition of the gj , since θ (α) ≤ θ (α), trivially. This proves the second part of the claim which itself concludes the proof.]

Projective Limits and Local Compactness When a projective limit is constructed from a given projective system, the bonding maps are given and the limit maps are constructed. We shall now observe that the surjectivity of all limit maps almost trivially implies the surjectivity of the bonding maps; the reverse implication is true in the presence of some compactness. Proposition 1.22. Assume that G = limj ∈J Gj for a projective system fj k : Gk → Gj of topological groups, j ≤ k in J , and denote the limit maps by fj : G → Gj . Consider the following statements: (i) All limit maps fj are surjective. (ii) All bonding maps fj k are surjective. Then (i) ⇒ (ii), and if all kernels ker fj k , j ≤ k in J are compact, then the statements are equivalent, and the kernels of the limit maps are also compact. Proof. (i) ⇒ (ii): Let j ≤ k. Then fj = fj k fk . Thus the surjectivity of fj implies that of fj k . (ii) ⇒ (i) when all ker fj k are compact: Fix i ∈ J . Let h ∈ Gi ; we must find an element g = (gj )j ∈J ∈ G with gi = fi (g) = h. By the Cofinality Lemma 1.21 (ii)

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we have limi≤j ∈J Gj ∼ = limj ∈J Gj . For our proof it is therefore no loss of generality  if we assume that J = ↑i. Now for all k ∈ J we define Ck ⊆ j ∈J fij−1 (h) by {(xj )j ∈J | (∀j ∈ J ) fij (xj ) = h and (∀j ≤ k) xj = fj k (xk )}. Since fik is surjective, there is an xk ∈ Gk such that fik (xk ) = h. For all j ≤ k we set xj = fj k (xk ). For all j ≤ k, since fij is surjective, fij−1 (h) contains an

element xj . Then (xj )j ∈J ∈ Ck , and thus Ck = Ø. Moreover, since ker fij = fij−1 (1)

is compact, the set fij−1 (h) = hj fij−1 (1) is compact as well and thus Ck is compact. If i ≤ k ≤ k  then we claim Ck  ⊆ Ck . Indeed (xj )j ∈J ∈ Ck  implies fj k (xk ) = fj k fkk  (xk  ) = fj k  (xk  ) = xj . Thus (xj )j ∈J ∈ Ck and the claim is established. Now o {C k | k ∈ J, i ≤ k} is a filter basis (see e.g. [25, Chap. I], §6, n 3) of compact sets in j ∈J Gj and thus has nonempty intersection. Assume that g = (gm )m∈J is in this intersection. Then, firstly, gi = h. Secondly, let j ≤ k. Since J is directed, there is a k  with i, k ≤ k  . Then (gm )m∈J ∈ Ck  . Hence gj = fj k  (gk  ) = fj k fkk  (gk  ) = fj k (gk ) by the definition of Ck  . Hence g ∈ limj ∈J Gj . Thus g is one of the elements which we sought. It remains to show that the compactness of all kernels ker fj k implies the compactness of the kernel ker fi for each i ∈ J . By the Cofinality Lemma 1.21 (ii) we assume again J = ↑i. Let g = (gk )k∈J ∈ ker fi . If j ∈ J , then fij (gj ) = fij (fj (g)) = def

fi (g) = 1 and thus g Kj are compact by assumption, j ∈ Kj = ker fij . These groups  and g = (gj )j ∈J ∈ j ∈J Kj . Hence ker fi ⊆ j ∈J Kj , and thus the compactness of ker fi is established. From Appendix 2, Definition A2.5 we recall that a real topological vector space L is called weakly complete iff it is the projective limit of the projective system of its finite-dimensional quotient spaces. Remark. If we assume that all Gj are weakly complete topological vector spaces and the morphisms fj k are continuous linear morphisms, then (i) and (ii) are equivalent. Indeed, in the proof of 1.22, the sets Ck form a filter basis of closed affine varieties in a weakly complete topological vector space, and such filter bases have a nonempty intersection in linearly compact spaces. We shall give an independent proof of this fact in Lemma 5.18 later. In order to clarify this situation further, we record that a continuous function f : X → Y between Hausdorff spaces is called proper or perfect if the inverse image f −1 (y) of each point in Y is compact and f maps closed sets to closed sets. (See e.g. [25, Chap. I, §10].) In particular, a proper map is a quotient map if and only if it is surjective. We recall that a Hausdorff space is said to be a k-space if the following condition holds: A subset U is open iff U ∩ C is open in C for each compact subspace C. Locally compact Hausdorff spaces and metrizable spaces are k-spaces. If the underlying space of a topological group G is a k-space we shall mildly abuse the language and say that G is a k-space. (For k-spaces see [3, 64ff.].)

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A topological space is called σ -compact if it is a countable union of compact subspaces. Proposition 1.23. Let f : G → H be a morphism of topological groups and consider the following conditions: (i) f is a proper map. (ii) ker f is compact and f is a quotient map onto its image f (G), and f (G) is closed in H . (iii) For each compact subset C of H the inverse image f −1 (C) is compact. (iv) ker f is compact. (v) f is a closed morphism. (A) Then (i) ⇔ (ii) ⇒ (iii) ⇒ (iv) and (i) ⇒ (v). (B) If H is a k-space, then (i) ⇔ (ii) ⇔ (iii), and these conditions together with the local compactness of H imply that G is locally compact. (C) If f is surjective, G is locally compact and σ -compact, and if H is a Baire space, then (iv) ⇒ (i). In particular, if f is a surjective morphism between σ -compact locally compact topological groups, then conditions (i)–(iv) are equivalent. (D) If f is surjective, G is locally compact and σ -compact, and H is nondiscrete and has a countable basis of the filter of its identity neighborhoods, then (ii) and (v) are equivalent. Proof. Exercise E1.12. The identity morphism f : Rd → R from the additive group of real numbers with the discrete topology onto the group of real numbers with its natural topology is a surjective morphism between locally compact groups with singleton, and thus compact, kernel and the range group is locally compact metrizable. But f is not a proper map. This shows that in the last part of 1.23 (C) one cannot dispense with the hypothesis of σ -compactness of G, and also one cannot dispense with the assumption that f be a quotient map in statement (ii). Exercise E1.12. Prove Proposition 1.23. [Hint. (i) ⇒ (ii): Note that a closed surjective morphism of topological groups is a quotient morphism. Trivially (i) implies that ker f = f −1 (1) is compact. (ii) ⇒ (i): Let h ∈ H . If h ∈ / f (G), then f −1 (h) = Ø which is trivially compact. If h ∈ f (G) then there is a g ∈ G with f (g) = h. Then f −1 (h) = gf −1 (1) = g · ker f is compact. Next let A be a closed set in G. Since ker f is compact, the set f −1 (f (A)) = A · ker f is closed. As f is a quotient morphism this implies that f (A) is closed in f (G). Since f (G) is closed in H , the image f (A) is also closed in H . (i) ⇒ (iii): Let C ⊆ H be compact and set K = f −1 (C). Since f is a proper map, the restriction and corestriction f  : K → C of f is proper and the constant map p : C → {1} is proper because of the compactness of C; hence the composition

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p  f  : K → {1} is proper, and thus, as the inverse image of a point under a proper map, K is compact. (iii) ⇒ (iv) is trivial. (iii) ⇒ (ii) in (B): Trivially (iii) implies that ker f is compact. Now assume that H is a k-space. We must show that f is a closed map. For a proof let A be closed in G and C compact in H . We must show that f (A) ∩ C is closed in C. By (iii), the set def

K = f −1 (C) is compact. We shall use this information to show that the restriction and corestriction f |K : K → C is a closed map. Let B be closed in K. Since K is compact and Hausdorff, B is compact and thus (f |K)(B) = f (B) is compact in the Hausdorff space H and thus is closed in C. Therefore f (A ∩ K) is closed in C. Clearly f (A ∩ K) ⊆ f (A) ∩ C. Let c ∈ f (A) ∩ C. Then there is an a ∈ A such that f (a) = c ∈ C. Hence a ∈ f −1 (C) = K and thus a ∈ A ∩ K whence c = f (a) ∈ f (A ∩ K). Thus f (A) ∩ C = f (A ∩ K) and this set is closed in C as claimed. Thus the corestriction g  → f (g) : G → f (G) of f , as a closed morphism of topological groups, is a quotient morphism. Local compactness of G in (B): Let H be locally compact and assume (iii). Let C be a compact identity neighborhood of H . Then f −1 (C) is compact by (iii). Since H is a k-space, f is a closed map by what we just saw and thus is open onto its image f (G). Since C ∩ f (G) is an identity neighborhood of f (G), the set f −1 (C) is an identity neighborhood of G. Thus f −1 (C) is a compact identity neighborhood of G and thus G is locally compact. (iv) ⇒ (ii) in (C): Now assume that H is a Baire space and that f is surjective. Then the Open Mapping Theorem for Locally Compact Groups (see e.g. [102, Exercise EA1.21 preceding Remark A1.60]) applies to show that f is open and hence a quotient morphism. The Baire Category Theorem shows that every locally compact space or completely metrizable space is a Baire space ([26, Chap. 9, §5, no 3 Théorème 1], [131, p. 200, Theorem 34], [79, p. 42, 5.28 and 5.29]). (ii) ⇒ (v): Let C be closed in G. Since ker f is compact, the set f −1 (f (C)) = C · ker f is closed; thus f (C) is closed since f is a quotient morphism. (v) ⇒ (ii) if H is metric and nondiscrete: If f is a closed map then it is a quotient map; we must show that ker f is compact. Suppose now that ker f is not compact. We shall derive a contradiction. We assume thatthe topology of H is given by a metric d. Also we claim that we can write G = ∞ n=1 Cn with an ascending family of compact subsets such that every compact subset is contained in some Cn : indeed  G= ∞ n=1 Kn with compact subsets Kn ; by replacing Kn by K1 ∪· · ·∪Kn if necessary we may assume that the Kn are increasing; now let V be an open identity neighborhood  so that V is compact. Then all Kn V = k∈Kn kV are open and one notes at once that def

every compact subset of G is contained in one of the sets Kn V ; the sets Cn = Kn V are compact and satisfy the claim. Since H is nondiscrete and satisfies the First Axiom of Countability, there is a sequence (hn )n∈N in H \ {1} converging to 1. Since ker f is not compact and f is surjective, the set f −1 (hn ) is not compact and thus there is a gn ∈ f −1 (hn ) \ Cn . Set A = {gn | n ∈ N}. If g ∈ A, let U be a compact neighborhood of g. Then

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U ⊆ Ck for some k and g ∈ A ∩ Ck . As the Cn form an ascending family, Ck ∩ {gk , gk+1 , gk+2 , . . . } = Ø and thus g ∈ {g1 , . . . , gk−1 } = {g1 , . . . , gk−1 } ⊆ A. Therefore A is closed in G but f (A) fails to be closed in H because limn∈N f (gn ) = / f (A).] limn∈N hn = 1 ∈ We also note that a topological group H is metrizable if and only if the filter of identity neighborhoods has a countable basis. (See e.g. [102, Theorem A4.16ff.]) For more subtle points on proper maps we refer to [25, Chap. I, §10]. The Definition of Strict and Strong Projective Limits Definition 1.24. A projective system of topological groups in which all bonding maps and all limit maps are surjective is called a strict projective system and its limit is called a strict projective limit. It will be called a strong projective system and its limit is called a strong projective limit if the bonding maps are surjective and proper. Corollary 1.25. For a projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of topological groups consider the following conditions: (i) The bonding maps fj k are all proper. (ii) For all j ∈ J and all compact subsets C ⊆ Gj the inverse images fj−1 (C) are compact and nonempty. (iii) The limit maps fj are all proper. Then (i) ⇒ (ii); if all groups Gj are k-spaces, then (ii) ⇔ (iii), and if the fj are surjective, then (iii) ⇒ (i). Therefore, if all Gj are k-spaces and all limit maps are surjective, all three conditions are equivalent. In particular, for a strong projective limit of topological groups which are k-spaces, the limit maps are proper and surjective. As a consequence, a projective limit of compact groups is strong if and only if it is strict. Proof. (i) ⇒ (ii): Assume that the bonding maps are proper. Let j ∈ J and assume that C ⊆ Gj is compact. We must show that K = fj−1 (C) is compact. By the Cofinality def

Lemma 1.21 (ii) we assume J = ↑j . Then for all k ∈ J the subset Kk = fj−1 k (C) is  compact as fj k is proper (see Proposition 1.23). Now K = fj−1 (C) ⊆ k∈J Kk since fj = fj k fk . Since the product is compact, the compactness of K follows. (iii) ⇒ (ii): This follows from 1.23 (A). (ii) ⇒ (iii): Assume that all Gj are k-spaces. Then Proposition 1.23 (B) applies and proves the claim. (iii) ⇒ (i): If the limit maps fj are proper, they are closed and have compact kernels. Then the relations fj = fj k  fk show that the fj k are closed. Now assume that the fk are surjective; then ker fj k = fk (ker fj ), and these groups are compact. Hence the bonding maps are proper. def

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Now assume that the projective limit is strong and all Gk are k-spaces. Then the limit maps are surjective by Proposition 1.22, and proper by the preceding results. The last assertion is immediate, because every continuous function between compact Hausdorff spaces is proper. Because of the remark in 1.9 (iv) in [102], we did not have to formulate the concept of a strong projective limit in addition to that of a strict one. All examples in 1.20 are strict projective limits with the exception of Example 1.20 (C) which is a strict projective limit if and only if the filter basis F has only one element. The Examples 1.20 (A) are strong projective limits. In Example 1.20 (B), if RI for an infinite set I is represented as a projective limit of finite-dimensional vector spaces, then it is a strict projective limit but not a strong projective limit. Corollary 1.26. (a) Assume that G = limj ∈J Gj is a projective limit of a projective system of topological groups such that all bonding maps are proper and all of the Gj are locally compact. Then G is locally compact. (b) For a strong projective limit the following statements are equivalent: (i) G is locally compact. (ii) All Gj are locally compact. (iii) There is a j ∈ J such that Gj is locally compact. (c) If {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} is a strong projective system in

LCGr, then its projective limit exists in LCGr and agrees with its limit in TopGr.

Proof. (a) Since the topological groups Gj are locally compact, they are k-spaces. Hence by 1.25, the limit maps fj : G → Gj are proper. Fix a j ; then Gj is locally compact and then 1.23 (B) implies that G is locally compact. (b) Consider a strong projective system of topological groups. (i) ⇒ (ii): All limit maps fj : G → Gj are quotient morphisms by 1.23 (A). Thus Gj is isomorphic to a factor group of G and is therefore locally compact. (ii) ⇒ (iii) is trivial. (iii) ⇒ (i): By the definition of strong projective limit, the morphism fj : G → Gj is surjective and proper and thus the implication is a consequence of 1.23 (B) again. (c) The limit L of the given strong projective system of locally compact groups taken in TopGr is locally compact by (a) and is therefore contained in LCGr. Since the universal property required by Definition 1.2 holds in TopGr, and since LCGr is a full subcategory of TopGr it holds in LCGr; therefore L is the limit in LCGr. Now we are better equipped to deal with Example E1.20 (A) above and, in particular, with the group of p-adic rationals. Exercise E1.13. As in Example 1.20 (A), consider a sequence ϕn : Gn+1 → Gn , n ∈ N of morphisms of topological groups, and assume that in each group Gn we have a subgroup Kn such that ϕn (Kn+1 ) ⊆ Kn . We define bonding maps fmn and limit maps

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fn as in 1.20 (A). Set G = limn∈N Gn and K = limn∈N Kn and obtain a commuting diagram with ψn being the restriction and corestriction of ϕn : K⏐1 ⏐ incl G1

ψ1

←−− K⏐2 ⏐ incl ←−− G2 ϕ1

ψ2

←−− K⏐3 ⏐ incl ←−− G3 ϕ2

ψ3

←−− K⏐4 ⏐ incl ←−− G4 ϕ3

ψ4

←−− · · · ←−− · · · ϕ4

K ⏐ ⏐ incl G.

−1 (K ) = K for all m ≥ n and that K is an open subgroup (i) Assume now that fmn m n n of Gn ; then K is an open subgroup of G. (ii) If, in addition, the Kn are all compact open subgroups and all the ϕn are surjective, then K is a compact open subgroup of G. In particular, G is locally compact. As a consequence, the additive group Zp of p-adic integers is a compact open subgroup of the additive group of p-adic rationals Qp . In particular, Qp is locally compact. [Hint (i) Use the hypotheses to show fn−1 (Kn ) = K. Then K is open in G if Kn is open. (ii) If the ϕn are surjective, then the bonding maps, being compositions of successive ϕn , are surjective. Since ker ϕn ⊆ Kn+1 , the kernels of the bonding maps are compact. Hence the bonding maps are proper surjective. The projective system of the Gn therefore is strong. Thus by Corollary 1.26, G is locally compact, and by Proposition 1.22, all limit maps are surjective. Hence by (i) above, K is an open subgroup. As a limit of compact groups, K is also compact. These conclusions apply directly to the groups of p-adic integers and p-adic rationals as introduced in 1.20 (A).]

It is rather obvious at this early stage, that strong projective limits are especially appropriate for dealing with locally compact groups.

The Fundamental Theorem on Projective Limits A filter basis F in a topological space X is said to converge to x ∈ X if for each neighborhood U of x there is an F ∈ F such that F ⊆ U . If X is a Hausdorff space, we shall then write lim F = x. Fundamental Theorem on Projective Limits Theorem 1.27. Let G = limj ∈J Gj be a projective limit of a projective system P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of topological groups and let Uj denote the filter of identity neighborhoods of Gj , U the filter of identity neighborhoods of G, and N the set {ker fj | j ∈ J }. Then (i) U has a basis of identity neighborhoods {fk−1 (U ) | k ∈ J, U ∈ Uk }.

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(ii) N is a filter basis of closed normal subgroups converging to 1. If M ⊇ N in N and if νMN : G/N → G/M is defined by νMN (gN ) = gM, then Q = {νMN : G/N → G/M | (M, N ) ∈ N × N , M ⊇ N} is a projective system of topological groups, and there is a unique isomorphism η : limN∈N G/N → G such that the following diagram commutes with j ≤ k, M = ker fj , N = ker fk , and with the morphisms fj : G/ ker fj → Gj induced by the limit map fj : G → Gj : ··· ···

G/M ⏐ ⏐ fj  Gj

νMN

←−− G/N ⏐ ⏐f   k ←−− Gk fj k

νN

←−− limP ∈N⏐(G) G/P ⏐η  fk ←−− G.

The limit maps νN are quotient morphisms. (iii) Assume that all bonding maps fj k : Gj → Gk are quotient morphisms and that all limit maps fj are surjective. Then the limit maps fj : G → Gj are quotient morphisms. (iv) Set Hj = fj (G) for each j ∈ J and let fj k : Hk → Hj the morphisms induced by fj k for j ≤ k. Then Q = {fj k : Hk → Hj | (j, k) ∈ J × J, j ≤ k} is a projective system of topological groups and G = limj ∈J Hj . The limit maps fj : G → Hj are corestrictions of the fj and they have dense images. Proof. (i) Let V ∈ U and consider the definition  of the projective limit as a subset of  G . We find an identity neighborhood W of j j ∈J j ∈J Gj such that W ∩lim j ∈J Gj ⊆  V and that W = j ∈J Wj with Wj ∈ Uj and with Wj = Gj except for j in some finite subset F of J . Since J is directed, there is an upper bound k ∈ J of F . There is a U ∈ Uk such that fj k (U ) ⊆ Wj for all j ∈ J . Then fk−1 (U ) ⊆ W ∩ limj ∈J Gj ⊆ V . (ii) Evidently, each ker fj is a closed normal subgroup. Since i, j ≤ k implies ker fk ⊆ ker fi ∩ ker fj and J is directed, N is a filter basis. For each j ∈ J we have ker fj = fj−1 (1) ⊆ fj−1 (Gj ) except for j in some finite subset F of J . Since J is directed, there is an upper bound k ∈ J of F . There is a U ∈ Uk such that fj k (U ) ⊆ Wj for all j ∈ J . Then fk−1 (U ) ⊆ W ∩ limj ∈J Gj ⊆ V . (ii) Evidently, each ker fj is a closed normal subgroup. Since i, j ≤ k implies ker fk ⊆ ker fi ∩ ker fj and J is directed, N is a filter basis. For each j ∈ J we have ker fj = fj−1 (1) ⊆ fj−1 (U ) for any U ∈ Uj . Since fj−1 (U ) is a basic neighborhood of the identity by (i), we are done. By the universal property of the limit G = limj ∈J Gj , the morphism η exists uniquely making the diagram commute. The cone of all quotient morphisms qN : G → G/N, N ∈ N over the projective system Q and the universal property of the limit limN∈N G/N, there is a morphism γ : G → limN ∈N G/N such that qN = νN  γ for all N. From the uniqueness in the limit property of G = limj ∈J Gj we get η γ = idG .

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We claim that η is injective: Indeed let x = (gN N )N ∈N and assume η(x) = 1. By the definition of η, we have 1 = fj (η(x)) = fj (νker fj (x)) = fj (gker fj ker fj ) = fj (gker fj ) for all j ∈ J and thus gker fj ∈ ker fj for all j i.e., gN N = N for all N ∈ N . Thus x = 1 and this establishes our claim. Now η is an injective retraction and thus is an isomorphism whose inverse is γ . Then qN = νN  γ shows that νN is a quotient morphism. (iii) Let j ∈ J and assume that U ∈ U. We must show that fj (U ) is an identity neighborhood of fj (G). By (i) we may assume that there is a k ∈ J and an identity neighborhood V of Gk such that U = fk−1 (V ). Since fj k is open it follows that fj k (V ) is an identity neighborhood of fj (G). Now let g ∈ fj k (V ); then g = fj k (v) with v ∈ V . By definition of U and the surjectivity of fk we have a u ∈ U such that v = fk (u) and thus g = fj k (fk (u)) ∈ fj (U ). Hence fj k (V ) ⊆ fj (U ) and therefore fj (U ) is an identity neighborhood as asserted. (iv) We have fj = fj k  fk and thus Hj = fj (G) = fj k (fk (G)) ⊇ fj k (Hk ) ⊇ fj k (fk (G)) =fj (G). This shows that fj k maps Hk densely into Hj . Since H = limj ∈J Hj ⊆ j ∈J Hj ⊆ j ∈J Gj we may assume that H  ⊆ G. If g = (gj )j ∈J ∈ G, then gj = fj (g) ∈ f (Gj ) ⊆ Hj and thus G ⊆ G ∩ j ∈J Hj = H . The limit maps fj : H = G → Hj are therefore simply the corestrictions of the fj and thus fj (H ) = fj (G). Hence Hj = fj (H ). We shall see in Theorem 1.33 below that if all Gj are complete, then η : lim G/N → G N ∈N

is an isomorphism. Example 1.20 (C) exhibits a typical case to which Theorem 1.27 (iii) above does not apply. But when all limit maps are quotient morphisms as in Theorem 1.27 (iii), then the above theorem shows how, in a geometric way, the projective limit G = limj ∈J Gj is approximated by factor groups G/N modulo small normal closed subgroups of N = ker fj , and it indicates that on each projective limit G of topological groups we are given a filter basis N of closed normal subgroups converging to 1 such that G is isomorphic to the apparently simpler limit limN ∈N G/N. The trouble with this “reduction” may be that while there is an injective morphism G/N → Gj with N = ker fj , we do not know that it is an isomorphism onto its image. After Theorem 1.27 (iv) we know that it is no loss of generality to assume that bonding maps and limit maps have dense images. Thus we are motivated to look more closely at topological groups equipped with filter bases of closed normal subgroups.

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91

The Internal Approach to Projective Limits Let M and N be two closed normal subgroups of a topological group G. If M ⊇ N , then there is a morphism νMN : G/N → G/M given by νMN (gN ) = gM; it is a quotient morphism with kernel ker νMN = M/N. A filter basis N of closed normal subgroups of G is a directed set with respect to the partial order ⊇. Hence every filter basis N of closed normal subgroups determines a projective system PN = {νMN : G/N → G/M | (M, N ) ∈ N × N , M ⊇ N }, which is special in so far as each bonding map is a quotient morphism. Definition 1.28. Let G be a topological group and N a filter basis of closed normal subgroups. Then the projective system PN is called the projective system associated def

with (G, N ). Its projective limit GN = limN ∈N G/N will be called the projective limit associated with (G, N ). The limit maps of the associated projective limit are called νN : GN → G/N and the simple quotient maps modulo N will be denoted . qN : G → G/N . The kernel of the limit map νN will be written N  We should recall that GN ⊆ N ∈N G/N is the set of all (gN N )N ∈N such that gN M = νMN (gN N) = gM M for all M ⊇ N; that is # $  −1 gN ∈ M for all M ⊇ N , (1) GN = (gN N)N ∈N ∈ N ∈N G/N | gM and the limit map νN is given by νN ((gM M)M∈N ) = gN N.

(2)

Theorem 1.29. Let N be a filter basis of closed normal subgroups of a topological group G. (i) There is amorphism γ = γG,N : G → GN given by γ (g) = (gN )N ∈N whose kernel is N and whose image is dense. (ii) νN  γ = qN : G → G/N for all N ∈ N . → (iii) νN : GN → G/N is a quotient map, i.e. induces an isomorphism GN /N G/N . In particular, the associated projective limit GN is strict. Moreover,  = γ (N). N (iv) The associated projective limit is strong if and only if M ⊇ N in N implies that M/N is compact. (v) The morphism γG,N : G → GN is natural in the following sense: Let f : G → H be a morphism of topological groups. Assume that on G and H we are given filter bases NG and NH of closed normal subgroups such that the following conditions are satisfied: (a) f (M) ∈ NH for each M ∈ NG , and (b) for each N ∈ NH there is an M ∈ NG such that N ⊇ f (M).

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1 Limits of Topological Groups

Then there is a unique morphism fN : GNG → HNH such that the diagram G ⏐ ⏐ f H

γG

−−−−−→ G⏐NG ⏐f  N −−−−−→ HNH γH

commutes. (vi) Condition (v)(b) is automatically satisfied if lim NG = 1 and if the identity element of G/N has a neighborhood which does not contain nonsingleton subgroups. Proof. (i) From (1) it follows that (gN )N ∈N ∈ GN . The function γ is therefore well defined; it is a morphism of groups and since qN = (g  → gN ) : G → G/N is continuous for each N it follows that γ iscontinuous. It is clear that g ∈ ker γ iff g ∈ N for all N ∈ N . Thus ker γ = N . Now we must show that im γ is dense in GN . For this purpose let  g = (gN N)N ∈N ∈ GN and let UN be an identity neighborhood in GN . By 1.27 (i) we find an identity neighborhood U of G and an −1 M ∈ N such that νM (U M/M) ⊆ UN . Now γ (gM ) = (gM N )N ∈N and −1 −1 νM (γ (gM )−1 gN N)N ∈N ) = gM gM M = M, g ) = νM ((gM

 ⊆ the identity of G/M. Thus γ (gM ) ∈  gM g UN . (ii) This is straightforward from (2) above. (iii) From (ii) it follows that νN is surjective; hence νN induces a bijective morphism  : G /N  → G/N, ν  ((gM M)M∈N N)  = gN N . Now the morphism  νN νN  γ : G → N N , where   is the quotient morphism, has kernel {g ∈ G | GN /N νN : GN → GN /N } = {g ∈ G | gN = N } = N and thus induces a morphism of (gM)M∈N ∈ N  mapping gN to (gM)M∈N N.  One verifies topological groups γN : G/N → GN /N   at once that νN and γN are inverses of each other. Hence νN is an isomorphism of topological groups, and consequently νN is a quotient morphism. Since ker νN is closed and contains γ (N) we have γ (N) ⊆ ker νN . An element (gM M)M∈N ∈ GN is in γ (N) iff there is a net (gj )j ∈J in N such that (gM M)M∈N = limj ∈J (gj M)M∈N . If a (gM M)M∈N ∈ GN is given, then for fixed M ∈ N , the net (gN M)N ∈N in G/M is eventually constant since gN ∈ gM M for N ⊆ M. Hence J = N and (gN )N ∈N yields such a net(gj )j ∈J . (iv) The limit is strong if all bonding maps νMN : G/N → G/M of the associated projective system are proper and surjective by definition. They are quotient maps and hence by 1.23 they are proper if their kernels M/N are compact. (v) We apply Example 1.15 with J1 = NG , GM = G/M and the natural bonding maps, and similarly with J2 = NH , HN = G/N . We take θ (M) = f (M) and note that our conditions make θ a cofinal map. For each M ∈ NG there is a morphism γM : G/M  → H /f (M) given by γM (gM) = f (gM) = f (g)f (M). Thus by Example 1.15, the morphism fN : GNG → HNH exists such that the following diagram commutes for all M ∈ NG :

Projective Limits and Completeness (G)

G ⏐ ⏐ f H

93

(G)

qM

νM

−−−−−→

G/M ←−− ⏐ ⏐ γM  −−−−−→ H /f (M) ←−− (H )

(H )

qf (M)

G⏐NG ⏐f  N HNH .

νf (M)

We obtain fN ((gM)M∈NG ) = (f (g)f (M))M∈NG ∈ lim H /f (M) = lim H /N M∈NG

N ∈NH

and this completes the proof of (v). (vi) Let U N/N be an identity neighborhood of G/N without nonsingleton subgroups, where U is a suitable identity neighborhood of G; since lim NG = 1 and f is continuous, there is an M ∈ NG such that f (M) ⊆ U . The subgroup f (M)N/N of G/N must be singleton because it is contained in U N/N. So f (M) ⊆ N. The situation may be illuminated by the following commutative diagram with exact rows and columns:

{1} 

NG

{1}

id

{1} ⏐ ⏐ 

−−−−−→

N ⏐ ⏐  −−−−−→ G ⏐ ⏐ qN 

−−−−−→

−−−−−→ G/N ⏐ ⏐  {1}

=  −−−−−→ {1} −−−−−→ GN /N γN ⏐ ⏐  {1}. −−−−−→

−−−−−→

γ |N

γ

−−−−−→ ∼

{1} ⏐ ⏐   N ⏐ ⏐  G⏐N ⏐f  N

id

Theorem 1.29 and Theorem 1.27 show how the “external” concept of a projective limit and the “internal” concept of a projective limit associated with a topological group and a filter basis of closed normal subgroups are related; Theorem 1.29 shows that the latter has very special features. Statements 1.29 (v) and (vi) could be easily elaborated by defining a suitable category of pairs (G, N ) and defining a functor (G, N )  → GN .

Projective Limits and Completeness Understandably, we would like to investigate the  question: When is γG : G → GN an isomorphism? This requires that ker γG = N = {1}. Ostensibly we shall not which is not already coded into be able  to extract any information from GN about G  G/ N . We shall therefore assume, as a rule, that N = {1} and thus that γG is injective, that is, that G is continuously injected into GN .

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We shall say that a function f : X → Y between topological spaces is an embedding if its corestriction to the image is a homeomorphism. If f is known to be injective and continuous, then it is obviously an embedding iff it is an open map onto its image. If X and Y are topological groups, and f is in addition a morphism of topological groups, then we call it an embedding morphism. A filter basis F on a topological group G is called a Cauchy filter basis if for each identity neighborhood U of G there is a member F ∈ F such that F F −1 ⊆ U . Theorem 1.30. Let G be a topological group with a filter N of closed normal subgroups having trivial intersection. Then the following statements are equivalent: (i) γG : G → GN is an embedding morphism. (ii) lim N = 1. (iii) For each element (gN N)N ∈N ∈ GN the set {gN N : N ∈ N } is a Cauchy filter basis. Proof. Condition (i) is tantamount to saying (i ) For every identity neighborhood U of G there is an identity neighborhood VN of GN such that VN ∩ γG (G) ⊆ γG (U ). In view of 1.27 (i) this is equivalent to (i ) For every identity neighborhood U of G there is an identity neighborhood V of −1 (V N/N ) ∩ γG (G) ⊆ γG (U ). G and an N ∈ N such that νN −1 However, if (gM M)M∈N ∈ νN (V N/N ) ∩ γG (G), then there is a g ∈ G such that (gM M)M∈N = (gM)M∈N with gN N = gN ⊆ V N, i.e. g ∈ V N, and consequently −1 νN (V N/N) ∩ γG (G) ⊆ γG (U ) is equivalent to V N ⊆ U . Thus (ii) can be rephrased as

(i ) For every identity neighborhood U of G there is an identity neighborhood V of G and an N ∈ N such that V N ⊆ U . On the other hand, (ii) is equivalent to (ii ) For every identity neighborhood U of G there is an N ∈ N such that N ⊆ U . Now the equivalence is an easy matter: (i ) ⇒ (ii ) is trivial, and if (ii ) is satisfied and U is given, find a V such that V V ⊆ U and then an N such that N ⊆ V to obtain V N ⊆ V V ⊆ U as desired. (iii) ⇒ (ii ) is trivial because by (iii), the filter basis N = {N : N ∈ N } is a Cauchy filter basis and this means that it converges to 1. Now assume (ii ): If (gN N)N ∈N ∈ GN and Q ⊇ P in N , then gP ∈ gQ Q and thus gP P ⊆ gQ Q. Hence F = {gN N : N ∈ N } is a filter basis. If U is an identity neighborhood in G, then by (ii ) there is an N ∈ N such that N ⊆ U . Set −1 F = gN N ∈ F . Then F F −1 = (gN N )(gN N )−1 = gN N N −1 gN = N ⊆ U. Hence F is a Cauchy filter basis.

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95

In order to find a reasonable condition for the surjectivity of γG when it is an embedding, we need to understand the concept of completeness in a topological group. A topological group is called complete if every Cauchy filter basis converges. A net (gj )j ∈J on a topological group G is a Cauchy net if for each identity neighborhood U of G there is an index i ∈ J such that gj gk−1 ∈ U for all j, k ≥ i. Completeness is equally characterized by the convergence of each Cauchy net. Remark 1.31. Every locally compact group is complete. Proof. Let U denote the filter of identity neighborhoods of G and let U be a compact member of U. Consider a Cauchy filter basis C on G; we must exhibit a limit of C. First we find a set C ∈ C such that CC −1 ⊆ U . Let c ∈ C, then C ⊆ U c. Thus def C|U c = {F ∩U c | F ∈ C and F ∩U c = Ø} is a filter on the compact space U c. Hence  there is a g ∈ F ∈C|U c F . Let V ∈ U and pick a W ∈ U with W W ⊆ V . Then there

is an F ∈ C|U c such that F F −1 ⊆ W ; now g ∈ F implies the existence of an h ∈ F such that h ∈ Wg. Hence F h−1 ⊆ F F −1 ⊆ W and thus F ⊆ W h ⊆ W Wg ⊆ V g. Hence C converges to g.

Clearly, the proof of this remark required only that we had an identity neighborhood U of G on which every Cauchy filter basis has a limit. Lemma 1.32. (i) A product of complete groups is complete. (ii) A closed subgroup of a complete group is complete. (iii) If G is a projective limit limj ∈J Gj and if all groups Gj are complete, then G is complete. (iv) A complete subgroup of a topological group is closed. (v) A locally compact subgroup of a topological group is closed. Proof. Exercise E1.14. Exercise E1.14. Prove Lemma 1.32. [Hint. Recall our general assumption that all topological groups we consider are Hausdorff. (i) Show that a morphism of topological groups maps Cauchy filter bases to Cauchy filter bases. Use this to show that a filter basis in a product is a Cauchy filter basis if and only if all of its projections are Cauchy filter bases. (ii) is straightforward. def  (iii) If all Gi are complete, so is the product P = j ∈J Gj by (i). Since the limit G is a closed subgroup of P , it is complete by (ii). (iv) Assume that H is a complete subgroup of a topological group G. Let g ∈ H . Consider the neighborhood filter U of 1 and set F = {H ∩ Ug | U ∈ U}. If U ∈ U is given, find a V ∈ U such that V V −1 ⊆ U . Set F = H ∩ V g then F F −1 ⊆ V gg −1 V −1 ⊆ U . Hence F is a Cauchy filter basis. Since H is complete, it has a limit in H . But lim F = g. Hence g ∈ H and H is closed. (v) By Remark 1.31 above, a locally compact subgroup of a topological group is complete and is therefore closed by (iv).

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An alternative and purely topological proof of Lemma 1.32 (v) is given in [102, Corollary A4.24]. For more information on matters of Cauchy filter bases and completeness we refer the reader to [25, Chap. III, §3] and [176]. Theorem on Complete Groups and Projective Limits Theorem 1.33. (i) Let G be a complete topological group and N a filter basis of closed normal subgroups converging to the identity. Then γG = γG,N : G → GN is an isomorphism. That is, G ∼ = limN ∈N G/N. (ii) Let everything be as in Theorem 1.27 and assume that all Gj are complete, then so is G and η is an isomorphism inverting γG . Proof. (i) By Theorem 1.30, γG is an embedding. Since G is complete, we know that γG (G) is complete. Then by 1.32 (iv), γG (G) is closed. By Theorem 1.29 (i), γG (G) is dense in GN . Thus γG (G) = GN and this proves (i). (ii) If all Gj are complete then G = limj ∈J Gj is complete and thus γG : G → limN∈N G/N is an isomorphism by Part (i) above. It is straightforward that ηγG = idG and since γG is an isomorphism this shows that η−1 = γG . This completes the proof. The relevance of Theorem 1.33 lies in the fact that we may be given a complicated topological group G plus a filter basis N of closed normal subgroups converging to the identity such that G/N is of a simpler nature than G for all N ∈ N and thus recover G as limN∈N G/N ; see for instance [102, Corollary 2.43 and Lemma 9.1].

The Closed Subgroup Theorem It will turn out to be very important to be able to understand how closed subgroups of projective limits are themselves to be described as projective limits. The Closed Subgroup Theorem for Projective Limits Theorem 1.34. Assume that N is a filter basis of closed normal subgroups of the complete topological group G and assume that lim N = 1 and that all quotient groups G/N are complete for N ∈ N . Let H be a closed subgroup of G. For N ∈ N set HN = H N . Then the following conclusions hold. (i) The isomorphism γG : G → limN ∈N G/N maps H isomorphically onto limN∈N HN /N. (ii) Under the present hypotheses, H ∼ = lim H /(H ∩ N) ∼ = lim H N/N ∼ = lim H N /N. N ∈N

N ∈N

N ∈N

The Closed Subgroup Theorem

97

(iii) The limit maps μM : limN ∈N H N/N → H M/M, M ∈ N , are quotient morphisms. (iv) The standard morphisms H /(H ∩ N) → H N/N are isomorphisms of topological groups. (v) Let {KN : N ∈ N } be any set of closed subgroups  of G such that N ⊆ KN , that N ⊆ M implies KN ⊆ KM , and that H = N ∈N KN . Then the set of factor groups KN /N and the natural quotient maps between them form a projective system, and H ∼ = limN ∈N KN /N. Proof. (i) We note HN /N = HN /N ⊆ G/N, (a) and thus HN /N , as a closed subgroup of a complete group is a complete group. Let U be the filter of identity neighborhoods of G; for U ∈ U find V ∈ U such that V V ⊆ U . Since lim N = 1 by hypothesis, there is an N ∈ N such that N ⊆ V . For any subset A of a topological group, the closure A is the intersection of the sets AW where W ranges through all identity neighborhoods. Thus HN = H N ⊆ H N V ⊆ H V V ⊆ H U whence    HN = HN ⊆ H U = H = H. (b) N ∈N

N ∈N

U ∈U

For M ⊇ N, the bonding map νMN : G/N → G/M induces a bonding map μMN : HN /N → HM /M by restriction and corestriction, and def

PN = {νMN : G/N → G/M | (M, N ) ∈ N × N , M ⊇ N}, def

QN = {μMN : HN /N → HM /M | (M, N ) ∈ N × N , M ⊇ N }

(c) (d)

are projective systems in which the bonding maps have dense image. (In the former system they are of course quotient morphisms.) The projective limits are written limN∈N G/N and limN ∈N HN /N, respectively. There is a unique morphism ε : lim HN /N → lim G/N, N ∈N

ε((gN N )N ∈N ) = (gN N )N ∈N

N ∈N

such that the following diagram commutes: ···

HM⏐/M ⏐ inclM  ··· G/M

μMN

←−−−− HN⏐/N ←−−−− · · · ⏐incl  N νMN ←−−−− G/N ←−−−− · · ·

limN ∈N⏐ HN /N ⏐ε  limN ∈N G/N.

(e)

Since G is complete, by Theorem 1.33, there is an isomorphism γG : G → lim G/N, N ∈N

and there is a morphism δH : H → limN ∈N HN /N defined by δH (h) = (hN )N ∈N such that the following diagram commutes: H ⏐ ⏐ incl G

δH

−−−−−→ limN ∈N⏐ HN /N ⏐ε  −−−−−→ limN ∈N G/N. γG

(f)

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1 Limits of Topological Groups

We claim that δH is an isomorphism. For this purpose we define a function σ : limN∈N HN /N → H which we shall show to be a morphism of topological groups and to invert δH . Let (gN N)N ∈N ∈ limN ∈N HN /N, that is, gN ∈ HN and M ⊇ N implies gM ∈ def

gN M, equivalently, gN ∈ gM M. Then F = {gN N : N ∈ N } is a Cauchy filter basis in G (see Theorem 1.30 (iii)), and F does not depend on the particular choice of the representatives gN of the cosets gN N, but only on the cosets. Since G is complete by hypothesis, g = lim F exists. Note that g is also the limit of the net (gN )N ∈N , irrespective of the choice of the representatives gN . We claim that g ∈ HN for all N ∈ N . Fix N ∈ N and consider N ⊇ P in N . Then gP P ⊆ gN N ⊆ HN , for all of these  P and thus g ∈ HN = HN for all N ∈ N ; this proves the claim. Therefore g ∈ N∈N HN = H by (b). We thus define a function σ : limN ∈N HN /N → H by setting (g) σ ((gN N)N ∈N ) = lim{gN N : N ∈ N }. From this definition it follows that    σ ((gN N)N∈N (gN N)N ∈N ) = σ ((gN gN N)N ∈N ) = lim gN gN   = lim gN lim gN = σ ((gN N )N ∈N )σ ((gN N )N ∈N ).

Thus σ is a morphism of groups. Next we show that σ is continuous at the identity. Let V ∈ U; pick a U ∈ U such that U U ⊆ V ; by Theorem 2.1 (i) we may assume that U = U N = NU for some N ∈ N . Now we define   UM ⊆ HM /M by UM = HM /M = for M = N and by UN = U/N and set U M∈N UM ∩ lim M∈N HM /M. Now . Then gN N ∈ UN = U/N . Hence for N ⊇ P we have let g = (gM M)M∈N ∈ U gP ∈ gN N ⊆ U . Thus σ (g) = limM∈N gM ∈ U ⊆ U U ⊆ V . This concludes the proof of the claim that σ : limN ∈N HN /N → H is a morphism of topological groups. For h ∈ H we have σ (δH (h)) = σ ((hN )N ∈N ) = limN ∈N h = h. Now let g = (gN N)N∈N , then δH (σ (g)) = δH (limN ∈N gN ) = (hN )N ∈N with h = limN ∈N gN . If now N ∈ N then N ⊇ P implies gP ∈ gN N whence h = limP ∈N gP ∈ gN N , and thus hN = gN N for all N ∈ N . We conclude δH (σ (g)) = g. Therefore σ and δH are inverses of each other. We have shown that H ∼ = limN ∈N HN /N where HN /N is a closed subgroup of G/N for each N from the filter basis N . (ii) The filter basis {H ∩ N : N ∈ N } in H converges to 1. We know that γH : H → limN ∈N H /(H ∩ N), γH (h) = (h(H ∩ N ))N ∈N is an isomorphism by Theorem 1.33. The bijective morphisms of topological groups H /(H ∩N ) → H N/N , N ∈ N , induce a bijective morphism j in the following sequence of morphisms γH

j

incl

σ

H −−−→ lim H /(H ∩ N) −−−→ lim H N/N −−−→ lim HN /N −−−→ H N ∈N

N ∈N

N ∈N

whose composition is the identity, i.e. σ  incl j γH = id, so that incl (j γH  σ ) = id. Hence the inclusion morphism incl is an isomorphism. (iii) We must show that the limit morphisms μM : limN ∈N H N/N → H M/M are quotient morphisms. Indeed, let U be an identity neighborhood of the limit;

The Closed Subgroup Theorem

99

since lim N = 1 by hypothesis, we may assume that there is an N ⊆ M such that U ker μN = U . Then μN (U ) is an identity neighborhood of H N/N. Since μMN : H N/N → H M/M is a quotient morphism and μM = μMN  μN we conclude that μM (U ) is open which establishes the claim. (iv) We must show that η : H /(H ∩ N) → H N/N,

η(h(H ∩ N )) = hN,

is an isomorphism. In the proof of (ii) we saw that δ = j  γH : H → limN ∈N H N/N is an isomorphism of topological groups. By that which we just saw, for each M ∈ N , the morphism μM j γH : H → H M/M is a quotient morphism. Its kernel, however, is H ∩ M. Hence in the canonical decomposition μM j γH

H −−−−−→ ⏐ ⏐ quot H /(H ∩ M) −−−−−→ η

H M/M  ⏐id ⏐ H M/M H M/M,

the morphism η is an isomorphism. (v) Since the family of {KN : N ∈ N } is filtered, the system of natural quotient maps gN → gM : KN /N → KM /M for N ⊆ M in N is a projective system and def

the projective limit L = limN ∈N KN /N is well defined. We have a natural map δ : H → L induced by γG and thus defined by δ(h) = (hN )N ∈N . Just as was done for the family {HN : N ∈ N } in (i) above we can define an inverse σ : L → H σ ((gN )N∈N ) = limN ∈N gN . The fact that σ is a morphism of topological groups and inverts δ is shown in a fashion that is completely analogous to that which we applied in (i). Let us record a first simple application of the Closed Subgroup Theorem.  Let (Gα : α ∈ A) be a family of topological groups and set G = α∈A Ga . For def  B ⊆ A we set GB = β∈B Gβ . Denote by J the set of finite subsets of A, directed with respect to ⊆. Corollary 1.35. Consider a family of complete topological groups Gα , α ∈ A and let H be a closed subgroup of G = α∈A Gα . For j ∈ J set Hj = pr j (H ) ⊆ Gj , where pr j : G → Gj is the projection. For j ⊆ k in J , the projection pr j k : Gk → Gj maps Hk into Hj , and thus via restriction and corestriction of the projections pr j k , a projective system {pj k : Hk → Hj : (j, k) ∈ J × J, j ⊆ k} arises. Then (hα )α∈A  → ((hα )α∈j )j ∈J : H → lim Hj j ∈J

is an isomorphism of topological groups. Proof. For j ∈ J we identify GA\j in a canonical way with a closed normal subgroup Nj of G. Define N = {Nj : j ∈ J } and notice that N is a filter basis of closed normal subgroups converging to 1. Since all factors Gα are complete, all products Gj are

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complete. Since pr j : G → Gj is a homomorphic retraction, the map (gα )α∈A Nj  → (gα )α∈j : G/Nj → Gj is an isomorphism mapping H Nj /Nj isomorphically onto pr j (H ) for each j ∈ J . In particular, all G/Nj are complete. From Theorem 1.5(b) we know that G ∼ = limj ∈J Gj . Thus Theorem 1.34 applies in such a fashion that the morphism γG is defined by γG ((gα )α∈A ) = ((gα )α∈j )j ∈J . Since H Nj /Nj = pr j (H ), the corollary now follows from Theorem 1.34 (i).

The Role of Local Compactness Finally let us consider once more how local compactness fits into this picture. From Theorem 1.29 (iv) and Corollary 1.26 we immediately obtain the following result: Proposition 1.36. Let N denote a filter basis of closed normal subgroups of a topological group G and assume that M/N is compact for all M, N ∈ N with M ⊇ N . Then the following statements are equivalent: (i) GN is locally compact. (ii) For all N ∈ N the factor group G/N is locally compact. (iii) There is an N ∈ N such that the factor group G/N is locally compact. If these conditions are satisfied, and N converges to 1, then there is a member N ∈ N  is compact and satisfies GN /N ∼ such that N = G/N . Proof. The equivalence of (i), (ii) and (iii) immediately follows from Theorem 1.29 (iv) and Corollary 1.26.  be a compact identity neighborhood of GN . Then Now assume lim N = 1. Let U −1  by Theorem 1.27 (i), U contains a neighborhood of the form νN (U N/N ) with some  = ker νN ⊆ ν −1 (U N/N ) and thus N  is identity neighborhood U of G. Thus N N compact. The remainder follows from Theorem 1.29 (iii). The prototype of the situation we have described is given by the following example: Example 1.37. Let G = Z and let N be the filter basis of all infinite cyclic subgroups. We may index N by N, where we partially order N by m|n (meaning that there is a k ∈ N such that n = km). Then GN = limn∈N Z/Zn is a compactgroup, into which Z is continuously and densely injected but not embedded. Indeed N = {0}, but N n does not converge. If Zm ⊇ Zn, then m|n and Zm/Zn = Z/Z m is finite, but none of the members of N is compact. With Pontryagin Duality as expounded e.g. in [102, Chapter 7 and 8], it is easy to see what the group GN is explicitly. The character group of Z/Zm may be identified with m1 · Z/Z and the dual of the quotient map Z/Zn → Z/Zm if m|n with the inclusion map m1 Z/Z → n1 Z/Z. Since GN = limn∈N Z/nZ, the dual is the di 1 ∞ ∞ rect limit n∈N n1 · Z/Z = Q/Z = p prime Z(p ), where Z(p ) = p∞ · Z/Z,

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101

#m

$  % · Z | m ∈ Z, n ∈ N . Hence GN ∼ /Z = p prime Zp . The dual of =Q % /Z and this is none the inclusion morphism Q/Z → R/Z = T is a morphism Z → Q other than the morphism γG : G → GN . 1 p∞

·Z =

pn

Exercise E1.15. Verify the statements concerning the structure of Z{Zm|m∈N} . [Hint. Use [102, Chapters 1, 7, 8]. From Theorem 1.30 we know the significance of the convergence of the filter basis N . The following proposition provides some elucidation. Proposition 1.38. Let N denote a filter basis  of closed normal subgroups of a locally compact topological group G. Assume that N = {1}. Then the following conditions are equivalent: (i) lim N = 1. (ii) N contains a compact member. Proof. (i) ⇒ (ii): Since G is locally compact we find a compact identity neighborhood U in G. By (i) there is an N ∈ N such that N ⊆ U . Then N is compact. (ii) ⇒ (i): This is a consequence of the following purely point set topological lemma: Lemma A. Let F be a filter basis of closed subsets of a Hausdorff topological space and assume that F contains an element which is a compact set. Then the following statements are equivalent: (a) lim  F = x. (b) F = {x}.

  Proof of Lemma A. (a) ⇒ (b): Since X is Hausdorff, we have F ⊆ UX (x) = {x}.  (b) ⇒ (a): Let N ∈ F be compact and assume F = {x}. Let U be any open def

neighborhood of x. Now the set G = {M ∈ F : M ⊆ N } is a filter basis of compact sets intersecting in {x}. There is at least one of these M contained in U , for if not, then {M \ U : N ⊇ M ∈ F } would be a filter basis of compact  sets which, accordingly, has a point g in its intersection. On the one hand, g ∈ G = F = {x}, on the other g ∈ G \ U , a contradiction. This proves Lemma A and thus completes the proof of the proposition. In fact, conditions (i) and (ii) in the preceding Proposition 1.38 are equivalent to (iii) There is an N ∈ F such that for all M ∈ N with N ⊇ M the factor group M/M0 (where M0 is the connected component of 1 in M) is compact. However, we cannot yet prove (iii) ⇒ (i) at this time. We note that a topological group G whose factor group G/G0 modulo its identity component is compact is called almost connected.

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Remarks 1.39. Assume  that G is a topological group and N is a filter of closed subgroups such that N = {1} and let U be the filter of identity neighborhoods of G. Then the filter UN of all supersets of the sets in the filter basis {U N | (U, N ) ∈ U × N } is the filter of identity neighborhoods of a group topology ON which is coarser than or equal to the topology of G; it is equal to it iff N converges to 1. The topology ON is that unique topology on G which makes γG : (G, ON ) → GN an embedding. If all G/N , N ∈ N are complete, then GN is a completion of (G, ON ). (Completions are unique up to isomorphism.) In particular, if (G, ON ) is complete, then (G, ON ) ∼ = GN . Proof. Exercise E1.16. Exercise E1.16. Prove Remarks 1.39.

The Role of Closed Full Subcategories in Complete Categories We shall have ample justification to consider a full closed subcategory A of a complete category B, that is, a subcategory such that for two objects A1 and A2 of A it contains all B morphisms A1 → A2 and which for any functor D : J → A from a small category J to A ⊆ B it contains the limit lim D formed in B, which exists by the completeness of B. Earlier in this chapter we encountered sufficient conditions for such a set-up to arise in Theorem 1.10 (iv). Specifically, we have remarked in Theorem 1.11 (i), (ii) that the category TopGr of topological groups and continuous morphisms between them is a complete category, and a full subcategory of it is complete if it is closed under the formation of arbitrary products and the passing to closed subgroups. In due course we will encounter full complete subcategories in all completeness theorems such as the Completeness Theorem of the Category of Groups with Lie Algebras 2.25, the Completeness Theorem for Projective Limits of Lie Groups 3.3, and, most crucially for this book, the Completeness Theorem for Pro-Lie Groups 3.36. It is therefore important to remind the reader that full complete subcategories are “retracts” in the following sense. Theorem 1.40 (The Retraction Theorem for Full Closed Subcategories). Let A be a full closed subcategory of the complete category B. Assume that the following condition is satisfied: (S) For each object B in the supercategory B there is a set S(B) of pairs (ϕ, M), ϕ : B → M with objects M of A, such that for every B-morphism f : B → A into an object A of A, there is some (ϕ, M) ∈ S(B), and some morphism f0 : M → A such that f = f0  ϕ.

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103

Then the inclusion functor A → B has a left adjoint L : B → A. In particular, L|A is naturally isomorphic to the identity functor of A. Proof. Since A is closed under the formation of all limits in the complete category B, the category A itself is complete and the inclusion functor A → B preserves limits. By the Adjoint Functor Existence Theorem (see e.g. [102, Theorem A3.60]) it has a left adjoint provided that the inclusions functor A → B so-called Solution Set Condition is satisfied ([102, Definition A3.58]). In our situation, however, this is exactly condition (S) above. For any object B in the category B the universal morphism θA : A → LA (the “front adjunction,” [102, Definition A3.37]) is characterized uniquely up to natural isomorphism by the universal property that for every B-morphism f : B → A for an object A in A there is a unique morphism f  : LB → A such that f = f   θB . Clearly if B is in A, then the identity morphism idB : B → B has this universal property. The most relevant consequence of Theorem 1.40 for this book is the following: The Retraction Theorem for Full Closed Subcategories of TopGr Corollary 1.41. For any closed full subcategory G of the category TopGr of topological groups and continuous morphisms that is closed in TopGr under the formation of products and the passage to closed subgroups, there is a left adjoint functor F : TopGr → G which on G agrees (up to a natural isomorphism) with the identity functor on G. In particular given any topological group G, there is a topological group F G in G and a morphism θG : G → F G with dense image such that for every morphism f : G → H into a G-group H there is a unique morphism f  : F G → H such that f = f   θG . Proof. The existence of F is immediate from Theorem 1.40 once we verify condition (S): Let G be a topological group. Let us say that morphisms ϕj : G → Mj , j = 1, 2 in TopGr are equivalent, if there is an isomorphism ψ : M1 → M2 such that ϕ2 = ψ  ϕ1 . From each equivalence class of morphisms ϕ : G → M such that M = ϕ(G) and M is a topological group in G, let us pick one representative. Let us call the class of all such representatives S(G). Then S(G) is a set, because there is, up to equivalence, only a set of images ϕ(G) under any morphism ϕ with domain G and a set of topologies on each ϕ(G) and that there is up to a natural equivalence (in an obvious sense) a set of topological spaces in which a topological space ϕ(G) is dense, since the cardinal of such a space is not bigger than the cardinal of the set of all filters on ϕ(G). Now let f : G → H be a morphism from G to a topological group H in G. Since G is closed under passing to closed subgroups, f (G) belongs to this category. Thus the corestriction f  : G → f (G) of f to the closure of its image has an equivalent representative in S(G). For brevity we may assume that f  ∈ S(G). Let f0 : f (G) → H be the inclusion map. Then f = f0  f  . Thus condition (S) is verified. It remains to verify that θG has dense image, but that is immediate from the assumption that G is closed under passing to closed subgroups and that the corestriction

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G → θG (G) of θG to the closure of its image has the universal property of θG and thus must agree with θG . G

TopGr

G ⏐ ⏐ ∀f  H

θG

−−−−−→ F⏐G ⏐  f −−−−−→ idH

H

F⏐G ⏐  ∃!f H

Typical examples are the categories CompGr of compact groups and ZCompGr of compact zero-dimensional groups. In the first case for each topological group G, the compact group F G is the Bohr compactification of G and θG : G → F G is the Bohr compactification morphism. In the second case the compact group Fz G = F G/(F G)0 is the zero-dimensional Bohr compactification with the zero-dimensional (z) Bohr compactification morphism zθG : G → Fz G. The notation varies; other names for the Bohr compactification and the zero-dimensional Bohr compactification are also current, such as for instance α(G), respectively, ζ (G). Other examples are the category AbTopGr of topological abelian groups and the category ComplTopGr of complete topological groups. In the first of these two examples, F G is the commutator factor group G/G where G is the algebraic commutator group and in the second, F G is a complete topological group compl(G) together with a universal morphism γG : G → compl(G) through which every morphism from G into a complete topological group factors uniquely. It is customary to say that the group G has a completion if γG is a dense embedding. The category LieGr of Lie groups is incomplete as is the category LCGr of locally compact groups and there is no universal construction producing a universal Lie group or a universal locally compact group for an arbitrary topological group. There will be, however, a universal projective limit of Lie groups attached to a topological group.

Postscript A key result in the theory of locally compact groups is that every connected locally compact group can be expressed as a projective limit of Lie groups. Therefore we begin this chapter with a study of limits, with a strong emphasis on projective limits, keeping in mind that a proof of this key result will be central to any general structure theory of locally compact groups. In order to apply category theory, it is necessary to determine an appropriate category in which to work. That category must be both well-behaved and large enough to include the objects in which we are interested. In this case we seek a category of topological groups which is complete, that is, it has arbitrary limits. The Limit Existence Theorem 1.10 tells us that it suffices that the category has products and equalizers. It

Postscript

105

is seen to be a very useful criterion that a full subcategory of a complete category is complete if it is closed under the formation of products and retracts. In this chapter we repeat some of the material on projective limits from [102], but the presentation here goes much farther. One reason for this is that projective limits of noncompact groups involve additional complexity. For the reader who wishes to work with projective limits of topological groups in a more traditional way than that of putting universal properties in the foreground, Proposition 1.18 and the paragraphs leading up to it provide a self-contained approach that is very concrete. Cofinality plays an important role in the theory of projective limits. The Cofinality Lemma 1.21 is one of those results on projective limits which themselves together with their proofs may appear very technical on the surface. Yet it is but an analog of the well-known fact in the elementary theory of metric spaces that a subsequence of a convergent sequence converges to the same limit. The Cofinality Lemma is important enough that we give a proof in the general context of functors in 1.13; for the case of projective limits of topological groups we not only specialize the general category theoretical version (see 1.21), but also give a separate proof in the concrete situation of topological groups: see Exercise E1.11. In the study of projective limits of compact groups in [102], we saw surjectivity of the bonding maps implies surjectivity of the limit maps. For locally compact groups, the same is true if we also assume that the kernel of each of the bonding maps is compact. In the compact group case, surjective bonding maps are quotient maps. But this is not necessarily so in the locally compact case. This leads one to make the additional assumption that the bonding maps are proper. The relevance of proper morphisms for projective limits is demonstrated in 1.25, precipitating the definitions of strict and strong projective limits in 1.24. In the context of compact groups, these concepts coincide, but in more general circumstances strong is stronger than strict. In 1.26, we give conditions for projective limits to be locally compact. The relevance of projective limits to the structure theory of locally compact groups is a “geometric” one: One wishes to “approximate” a given, possibly complicated group G by better and better approximations through groups H with better known properties; this could happen by finding that one has factor groups H = G/N modulo very small normal subgroups N. Then every element of G is contained in a very small coset gN which is a member of the group H = G/N having the desired better properties. In [102] this is seen by the fact that every compact group is a projective limit of compact Lie groups ([102, Chapters 2 and 9]). The “geometric” aspect of approximating groups given as projective limits is introduced in Theorem 1.27 and it motivates us to consider, in a second subsection, topological groups equipped with a filter basis of closed normal subgroups and their relationship with projective limits. This relationship is presented in Theorems 1.29 and 1.30 which tell us that projective systems and limits associated with topological groups equipped with filter bases N of normal subgroups are much better behaved than projective systems and limits in general. Most of the information is condensed in a natural morphism γG from thegiven topological group G to the associated projective limit GN . It is injective iff N = {1}. It is an algebraic and topological embedding if and only if lim N = 1. But when is it an

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isomorphism? The only reasonable sufficient condition is that of completeness. We present the required fragments of completeness theory in 1.31 (saying that every locally compact group is complete), 1.32 (giving some very basic facts), and 1.33 saying finally that γG is an isomorphism iff G is complete and lim N = 1. Under such circumstances, local compactness plays a special role as we see in 1.36, 1.37, and 1.38, and in 1.39 we make some complementary comments on this geometric approach to approximation. Some pieces of information are taken from [107] where one finds more information related to this context. As one of the essential features of our discourse we will show in Chapter 3 that every member G of the category of projective limits of finite-dimensional Lie groups can be approximated by Lie groups in the internal sense that it has arbitrarily small closed normal subgroups N such that G/N is a finite-dimensional Lie group. This is by no means trivial since G need not be locally compact. Theorem 1.34 discusses the important issue of closed subgroups of projective limits. It clarifies the question: when can we expect a closed subgroup to behave decently in relation to a given filter basis of closed normal subgroups? The perennial difficulty is that the algebraic isomorphism H /(H ∩ N ) → H N/N is an isomorphism of topological groups only under special circumstances. As a first application we see in Corollary 1.35 that a closed subgroup of a product of complete groups is a projective limit of a particular type that will be useful in Chapter 3 when all Gα are taken to be finite-dimensional Lie groups. In Corollary 1.41 we apply our functorial approach to prove that complete full subcategories of the category of topological groups have a retracting functor which behaves very much like the special case of the Bohr compactification in the case that the subcategory is the category of compact groups. We will invoke that theorem in several spots in the book. Example 1.15 investigates the behavior of projective systems of topological groups under morphisms. The category theoretical approach will surface often in later chapters. Projective limits of topological groups are treated in [25, Chap. III, §7].

Chapter 2

Lie Groups and the Lie Theory of Topological Groups

The essential attributes of a Lie group G are the associated Lie algebra L(G) and the exponential function exp : L(G) → G; this was explained in detail in Chapter 5 of [102] for linear Lie groups. In this chapter we present a definition of a Lie group that places these attributes in the foreground and so is different from that found in most textbooks. However, many features of Lie theory are shared by classes of topological groups which are much larger than that of Lie groups; indeed they include the class of compact groups and connected locally compact groups. In the second part of this chapter we therefore introduce a general Lie theory for topological groups and provide information which we shall use in subsequent chapters. For obvious categorical reasons we will restrict our attention to the category of topological groups having a Lie algebra and focus on the functor L from this category to the category of topological Lie algebras. In this chapter we shall then be able to put Lie’s Third Fundamental Theorem into a general framework by observing that there is a left adjoint functor  to L. Prerequisites. Even though we begin this chapter with the definition of a Lie group, it is advisable that the reader have some familiarity with Lie theory, as can be found for example in Chapter 5 of [102]. We also use material from Appendix 1. As in Chapter 1 we shall utilize at a late point in this chapter some basic category theory such as concerns the nature and existence of adjoint functors, but not more than is made explicit for instance in Appendix 3 of [102].

The General Definition of a Lie Group From Appendix 1, Definition A1.1 (ii) we know that a Dynkin algebra is a completely normable topological vector space with a Lie algebra structure such that the Lie bracket [·, ·] : L × L → L is continuous. By the Fundamental Theorem on Dynkin Algebras A1.7, every Dynkin algebra has a Campbell–Hausdorff neighborhood B for which there is a Campbell–Hausdorff multiplication (see Definition A1.8). These facts permit us to introduce the following definition. The Definition of Lie Groups Definition 2.1. A topological group G is called a Lie group if there is a Dynkin algebra g and a homeomorphism e : B → U from a Campbell–Hausdorff neighborhood

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B of g (see Definition A1.8 of Appendix 1) onto an open identity neighborhood U of G such that the following condition is satisfied (∀X, Y ∈ B)

(X ∗ Y ∈ B) ⇒ e(X ∗ Y ) = e(X)e(Y ).

Clearly, every topological group that is isomorphic (as a topological group) to a Lie group is a Lie group. The classical matrix groups Gl(n, R) and all of their closed subgroups are Lie groups. Indeed, we see in Proposition 2.5 below that every linear Lie group (see Definition 2.4 below) is a Lie group thereby providing a wealth of further examples. Our definition of Lie group is not that found in most textbooks. For our purposes, however, this is the ideal definition as will become clear in the course of this book. (The impatient reader may wish to see [102], Chapter 5, for linear Lie groups.) In many texts, a Lie group is defined to be a group on an analytic manifold for which the group operations of multiplication and inversion are analytic maps. The proof in [102, Corollary 5.34ff.] that linear Lie groups are analytic groups, works in the present context to show that a Lie group in the sense of 2.1 above is analytic. Any analytic group is smooth, and in standard sources such as Bourbaki [18] it is shown, not without effort, that any smooth Lie group is a Lie group in our sense. Restrictions on the dimension are not needed, but classical texts deal with the special case of analytic groups on finite-dimensional manifolds. If we use the terminology introduced in and after Appendix A1.8 we can reformulate our definition using the concept of local group from A1.9. Proposition 2.2. For a topological group, the following statements are equivalent: (i) G is a Lie group. (ii) There is a Dynkin algebra g, a Campbell–Hausdorff neighborhood B of g, and an open identity neighborhood U of G such that the local group (B, DB ) associated with g and B is isomorphic to the local group (U, DU ) associated with G and U . Proof. (i) ⇒ (ii): Let g be a Dynkin algebra and B0 a Campbell–Hausdorff neighborhood with a homeomorphism e : B0 → U0 onto an open identity neighborhood of G satisfying e(X ∗ Y ) = e(X)e(Y ) for all X, Y ∈ B0 with X ∗ Y ∈ B0 . Next let B be a Campbell–Hausdorff neighborhood of g contained in B0 such that B ∗ B ⊆ B0 . Set U = e(B). Since B is open and e is a homeomorphism, U is open in U0 and since U0 is open in G, the set U is open in G. Obviously, the restriction and corestriction f : B → U of e is a homeomorphism satisfying f (X ∗ Y ) = f (X)f (Y ) for all X, Y, X ∗ Y ∈ B. Let g, h ∈ U such that gh ∈ U . Then there are X, Y ∈ B such that g = f (X) and h = f (Y ). Now X ∗ Y ∈ B ∗ B ⊆ B0 and thus e(X ∗ Y ) = e(X)e(Y ) = f (X)f (Y ) = gh ∈ U ⊆ U0 and so X ∗Y ∈ B since the homeomorphism e : B0 → U0 maps B to U . Then f −1 (gh) = X ∗ Y = f −1 (g) ∗ f −1 (h). This completes the proof of (i) ⇒ (ii). (ii) ⇒ (i): Let g, B, and U be as in (ii), naming e the homeomorphism that implements the isomorphism of local groups. Then e : B → U is a homeomorphism

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109

satisfying e(x ∗ y) = e(x)e(y) for all x, y ∈ B for which x ∗ y ∈ B. This proves (ii) ⇒ (i). Let G be a Lie group. If g a Dynkin algebra satisfying the conditions of Definition 2.1 and of Proposition 2.2, we shall call it a concomitant Dynkin algebra. Example 2.3. Let T = R/Z be the circle group. Then G = T is a Lie group: Set g = R which is a Dynkin& algebra ' with [X, Y&] =1 01and ' hence  X ∗ Y = X + Y for all 1 1 X, Y ∈ R, and take B = − 2 , 2 and U = − 2 , 2 + Z /Z; then B is a Campbell– Hausdorff neighborhood of g and U is an open symmetric identity neighborhood of G such that the function e : B → U , e(X) = X + Z is a homeomorphism and satisfies e(X ∗ Y ) = X + Y + Z = (X + Z) + (Y + Z) = e(X) + e(Y ). Now let g = h = 1 1 −1 −1 gh = 23 + Z = − 13 + Z, we have 3 + Z; then e (g) = e (h) = 3 and since   e−1 (g) ∗ e−1 (h) = 13 + 13 = 23 = − 13 = e−1 − 13 + Z = e−1 (gh). This example shows that we have to be careful with the proof of (i) ⇒ (ii) in Proposition 2.2; it is instructive to read it again with this example in mind. In [102] we defined the concept of linear Lie group and we based the entire analysis of the structure of compact groups on it. Let us recall the definition of the exponential function from Appendix 1, Proposition A1.4, or from [102, Definition 5.1] and reproduce the definition of a linear Lie group here: The Definition of Linear Lie Groups Definition 2.4. A topological group G is called a linear Lie group if there is a Banach algebra A with identity and an isomorphism of topological groups from G onto a subgroup G of the multiplicative group A−1 of A such that there is a closed Lie subalgebra g of (A, [· , ·]) with the property that exp maps some 0-neighborhood of g homeomorphically onto a 1-neighborhood of G . Naturally, we wish to understand at once the following proposition: Proposition 2.5. (i) Every linear Lie group is a Lie group. (ii) A locally compact subgroup of a Lie group is a Lie group. (iii) A closed subgroup of a finite-dimensional Lie group is a Lie group. Proof. (i) Assume that the topological group G satisfies the conditions of a linear Lie group specified in Definition 2.4. By Proposition A1.5 in Appendix 1 there is an open ball B0 around 0 in A such that the exponential function maps B0 homeomorphically onto an open identity neighborhood U0 in the group of invertible elements A−1 of A such that for all x, y ∈ B0 we have exp(x ∗ y) = (exp x)(exp y). If x, y ∈ g, then x ∗ y ∈ g: this follows from A1.6 or from [102, Theorem 5.21 (iv)], the proof of which does not use A1.6. Now let B be some open ball around 0 in g which is mapped homeomorphically by exp onto an open identity neighborhood U of G . Clearly there is no harm in assuming that B ⊆ B0 and thus U ⊆ U0 . Let e : B → U be the restriction and corestriction of exp. If now x, y ∈ B such that x ∗ y ∈ B then

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e(x ∗ y) = exp(x ∗ y) = (exp x)(exp y) = e(x)e(y). This shows that the conditions of Definition 2.1 are satisfied for G . Thus G is a Lie group. Since G is isomorphic to G , the group G is a Lie group as well. (ii) Let G be a Lie group and H a locally compact subgroup. Give the Dynkin algebra L(G) a norm and let B be a Campbell–Hausdorff neighborhood such that B ∗ B is def

defined and exp(X ∗ Y ) = exp X exp Y for X, Y ∈ L(G). Set  = B ∩ exp−1 G. Then  is a locally compact local subgroup of B such that  ∗  ∩ B ⊆ . Set def

h = T() then by [102, Theorem 5.28], h is a Dynkin algebra and h ∩ B is open in . Accordingly, expG maps h into H and induces a homeomorphism of h ∩ B onto an open identity neighborhood of H . Hence H is a Lie group with exponential function expG |h : h → H . (iii) This is now immediate from (ii) since a finite-dimensional Lie group is locally compact and thus a closed subgroup of it is locally compact. A closed subgroup of an infinite-dimensional Lie group need not be a Lie group. Typically, the additive group G of the real Banach space L1 ([0, 1], λ) for Lebesgue measure on the unit interval contains a closed contractible, hence arcwise connected, subgroup H = {class(f ) ∈ L1 ([0, 1], λ) : f (r) ∈ Z for almost all r ∈ [0, 1]}, and this subgroup has no nonzero one parameter subgroups; hence fails to be a Lie group.

The Exponential Function of Topological Groups Linear Lie groups already come with an exponential function. We shall now see that this extends to Lie groups. The Definition of One Parameter Subgroups, the Exponential Function, and L(G) Definition 2.6. (i) Let G be a topological group. A one parameter subgroup is a morphism X : R → G of topological groups. The set Hom(R, G) of all one parameter subgroups is given the topology of uniform convergence on compact sets, that is, basic neighborhoods of a one parameter subgroup X : R → G are of the form W (X; n, U ), where n ∈ N and U ranges through the open identity neighborhoods of G and where def

W (X; n, U ) = {Y ∈ Hom(R, G) | (∀r ∈ R, |r| ≤ n)

Y (r)X(r)−1 ∈ U }.

The set Hom(R, G) endowed with this topology is denoted by L(G). The continuous evaluation function X  → X(1) : L(G) → G will be denoted by expG and will be called the exponential function of the topological group G. We note expG X = X(1). (ii) A partial one parameter subgroup of a group G is a function f : I → G from an interval I ⊆ R which is a neighborhood of 0 such that f (x + y) = f (x)f (y) whenever x, y, x + y ∈ I . A local one parameter subgroup of a topological group is a partial one parameter subgroup which is, in addition, continuous.

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111

(iii) A filter basis F on Hom(R, G) is called a Cauchy filter base if for each natural number n and each identity neighborhood U of G there is an F ∈ F such that X, Y ∈ F implies (∀r ∈ R, |r| ≤ n) Y (r)X(r)−1 ∈ U. A few immediate observations are in order. Firstly, let us say that an action (r, x)  → r · x : R × X → X of R on a Hausdorff topological space X with base point x0 is a scalar multiplication if the following conditions are satisfied: (i) (ii) (iii) (iv)

The action is continuous. (∀x ∈ X) 0 · x = x0 and (∀x ∈ X) 1 · x = x. (∀r, s ∈ R, x ∈ X) (rs) · x = r · (s · x). For each x ∈ X, the orbit R · x is an abelian group with respect to an operation + which satisfies (∀r, s ∈ R) (r · x) + (s · x) = (r + s) · x.

Exercise E2.1. Show that the following statement holds: Assume that X is a Hausdorff topological space with scalar multiplication and base point x0 . If x0 = x ∈ X, then the function r  → r · x : R → R · x is a bijective morphism of abelian groups. The base point x0 is the neutral element of all abelian groups R · x, x ∈ X. The multiplicative group R× = (R \ {0}, ·) acts on X and has the orbits {x0 } and R× · x with x = x0 . [Hint. The function r  → r · x : R → R · x is a surjective morphism of abelian topological groups. It maps the zero 0 of R to the neutral element 0 · x = x0 of R · x. We claim that it is bijective if x = x0 ; indeed, if r · x = x0 for r = 0, then x = 1·x = (r −1 r)·x = r −1 ·(r ·x) = r −1 ·x0 = r −1 ·(0·x0 ) = (r −1 0)·x0 = 0·x0 = x0 . If x = x0 then R · x = R · x0 = R · (0 · x0 ) = {x0 }. Clearly, the multiplicative group R× acts on X, and by the preceding the action has the orbits {x0 } and R× · x, x = x0 . The preceding exercise shows that a topological space with a scalar multiplication looks a lot like a topological vector space without addition. Exercise E2.2. Verify the following assertion. Let Rn be a euclidean space and let S be any closed nonempty subset of the unit sphere. Then R · S is a locally compact space with scalar multiplication. The following remark will show that the topological space L(G) has a scalar multiplication for every topological group G. Remark 2.7. (i) The constant morphism O : R → G, O(r) = 1, is a member of L(G) which we consider distinguished; so L(G) is a pointed space, i.e. a space with a base point. (ii) Define a continuous action R × L(G) → L(G) by (r · X)(t) = X(tr) for r, t ∈ R, X ∈ L(G). Then 0 · X = O and (rs) · X = r · (s · X). def

(iii) For each X ∈ L(G), the image A = X(R) is an abelian subgroup of G, and thus L(A) = Hom(R, A) is an abelian group under pointwise multiplication. Each

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r · X may be considered as an element of L(A) such that (r + s) · X = (r · X)(s · X). The last identity can be rephrased as exp(r + s) · X = (exp r · X)(exp s · X). (iv) If G is a complete topological group, then every Cauchy filter base in L(G) converges. The action (r, X)  → r · X : R × L(G) → L(G) will be called the scalar multiplication of L(G). Exercise E2.3. Prove Remark 2.7. [Hint. (i) and (iii) are clear. (ii): Firstly, (0 · X)(t) = X(t · 0) = X(0) = 1 whence 0 · X = O. Next (rs) · X(t) = X(trs) = (s · Y )(t) where Y (t) = X(tr) = (r · X)(t). Hence (rs)·X = r ·(s ·X). Finally verify continuity of the action: Let W (r0 ·X0 ; n, U ) be a basic neighborhood of r0 · X0 according to Definition 2.6 (i). Then we have Y ∈ W (r0 · X0 ; n, V ) iff Y (t) ∈ V X0 (tr0 ) for all t ∈ [−n, n]. Let U be an open identity neighborhood of G such that U U ⊆ V . Let δ ∈ ]0, 1] be such that |r − r0 | < δ implies X0 (tr) ∈ U X0 (tr0 ) for t ∈ [−n, n]. Find m ∈ N so that |r −r0 | ≤ 1 and |t| ≤ n imply |tr| ≤ m. Then (r, X) ∈ ]r0 − δ, r0 + δ[ × W (X0 ; m, U ) and |t| ≤ n imply |tr| ≤ m and thus (r · X)(t) = X(tr) ∈ U X0 (tr) ⊆ U U X0 (tr0 ) ⊆ V (r0 · X0 )(t), that is, r · X ∈ W (r0 · X0 ; n, V ). (iv) Let F be a Cauchy filter base of L(G). Then for each r ∈ R, the set {F (r) : F ∈ F }, where F (r) = {X(r) : X ∈ F } is a Cauchy filter basis in G. Since G is a complete topological group, it converges to a limit Z(r) in G. Let n ∈ N, U a closed identity neighborhood of G then there is an F0 ∈ F such that X(r)Y (r)−1 ∈ U for all X, Y ∈ F0 and r with |r| ≤ n. Since (F (r))f ∈F converges to Z(r) for all r, and since U is closed, we conclude Z(r)Y (r)−1 ∈ U for |r| ≤ n. Since Z is a pointwise limit of one parameter groups, Z is a group homomorphism. Use the information provided to show that Z, given an identity neighborhood V of G, there is an ε > 0 such that Z(] − ε, ε[) ⊆ V . Thus Z ∈ Hom(R, G) = L(G). Finally show that F converges to Z in the topology of L(G).] Among other things we have shown that L(G) is a uniform space which is complete if G is complete. The following remarks are still quite elementary. Remark 2.8. (i) Every partial one parameter subgroup I → G of a group extends uniquely to a group morphism R → G. (ii) Every local one parameter subgroup of a topological group extends uniquely to a one parameter subgroup, i.e. to a member of L(G). def

(iii) If X, Y ∈ L(G), then the equalizer E = {r ∈ R : X(r) = Y (r)} is a closed subgroup of R, and thus is either R (in which case X = Y ) or is of the form Zr for some r ∈ Z. (For a category theoretical definition of equalizers, see Definition 1.4 in the previous chapter.) In particular, if two one parameter subgroups agree on some nonempty open set, then they are equal.

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Exercise E2.4. Prove Remark 2.8. [Hint. (ii) is a consequence of (i) in view of the fact that a group homomorphism between topological groups is continuous iff it is continuous at the identity. For a proof of (i) consider a partial one parameter subgroup f : I → G. For r ∈ R find a natural number m such that m1 · r ∈ I and show that for two such numbers m and n m   n   = f n1 · r ; define this group element to be F (r) and show we have f m1 · r that F : R → G is the desired extension. See [102, Lemma 5.8]. Condition (iii) is readily verified; one must know the elementary fact that a closed proper subgroup of R is cyclic.] Lemma 2.9. Let G be a Lie group and let g, B, e : B → U be as in Definition 2.1. (i) For each x ∈ g, set Ix = {r ∈ R | r · x ∈ B}. Then ex : Ix → G, ex (r) = e(r · x) is a local one parameter subgroup. (ii) For each x ∈ g there is a unique one parameter subgroup η(x) : R → G such that η(x)|Ix = ex . Proof. (i) Since B is a convex open symmetric neighborhood of 0 in g and r  → r · x : R → g is a continuous linear map, the set Ix is an open symmetric interval about 0. If r, s, r + s ∈ Ix , then ex (r + s) = e((r + s) · x) = e(r · x ∗ s · x) = e(r · x)e(s · x) = ex (r)ex (s). (ii) For x ∈ g, let η(x) : R → G be the unique extension of ex according to 2.8 (ii). Proposition 2.10. Let G be a Lie group and let g, B, e : B → U be as in Definition 2.1 for a concomitant Dynkin algebra g. Then η : g → L(G) is a homeomorphism satisfying t · η(x) = η(t · x) for all t ∈ R. Proof. Choose η(x) = ex as in 2.9 (ii). Recall that (∀r ∈ Ix )

η(x)(r) = e(r · x).

(1)

Firstly, we show that t · η(x) = η(t · x) for t ∈ R. For all sufficiently small r ∈ R we have (t · η(x))(r) = η(x)(rt) = e(rt · x) and η(t · x)(r) = e(r · (t · x)) = e(rt · x) and this proves the claim in view of 2.8 (iii). Secondly, we show that η is injective. Assume η(x) = η(y). Then e(r ·x) = e(r ·y) for all sufficiently small r ∈ R. Since e : B → U is a homeomorphism, r · x = r · y follows for all sufficiently small r, which implies x = y. Thirdly, we show that η is surjective. Let X ∈ L(G). We must show that there is an x ∈ g such that X = η(x). Since X : R → G is continuous, there is an open symmetric interval J around 0 in R such that X(J ) ⊆ U . Let j : B → g denote the def

inclusion map. Then f = j  e−1  X|J : J → g is a continuous function satisfying f (r + s) = f (r) ∗ f (s) for r, s ∈ J . We fix a positive t ∈ J . For 0 < m ≤ n ∈ N we set q = m n and compute −1 f (q · t) = f (n · · + n−1 · +t ) = f (n−1 · t) ∗ · · · ∗ f (n−1 · t) = m · f (n−1 · t). ( · t + ·)* )* + ( m times

m times

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The case m = n shows f (t) = n · f (n−1 · t) i.e. f (n−1 · t) = n−1 · f (t). Thus f (q · t) = m n f (t) = q · f (t). Since f is continuous, we have f (rt) = r · f (t) for all r ∈ [0, 1]. Hence e(r · f (t)) = X(rt) = t · X(r) for all 0 ≤ r ≤ 1. On the other hand η(f (t))(r) = e(r · f (t)) for all r ∈ If (t) . Thus η(f (t)) = t · X and therefore X = t −1 · η(f (t)) = η(t −1 · f (t)). The claim follows with x = t −1 · f (t). Thus η is a bijection. It now remains to show that it is a homeomorphism. Since both R and G satisfy the First Axiom of Countability (that is the statement that each point has a countable basis for the filter of its neighborhoods) it suffices to show that for each sequence (xn )n∈N in g and each x ∈ g the relations x = lim xn and

(2)

n∈N

η(x) = lim η(xn )

(3)

n∈N

are equivalent. Statement (3) is equivalent to (∀p, q ∈ N)(∃N ∈ N)(∀n > N, |r| ≤ p)

η(xn )(r) ∈ η(x)(r)e

1 q

 ·B .

(3 )

  1 Assume   1that (3 ) is satisfied. Remember (1) above and let p0 and q0 be such that p0 · x ∗ q0 · B ⊆ B. Then for all q ≥ q0             η(x) p10 e q1 · B = e p10 · x e q1 · B = e p10 · x ∗ q1 · B ⊆ e(B).       So for all n > N with η(xn ) p10 ∈ η(x) p10 e q1 · B and q ≥ q0       e p10 · xn ∈ e p10 · x ∗ q1 · B ,

i.e. 1 p0

We conclude that limn∈N

1 p0

· xn ∈

· xn =

1 p0



1 p0

   · x ∗ q1 · B

· x and thus x = limn∈N xn , proving (2).

Conversely, assume (2) holds. Pick p0 ∈ N so that p10 · x ∈ B. Then pr0 · xn converges to pr0 · x uniformly for r ∈ [0, 1] on B ⊆ g (where we consider only     sufficiently large n). Then p10 · η(xn )(r) = η p10 · xn (r) = e pr0 · xn converges   to e pr0 · x = · · · = p10 · η(x)(r) uniformly for r ∈ [0, 1] on G. Since scalar   multiplication is continuous on L(G) by 2.7 (ii), η(xn )(r) = pp0 · p10 · η(xn ) pr   converges to η(x)(r) = pp0 · p10 · η(x) pr uniformly for r ∈ [0, p]. The same argument shows that convergence is uniform for r ∈ [−p, 0] and hence is uniform on [−p, p]. This completes the proof. If X and Y are two topological spaces with scalar multiplication, then a function f : X → Y is called a scalar morphism if it is continuous and satisfies f (t ·x) = t ·f (x) for all t ∈ R and x ∈ X. With this concept we can phrase Proposition 2.10 as follows: For each Lie group G with a concomitant Dynkin algebra g there is a scalar isomorphism from g to L(G) as spaces with scalar multiplication.

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The Lie Algebra of a Topological Group In any group G we define the commutator comm(g, h) of two elements g, h ∈ G to be ghg −1 h−1 . For a topological group G and one parameter subgroups X, Y ∈ L(G) we define continuous functions XY, comm(X, Y ) : R → G by (XY )(r) = X(r)Y (r) and comm(X, Y )(r) = comm(X(r), Y (r)). Of course these functions are not one parameter subgroups in general except when comm(X(r), Y (r)) = 1 for all r, which is certainly the case if G is commutative. For the following theorem let us denote by  : R → R the function given by (t) = t 2 . For a subset A of a (not necessarily topological) group G we shall write A for the subgroup algebraically generated by A in G. The Definition of a Topological Group with a Lie Algebra and of a Generating Lie Algebra Definition 2.11. Let G be a topological group. Then it is said that G has a Lie algebra or, equivalently, that G is a topological group with a Lie algebra if the following conditions hold: (i) For all X, Y ∈ L(G), the following limits exist pointwise: ,, -, --n 1 1 def X + Y = lim , ·X ·Y n→∞ n n , -n2 1 1 def [X, Y ]   = lim comm · X, · Y n→∞ n n

(4) (5)

and X + Y, [X, Y ] ∈ L(G). (ii) Addition (X, Y )  → X + Y : L(G) × L(G) → L(G) and bracket multiplication (X, Y )  → [X, Y ] : L(G) × L(G) → L(G) are continuous. (iii) With respect to scalar multiplication ·, addition +, and bracket multiplication [· , ·] the set L(G) is a real Lie algebra. The Lie algebra L(G) of a topological group is said to be generating, if the closed subgroup exp L(G) generated by the image is the identity component G0 of G. In particular, if G has a Lie algebra, then L(G) is a topological Lie algebra. Note that a topological group G has a Lie algebra if and only if G0 has a Lie algebra. We shall amplify this remark later in Proposition 3.5. Theorem 2.12. Every Lie group has a Lie algebra, and the Lie algebra of a Lie group is a Dynkin algebra. Proof. First we verify condition 2.11 (i). Let G be a Lie group, g a Dynkin algebra and B a Campbell–Hausdorff neighborhood such that there is a homeomorphism e : B → U

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onto an open identity neighborhood U of G as in Definition 2.1. Let X, Y ∈ L(G) and determine x, y ∈ g with the help of Proposition 2.10 in such a fashion, that η(x) = X and η(y) = Y . By the Fundamental Theorem on Dynkin Algebras A1.7 (iii), for each t ∈ R we have , t t ·x∗ ·y , (6) t · (x + y) = lim n · n→∞ n n , t t 2 2 t · [x, y] = lim n · comm∗ · x, · y . (7) n→∞ n n Since η is continuous and preserves scalar multiplication by Proposition 2.10, we have η(x + y)(t) = η(t · x + t · y)(1) -n , , t t t t · x ∗ · y (n) = lim e ·x∗ ·y = lim η n→∞ n→∞ n n n n , , , , - , --n , --n t t t t = lim e = lim η(x) ·x e ·y η(y) n→∞ n→∞ n n n n , , - , --n ,, - , - -n t 1 t 1 = lim X = lim Y · X (t) · Y (t) n→∞ n→∞ n n n n ,., -, -/ -n 1 1 = lim ·X ·Y (t) , n→∞ n n and this limit is (X + Y )(t) by definition. An analogous computation shows , , - - n2 1 1 2 , η([x, y])(t ) = lim comm · X, · Y (t) n→∞ n n and this limit is [X, Y ](t 2 ) = ([X, Y ] )(t). Thus (i) is proved and we have established (∀x, y ∈ g) η(x + y) = η(x) + η(y), (∀x, y ∈ g) η([x, y]) = [η(x), η(y)].

(8) (9)

Condition 2.11 (ii) and (iii) now follow from the fact that η is a homeomorphism and from (8) and (9) above which show that η transports the topological Lie algebra structure from g to L(G). This completes the proof of the theorem. Let G be a Lie group and let g, B, e : B → U be as in Definition 2.1 with a concomitant Dynkin algebra g. Let η be as in 2.9 and 2.10. In summing up what has been said we find the following commutative diagram helpful: B ⏐ ⏐ e V

incl

−−−−−→

g

η

−−−−−→ ⏐ L(G) = Hom(R, G) ⏐exp  G −−−−−→ G −−−−−→ G. incl

idG

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117

Theorem 2.13. A Lie group G determines the Dynkin algebra g which occurs in Definition 2.1 uniquely up to isomorphism. Indeed every such Dynkin algebra is isomorphic to L(G). The exponential function expG : L(G) → G maps a suitable Campbell– Hausdorff neighborhood B of L(G) homeomorphically onto an open identity neighborhood of G in such a fashion that (∀X, Y ∈ B)

exp(X ∗ Y ) = (exp X)(exp Y ).

The image of the exponential function of a Lie group is an identity neighborhood of G; in particular, the subgroup expG L(G) algebraically generated by this image agrees with the identity component G0 of G. Proof. The Dynkin algebra L(G) is uniquely determined by G through its definition and by Theorem 2.12. That η : g → L(G) is the desired isomorphism of Dynkin algebras also follows from 2.12. From A1.9 we deduce that η(x ∗ y) = η(x) ∗ η(y) for all sufficiently small x, y ∈ g. For all sufficiently small x ∈ g we recall η(x)(1) = e(x) and so, setting X = η(x), Y = η(y) we have exp X = X(1) = e(x), exp Y = Y (1) = e(y) and thus exp(X ∗Y ) = (X ∗Y )(1) = (η(x)∗η(y))(1) = η(x ∗y)(1) = e(x ∗y) = e(x) ∗ e(y) = η(x)(1) ∗ η(y)(1) = (exp X) ∗ (exp Y ) for all sufficiently small x and y. Since the identity neighborhood e(B) = expG (B) is contained in the image of expG , the latter is an identity neighborhood. Hence expG L(G) is an open subgroup of G, and is therefore closed (see [102, A4.25 (ii), (iii)]). Thus it contains the connected component G0 of the identity. On the other hand, the set expG L(G) is arcwise connected, and so is the group it generates; therefore expG L(G) ⊆ G0 . Theorem 2.14. Every abelian topological group G has a Lie algebra L(G), and L(G) is commutative. If G is a complete topological group, then L(G) is a complete topological vector space. Proof. If X, Y ∈ L(G) and comm(X(r), Y (r)) = 0 for all r ∈ R, then , , - , --n t t X = X(t)Y (t) Y n n and thus 2.11 (i) is clearly satisfied with X + Y being the pointwise product. Obviously the bracket product vanishes. The assertion about completeness follows from Remark 2.7 (iv). Proposition 2.15. If G is the additive group of a topological vector space E, then the function ϕ : E → L(G) which is defined by ϕ(x)(t) = t · x is a linear map which is the inverse of the exponential function exp : L(G) → G. Therefore, L(G) ∼ = E as topological vector spaces. In particular, every topological vector space occurs (up to isomorphism of topological vector spaces) as the Lie algebra of a topological group, and every complete topological vector space occurs as the Lie algebra of a complete topological group. Proof. Exercise E2.5.

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Exercise E2.5. Prove Proposition 2.15. Next we observe that Lie groups have a property which is expressible in terms of simple group theoretical and simple topological concepts. Definition 2.16. A topological group G is said to have no small subgroups if it has an identity neighborhood U such that for any subgroup H of G the relation H ⊆ U implies H = {1}. A subgroup of a group having no small subgroups has no small subgroups. Proposition 2.17. A Lie group has no small subgroups. Proof. Let G be a Lie group and B an open convex symmetric zero neighborhood of L(G) which is mapped homeomorphically onto an open identity neighborhood V of G   def by the exponential function. Set U = expG 21 · B and let H be a subgroup of G  with H ⊆ U . Now assume that h ∈ H . Then since expG  21 · B : 21 · B → U is a homeomorphism, there is a unique X ∈ 21 · B such that h = expG X. Suppose now that h = 1. Then X = 0 and there is a smallest natural number n > 1 such that n · X ∈ B \ 21 · B. It follows that hn = (expG X)n = expG n · X ∈ H ∩ (V \ U ) = Ø. This contradiction shows h = 1 and thereby H = {1}. In [102, Exercise E7.17] we have seen a closed arcwise connected subgroup G of the additive group of a separable Banach space E such that L(G) = {0}; in particular, G is an arcwise connected complete group without small subgroups which fails to be a Lie group. We also saw in [102, Proposition 5.33 (iv)], that a compact group without small subgroups is a (linear) Lie group; this remains true for locally compact groups as we shall see in due course, but the proof is much harder. At this point we observe, however, that it is easy to see that a locally compact group with no small subgroups is metrizable. Exercise E2.6. Let G be a topological group with no small subgroups. Then G has a metrizable topology which is coarser than or equal to the given one, and if G is locally compact, then they are equal. [Hint. Let U0 be a symmetric identity neighborhood of G containing no subgroups other than {1}. Construct inductively a descending sequence Un of symmetricidentity neighborhoods Un such that Un Un ⊆ Un−1 , n = 1, 2, . . . . Then D = ∞ n=1 Un is a group contained in U0 and thus is {1}. If U0 can be chosen to be compact, then the filter basis {Un | n ∈ N} converges to 1 and thus forms a basis of identity neighborhoods of 1; i.e. the topology of G is first countable and thus metrizable (see e.g. [102, Lemma A4.10ff., notably Theorem A4.16]). If G is not locally compact, the filter basis {Un | n ∈ N} defines a left (or right) Hausdorff uniform structure with a countable basis which is coarser than or equal to the left (respectively right) uniform structure determined by the given group topology. Uniform spaces with a countable basis are metrizable.]

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119

Let U be the group of unitary operators on a separable infinite-dimensional Hilbert space H . With respect to the operator norm topology, U is a Lie group, and its Lie algebra is (up to a natural identification) the Lie algebra of all skewadjoint bounded operators with the bracket [X, Y ] = XY − Y X and the operator norm topology. However, if we give U the strong operator topology, that is, the topology of pointwise convergence, then U is a topological group in which L(U ) may be identified with the set of all not necessarily bounded, but closed skewadjoint operators X (for which the domains of definition of X and X∗ agree). This is the content of Stone’s Theorem (see e.g. [69, p. 32, Theorem, 4.7]). However, L(G) is not a vector space because X + Y in general fails to be a closed operator with dense domain of definition. The unitary groups in the strong operator topology thus provide examples of topological groups which do not have a Lie algebra. However, if the closure X + Y is a closed skewadjoint operator, then the formula (4) does hold (Trotter Product Formula); see e.g. [69, p. 53, Theorem 8.12]. A central tool for the analysis of the structure of connected pro-Lie groups will be the associated Lie algebras or, more specifically, topological Lie algebras. Thus we use the machinery of Lie algebras and exponential functions on the background of topological groups in general. Accordingly we study the category LieAlgGr of topological groups which have Lie algebras and a number of its interesting subcategories.

The Category of Topological Groups with Lie Algebras We shall review and record the names of several categories of interest to us. Definition 2.18. (i) The category of topological groups (all of which are assumed to be Hausdorff) and continuous group homomorphisms as morphisms is denoted by TopGr. (ii) If C is a category, a subcategory A is called full if A(A1 , A2 ) = C(A1 , A2 ) for each pair  of objects A1 , A2 in ob(A). If C is any class of objects of a category C, then A = (A1 ,A2 )∈C×C C(A1 , A2 ) is a full subcategory of C called the full subcategory of C-objects. (iii) The full subcategory of TopGr of all topological groups having a Lie algebra (in the sense of Definition 2.11) is denoted by LieAlgGr. The full subcategory of LieAlgGr of all Lie groups will be called LieGr. (iv) The full subcategory of TopGr of abelian topological groups is denoted by AbTopGr. (v) The category of Hausdorff topological spaces with continuous functions as morphisms is denoted by Top, and the category of Hausdorff topological spaces with base point and base point preserving continuous functions as morphisms is denoted by Top∗ . (vi) The category of Hausdorff topological spaces with scalar multiplication and scalar morphisms between them is denoted by Scal (see discussion following 2.6).

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It is worth taking note of the fact that a morphism f : G → H between two groups having a Lie algebra is simply a continuous group homomorphism. Theorem 2.19. Let f : G → H be a TopGr-morphism. Then there is a unique scalar morphism L(f ) : L(G) → L(H ) such that L(f )

L(G) −−−−−→ L(H ⏐ ⏐ ) ⏐ ⏐exp expG   H G −−−−−→ H

(10)

f

commutes. It is defined as L(f )(X) = f  X. Proof. Since f is continuous, if U is an identity neighborhood of G we find an identity neighborhood V of G such that f (V ) ⊆ U ; if now C is a compact subset of R, then for X, Y ∈ L(G), the relation Y (r)−1 X(r) ∈ V for all r ∈ C implies that L(f )(Y )(r)−1 L(f )(X)(r) = (f Y )(r)−1 (f X)(r) = f (Y (r)−1 X(r)) ∈ f (V ) ⊆ U for all r ∈ C. It follows that L(f ) is continuous. Let r ∈ R. Then L(f )(r · X)(t) = f ((r · X)(t)) = f (X(tr)) = (f  X)(tr) = (r · (f  X))(t) = (r · L(f )(X))(t). Thus L(f ) is a scalar morphism. If x ∈ L(G), then expH L(f )(X) = L(f )(X)(1) = f (X(1)) = f (expG X). Thus (10) is commutative. It remains to show uniqueness. Assume that  : L(G) → L(H ) is a scalar morphism such that expH  = f  expG . Let X ∈ L(G). Then expH ((r · X)) = expH (r · (X)) = (X)(r) on the one hand and f (expG r · X) = f (X(r)), that is (X) = f  X = L(f )(X). Let I : Scal → Top∗ and J : TopGr → Top∗ be the forgetful functors which assign to a space with scalar multiplication the underlying pointed space, respectively, to a topological group the underlying pointed space. Theorem 2.20. The assignments G  → L(G) and f  → L(f ) define a continuous functor L : TopGr → Scal, and exp : I L → J is a natural transformation of functors: TopGr → Top∗ . Proof. We have to verify several statements. (a) L is a functor: If f : G → G is the identity map, then L(f )(X) = f  X = X shows that L(f ) is the identity morphism. If f1 : G1 → G2 and f2 : G2 → G3 are morphisms of topological groups, then L(f2 f1 )(X) = (f2 f1 )X = f2 (f1 X) = (L(f2 )  L(f1 ))(X). Thus L(f2  f1 ) = L(f2 )  L(f1 ). (b) exp is a natural transformation I L → J : This is immediate from the definition of a natural transformation of functors in 1.1 and the commutativity of the diagram (10) above. (c) The functor L is continuous, i.e., preserves limits: (See 1.2 and 1.16.) We shall apply Theorem 1.17 and show that L preserves products and equalizers. def

groups. Let pr j : P =  Products. Let {Gj | j ∈ J } be a family of topological  G → G denote the projections. Define ϕ : L(P ) → j k∈J k j ∈J L(Gj ) by ϕ(X) = (pr j X)j ∈J ; conversely, if (Xj )j ∈J ∈ P , then the universal property of the product gives a unique X : R → P such that Xj = pr j X. If we write X = ψ((Xj )j ∈J ),

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 then ψ : j ∈J L(Gj ) → P is an inverse of ϕ. Since both ϕ and ψ preserve scalar multiplication (which is componentwise on products) we have    L Gj ∼ L(Gj ). = j ∈J

j ∈J

Equalizers. Let f1 , f2 : G → H be two morphisms of topological groups, set E = {g ∈ G | f1 (g) = f2 (g)} and e : E → G the inclusion, then E is a closed subgroup of G and e is the equalizer of f1 and f2 . We claim that L(e) : L(E) → L(G) is the equalizer of L(f1 ), L(f2 ) : L(G) → L(H ). For this purpose, let X ∈ L(E), then L(f1 )(X)(t) = f1 (X(t)) = f2 (X(t)) = L(f2 )(X)(t). Thus X equalizes L(f1 ) and L(f2 ). Conversely assume that X ∈ L(G) equalizes L(f1 ) and L(f2 ). Then f1 (X(t)) = L(f1 )(X)(t) = L(f2 )(X)(t) for all t ∈ R and thus X(t) ∈ E for all t, and thus X ∈ L(E). This concludes the proof that L preserves products and equalizers and thus arbitrary limits by 1.17. To the category theorists, the continuity of L is not a surprise because L = Hom(R, −) : TopGr → Scal is a hom-functor. Definition 2.21. Let G be a topological group, let G0 denote the connected component of 1 in G, called the identity component of G, and let Ga denote the arc component of 1 in G, called the identity arc component of G. Further, denote by E(G) the subgroup expG L(G) algebraically generated by all points lying on a one parameter subgroup. Finally, let E(G) denote the smallest closed subgroup of G containing the images of all one parameter subgroups; that is 1 0 E(G) = expG (L(G)) = E(G).

(11)

Observe that G0 and Ga are mapped into themselves by any continuous map G → G which fixes 1, and that G0 , Ga , E(G), and E(G) are mapped into themselves by all endomorphisms of G. That is, each of these four groups is a fully characteristic subgroup of G (see [102, Exercise E1.12 following Theorem 1.34]). Notice that Ga and E(G) may not be closed as the example of the p-adic solenoid (in 1.4(A) (ii)) shows; see also [102], 1.28, 1.38, and Chapter 8 of [102] in general. In the paragraph following Proposition 2.5 we described an arcwise connected closed additive subgroup G of a Banach space where G has no nontrivial one-parameter subgroup, whence E(G) = E(G) = {0} = Ga = G0 = G. Recall from Definition 2.11 (iii) that we say that the Lie algebra L(G) is generating if G has a Lie algebra and G0 = E(G). Since expG L(G) is arcwise connected, so is expG L(G) and hence the closure of this group is connected. If G is a Lie group, then by Theorem 2.13 we have E(G) = G0 and thus the Lie algebra of any Lie group is generating. The universe in which we shall work is that of topological groups which have a generating Lie algebra. Later in this chapter we will identify a smaller universe that is just right for us.

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In [102], Exercise E7.17 following Example 7.39, we encountered a closed contractible (hence arcwise connected) nonsingleton subgroup G of the additive Lie group of a Banach space which has no one parameter subgroups, hence satisfies E(G) = {0}. If X is an arcwise connected compact pointed space and F (X) is the free compact abelian group on X (see [102, Definition 8.51ff.]) the subgroup X of F (X) is free as an abelian group (see [102, Proposition 8.52]) and thus as a topological group satisfies E(X) = {0} while being arcwise connected. Similar comments apply to the free (nonabelian) compact group. Thus there is an abundance of connected topological groups (even abelian ones) which have a Lie algebra due to the fact that they have no nontrivial one parameter subgroups. This confirms that no structural information via the exponential function is to be obtained unless the Lie algebra is generating. We note that for every one parameter subgroup X and every endomorphism f of G the composition f  X is again a one parameter subgroup. Thus f (E(G)) ⊆ E(G) and so again E(G) is a fully characteristic subgroup. If G has a Lie algebra, the group E(G) has a generating Lie algebra by definition. If H is a subgroup of a topological group G, then the coextension of any one parameter subgroup X : R → H to G gives a one parameter subgroup R → G which we shall also write as X by a slight abuse of notation; that is, if incl : H → G is the inclusion, we identify X and incl  X. Thus we write L(H ) ⊆ L(G). Proposition 2.22. For any topological group G, one has L(expG L(G)) = L(Ga ) = L(G0 ) = L(G) = L(E(G)).

(12)

The following statements are equivalent: (i) (ii) (iii) (iv)

G has a Lie algebra. G0 has a Lie algebra. E(G) has a Lie algebra. Ga has a Lie algebra.

Proof. Since R is arcwise connected and every one parameter subgroup X : R → G is a continuous map mapping 0 to 1 we have X(R) ⊆ expG L(G) ⊆ Ga ⊆ G0 and X(R) ⊆ E(G). Hence L(G) ⊆ L(expG L(G)) ⊆ L(Ga ) ⊆ L(G0 ) and L(G) ⊆ L(E(G)) ⊆ L(G0 ). But since trivially L(G0 ) ⊆ L(G), all of these containments are equalities. We shall see later that for locally compact groups G we have Ga = exp L(G) and G0 = E(G), but that is not obvious at this point. In [102] it was shown that for any locally compact abelian group G one has Ga = exp L(G) (see [102, Theorem 8.30 (ii)]) and that for a compact abelian group  Z) G, the factor group G/Ga , as an abstract abelian group, is isomorphic to Ext(G, (loc. cit. (iii)). In the case of the p-adic solenoid Tp of Example 1.20 (A)(ii) we have L(Tp ) ∼ = R, and expTp : L(Tp ) → Tp is injective. Thus (Tp )a is a copy of R endowed with a properly coarser topology. By the preceding proposition we have

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123

R∼ = L(Tp ) = L((Tp )a ). In particular, (Tp )a is a topological group with a Lie algebra due to Theorem 2.14 and its Lie algebra is R. It should be clear that, in the spirit of universal topological algebra, a topological Lie algebra is a real Lie algebra L which is at the same time a Hausdorff topological space such that scalar multiplication R × L → L, addition L × L → L and Lie bracket [·, ·] : L × L → L are continuous. Thus a Dynkin algebra is a topological Lie algebra.

Definition 2.23. The category of topological Lie algebras and continuous Lie algebra morphisms is denoted by LieAlg. Proposition 2.24. (i) The category LieAlg of topological Lie algebras is complete. (ii) The functor L : TopGr → Scal maps the category LieAlgGr of topological groups with Lie algebras into the category LieAlg of topological Lie algebras. Proof. Part (i) is Exercise E2.7. (ii) If G is a topological group with a Lie algebra, then by Definition 2.11, L(G) ∈ ob(Scal) is a topological Lie algebra and thus belongs to the subcategory LieAlg. Now let f : G → H be a LieAlgGr-morphism. We know that L(f ) : L(G) → L(H ) is a Scal-morphism. We must show that for X, Y ∈ L(G) we have f (X + Y ) = f (X) + f (Y ) and f [X, Y ] = [f (X), f (Y )]. Firstly we deal with addition: By 2.11(4) we have , , - , --n t t . Y (X + Y )(t) = lim X n→∞ n n Since f is continuous and a group morphism we have , , , -- , , ---n t t L(f )(X + Y )(t) = f ((X + Y )(t)) = lim f X f Y n→∞ n n , -n 1 1 = lim · (f  X) · (f  Y ) (t) = (L(f )(X) + L(f )(Y ))(t). n→∞ n n Next we treat the Lie bracket: By 2.11 (5) we have , , , --n2 t t [X, Y ](t ) = lim comm X . ,Y n→∞ n n 2

Accordingly, we get this time , , , -, , ---n2 t t L(f )[X, Y ](t) = f ([X, Y ](t)) = lim comm f X ,f Y n→∞ n n - n2 , 1 1 · (f  X), · (f  Y ) = lim comm (t) n→∞ n n = [L(f )(X), L(f )(Y )](t). Thus L(f ) is a Lie algebra morphism, and the proof of the proposition is complete.

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Exercise E2.7. Prove Proposition 2.24 (i). [Hint. Prove that the cartesian product of topological Lie algebras with the product topology is a topological Lie algebra and that the equalizer E = {x ∈ L | f1 (x) = f2 (x)} of two morphisms fj : L → K, j = 1, 2 of topological Lie algebras is a topological Lie algebra.] The Completeness Theorem of the Category of Groups with Lie Algebras Theorem 2.25. (i) The category LieAlgGr of topological groups having a Lie algebra is closed in the category TopGr of topological groups under the formation of arbitrary limits and passage to closed subgroups. In particular, LieAlgGr is a complete category. The full subcategory LieAlgGenGr of all topological groups having a generating Lie algebra is closed under the formation of arbitrary products and passage to retracts. (ii) The functor L : LieAlgGr → LieAlg is continuous, i.e. preserves all limits. (iii) There exists a functor L : TopGr → LieAlgGr such that given any topological group G, there is a topological group LG with Lie algebra and a morphism ηG : G → LG with dense image, such that for every morphism f : G → H into a topological group H with Lie algebra, there is a unique morphism f  : LG → H such that f = f   ηG . TopGr

G ⏐ ⏐ ∀f  H

LieAlgGr

ηG

−−−−−→ LG ⏐ ⏐  f

LG ⏐ ⏐  ∃!f

−−−−−→

H

idH

H

Proof. Once it is shown that LieAlgGr is closed under the formation of limits, from Theorem 2.20 and Proposition 2.24 we conclude that L : LieAlgGr → LieAlg is a continuous functor. Thus, by Theorem 1.11 (ii) it suffices to show that LieAlgGr and LieAlgGenGr are closed under the formation of products and the passing to intersections of coherent retracts in TopGr. def  Products. Let {Gj | j ∈ J } be a family of topological groups and P = j ∈J Gj its product. From Theorem 2.20 we know that we can write 

L

j ∈J

  Gj = L(Gj ) j ∈J

such that expP (Xj )j ∈J = (expGj Xj )j ∈J .

The Category of Topological Groups with Lie Algebras

Since each L(Gj ) is a topological Lie algebra, so is



j ∈J

125

L(Gj ) by 2.24 (i). Moreover,

(Xj )j ∈J + (Yj )j ∈J = (Xj + Yj )j ∈J ,, -n 1 1 = lim · Xj · Yj n→∞ n n j ∈J , -n 1 1 = lim · (Xj )j ∈J · (Yj )j ∈J . n→∞ n n This shows that condition 2.11(i)(4) holds. Analogously one   proves condition 2.11(i)(5). Gj 0 = j ∈J (Gj )0 (Exercise E2.8.) One observes straightforwardly that j ∈J    and E = j ∈J Gj j ∈J E(Gj ) and concludes readily from these facts that LieAlgGenGr is closed in TopGr under the formation of products. Passage to closed subgroups. Let H be a closed subgroup of G. We may assume that L(H ) ⊆ L(G), as every one parameter subgroup of H may be considered as one of G. The relations ,, -, --n 1 1 def X + Y = lim ·X ·Y , (∗) n→∞ n n , - n2 1 1 def [X, Y ]   = lim comm (∗∗) · X, · Y n→∞ n n and X + Y, [X, Y ] ∈ L(H ) hold as they hold in G and since H is closed. Thus H is a topological group with Lie algebra and thus belongs to LieAlgGr. Retracts. Let p : G → H be a retraction in TopGr, i.e. there is a morphism j : H → G in TopGr such that pj = idH . There is no loss in generality in assuming that H is a subgroup of G and that j : H → G is the inclusion map (see E1.5), and since H is a retract, it is in fact a closed subgroup. Assume that G belongs to LieAlgGenGr. Then H belongs to LieAlgGr by the preceding arguments. Functors preserve retractions and coretractions; specifically, L(p) : L(G) → L(H ) is a retraction with L(p)  L(j ) = L(pj ) = L(idH ) = idL(H ) . Thus L(H ) is a retract of L(G). We notice that p(E(G)) ⊆ E(H ). Hence, if G ∈ ob LieAlgGenGr, then p(G0 ) = p(E(G)) ⊆ E(H ). Now p|H = pj = idH , whence H0 = p(H0 ) ⊆ p(G0 ). Thus H0 ⊆ E(H ) ⊆ H0 , and equality follows. Thus LieAlgGenGr is closed in TopGr under passing to retracts, and this completes the proof. Finally, (iii) is an immediate consequence of the completeness of the subcategory LieAlgGr and the Retraction Theorem for Full Closed Subcategories of TopGr 1.41. In the proof we used the fact that a retraction p : G → H of topological groups maps G0 onto H0 . This is a special situation for retractions. In general, even quotient maps between locally compact abelian groups fail to map components onto components. In [102, E1.11 following Definition 1.30], one observes a quotient morphism Zp × R → Tp ;

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the identity component of the domain is {0}×R and the p-adic solenoid Tp is connected. (This is only a special case of a general theorem on abelian topological groups: see [102, Theorem 8.20].) We also recall that there are examples of additive groups of separable Banach spaces G having closed arcwise connected subgroups H with zero Lie algebra (see [102, Exercise E7.17 following Example 7.39]). Then G is in LieAlgGenGr while H is not. Since G and G/H are in LieAlgGenGr the subgroup H , as a kernel is a limit. Thus the category LieAlgGenGr is not closed under the passage to all limits and thus fails to be complete. Exercise E2.8. Fill in the steps left out in the proof of the claim that a product of topological groups with Lie algebras is a topological group with a Lie algebra. The category LieGr of Lie groups, which by Theorem 2.12 is a subcategory of LieAlgGr, has finite products and equalizers, hence finite limits. Theorem 2.25 shows that all limits, in particular, arbitrary products and projective limits of Lie groups, are topological groups with Lie algebras. In Exercise E4.4 in Chapter 4 we shall give an explicit construction for the functor L : TopGr → LieAlgGr.

The Lie Algebra Functor Has a Left Adjoint Functor In Theorem 2.25 (iii), which rests on the Retraction Theorem for Full Closed Subcategories 1.41, we have, more or less implicitly, established the existence of a left adjoint functor, but we did not talk about this fact at that time. Talking about left adjoint functors explicitly now becomes inevitable. We recall that a functor U : A → B is said to have a left adjoint functor F : B → A if there is a natural transformation ηB : B → U (F (B)) such that for each morphism f : B → U (A) in B there is a unique morphism f  : F (B) → A such that f = U (f  )  ηB . In diagram form: B B ⏐ ⏐ ∀f 

A

ηB

−−−−−→ U (F⏐(B)) ⏐  U (f )

U (A) −−−−−→ idU (A)

U (A)

F (B) ⏐ ⏐  ∃!f

()

A

For details we refer to any basic text on category theory or to [102, Appendix 3]. The relation between the two adjoint functors may be expressed by saying that there is a natural bijection between the sets B(B, U (A)) and A(F (B), A). In fact, in our notation, the bijection is implemented by the function f  → f  and its inverse g  → U (g)  ηB : A(F (B), A) → B(B, U (A)).

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127

The natural transformation η is called the front adjunction. There is, dually, a back adjunction πA : F (U (A)) such that for each morphism g : F (B) → A there is a unique morphism g  : B → U (A) such that g = πA  F (g  ) (see for instance [102, Proposition A3.36]). B U (A)  ⏐ ∃!g  ⏐ B

A πA

F (U(A)) −−−−−→ ⏐ F (g  )⏐ F (B)

A  ⏐∀g ⏐

−−−−−→ F (B) idF (B)

Among the characteristic properties of adjunctions the following one will be relevant in the present context: If B is an object in B, then πF (B)  F (ηB ) : F (B) → F (B) is the identity map of the A-object F (B). Similarly, if A is an object of A, then U (πA )  ηU (A) : U (A) → U (A) is the identity map of the B-object U (A). (See for instance [102, Proposition A3.38].) Let Cat be a complete full subcategory of the category LieAlgGr of topological groups with Lie algebra. Then we have a continuous functor L : Cat → LieAlg. The following theorem will provide a left adjoint functor Cat : LieAlg → Cat. We alert the reader beforehand that this functor will depend on the subcategory Cat. We can take Cat = LieAlgGr right away, but later we shall be interested in the category Cat of all projective limits of Lie groups, and the left adjoints Cat and LieAlgGr are likely to be quite different; the former we shall identify rather accurately and explicitly via projective limits. The Adjunction Theorem for L Theorem 2.26. (i) The functor L : Cat → LieAlg possesses a left adjoint functor Cat : LieAlg → Cat. (i ) In other words, for each topological Lie algebra g there is a functorially associated topological group Cat (g) with a Lie algebra and there is a natural transformation ηg : g → L(Cat (g)) such that for each morphism f : g → L(H ) of topological Lie algebras, there is a

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unique morphism f  : Cat (g) → H such that f = L(f  )  ηg . In diagram form: LieAlg

g ⏐ ⏐ ∀f 

Cat

ηg

−−−−−→ L(Cat ⏐ (g)) ⏐  L(f )

L(H ) −−−−−→ idL(H )

Cat ⏐(g) ⏐  ∃!f

L(H )

()

H

(i ) Let G be a group with a Lie algebra. Then there is a natural transformation  → G, πG : G

 def G = Cat (L(G)),

such that for each topological Lie algebra g and each morphism f : Cat (g) → G there is a unique morphism f  : g → L(G) of topological Lie algebras such that f = πG  Cat (f  ). LieAlg

L(G)  ⏐ ∃!f  ⏐ h

Cat πG

Cat (L(G)) −−−−−→  ⏐  Cat (f )⏐ Cat (h)

G  ⏐∀f ⏐

(⊥)

−−−−−→ Cat (h) idCat (h)

(ii) The group Cat (g) has a generating Lie algebra and is therefore a member of

LieAlgGenGr. Thus Cat maps LieAlg into LieAlgGenGr.

(iii) For a topological Lie algebra, g, abbreviate Cat (g) by G. Then there are two inverse isomorphisms πG : Cat (L(G)) → G and Cat (ηg ) : G → Cat (L(G)).

Proof. (i) The proof is almost pure category theory; we will use the core existence result for adjoint functors which we have already invoked in the proof of Theorem 1.40. By the Left Adjoint Functor Existence Theorem (see for instance [102, Theorem A3.60]), we have to verify that L satisfies the Solution Set Condition (see for instance [102, Definition A3.58]), which in our case reads as follows: For each topological Lie algebra g, there is a set S(g) (and not a proper class) of pairs (f, H ), where f : g → L(H ) is a morphism of topological Lie algebras, such that for any pair (F, K), F : g → L(K), there is a pair (f, H ) ∈ S(g) and a morphism f0 : H → K such that F = L(f0 )  f . As is usual in such a situation, this condition is verified by establishing cardinality estimates. (a) There are, up to equivalence, only a set of homomorphic surjective homomorphisms f : g → h of topological Lie algebras since there is a cardinality bound on the set of closed ideals i of g and a cardinality bound on the set of topologies on each quotient g/i.

The Lie Algebra Functor Has a Left Adjoint Functor

129

(b) Given a topological space T there is a cardinality bound on all equivalence classes of dense embeddings of T into some space T, because there is a cardinality bound on the set of all filters on T , and because every point in a space T, in which T is contained densely, is the limit of a filter on T . (c) Given a topological Lie algebra h, there is, up to equivalence, only a set of continuous functions e : h → S onto a Hausdorff space S up to equivalence; next there is for each space S, up to isomorphy, at most a set of groups H which are algebraically generated by S, and there is at most a set of group topologies on H . Hence there is at most a set of topological groups H which have h as their Lie algebra and satisfy H = expH h. Moreover, by (b) above, there is at most a set of topological groups in which expH h is dense. We say that two pairs (fj , Hj ), fj : g → L(Hj ), j = 1, 2 are equivalent if there is an isomorphism ϕ : H1 → H2 such that f2 = L(ϕ)  f1 . Now we consider the 0 1 class of all pairs (f, H ), f : g → L(H ) such that H = expH (f (g)) . The preceding cardinality considerations show that we have a set S(g) of representatives (f, H ) for the equivalence classes of such pairs. Now let K be a topological group with a Lie algebra and F : g → L(K) a morphism of topological Lie algebras. Let H = expK f (g). Then H is a closed subgroup of K and thus has a Lie algebra by Theorem 2.25 (i). Moreover, the corestriction f : g → L(H ) gives a pair (f, H ) which is equivalent to a member of S(g). If f0 : H → K is the inclusion morphism we have F = L(f0 )  f , and since (f, H ) is equivalent to a member of S(g), this proves that L satisfies the Solution Set Condition. Since L is continuous by Theorem 2.25 (ii), the LeftAdjoint Functor Existence Theorem applies and proves the existence of a left adjoint functor Cat for L. (i ) and (i ): The universal properties expressed in (i ) and (i ) are equivalent and express the fact that Cat is a left adjoint functor of L. See for instance [102, Proposition A3.36]. (ii) This assertion is a consequence of the selection of the solution set and the construction of the left adjoint functor from the solution set in the Left Adjoint Functor Existence Theorem; indeed, (ηg , Cat (g)) is a member of the solution set S(g). def

(iii) Let g be a topological Lie algebra and set G = Cat (g). By (ii), G =  def  = expG ηg (g). Set G = Cat (L(G)) and note that in a similar vein, we have G 1 0 expG˜ ηL(G) (L(G)) . The situation is described by the following diagram: ηg

ηL(G)

 −−−−−→ L(⏐G) g −−−−−→ L(G) ⏐ ⏐ ⏐ expG  expG˜  G. G −−−−−→ Cat (ηg ) 0 1 0 1 Thus expG˜ ηL(G) (L(G)) = Cat (ηg ) (expG L(G)) is dense in Cat (ηg )(G) on the  on the other. Hence Cat (ηg ) has a dense image. We recall from one hand and in G our review preceding the theorem, that πCat (g)  Cat (ηg ) : Cat (g) → Cat (g) is the identity of Cat (g) = G, that is πG  Cat (ηg ) = idG . We saw that the coretraction

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Cat (ηg ) has a dense image; but then it must be surjective and thus an isomorphism whose inverse is πG . According to this Theorem 2.26 (i) we have a natural bijection between the sets LieAlg(g, L(G))

and Cat(Cat (g), G).

 = Cat ◦ L does so Since Cat depends on the category Cat, the functor G  → G as well, but notation indicating this dependence would be clumsy.

Sophus Lie’s Third Fundamental Theorem Once a complete full subcategory Cat of LieAlgGr containing all finite-dimensional Lie groups is fixed we often simply write  = Cat and note that Theorem 2.26 (ii ) is a very general form of Lie’s Third Theorem. This becomes more evident if one restricts one’s attention to the class of topological Lie algebras, for which ηg : g → L((g)) is an isomorphism. In that case (g) realizes a group whose Lie algebra is the given Lie algebra g. If g is a finite-dimensional Lie algebra, then (g) is the simply connected Lie group whose Lie algebra is the given one. There are Banach Lie algebras g for which ηg fails to be an isomorphism. In Chapter 6 we shall show that for all Lie algebras g which are projective limits of finite-dimensional ones the morphism ηg is an isomorphism. For Lie groups G, the morphism πG : (L(G)) → G is the universal covering  def = (L(G)) is the simply connected morphism of the identity component, and G covering group of the identity component G0 of G. One may wish to consider the two adjunction theorems 2.25 (iii) and 2.26 side by side. If G is a topological group, then there is a left reflection LG, that is a universally attached topological group with a Lie algebra; its Lie algebra L(LG) is the image in LieAlg under a right adjoint functor, and (L(LG)) is the image of it under the left adjoint functor . We have natural maps

πG

(L(LG)) −−−−−→

G ⏐ ⏐η G LG.

On the lower level we are dealing with topological groups with Lie algebras, and the left lower corner is a group in LieAlgGenGr.

The Adjoint Representation of a Topological Group with a Lie Algebra

131

The Adjoint Representation of a Topological Group with a Lie Algebra Let G be a topological group with a Lie algebra and let expG : L(G) → G be its exponential function. Let g ∈ G and denote by Ig : G → G, Ig (x) = gxg −1 , the inner automorphism implemented by g. Let Aut(G) denote the group of automorphisms of G. In the absence of any special information on G we do not topologize Aut(G). The function g  → Ig : G → Aut(G) is a homomorphism of groups. Definition 2.27. Let G be a topological group. The morphism L(Ig ) : L(G) → L(G) of spaces with scalar multiplication induced by Ig according to Theorem 2.20 is called the adjoint morphism and is denoted by Ad(g). By Proposition 2.24 (ii), it is a morphism of topological Lie algebras if G has a Lie algebra. The morphism of (abstract) groups Ad : G → Aut(L(G)) is called the adjoint representation of G. We note that for each g ∈ G the following diagram is commutative: Ad(g)

L(G) −−−−−→ L(G) ⏐ ⏐ ⏐exp expG ⏐   G G −−−−−→ G.

(13)

Ig

This fact may be rephrased in terms of the following formula: (∀g ∈ G, X ∈ L(G))

g(expG X)g −1 = expG Ad(g)(X).

(14)

We note that if a topological group G acts on a topological vector space E by continuous linear operators, then we have g · X − g0 · X0 = g0 · X − g0 · X0 + g · X0 − g0 · X0 + g · X − g0 · X0 − g0 · X + g0 · X0 = g0 · (X − X0 ) + g0 · (g0−1 g · X0 − X0 ) + g0 · (g0−1 g · (X − X0 ) − (X − X0 )). Thus the action (g, X)  → g · X : G × E → E is continuous if it is continuous at (1, 0) and the function g  → g · X0 : G → E is continuous for each X0 . Proposition 2.28. For a topological group G, the function (g, X)  → Ad(g)(X) : G × L(G) → L(G) is continuous. Proof. First we consider an X0 ∈ L(G) and show that g  → g · X0 : G → L(G) is continuous. Let W (Ad(g0 )(X0 ); n, V ) = {X ∈ L(G) : (∀|r| ≤ n)X(r) ∈ V Ad(g0 )X0 (r)} be a basic neighborhood of Ad(g0 )(X0 ) in L(G). We have to find an identity neighborhood U of G such that Ad(Ug0 )(X0 ) ⊆ W (Ad(g0 )X0 ; n, V ). def

Now Ad(Ug0 ) = Ad(U ) Ad(g0 ). Setting Y = Ad(g0 )(X0 ) we have to find U such that Ad(U )Y ⊆ W (Y ; n, V ), i.e. that Ad(u)Y (t) ∈ V Y (t) for all t ∈ R with |t| ≤ n and all u ∈ U , and that is equivalent to (∀u ∈ U, t ∈ [−n, n])

uY (t)u−1 ∈ V Y (t).

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Since G is a topological group and Y is continuous, for each t ∈ [−n, n] there is an identity neighborhood U (t) of G and an open interval It around t in R such that U (t)Y (It )U (t)−1 ⊆ V Y (t). Since [−n, n] is compact, there is a finite sequence t1 , . . . , tk ∈ [−n, n] such that [−n, n] ⊆ It1 ∪ · · · ∪ Itk . Set U = U (t1 ) ∩ · · · ∩ U (tk ). Then U is an open identity neighborhood of G and u ∈ U implies uY (t)u−1 ∈ V Y (t) for all t ∈ [−n, n] which is what we had to show. Finally we have to show that (g, X)  → Ad(g)X : G × L(G) → L(G) is continuous at (1, 0). So let W (0; n, V ) = {X ∈ L(G) : (∀|r| ≤ n)X(r) ∈ V } be a basic neighborhood of Ad(1)(0) = 0 in L(G). We have to find an identity neighborhood U in G and a basic neighborhood W (0; n , V  ) of 0 in L(G) such that Ad(U )(W (0; n , V  )) ⊆ W (0; n, V ), that is, such that for all u ∈ U , the relation X(r) ∈ V  for all |r| ≤ n implies uX(r)u−1 ∈ V for all |r| ≤ n. This can clearly be satisfied by taking n = n and picking U and V  so that U V  U −1 ⊆ V . Corollary 2.29. Let G be a topological group with a Lie algebra. If the group Aut(L(G)) of linear operators on the topological vector space L(G) is given the topology of pointwise convergence, then the representation Ad : G → Aut(L(G)) is continuous. In the context of linear operators on Banach spaces or Hilbert spaces, the topology of pointwise convergence is called the strong operator topology, and Corollary 2.29 says that the adjoint representation of G is strongly continuous. Proposition 2.30. Assume that G is a topological group with a Lie algebra. Then the kernel of the adjoint representation Ad : G → Aut(L(G)) is the centralizer Z(E(G), G) of the closed subgroup generated by the image of the exponential function in G, and the adjoint representation of G induces a faithful representation of G/Z(E(G), G) into Aut(L(G)). Proof. An element g ∈ G is contained in ker Ad iff Ad(g)(X) = X for all X ∈ L(G), i.e. gX(t)g −1 = (Ad(g)(X))(t) = X(t) for all X ∈ L(G) and all t ∈ R. This is saying that gxg −1 = x for all x ∈ expG L(G), and it means that g ∈ Z(expG (L(G)), G). The centralizer Z(A, G) of any subset A of a topological group G is a closed subgroup as an intersection of centralizers of singletons. Hence it contains A ⊆ Z(Z(A, G), G), whence Z(A, G) ⊇ Z(Z(Z(A, G), G), G) = Z(A, G) ⊇ Z(A, G). Thus Z(A, G) = Z(A, G). Therefore we see ker Ad = Z(E(G), G). The canonical decomposition of the morphism Ad (see e.g. [102, passage preceding Definitions 1.9]) gives an injective morphism G/ ker Ad → Aut(L(G)) which is continuous since Ad is continuous. Notice that the kernel is closed, since the topology of pointwise convergence on Aut(L(G)) is Hausdorff. In the next chapter we shall observe that LieAlgGenGr is a rich category by proving that e.g. all compact groups have Lie algebras.

Postscript

133

Postscript There are many books on classical Lie groups. Their approach is quite different from ours primarily because their aims are different. We target the topological group structure of locally compact groups. As will become evident in later chapters (and was evident in [102]), the basic tools are the Lie algebra and the exponential map; this will be the case even when we are well outside the class of Lie groups. Therefore our definition of Lie group is not built around the concept of manifold, but rather that of Lie algebra and, in particular, the formalism of the Baker–Campbell–Dynkin–Hausdorff series. This formalism has been summarized for this purpose in Appendix 1. As we remarked after Definition 2.1, while our definition of a Lie group is not that commonly found in the literature, it is equivalent to other definitions in the literature. Our definition applies at once to linear Lie groups which permitted us to access the classical body of Lie group theory in [102]. The Lie algebra machinery will of course be useful only within a class of topological groups whose members have a Lie algebra. This class is introduced in Definition 2.11 and it is surprisingly large. From [102] we know that it includes all compact groups and all locally compact abelian groups; later we shall see that it includes all locally compact groups. Most importantly we shall see in Chapter 3 that it includes the class of pro-Lie groups; these groups are introduced in that chapter and will be shown to form a complete subcategory of LieAlgGr containing all finite-dimensional Lie groups. We readily see that the categories TopGr of all topological groups and continuous homomorphisms and its full subcategory CompGr of all compact groups are complete. However, the category LCGr of all locally compact groups and continuous homomorphisms is not complete as it fails to have arbitrary products (see 1.11 (iii)). So the category we seek should be better behaved than LCGr and should at least contain all connected locally compact groups. Chapters 8, 9 and 10 of [102] demonstrated how powerful a pair is the Lie algebra of a compact group and the exponential function mapping the Lie algebra of the compact group into the compact group. This pair will also turn out to be powerful in the context of locally compact groups. So we focus our attention on subcategories of LieAlgGr; this is the full subcategory of the category TopGr whose objects are topological groups which have Lie algebras. We showed that the full category LieAlgGr of all topological groups having Lie algebras is closed in the category TopGr of all topological groups, that the property of having a Lie algebra is preserved by passing to closed subgroups, and that the functor L : LieAlgGr → LieAlg assigning to a topological group having a Lie algebra its topological Lie algebra preserves all limits. We also showed that the restriction of L to any full complete subcategory Cat of LieAlgGr has a left adjoint functor  = Cat : LieAlg → Cat which attaches to a topological Lie algebra g in a universal def

fashion a topological group G = (g) in Cat, depending on Cat, such that for any morphism f : g → L(H ) of topological Lie algebras there is a unique morphism f  : G → H in Cat such that f = L(f  )  ηg . For the class of topological groups

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which we shall consider in the book, namely, projective limits of finite-dimensional Lie groups we shall show in Chapter 6, that ηg : g → L(G) is an isomorphism of topological Lie algebras. The existence of the functor  is therefore a novel expression of Lie’s Third Fundamental Theorem saying that every Lie algebra is realized as the Lie algebra of a group. The generality at which we approached Lie theory in the later portions of this chapter thus led us to the uncovering of a functorial set-up that escaped notice in the classical context where one rather straightforwardly aims for the universal  of a connected finite-dimensional Lie group H which is none other covering group H than (L(H )) with the universal covering morphism being none other than the back adjunction πH : (L(H )) → H . One should alert the reader to the possibility that a topological group having no one parameter subgroups whatsoever is a topological group with a Lie algebra, albeit trivial. Lie theory as introduced in this chapter gives information on a topological group only to the extent of the fully characteristic closed subgroup E(G) generated algebraically and topologically by all one parameter subgroups. If G is of the form (g) for some Lie algebra g, then indeed G = E(G).

Chapter 3

Pro-Lie Groups

With the concepts introduced in Chapter 1 we establish that within the category of topological groups TopGr, the full subcategory of projective limits of finite-dimensional Lie groups is closed under the formation of all limits and is therefore a complete category in its own right. We see that, in the spirit of Chapter 2, its members have a decent Lie theory. For a topological group G we define N (G) to be the set of all normal subgroups N of G such that G/N is a finite-dimensional Lie group. We shall call G a pro-Lie group if, firstly, G is a complete topological group, secondly, N (G) is a filter basis, and thirdly, every identity neighborhood of G contains some N ∈ N (G). It is then easy to see that every pro-Lie group G is a projective limit limN ∈N (G) G/N . The converse emerges as a difficult question, but it is shown in Theorem 3.34 of this chapter that any projective limit of finite-dimensional Lie groups is in fact a pro-Lie group. In particular then, the category of pro-Lie groups is complete. In Theorem 3.39 we shall summarize three equivalent ways of defining pro-Lie groups. Another result which sounds obvious, but is not, will be proved in Theorem 3.35; it says: A closed subgroup of a pro-Lie group is a pro-Lie group. Prerequisites. We shall use the Lie theory for topological groups developed in Chapter 2 and apply it to projective limits of Lie groups. Throughout the entire chapter we shall make frequent reference to the theory of limits presented in Chapter 1.

Projective Limits of Lie Groups The Definition of a Projective Limit of Lie Groups Definition 3.1. A topological group G is called a projective limit of Lie groups if there is a projective system of finite-dimensional Lie groups P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} whose projective limit limj ∈J Gj is isomorphic to G. The full subcategory in TopGr of projective limits of Lie groups will be called LieProGr. Proposition 3.2. For a topological group G, the following conditions are equivalent: (i) G is a projective limit of Lie groups. (ii) G is isomorphic to a closed subgroup of a product of a family of finite-dimensional Lie groups. (iii) G is isomorphic to a closed subgroup of a projective limit of Lie groups.

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Proof. (i) ⇒ (ii): This is an immediate consequence of the explicit definition of a projective limit (see Proposition 1.18 and the discussion preceding 1.18). (ii) ⇒ (iii): Every product is a projective limit of its finite partial products (see Theorem 1.5(b)). A finite product of finite-dimensional Lie groups is a finite-dimensional Lie group. Thus a product of a family of finite-dimensional Lie groups is a projective limit of Lie groups. (iii) ⇒ (i): A projective limit of Lie groups is a closed subgroup of a product of finite-dimensional Lie groups (see 1.18). Thus G is isomorphic to a closed subgroup H  of a product α∈A Gα of finite-dimensional Lie groups Gα . Then by Corollary 1.35 of ∼ the Closed Subgroup Theorem for Projective Limits 1.34 we know that H = limj ∈J Hj for a projective system of closed subgroups of finite products α∈j Gα of finite-dimensional Lie groups. Since closed subgroups of finite-dimensional Lie groups are Lie groups, H is a projective limit of Lie groups and since G ∼ = H , assertion (i) follows. Completeness Theorem for Projective Limits of Lie Groups Theorem 3.3. (i) The category LieProGr of projective limits of Lie groups is closed in TopGr under the formation of all limits and is therefore complete. (ii) LieProGr is the smallest full subcategory of TopGr that contains all finite-dimensional Lie groups and is closed under the formation of all limits. Proof. (i) By Proposition 3.2, the full subcategory LieProGr of the category TopGr of topological groups is closed under the passage to closed subgroups, hence the passing to kernels of continuous cocycles and under the formation of projective limits. Hence it is closed under the formation of all limits in TopGr and is therefore a complete category by Proposition 1.11 (b). (ii) Let C be any full subcategory of TopGr which contains all finite-dimensional Lie groups and is closed in TopGr under the formation of all limits and the passing to isomorphic objects. Let G be a projective limit of Lie groups. Then G = limj ∈J Gj for a projective system of finite-dimensional Lie groups Gj . Then all Gj are contained in C and since C is closed under the formation of all limits, G is in C. Thus LieProGr ⊆ C. By this result we have identified the smallest full subcategory of TopGr which is closed under the formation of all limits and contains the full subcategory of all finitedimensional Lie groups. This closure of the category of finite-dimensional Lie groups is precisely LieProGr. We shall show in Theorem 3.35 below that LieProGr is also closed under passing to closed subgroups. By Theorem 3.3, in particular every limit of projective limits of Lie groups is a projective limit of Lie groups. Exercise E3.1. Provide a direct proof of the fact that a product of projective limits of Lie groups is a projective limit of Lie groups.

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137

[Hint. Let {fjαk : Gαk → Gαj | (j, k) ∈ J α × J α , j ≤ k},

α∈A

be a family of projective systems of finite-dimensional Lie groups. Then we have a family {Gα : α ∈ A}, Gα = limj ∈J α Gαj of groups that are themselves projective def  limits of Lie groups. We shall show that G = α∈A Gα is a projective  limit of Lie groups. We let F be the poset of finite subsets of A and set J = F × α∈A J α . For j = (F1 , (j1α )α∈A ) and k = (F2 , (j2α )α∈A ) we set j ≤ k iff F1 ⊆ F2 and j1α ≤ j2α for  all α ∈ F1 . Then J is directed. For j = (F, (j α )α∈A ) we write Gj = α∈F Gαjα . For j ≤ k we define fj k : Gk → Gj by fj k ((gjαα )α∈F2 ) = (fj1α j2α (gjαα ))α∈F1 . 2

2

Then {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} is a projective system of finite-dimensional Lie groups. For j ∈ J , j = (F, (j α )α∈A ) we find a morphism pj : G → Gj , pj (((gjαα )j α ∈J α )α∈A ) = (gjαα )α∈F , giving us a morphism p : G → limj ∈J Gj . Show that p has an inverse.] Examples 3.4. If G belongs to any class of topological groups in the following list, then the set N (G) of all closed normal subgroups N such that G/N is a finite-dimensional Lie group is a filter basis converging to 1 and G ∼ = limN ∈N (G) G/N ; in particular, G is a projective limit of Lie groups, and G is indeed a limit of a very special projective system of finite-dimensional Lie groups with all bonding maps being quotient morphisms. (i) (ii) (iii) (iv)

All finite-dimensional Lie groups. Products of any family of finite-dimensional Lie groups. Compact groups. Locally compact abelian groups.

Exercise E3.2. Prove the assertions made in 3.4. [Hint. (iii) See Fundamental Theorem of Projective Limits 1.27 and [102, Corollary 2.43]. (iv) See Fundamental Theorem 1.27 and [102, Theorem 7.57]. Note that a locally compact abelian group H with a compact open subgroup K is a projective limit of Lie groups because K is a projective limit of Lie groups by (iii), and because a topological group having an open Lie subgroup is a Lie group.] In particular, for any set X the power RX is a projective limits of Lie groups.

The Lie Algebras of Projective Limits of Lie Groups We now merge the developments of topological groups with Lie algebras on the one hand and of projective limits of Lie groups on the other. In Definition 2.11 it was explained what it means when we say that a topological group has a Lie algebra.

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Theorem 3.5. (i) Every projective limit of Lie groups has a Lie algebra; that is, LieProGr ⊆ LieAlgGr.

(ii) The subcategory LieProGr is closed in LieAlgGr under the formation of all limits. (iii) The functor L : LieProGr → LieAlg is continuous, that is, preserves all limits. Proof. Let G ∈ ob LieProGr. Then G ∼ = limj ∈J Gj for a suitable projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of finite-dimensional Lie groups Gj . Theorem 2.12 shows that Gj ∈ LieAlgGenGr. Hence by the Completeness Theorem for Groups with Lie Algebras 2.25 we have G ∈ LieAlgGr. The remainder follows. By 3.4, every compact and every locally compact abelian group is a projective limit of Lie groups; all those groups have a generating Lie algebra, see [102, Chapter 9, Definition 9.44ff., and Chapter 7, Theorem 7.71].

Pro-Lie Algebras The Definition of Pro-Lie Algebras Definition 3.6. Let L be a topological Lie algebra and  (L) the filter basis of closed ideals I of L such that L/I is finite-dimensional. Then L is called a pro-Lie algebra if the following conditions are satisfied: (i)  (L) converges to 0. (ii) As a topological vector space, L is complete. The category of pro-Lie algebras and continuous Lie algebra morphisms is denoted by proLieAlg. It would have been a bit more accurate to call a pro-Lie algebra a profinite-dimensional Lie algebra, but in the end we opted for brevity. Recall from Appendix 2, Definition A2.5, that a real topological vector space L is called weakly complete iff it is the projective limit of the projective system of its finitedimensional quotient spaces. Weakly complete topological vector spaces are complete locally convex topological vector spaces. Thus we have the following proposition. Proposition 3.7. Let L be a real topological vector space, and consider it as a commutative topological Lie algebra. Then the following statements are equivalent. (i) L is a pro-Lie algebra. (ii) As a topological vector space, L is weakly complete. (iii) The natural evaluation map η : L → (L )∗ is an isomorphism of topological vector spaces, where L is the topological dual of L and (L )∗ is the algebraic dual of L endowed with the weak *-topology.

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(iv) L is isomorphic as a topological vector space and as a topological Lie algebra with zero brackets to a product RJ for some set J . Proof. Exercise E3.3. Exercise E3.3. Prove Proposition 3.7. [Hint. Use Theorems 1.13 and 1.14 to establish the equivalence of (i) and (ii). The equivalence of (ii) and (iii) is established by applying Theorem A2.8 of the Appendix. (iii) implies (iv): The real vector space L has a basis and thus as a vector space is isomorphic to a vector space R(J ) . Now by (iii) we have the topological vector space isomorphisms L ∼ = (R(J ) )∗ ∼ = (R )J ∼ = RJ . = (L )∗ ∼ (iv) implies (ii): If L = RJ then L is the projective limit of the finite-dimensional partial products (see Example 1.20 (B)). The filter basis of partial products RJ \F with F finite in J and where R J \F is considered as a subspace of RJ in the obvious fashion is cofinal in the filter basis N0 (L) of all cofinite-dimensional closed vector subspaces. Thus L ∼ = limN ∈N0 (L) L/N. Alternative proof: The categories of real vector spaces and that of weakly complete topological vector spaces are dual to each other (see Appendix 2, Theorem A2.8.) The category of real vector spaces is cocomplete; by duality, the category of weakly complete topological vector spaces is complete, and thus is closed under the formation of products in the category of all topological vector spaces; thus, since R is weakly complete, so is RJ . See also Appendix 2, Corollary A2.9.] Proposition 3.8. The underlying topological vector space of a pro-Lie algebra is weakly complete. Proof. Let  (L) again denote the filter basis of closed ideals of the pro-Lie algebra L such that dim L/I < ∞ and let N denote the filter basis of all closed vector subspaces M such that dim L/M < ∞. Clearly,  (L) ⊆ N . In view of the Cofinality Lemma 1.21 it is sufficient to show that  is cofinal in N . For this purpose let N ∈ N . In the finite-dimensional vector space L/N find an open identity neighborhood V in which {0} is the only vector subspace. Let p : L → L/N be the quotient morphism of topological vector spaces and set U = p −1 (V ). Then U is a zero neighborhood of L. Since lim  (L) = 0 by 3.12 (i) there is an I ∈  such that I ⊆ U . Then p(I ) is a vector subspace contained in V and therefore is {0} by the selection of V . Thus I ⊆ M, and this shows that  (L) is cofinal in N . By Definition 3.12 and Theorem 1.30 we have L ∼ = limI ∈ (G) L/I . Then limI ∈ (L) L/I = limM∈N L/M by the Cofinality Lemma 1.21. Therefore L is weakly complete by definition. For an overview of the linear algebra of weakly complete topological vector spaces see Appendix 2, notably Theorems A2.8, A2.11, and A2.12. We immediately record an exact analog of Proposition 1.40 on topological groups. Proposition 3.9. Let L be a topological Lie algebra. Then the following conditions are equivalent. (i) L is a pro-Lie algebra.

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(ii) There is a filter basis J of ideals converging to 0 such that L/I is finite-dimensional for each I ∈ J. (iii) There is a filter basis J of ideals contained and cofinal in  (L). Proof. Since (i) ⇒ (ii) is trivial by Definition 3.6 of a pro-Lie algebra; (iii) ⇒ (i) is a consequence of the Cofinality Lemma 1.21. Therefore we must prove (ii) ⇒ (iii). Clearly, J ⊆  (L). We claim that (∀I ∈  (L))(∃J ∈ J) I ⊇ J.

(∗)

This means that J is cofinal in  (L). The proof of (∗) remains, and we leave it as Exercise E3.4. Exercise E3.4. Finish the proof of Proposition 3.9. [Hint. Emulate the last portion of the proof of 1.29 (vi), see also 3.27 below.]

The Definition of a Topological Group with a Pro-Lie Algebra and with a Lie projective component Definition 3.10. A topological group G is said to have a pro-Lie algebra if it has a Lie algebra (i.e. if it belongs to LieAlgGr) and if L(G) is a pro-Lie algebra. The full subcategory of LieAlgGr of all topological groups having pro-Lie algebras and morphisms of topological groups between them is denoted by proLieAlgGr. A topological group G is said to have a Lie projective component if G0 , the identity component, is a projective limit of Lie groups. The full subcategory of TopGr of all topological groups with a Lie projective component is called proLieComGr. Instead of using the very compact phrase that a topological group “has a pro-Lie algebra” we shall also say that a topological group “has a pro-Lie algebra as Lie algebra”. Proposition 2.22 yields at once a simple observation, which should nevertheless be kept in mind: Proposition 3.11. For a topological group G, the following statements are equivalent: (i) G has a pro-Lie algebra. (ii) The identity component, G0 , of G has a pro-Lie algebra. (iii) The identity arc component, Ga , of G has a pro-Lie algebra. As a first indication of the significance of this concept we record Theorem 3.12. Every topological group whose identity component is a projective limit of Lie groups has a pro-Lie algebra as Lie algebra. In particular, every projective limit of Lie groups has a pro-Lie algebra as Lie algebra. So proLieGr ⊆ LieProGr ⊆ proLieAlgGr.

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141

Proof. In view of 3.11 it suffices to show that a projective limit of Lie groups has a pro-Lie algebra. Let G be a projective limit of Lie groups. Then G ∼ = limj ∈J Gj for a suitable projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of finitedimensional Lie groups Gj . By the continuity of L from 2.25 (ii) we have L(G) ∼ = limj ∈J L(Gj ). Let pj : L(G) → L(Gj ) denote the limit morphisms and set M = {ker pj | j ∈ J }. Since each ker pj is an ideal, and dim L(G)/ ker pj ≤ dim L(Gj ), the set M is a filter basis of ideals i of L(G) such that L(G)/i is finite-dimensional, and lim M = 0 by the Fundamental Theorem of Projective Limits 1.27 (ii). Now Proposition 3.9 shows that L(G) is a pro-Lie algebra. From 3.4 we get immediately: Corollary 3.13. Every compact group and every locally compact abelian group is a topological group with pro-Lie algebra. Proof. Exercise E3.5. Exercise E3.5. Prove Corollary 3.13. [Hint. Every compact group and every locally compact abelian group is a projective limit of Lie groups.] So the category proLieComGr (and thus also the category proLieAlgGr) contains CompGr and LCAbGr. Theorem 3.14 (Completeness Theorem for the Category of Topological Groups with Pro-Lie Algebras). The categories proLieAlgGr and proLieAlg are complete, and the functor L : proLieAlgGr → proLieAlg is continuous. Proof. From 3.8 we know that the category LieAlgGr of topological groups with Lie algebras and the category LieAlg of topological Lie algebras are complete, and that L : LieAlgGr → LieAlg is a continuous functor. We claim that an arbitrary product def  L = j ∈J Lj of pro-Lie algebras Lj is a pro-Lie algebra. Let j denote the filter basis of all closed ideals of Lj such that Lj /I is finite-dimensional for all I ∈ Jj . def  Now we let F denote all products I = j ∈J Ij such that Ij ∈ j and all but finitely many of the Ij agree with Lj . Then each member I of  is a closed ideal and L/I is finite-dimensional. Thus F is contained in the filter basis  of all closed ideals I of L such that L/I is finite-dimensional. Since all filter bases j converge to 0, the filter basis F and thus the finer filter basis  converge to 0. Since L is complete as the product of complete topological vector spaces, L is a pro-Lie algebra, as asserted. Now let L be a pro-Lie algebra and  the filter basis of ideals I such that L/I is finite-dimensional. Let E be a closed subalgebra of L; then {K ∩ I | I ∈  } is a filter basis of ideals of K which converges to 0, and K is complete as a closed vector subspace of a complete topological vector space. Finally, let I ∈  . Then K/(K ∩ I ) ∼ = (K + I )/I ⊆ L/I . Hence K/(K ∩ I ) is finite-dimensional. This shows that K is a pro-Lie algebra. In particular, this applies to the equalizer E ⊆ L1 of two morphisms L1 → L2 of pro-Lie algebras. Thus proLieAlg has products and equalizers

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and is therefore complete (see Theorem  1.10). Since L is continuous on LieAlgGr we conclude that for a product G =  j ∈J Gj of Lie groups Gj with pro-Lie algebras L(Gj ) has a Lie algebra L(G) ∼ = j ∈J L(Gj ) which is a pro-Lie algebra. A similar argument applies to equalizers. Thus proLieAlgGr has products and equalizers and thus is complete by 1.8. Theorem 3.15 (Completeness Theorem for the Category of Topological Groups with Lie Projective Component). The category proLieComGr of topological groups whose identity component is a projective limit of Lie groups is complete, and the functor L : proLieComGr → proLieAlg is continuous. Proof. From 2.24 and 2.25 we know that the categories LieAlg of topological Lie algebras and LieAlgGr of topological groups with Lie algebras are complete, and that L : LieAlgGr → LieAlg is a continuous functor. By Theorem 3.5, every topological group with a pro-Lie component has a pro-Lie algebra. What remains to show is that proLieComGr is closed in TopGr under the passing to products and closed subgroups. Since the identity component of a product of topological groups is the product of their identity components, and since the category of projective limits of Lie groups is closed under the formation of products by Theorem 3.3, the category proLieComGr is closed under the formation of products in TopGr. Now let G be a topological group such that G0 is a projective limit of Lie groups. Let H be a closed subgroup of G. Then H0 ⊆ G0 , and H0 is closed in G0 . By Proposition 3.2 it follows that H0 is a projective limit of Lie groups. This completes the proof of the theorem. The full subcategory of connected topological groups in TopGr was denoted ConnGr in Chapter 1, just before Theorem 1.11. It is relevant in our present context that the assignment G  → G0 : TopGr → ConnGr is a functor with good properties. We recall from Theorem 1.11 (v) that the full category ConnGr is complete while it is not closed under passing to limits in TopGr. Proposition 3.16. The functor G  → G0 : TopGr → ConnGr is right adjoint to the inclusion functor and is therefore continuous. Proof. Let |•| : ConnGr → TopGr denote the inclusion functor and let εG : |G0 | → G be the inclusion morphism. If C is a connected topological group and |C| the underlying topological group, then for every morphism f : |C| → G of topological groups, since f (C) is connected and contains 1, we have f (C) ⊆ G0 and thus there is a unique corestriction f  : C → G0 such that f = εG  |f  |. This shows that (•)0 : TopGr → ConnGr is right adjoint to | • |. All right adjoints preserve limits. (See e.g. [102, Theorem A3.52.].) Because of the equation L(G) = L(G0 ), the functor L : proLieComGr → proLieAlg factors in the form (•)

L

proLieComGr −−−→ proLieComGr ∩ ConnGr −−−→ proLieAlg.

Weakly Complete Topological Vector Spaces and Lie Algebras

143

Weakly Complete Topological Vector Spaces and Lie Algebras Information on weakly complete topological vector spaces is collected in Appendix 2. One way of explaining a weakly complete topological vector space is declaring a topological vector space as weakly complete if there is an isomorphism of topological vector spaces to some product vector space RX (see Appendix 2, Corollary A2.9). Lemma 3.17. (a) Let f : V → W be a morphism of weakly complete topological vector spaces. Then f (V ) is a closed vector subspace of W , and the natural bijection V / ker f → f (V ) is an isomorphism of topological vector spaces. (b) If V and W are closed vector subspaces of a weakly complete topological vector space U , then V + W is closed, and the function f : V /(V ∩ W ) → (V + W )/W , f (v + (V ∩ W )) = v + W is an isomorphism of topological vector spaces. Proof. See Appendix 2, Theorem A2.12. Lemma 3.18. Let g = limk∈J gk be a projective limit of a projective system {γj k : gk → gj | j ≤ k, (j, k) ∈ J × J } of finite-dimensional real vector spaces in the category of weakly complete topological vector spaces. Let γj : g → gj denote the limit maps. Then for each j ∈ J there is an index kj ≥ j such that γj kj (gk ) ⊆ γj (g). Proof. See Appendix 2, Corollary A2.10. We shall have to deal with topological groups G for which we make some standard assumptions: Notation 3.19 (General Assumptions). For a projective limit G of Lie groups there is a projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of finite-dimensional Lie groups Gj such that G = limj ∈J Gj . The limit maps are denoted fj : G → Gj , the kernels ker fj of the limit maps will be abbreviated by Kj . The finite-dimensional Lie algebras L(Gj ) will be written gj . Let us write def

def

fj k = L(fj k ) and fj = L(fj ). By Theorem 2.25 (ii), the functor L : LieAlgGr → LieAlg from the category of all topological groups having a Lie algebra and continuous group morphisms between them to the category of topological Lie algebras is continuous and thus, in particular preserves projective limits. Hence L(G) ∼ = limj ∈J L(Gj ), and we may identify the two profinite-dimensional Lie algebras. Thus {fj k : gk → gj | (j, k) ∈ J × J, j ≤ k}

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3 Pro-Lie Groups

is a projective system of finite-dimensional real Lie algebras and Lie algebra morphisms such that L(G) = lim gj j ∈J

and that the continuous Lie algebra morphisms fj : L(G) → gj are the limit morphisms. We set aj = fj (L(G)) ⊆ gj for each j ∈ J , and let αj k : ak → aj be the morphism of finite-dimensional Lie algebras induced by fj k for j ≤ k. Lemma 3.20. The system L = {αj k : ak → aj | (j, k) ∈ J × J, j ≤ k} def

is a projective system of finite-dimensional Lie algebras and surjective bonding maps. Then L(G) = lim aj . j ∈J

The limit maps αj : L(G) → aj are quotient morphisms. Proof. We apply the Fundamental Theorem on Projective Limits 2.1 (iv) to the system L and conclude that lim L = lim L. The limit maps αj : L(G) → aj are surjective and thus are quotient maps (Appendix 2, Theorem A2.8 (b)). It also follows that the bonding maps αj k : ak → aj are surjective. The following diagram illustrates the situation: a⏐j ⏐ inclj  gj

αj k

←−− a⏐k ⏐ inclk  ←−− gk fj k

··· ··· ···

←− L(G) = limj ∈J aj ⏐ ⏐id  L(G) ←− L(G) = limj ∈J gj .

Now for each j ∈ J , the subalgebra aj of the finite-dimensional Lie algebra gi dedef

termines an analytic subgroup Aj = expGj aj  of Gj such that L(Aj ) = aj . (For linear Lie groups a reference is [102, Theorem 5.52]. The proof there does not depend on the assumption that G is a linear Lie group.) Lemma 3.21. Under our general assumptions 3.19 for G = limj ∈J Gj we have (∀j ∈ J )(∃kj ≥ j, kj ∈ J ) fj kj ((Gkj )0 ) ⊆ Aj . Proof. From 3.18 we have (∀j ∈ J )(∃kj ≥ j, kj ∈ J ) fj kj (gkj ) ⊆ aj . The assertion now follows from the fact that as a finite-dimensional connected Lie group, (Gkj )0 is algebraically generated by exp gkj and Aj is algebraically generated by aj . Thus fj kj ((Gkj )0 ) = fj kj (exp gkj ) = exp L(fj kj )(gkj ) ⊆ exp aj  = Aj .

Weakly Complete Topological Vector Spaces and Lie Algebras

145

def

The morphisms fj k : Gk → Gj induce morphisms ψj k = fj k |Ak : Ak → Aj with L(ψj k ) = αj k and fj0k : (Gk )0 → (Gj )0 . Then {ψj k : Ak → Aj | (j, k) ∈ J × J, j ≤ k} def

is a projective system of analytic groups; let A = limj ∈J Aj be its limit. Each analytic group carries a topology which is in general finer than the induced topology, making the subgroup Aj into a connected Lie group Hj such that L(Hj ) = L(Aj ) = aj and that the morphisms ψj k : Aj → Ak induce morphisms of Lie groups ϕj k : Hk → Hj such that L(ϕj k ) = αj k . We have injective morphisms incl(Gj )0

inclAj

εj

Hj −−−−−→ Aj −−−−−→ (Gj )0 −−−−−→ Gj where εj is the bijective morphism of topological groups given by εj (h) = h and incl denotes the respective inclusion morphisms. We consider the projective system def

H = {ϕj k : Hk → Hj | (j, k) ∈ J × J, j ≤ k} of finite-dimensional Lie groups and let H = limj ∈J Hj denote its limit; we note that due to the continuity of the functor L we have L(H ) = lim L(Hj ) = lim aj = L(G). j ∈J

j ∈J

(L)

It is not at all clear at this time that the limit of a projective system of connected Lie groups is connected if all bonding maps are surjective and all groups in the system are connected. However, we observe the following lemma which we shall presently apply to H = limj ∈J Hj . Lemma 3.22. Assume that H is a projective limit limj ∈J Hj of finite-dimensional Lie groups satisfying the following two hypotheses: (i) For all j ∈ J the Lie group Hj is connected, and (ii) the limit maps ϕj : H → Hj , j ∈ J induce surjective morphisms L(ϕj ) : L(H ) → L(Hj ). Then H is connected. Proof. Let h be an arbitrary element of H . We shall show that arbitrarily close to h there are elements from the arc component of the identity of H ; thus the arc component of the identity is dense in H and so H is indeed connected. For a proof let U be any identity neighborhood of H . By 1.27 (i) we may assume that U = ϕj−1 (V ) for some identity neighborhood V of Hj . Since Hj is connected by hypothesis (i), and since any connected finite-dimensional Lie group is algebraically generated by the image of its exponential function, there are elements X1 , . . . , Xn ∈ L(Hj ) such that

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3 Pro-Lie Groups

ϕj (h) = exp X1 . . . exp Xn . By hypothesis (ii) the morphism L(ϕj ) : L(H ) → L(Hj ) is surjective, and thus we find elements Ym ∈ L(H ), such that Xm = L(ϕj )(Ym ) for all m = 1, . . . , n. Accordingly, expHj Xm = expHj L(ϕj )(Ym ) = ϕj (expH Ym ) in Hj for all m and so ϕj (h) = ϕj (expH Y1 ) . . . ϕj (expH Yn ) = ϕj (expH Y1 . . . expH Yn ). Let α : [0, 1] → H denote the arc in H given by α(t) = expH (t · Y1 ) . . . expH (t · Yn ). Then α(0) = 1 and α(1) = expH Y1 . . . expH Yn ∈ ϕj−1 (ϕj (h)) = h ker ϕj ⊆ hU . This proves our claim and thus finishes the proof of the lemma. Lemma 3.23. The system def

H = {ϕj k : Hk → Hj | (j, k) ∈ J × J, j ≤ k} is a projective system of quotient morphisms between finite-dimensional connected Lie groups and its limit H = limj ∈J Hj is a connected projective limit of Lie groups. The limit maps ϕj : H → Hj are quotient morphisms. Proof. Since all L(ϕj k ) = αj k are surjective, the morphisms ϕj k are surjective, and since Hk as a connected finite-dimensional Lie group is σ -compact and locally compact and Hj is locally compact, by the Open Mapping Theorem (see e.g. [102, Appendix 1, Exercise EA1.21 preceding Remark A1.60]) the morphisms ϕj k are quotient morphisms. Therefore, the limit maps ϕj : H → Hj are quotient morphisms by Theorem 1.27 (iii). The preceding Lemma 3.22 applies to show that H is connected. We illustrate the situation in the following diagram showing the limits of the various projective systems we consider: H⏐j εj ⏐  A⏐j inclAj ⏐  (G⏐j )0 incl(Gj )0 ⏐  Gj

ϕj k

←−−−− fj k

H⏐k ⏐

εk 

←−−−−

A⏐k ⏐ inclAk  fj0k ←−−−− (G⏐k )0 ⏐ incl(Gk )0  fj k ←−−−− Gk

··· ··· ··· ··· ··· ··· ···

H = lim ⏐ j ∈J Hj ⏐ε  A = lim ⏐j ∈J Aj ⏐incl  A G0 = (lim⏐j ∈J (Gj )0 )0 ⏐incl  G0 G = limj ∈J Gj .

(1)

The universal property of the limit G gives us the morphisms ε : H → A and the various inclusion morphisms incl filling in diagram (1). Notice that L(H ) = L(A) = L(B) = L(G) and we may identify L(ε) and the various maps L(incl) with idL(G) . By the concrete construction of the limits we have    Gj | (∀j ≤ k in J ) fj k (gk ) = gj , G = (gj )j ∈J ∈ 

A = (aj )j ∈J ∈ 

H = (hj )j ∈J ∈

j ∈J



j ∈J



j ∈J

 Aj | (∀j ≤ k in J ) fj k (ak ) = aj ,  Hj | (∀j ≤ k in J ) fj k (hk ) = hj .

Weakly Complete Topological Vector Spaces and Lie Algebras

147

Thus A is a subgroup of G and we may identify H with A except that its topology may be finer than the topology induced from G on A. The situation is further illustrated by the following diagram: =

=

εj

inclAj

−−−−−→ L(G) L(H ⏐ ) −−−−−→ L(A) ⏐ ⏐ ⏐ ⏐exp expH ⏐ exp A   G H − − − − − → A − − − − − → G ⏐ ⏐ ⏐ ε inclA ⏐ ⏐f ϕj ⏐ ψ j    j Hj −−−−−→ Aj −−−−−→ Gj , where ε and all εj are bijective and all incl are embeddings. For a given projective limit of Lie groups G = limj ∈J Gj , a very special type of connected projective limit of Lie groups H with quotient morphisms as limit maps emerged almost out of nowhere and it is mapped under the bijective morphism ε onto the subgroup A of G. Clearly we must identify this subgroup of G0 . Lemma 3.24. H = G0 . Proof. By Lemma 3.21, (∀j ∈ J )(∃j ≤ kj ∈ J )

fj kj ((Gkj )0 ) ⊆ Aj .

Now we notice that (Gkj )0 is locally arcwise connected and Hj is Aj equipped with the arc component topology (see [102, Theorem 5.52 (iv) and Lemma A4.1ff.]). Hence the restriction and corestriction fj kj |(Gjk )0 : (Gjk )0 → Aj factors through εj : Hj → Aj for a morphism fj kj : (Gkj )0 → Hj such that def

fj0kj = inclAj  εj  fj kj : (Gkj )0 → (Gj )0 . Temporarily, set def

G0 = lim (Gj )0 ⊇ G0

(∗)

j ∈J

in the category of topological groups and continuous morphisms. Thus for each j ∈ J there is a kj ≥ j and a commutative diagram (Gj )0 o

fj0k

j

(Gkj )0 w w w ww ww w ww ww fj kj {ww Hkj Hj o ϕ j kj

fj0k

j

 (Gkj )0

G0 β

···

inclkj εkj

inclj εj

 (Gj )0 o

···

 H inclkj εkj

···

 G0 .

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3 Pro-Lie Groups def

It follows that there is a morphism βj = fj kj |(Gkj )0  fkj |G0 : G0 → Hj which is independent of the choice of kj in as much as it agrees with fj kj  fkj k  fk |G0 for k ≥ kj . We notice that for j ≤ j  we get βj = ϕjj  βj : G0 → Hj . Thus the universal property of H = limj ∈J Hk implies the existence of a unique morphism β : G0 → H such that βj = ϕj  β. From inclA  ε  fj kj = fj0kj we conclude that inclA  ε  β = idG0 . Thus inclA  ε : H → G0 is a retraction, and since it is injective, it is an isomorphism. As it is also an inclusion map (except for continuity), we now see that it is an isomorphism. This shows G0 = H . Thus G0 is connected and so H = G0 ⊆ G0 .

(∗∗)

Now (∗) and (∗∗) imply H = G0

Pro-Lie Groups Among the projective limits of Lie groups there is a class of groups for which a projective limit representation is particularly nice. We discuss these now. Firstly, we introduce some terminology. Let G be a topological group and N (G) the set of all normal subgroups such that G/N is a finite-dimensional Lie group. Note that a normal subgroup N of a topological group is closed if and only if the quotient group G/N is Hausdorff – which is certainly the case if G/N is a Lie group. We shall occasionally call a subgroup N of a topological group G a co-Lie subgroup if it is normal and G/N is a Lie group. Thus N (G) is the set of all co-Lie subgroups; certainly G ∈ N (G). We shall say that a topological group G has arbitrarily small co-Lie subgroups if every identity neighborhood of G contains a member of N (G). We claim that in a topological group with arbitrarily small co-Lie subgroups the set N (G) is closed under finite intersections and thus is a filter basis (Glöckner’s Lemma). For a proof, let M, N ∈ N (G) and let δ : G → G/M × G/N be defined by δ(g) = (gM, gN ). Since G/M and G/N are Lie groups, they contain identity neighborhoods VM and VN , respectively, which contain no subgroups except the singleton def

ones. Then U = δ −1 (VM × VN ) is an identity neighborhood of G such that any subgroup of G contained in U is mapped to {(1, 1)} and thus is contained in ker δ = M ∩N . But G has arbitrarily small co-Lie subgroups. Thus there is a co-Lie subgroup D of G contained in U such that G/D is a Lie group. By what we just saw, D ⊆ M ∩ N . Since G/D ∼ = (G/D)/((M ∩ N )/D), the factor group G/(M ∩ N ) is a quotient group

Pro-Lie Groups

149

of G/D and is therefore a Lie group. Hence M ∩ N ∈ N (G), and this completes the proof. In any topological group with arbitrarily small co-Lie subgroups it makes sense to talk about the filter basis N (G), and the condition that G has small co-Lie subgroups means exactly that this filter basis converges to the identity. The Definition of Proto-Lie Groups and Pro-Lie Groups Definition 3.25. A topological group G is called a proto-Lie group if it has arbitrarily small co-Lie subgroups, that is, if (i) every identity neighborhood of G contains a normal subgroup such that G/N is a Lie group, and it is called a pro-Lie group if, in addition, (ii) G is complete. The group G is called protodiscrete, respectively, prodiscrete if the filter of identity neighborhoods has a basis of open normal subgroups, respectively, G is complete and protodiscrete. The category of all pro-Lie groups and all continuous group homomorphisms between them will be called proLieGr. By the remarks preceding the definition, condition (i) is equivalent to saying (i ) N (G) is a filter basis converging to 1. Warning. Many authors call a topological group a pro-Lie group if the set N (G) of closed normal subgroups N such that G/N is a finite-dimensional Lie group is a filter basis converging to 1 and contains compact members. Such groups are locally compact by Proposition 1.34. Pro-Lie groups in the sense of this book comprise a much larger class of topological groups; indeed we shall see presently that they form a complete subcategory of TopGr so that for instance all powers RX are pro-Lie groups whether X is finite or not. We alert the reader to the fact that in our paper [107] with M. Stroppel we used the more narrow terminology which causes all pro-Lie groups to be locally compact. At the end of the chapter we shall survey the most relevant statements which are equivalent to the definition of a pro-Lie group and thus may equally well serve as definition. In that context we shall exhibit an example of a complete abelian group G for which N (G) is not a filter basis; indeed N (G) will contain two members M and N such that M ∩ N = {1} while G is not a Lie group. Proposition 3.26. Let G be a topological group such that N (G) is a filter basis. Then the following conclusions hold:

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3 Pro-Lie Groups

(i) The map γG : G → GN (G) =

lim

N ∈N (G)

G/N

of Theorem 1.29 is an embedding morphism if and only if G is a proto-Lie group. (ii) The map γG : G → GN (G) is an isomorphism if and only if G is a pro-Lie group. (iii) For any proto-Lie group G, the set UN (G) of open identity neighborhoods U for which there is an N ∈ N (G) such that U N = N U = U is a basis of the filter of identity neighborhoods of G. Proof. Assertion (i) is an immediate consequence of the definition of a proto-Lie group (Definition 3.25), of Theorem 1.29, and of Theorem 1.30. Statement (ii) then follows from Theorem 1.33 and the fact that a pro-Lie group is always complete. Claim (iii) is then a consequence of the Fundamental Theorem of Projective Limits 1.27 (i). Proposition 3.26, in particular, shows that every proto-Lie group has a completion, and that this completion is a pro-Lie group. The example of the group of rationals in its natural topology shows that a dense subgroup of a Lie group need not be a proto-Lie group. We recall that for any family {Aj : j ∈ J } of groups the restricted product is the subgroup of j ∈J Aj consisting of all (aj )j ∈J with aj = 1 for all but a finite number of indices j ∈ J . The restricted product of a constant family {Aj : j ∈ J }, Aj = A for all j ∈ J , is denoted A(J ) . Exercise E3.6. Verify the following examples of proto-Lie groups. Example. (a) The additive group of any dense vector subspace of a weakly complete topological vector space is a proto-Lie group. (b) In particular, for any set X, the restricted product (in this case the direct sum) R(X) is a proto-Lie group when it is equipped with the topology induced from the product RX . (c) The additive group of any hyperplane in a weakly complete topological vector space (closed or not) is a proto-Lie group. (d) The  restricted product of any given family {Lj | j ∈ J } of Lie groups in the product j ∈J Lj is a proto-Lie group. [Hint. (a) Let V be a dense vector subspace of a weakly complete topological vector space W . If N is a closed vector subspace such that dim W/N < ∞, then V /(V ∩N ) ∼ = (V + N)/N is a dense vector subspace of W/N and thus agrees with W/N . Therefore limN V /(V ∩ N ) = limN W/N = W . (b) and the nonclosed case of (c) follow from (a) since a nonclosed hyperplane is dense. The closed case of (c) follows from 3.35. (d) is straightforward from the fact that the cofinite partial products are co-Lie subgroups.] Before we look at further examples, we provide an elementary, but useful tool.

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151

Proposition 3.27. For a topological group G, the following conditions are equivalent: (i) G is a proto-Lie group. (ii) There is a filter basis M of closed normal subgroups converging to 1 such that G/M is a finite-dimensional Lie group for each M ∈ M. If these conditions hold, then M is cofinal in N (G). Moreover, if G is complete, then these conditions are equivalent to (iii) G is a pro-Lie group. If (iii) holds then G ∼ = limM∈M G/M. Proof. Since (i) ⇒ (ii) is trivial by Definition 3.25 of a proto-Lie group, we prove (ii) ⇒ (i). Clearly, M ⊆ N (G). We claim that (∀N ∈ N (G))(∃M ∈ M)

N ⊇ M.

(2)

Let us begin by assuming that condition (2) is satisfied. Then we claim firstly that N (G) is closed under finite intersections and hence is a filter basis: Let N1 , N2 ∈ N (G), then by (2) there are subgroups M1 , M2 ∈ M with Nj ⊇ Mj for j = 1, 2. Since M is a filter basis, there is an M ∈ M such that M1 ∩ M2 ⊇ M. Hence N1 ∩ N2 ⊇ M. Therefore G/(N1 ∩N2 ) is a quotient group of the finite-dimensional Lie group G/M and is therefore itself a finite-dimensional Lie group. Hence N1 ∩ N2 ∈ N (G). Secondly, since M ⊆ N (G), and since M converges to 1, the filter basis N (G) converges to 1 as well. And finally, by (2), M is cofinal in N (G), whence G ∼ = limM∈M G/M = limN∈N (G) G/N by the Cofinality Lemma 1.21. Thus it remains to prove (2). So let N ∈ N (G) be given. Let U = U N be an open identity neighborhood of G such that U N/N is an identity neighborhood of the finitedimensional Lie group G/N which contains no subgroups other than the singleton one. If p : G → G/N is the quotient map, then the image filter basis p(M) converges to the identity in G/N. Hence there is an M such that p(M) ⊆ U N/N. Then the subgroup p(M) is singleton, that is M ⊆ N , which is what we had to show. If G is complete, then (i) shows that G is a pro-Lie group and by Theorem 1.33 we then know that G ∼ = limM∈M G/M. The argument used to prove (2) occurred in a similar fashion in the proof of part (vi) of Theorem 1.29. We draw attention to the fact that in the proof of the preceding result we have used the information that the quotient of a finite-dimensional Lie group is a Lie group. This is not true for Lie groups in general; there are examples of additive groups of separable Banach spaces having closed arcwise connected subgroups with zero Lie algebra (see [102, Exercise E7.17 following Example 7.39]); the respective quotients fail to be Lie groups. Examples 3.28. All classes of topological groups listed in 3.4 are in fact classes of pro-Lie groups. This is an easy exercise from Definition 3.25 of a pro-Lie group and the discussion of 3.4 in Exercise E3.2.

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3 Pro-Lie Groups

Using the terminology we have available now, the results we have proved in this section may be summarised as follows Lemma 3.29 (The First Fundamental Lemma on Pro-Lie Groups). Let G be a projective limit limj ∈J Gj of Lie groups. Then the following conclusions hold: (i) The identity component G0 is a pro-Lie group and thus G0 ∼ =

lim

M∈N (G0 )

G0 /M.

def

(ii) Set M = {G0 ∩ Kj : j ∈ J } for the kernels Kj of the limit morphisms fj : G → Gj . Then M is a cofinal subset of N (G0 ); that is, for each M ∈ N (G0 ) there is a j ∈ J such that G0 ∩ Kj ⊆ M. (iii) For each j ∈ J , the natural map G0 /(G0 ∩ Kj ) → (G0 Kj )/Kj is an isomorphism, the group G0 Kj /Kj is a Lie group and a closed subgroup of G/Kj , and G0 = lim (G0 Kj )/Kj . j ∈J

(iv) For each M ∈ M, there is an injective morphism of topological groups from G/M into a Lie group. Proof. (i) Since G0 is a pro-Lie group by Proposition 3.27 and Lemmas 3.23 and 3.24, we have G0 ∼ = limM∈N (G0 ) G0 /M. (ii) Since lim N (G0 ) = 1 and fj : G → Gj is continuous for each j ∈ J , we have lim fj (N (G0 )) = 1. But Gj is a Lie group and thus has no small subgroups. Hence there is an M ∈ N (G0 ) such that fj (M) = {1}, that is, M ⊆ Kj . Thus we have a quotient morphism G0 /M → G0 /(G0 ∩ Kj ). Since quotients of finite-dimensional Lie groups are Lie groups, G0 /(G0 ∩ Kj ) is a Lie group whence G0 ∩ Kj ∈ N (G0 ) by Definition 3.25. Hence M ⊆ N (G0 ). By Theorem 1.27 (i) we know that limj ∈J Kj = 1. Then limj ∈J G0 ∩ Kj = 1. Let M ∈ N (G0 ). Then G0 /M is a Lie group, and thus there is an open identity neighborhood U of G0 such that U M = U and U/M has no nonsingleton subgroups. Then there is a j ∈ J such that G0 ∩ Kj ⊆ U . Since (G0 ∩ Kj )M/M is a subgroup of G0 /M contained in U/M we have G0 ∩ Kj ⊆ M. def

(iii) By (ii) above, G0 /(G0 ∩ Kj ) is a finite-dimensional Lie group. We set N = {Kj : j ∈ J }. By 1.27 we know that lim N = 1. So we can apply the Closed Subgroup Theorem for Projective Limits 1.34 with H = G0 . In particular, 1.34 (iv) yields the assertions of (iii). (iv) According to our conventions 3.19 we have G = limj ∈J Gj for a projective system of finite-dimensional Lie groups Gj . Each M ∈ M is of the form M = ker fj for some j ∈ J for the limit morphism fj : G → Gj . This morphism induces an injective morphism of topological groups Fj : G/Kj → Gj such that fj = Fj  qj for the quotient map qj : G → G/Kj = G/M.

Pro-Lie Groups

153

Note that we have shown, in particular, that every connected projective limit of Lie groups is a pro-Lie group. Proposition 3.30. (a) For a projective limit of Lie groups G, the following statements are equivalent: (i) (ii) (iii) (iv)

G is prodiscrete. G is zero-dimensional. G is totally disconnected. L(G) = {0}.

(b) A quotient of a protodiscrete group is protodiscrete. Proof. First we prove (a). (i) ⇒ (ii): By (i) G is a closed subgroup of a product of discrete groups and therefore the filter of its identity neighborhoods has a basis of open subgroups. (ii) ⇒ (iii) ⇒ (iv): This is clear. (iv) ⇒ (i): Let G = limj ∈J Gj for a projective system as in 3.19. By 1.27 (iv) we may and will assume that the limit maps fj : G → Gj have dense images. Let Dj be the discrete group Gj /(Gj )0 and let D = {Fj k : Dk → Dj | j ≤ k, (j, k) ∈ J × J } be the projective system induced by P and let D = limj ∈J Dj . Since Gj = fj (G) and the groups Dj are discrete, the limit maps Fj are surjective. Then each quotient D/ ker Fj for the limit maps Fj : D → Dj is discrete, and D = limj ∈J D/ ker Fj by Theorem 1.27 (ii). Hence D is a prodiscrete group. Now by hypothesis (iv) we have {0} = L(G) = limj ∈J L(Gj ). Then by Lemma 3.21, for each j ∈ J , there is a kj ≥ j such that fj kj ((Gkj )0 ) = {1}. Thus fj kj factors through a morphism Fj kj : Dj kj → Gj . We have a diagram Gj o

fj kj

qj

Gkj qk

 Dkj }} }} } }} }} Fj k } j } } ~} Gkj Gj o  Dj o

···

Fj kj

fj kj

G q

···

 D p

···

 G.

By an argument analogous to that used in the proof of 3.10 we conclude the existence def

of a morphism πj = Fj kj  Fkj : D → Gj which is independent of the choice of kj

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3 Pro-Lie Groups

in as much as it agrees with Fj kj  Fkj k  Fk for k ≥ kj . We notice that for j ≤ j  we get πj = fjj   πj : D → Gj . Thus the universal property of D = limj ∈J Dk implies the existence of a unique morphism p : D → G such that πj = fj  p. Hence G is a retract of D. But retracts of prodiscrete groups are easily seen to be prodiscrete. This completes the proof. Proof of (b): If G is protodiscrete, N (G) is a filter basis of open normal subgroups which converges to 1. Now let N be a closed normal subgroup of G. Define U = {N U/U : U ∈ N (G)}. Each NU is an open and hence closed normal subgroup of G and thus the N U/U are open-closed subgroups of G/N , and we claim that U converges to the identity of G/N. Let W be an open identity neighborhood of G/N and V its full inverse image in G. Then V is an open identity neighborhood of G such that NV = V . Since N (G) converges to 1, there is a U ∈ N (G) such that U ⊆ V . Then NU ⊆ NV = V and thus NU/N ⊆ W . This proves the claim and establishes (b) in view of 3.27. Lemma 3.31 (The Second Fundamental Lemma on Pro-Lie Groups). For any G that is a projective limit of Lie groups, the component factor group G/G0 is protodiscrete; if it is complete, then it is prodiscrete. Proof. We retain the notation of the proof of Proposition 3.30 and consider the commutative diagram (G⏐j )0 ⏐ incl G⏐j ⏐ quot Dj

fj0k

←−−−− (G⏐k )0 ⏐ incl fj k ←−−−− G⏐k ⏐ quot Fj k ←−−−− Dk

···

G⏐0 = limj ∈J (Gj )0 ⏐ incl G = limj ∈J Gj ⏐ ⏐q  D = limj ∈J Dj .

··· ···

The morphism q : G → D is the fill-in map given by the universal property of the limit in the last row. (In passing we note that the first row is a limit diagram in the category of connected topological groups in view of 3.16.) Since the composition incl

quot

(Gj )0 −−−→ Gj −−−→ Dj is constant, so is the composition incl

q

G0 −−−→ G −−−→ D. Hence we have a unique morphism p : G/G0 → D, p(gG0 ) = q(g). Assume that g = (gj )j ∈J ∈ G is such that p(gG0 ) = 1, i.e., (gj (Gj )0 )j ∈J = q(g) = 1 in  limj ∈J Dj ; thus g ∈ j ∈J fj−1 ((Gj )0 ) = limj ∈J (Gj )0 = G0 . This shows that p is injective. The sets Fj−1 (1) are basic identity neighborhoods of D by 1.27 (i). As p−1 Fj−1 (1) = fj−1 ((Gj )0 )/G0 and this is an open-closed subgroup we see that p is

Pro-Lie Groups

155

def

an embedding. Therefore G/G0 may be identified with the subgroup S = im q = {(gj (Gj )0 )j ∈J : (gj )j ∈J ∈ G} of D. Let Nj = Fj−1 (1). Then Nj is an open-closed normal subgroup of D and S∩Nj is an open-closed normal subgroup of S. Since limj ∈J Nj = 1 we have limj ∈J S ∩ Nj = 1.  Hence G/G0 ∼ = S is a protodiscrete group and S = j ∈J SNj is prodiscrete. If G/G0 is complete, then G/G0 ∼ = S and G/G0 is prodiscrete. Before we continue, we record an independent elementary lemma. Lemma 3.32. (i) Let f : A → B be a quotient morphism of topological groups with discrete kernel. Then there is an open symmetric identity neighborhood V of A and an open symmetric identity neighborhood W of B such that f |V : V → W is a homeomorphism, and for every subgroup K of B contained in W there is a subgroup S of A contained in V such that f (S) = K. (ii) If A is connected and B is a pro-Lie group, then A is a pro-Lie group. Proof. (i) Let U be a symmetric open identity neighborhood of A such that U 2 ∩ ker f = {1}. Then f (U ) is an open symmetric identity neighborhood of B. So f |U : U → f (U ) is continuous, open and surjective; if u1 , u2 ∈ U and f (u1 ) = f (u2 ), then u1 u−1 ∈ (ker f ) ∩ U 2 . Thus f |U is a homeomorphism. Now let 2 V be an open symmetric identity neighborhood in A such that V 2 ⊆ U , and set def

W = f (V ). Then f |V : V → W is a homeomorphism onto an open identity neighborhood of B. Define ϕ : W → V to be its inverse and take w1 , w2 ∈ W such that w1 w2 ∈ W . Set vj = ϕ(wj ), j = 1, 2 and v = ϕ(w1 w2 ). Then (f |U )(v) = (f |V )ϕ(w1 w2 ) = w1 w2 . Further v1 v2 ∈ V 2 ⊆ U . Then (f |U )(v1 v2 ) = f (v1 )f (v2 ) = (f |V )ϕ(w1 )(f |V )ϕ(w2 ) = w1 w2 = (f |U )(v). Since (f |U ) is injective, we conclude v = v1 v2 , that is, ϕ : W → V is a homeomorphism such that (∀w1 , w2 ∈ W ) (w1 w2 ∈ W ) ⇒ (ϕ(w1 w2 ) = ϕ(w1 )ϕ(w2 ).

(∗)

In particular, if w ∈ W then w−1 ∈ W and ww −1 = 1 ∈ W and thus ϕ(w)ϕ(w −1 ) = ϕ(1) = 1; therefore ϕ(w−1 ) = ϕ(w)−1 . Now let K be a subgroup of B contained in W . Let g1 , g2 ∈ ϕ(K). Then there are elements w1 , w2 ∈ K ⊆ W such that gj = ϕ(wj ), j = 1, 2 and w1 w2−1 ∈ K ⊆ W . Hence g1 g2−1 = ϕ(w1 )ϕ(w2 )−1 = ϕ(w1 )ϕ(w2−1 ) = ϕ(w1 w2−1 ) ∈ ϕ(K). It follows that ϕ(K) ⊆ V is a subgroup of A. (ii) Now let N be a normal subgroup of B contained in W ; set M = ϕ(N ). Since N is normal in B, in particular wNw−1 = N for all w ∈ W . Applying ϕ we obtain vMv −1 = M for all v ∈ V . Since A is connected, V generates A algebraically and thus M is normal in A. As f : A → B is an open morphism, it induces an open morphism P : A/M → B/N via P (aM) = f (a)N, whose kernel consists of all aM such that f (a) ∈ N . Let m ∈ V ⊆ A be the unique element such that f (m) = f (a). Then am−1 ∈ ker f . We write D = ker f shortly for the discrete kernel of f . Then ker P = DM/M. We claim that ker P is discrete. For this purpose we note that DV M/M is an open identity neighborhood of A/M. An element dM ∈ ker P is in

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3 Pro-Lie Groups

DV M/M iff dM = δvN for some elements δ ∈ D and v ∈ V , that is, iff d = δvm for δ and v as before and m ∈ M. Then δ −1 d = vm ∈ D ∩ V M ⊆ D ∩ V 2 = {1}. Thus v = m−1 ∈ M and d ∈ M ⊆ V ; and since D ∩ V = {1} this entails d = 1. We have shown that DV M/M ∩ ker P = {M}, and this proves the claim that ker P is discrete. Hence P implements a local isomorphism, and so A/M and B/M are locally isomorphic. Therefore, if B is a pro-Lie group, then the filter basis N (B) converges to 1 whence all sufficiently small members N ∈ N (B) are contained in W . Then for all sufficiently small N ∈ N (B), the normal subgroups ϕ(N ) of A form a filter basis converging to 1 in A, and are contained in N (A). If M ∈ N (A), then A/M is a Lie group which has no small subgroups, and so A has an identity neighborhood UM such that all members ϕ(N) ∈ N (A) contained in UM are contained in M. Thus {ϕ(N ) : N ∈ N (B), N ⊆ W } is cofinal in N (A). So A is a proto-Lie group. Now A/D ∼ = f (A) is an open subgroup of B and thus is a pro-Lie group; moreover D, as a discrete group, is complete. Therefore A is complete (see [176], p. 225, 12.3). Hence A is a pro-Lie group as asserted. Lemma 3.33 (The Third Fundamental Lemma on Pro-Lie Groups). Let G be a topological group such that G0 is a finite-dimensional Lie group and assume that there is an injection f : G → L into a finite-dimensional Lie group L. If G/G0 is a protodiscrete group, then G is a finite-dimensional Lie group. Proof. We must show that G0 is open in G. First we make some reductions. Since f (G) is a Lie group as a closed subgroup of a Lie group, we may and will assume that L = f (G). Next, since f −1 (L0 ) is open in G there is no loss in assuming that L = L0 , i.e. that L is connected. Let M = f (G0 ). Then M is a closed normal subgroup of L and f induces an injective map G/f −1 (M) → L/M. Now G/f −1 (M), being a quotient of the protodiscrete group G/G0 is protodiscrete by 4.3 (b) and is, at the same time, without small subgroups. Hence it is discrete, that is, f −1 (M) is open. We may therefore assume that G = f −1 (M), i.e. that M = L. Thus we may assume that f (G0 ) is dense in L.  → L and form the pullback Now we consider the universal covering q : L F

P ⏐ −−−−−→ ⏐ Q G −−−−−→ f

 L ⏐ ⏐q  L.

 : f (g) = q()}. ˜ ∈ G× L ˜ If p = (g, ) ˜ ∈P In terms of elements, we have P = {(g, ) ˜ = 1 and thus g = 1 as f is injective. and F (p) = 1, then ˜ = 1, whence f (g) = q() Thus p = (1, 1) and this shows that F is injective. ˜ ∈ ker Q. Next F maps ker Q isomorphically onto ker q. Indeed let p = (g, ) ˜ Then 1 = Q(p) = g and then qF (p) = q() = f (g) = 1, that is, F (p) ∈ ker q. ˜ = f (1), whence p def Conversely, if ˜ ∈ ker q, then 1 = q() = (1, q) ∈ P , and Q(p) = 1, i.e. p ∈ ker Q and F (p) = q. Now let V be an identity neighborhood of

Pro-Lie Groups

157

 such that V ∩ ker q = {1} and assume that p = (g, ) ˜ ∈ ker Q ∩ (G × V ); then g = L ˜ thus ˜ ∈ V ∩ ker q = {1}. Thus p = 1 and ker Q is Q(p) = 1 and 1 = f (g) = q(); discrete in P . If (U ×V )∩P is an identity neighborhood and p = (u, v) ∈ (U ×V )∩P , then Q(p) = u and f (u) = q(v), whence Q((U × V ) ∩ P ) = U ∩ f −1 q(V ), and this set is an identity neighborhood. Thus the morphism Q is open and thus, since its kernel is discrete, it implements a local isomorphism. Therefore G is a Lie group if and only if P is a Lie group. Thus we must show that P is a Lie group, that is, that P0 is open.  and normal analytic subgroups in Now F (P0 ) is a normal analytic subgroup of L,  is F (P0 ) ker q, simply connected Lie groups are closed. The full inverse of f (G0 ) in L  (P0 ). Since ker q and thus this group is dense, and F (P0 ) ker q/F (P0 ) is dense in L/F  the group L/F  (P0 ) is abelian and simply connected, hence is isomorphic is central in L, to a vector group Rn . Thus F induces an injective morphism of P /P0 into the vector  (P0 ) and so P /P0 has no small subgroups. The quotient morphism group L/F quotG

Q

P −−−−−→ G −−−−−→ G/G0 vanishes on P0 and therefore factors through P /P0 : Q∗

quotP

quotG Q = (P −−−−−→ P /P0 −−−−−→ G/G0 ). We have ker Q∗ = P1 /P0 with P1 = Q−1 G0 . The following is a diagram of abelian topological groups (ι an injective morphism): P /P0 ⏐ ⏐ ∗ Q

ι

−−−→ Rn

G/G0 . def

The morphism Q = Q|P1 : P1 → G0 is a covering morphism of the Lie group G0 with kernel ker Q ∼ = ker q and thus is a Lie group containing the closed normal subgroup P0 = (P1 )0 . Then ker Q∗ = P1 /P0 is a totally disconnected Lie group and is therefore discrete. Since G/G0 has arbitrarily small open subgroups by the hypothesis of protodiscreteness, Lemma 4.5 applies to Q∗ and shows that P /P0 has arbitrarily small open subgroups (that is, P /P0 is a protodiscrete group). But ι injects P /P0 into Rn , and thus P /P0 has an identity neighborhood in which the singleton group {P0 } is the only subgroup; this subgroup, therefore, is open and so P0 is open which is what we had to show. Now we are ready for the principal result of this entire chapter. The Pro-Lie Group Theorem Theorem 3.34. Every projective limit of Lie groups is a pro-Lie group.

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3 Pro-Lie Groups

Proof. By the First Fundamental Lemma 3.29, a group G that is a projective limit of Lie groups has a filter basis M of closed normal subgroups M converging to 1 such that G0 M/M is a connected Lie subgroup of G/M, and that there is an injective morphism of G/M into a finite-dimensional Lie group. By the Second Fundamental Lemma 3.31, G/G0 has a basis O of open normal subgroups converging to the identity. It follows that for each M ∈ M, the factor group G/M has a filter basis of open normal subgroups U/M such that every open set V containing G0 M/M contains one of the U/M, U ∈ O. Thus every G/M, M ∈ M, satisfies the hypotheses of the Third Fundamental Lemma 3.33. As a consequence of 3.33, G/M is a finite-dimensional Lie group. Then by Proposition 3.27 it follows that G is a pro-Lie group. The Closed Subgroup Theorem for Pro-Lie Groups Theorem 3.35. A closed subgroup of a pro-Lie group is a pro-Lie group. Proof. The proof is immediate from Theorem 3.34 and Proposition 3.2. The Completeness Theorem for Projective Limits of Lie Groups 3.3 may now be reformulated as saying that the full subcategory of pro-Lie groups in the category of topological groups is closed under the formation of all limits and the passage to closed subgroups. We may record this as follows. Completeness Theorem for Pro-Lie Groups Theorem 3.36. (i) The full subcategory proLieGr of pro-Lie groups in the category TopGr of topological groups is closed under the formation of all limits and is therefore complete. (ii) Every limit of pro-Lie groups, in particular, every limit of finite-dimensional Lie groups is a pro-Lie group. (iii) There is a functor F : TopGr → proLieGr such that given any topological group G, there is a pro-Lie group F G and a morphism ηG : G → F G with dense image such that for every morphism f : G → H into a pro-Lie group H there is a unique morphism f  : F G → H such that f = f   ηG . TopGr

G ⏐ ⏐ ∀f  H

proLieGr

ηG

−−−−−→ F⏐G ⏐  f −−−−−→ idH

H

F⏐G ⏐  ∃!f H

(iv) Free pro-Lie groups exist. Specifically: there is a functor FproLieGr : Top0 → proLieGr from the category of pointed topological spaces into the category of pro-Lie groups such that there is a natural morphism ϕX : X → |FproLieGr (X)| of pointed topological spaces mapping the base point X0 of X to the identity of the topological group

Pro-Lie Groups

159

FproLieGr (X) such for every base point preserving continuous function F : X → H into a pro-Lie group H there is a unique morphism of pro-Lie groups f  : FproLieGr (X) → H such that f = |f  |  ϕX . Top0

X ⏐ ⏐ ∀f 

proLieGr

ϕX

−−−−−→ |FproLieGr ⏐ (X)| ⏐  f

|H | −−−−−→ idH

|H |

FproLieGr ⏐ (X) ⏐  ∃!f H

The subgroup generated in FproLieGr (X) generated algebraically by ϕX (X) is free. If X is completely regular, then ϕX is a topological embedding. Proof. (i) and (ii) are immediate consequences of the preceding results. See also Theorem 1.11 (i), (ii). Condition (iii) is an immediate consequence of (i) and the Retraction Theorem for Full Closed Subcategories of TopGr 1.41. (iv) The category TopGr of topological groups has a free functor FTopGr : Top0 → TopGr (see [71]) whose universal property is encapsulated in the following diagram. Top0

X ⏐ ⏐ ∀f 

TopGr

ψX

−−−−−→ |FTopGr ⏐ (X)| ⏐  f

|H | −−−−−→ idH

|H |

FTopGr ⏐ (X) ⏐  ∃!f H

We recall the retraction functor F : TopGr → proLieGr of (iii) above. Then the composition FproLieGr = F  FTopGr satisfies the requirement. Now let X be completely regular. Then X is topologically embedded into the free compact group via a natural map of pointed spaces ρX : X → |FCompGr (X)| (see for instance [102, Lemma 11.3]). Since the free compact group is a pro-Lie group, the universal property provides us with a morphism ρ  : FproLieGr (X) → FCompGr such that ρX : X = ρ   ϕX , and if ρX is an embedding, ϕX is an embedding. Since the group FTopGr (X) is algebraically free over the image of X, the subgroup ϕX (X) of FproLieGr (X) generated by the image of X under ϕX is free. Theorem 3.36 (iii) should be compared with the more familiar situation of the left adjoint to the inclusion functor CompGr → TopGr, where the Bohr compactification of G takes over the role of F G. Indeed the inclusion CompGr → proLieGr gives rise to a left adjoint mapping each pro-Lie group to its Bohr compactification.

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For results concerning free compact and free compact abelian groups see [94], [95], [96], [97], [100], and [101]; much, but not all, of the content of these sources is summarized in [102].

Some Examples After the Completeness Theorem 3.36 and the Closed Subgroup Theorem 3.35 it is not hard to realize that the class ofpro-Lie groups is quite large, because every closed subgroup of an arbitrary product j ∈J Lj of finite-dimensional Lie groups Lj is a pro-Lie group. The subclass of abelian pro-Lie groups alone is rather big, and we shall devote the entire Chapter 5 to them. After having discussed much of the general foundations of pro-Lie groups rather extensively, the reader might wish to be reminded of some simple examples which fail to be pro-Lie groups while being, say, locally compact. Both of the groups listed in the following two examples are totally disconnected but fail to have arbitrarily small open normal subgroups Example 3.37. Let L be a compact Lie group, for instance L = Z/2Z or L = R/Z. The classical example of a semidirect product LZ σ Z with the shift action of Z on the product is a locally compact group which is not a pro-Lie group because LZ × {0} is the unique smallest normal open subgroup. Example 3.38. The group Sl(2, Qp ) is a p-adic Lie group and thus is a totally disconnected locally compact group which is not a pro-Lie group (in the sense of this book) because it has no nontrivial normal subgroups except {±1}

An Overview of the Definitions of a Pro-Lie Group The mathematical content of this chapter is highly technical, but it is crucial for any deeper understanding of the definition of a pro-Lie group. It therefore serves a useful purpose if we summarize the possible definitions and comment on their relative merits. We recall that with any topological group G we associate canonically the set N (G) of all normal subgroups N such that G/N is a Lie group; occasionally we call the members of N (G) the co-Lie subgroups. We say that a topological group G has arbitrarily small co-Lie subgroups if every identity neighborhood of G contains a member of N (G). The First Definition of Pro-Lie Group Definition A. A topological group is a pro-Lie group if it is complete and has arbitrarily small co-Lie subgroups. It was this definition that we chose in the present chapter to define pro-Lie groups in Definition 3.6.

An Overview of the Definitions of a Pro-Lie Group

161

The Second Definition of Pro-Lie Group Definition B. A topological group G is a pro-Lie group if there is a projective system {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of morphisms between Lie groups such that G is isomorphic to its projective limit limj ∈J Gj . In this chapter we presented this definition in Definition 3.1 in order to define projective limit of Lie groups. The Third Definition of Pro-Lie Group Definition C. A topological group G is a pro-Lie group if it is isomorphic (algebraically and topologically) to a closed subgroup of a product of Lie groups. The major outcome of this chapter is The Equivalence Theorem of the Definitions of Pro-Lie Groups Theorem 3.39. The First, Second, and Third Definitions of Pro-Lie Groups are equivalent. Proof. A ⇒ B: In the circumstances of the first definition, by Glöckner’s Lemma preceding Definition 3.25, the set N (G) is a filter basis. Then Proposition 3.26 (ii) shows that G is isomorphic to the projective limit of Lie groups limN ∈N (G) G/N . B ⇒ A: This was established in the Pro-Lie Group Theorem 3.34. B ⇒ C: This is a consequence of Proposition 1.18 which made the general definition of a projective limit concrete in the category of topological groups.  C ⇒ A: If P = j ∈J Gj for a family of Lie groups Gj , then P is a pro-Lie group in the senseof Definition A, because P is complete and has arbitrarily small co-Lie subgroups j ∈J \F Gj , F ⊆ J finite. Let G be a closed subgroup of P . Then the Closed Subgroup Theorem 3.35 shows that G is a pro-Lie group in the sense of Definition A. In this chapter we chose Definition A for defining the concept of a pro-Lie group. This definition is the group theoretician’s choice because it provides the most immediate access to the information provided to us by the theory of finite-dimensional Lie groups. How powerful this approach is we shall see in the next chapter when we study the behavior of the Lie algebra functor on the quotient morphism G → G/N . However, using merely the information given in Definition A we find it impossible to conclude directly that the category of pro-Lie groups is complete. A proof of the completeness of the category of pro-Lie groups is easy if it is based on Definition B. On the other hand it may be very hard to draw any conclusion on the structure of a topological group G given as a projective limit G = limj ∈J Gj .

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Finally, Definition C formulates the concept of a pro-Lie group very much in the spirit of universal algebra and the theory of varieties. It looks rather simple, but experience shows that knowing a topological group (or any topological algebra, for that matter) to be a subgroup (or subalgebra) of a large product in general says very little about its structure. But Definition C may motivate one to look for product structures when we unfold the structure theory of pro-Lie groups. The two Definitions B and C are rather close – closer in fact that one might surmise at first glance; the Closed Subgroup Theorem for Projective Limits 1.34 is a tool that appears to be needed in one form or another. It is clear from the definition of a pro-Lie group that at the center is an important invariant attached to every topological group G, namely, the set N (G) of all normal subgroups N such that G/N is a Lie group. This set contains G and indeed all open normal subgroups if there are any. The set is partially ordered with respect to the relation ⊆. If S is any nonempty subset 0 1 of N (G), we may set N = S which is the smallest closed subgroup containing all members of S. This subgroup is normal, and since S = Ø we find an S ∈ S such that G/S is a Lie group and S ⊆ N. Then G/N ∼ = (G/S)/(N/S) is a quotient group of a Lie group and is therefore a Lie group, that is, N ∈ N (G). Thus every nonempty subset of N (G) has a least upper bound in N (G). This looks suspiciously as though N (G) should be a complete lattice, that is a partially ordered set in which every subset has a least upper bound and thus, as a consequence, every subset has also a greatest lower bound. But this is not the case, as the following example due to H. Glöckner (see [65, Appendix p. 452]) will show. Recall first that we say that two subgroups M and N of a group are said to be disjoint if M ∩ N = {1}. Example 3.40 (H. Glöckner). There is a complete topological abelian group G with two disjoint closed subgroups Nk , k = 1, 2 such that both quotient groups G/Nk are Lie / N (G). groups, while G itself fails to be a Lie group. So N1 , N2 ∈ N (G) but N1 ∩N2 ∈ Proof. We use the fact, that the additive group of rational numbers carries a grouptopology τ such that it is a nondiscrete complete topological group Qτ with respect to this topology (see [141]). It is nondiscrete and countable and thus cannot be locally compact because of the Baire Category Theorem; in particular, it is not a Lie group. Let  = {(r, r) : r ∈ R} ⊆ R2 and set G = (Qτ ×{0})+. We let b : Qτ × R → G be the bijective map b(q, r) = (q + r, r) and let j : G → R2 be the inclusion morphism. We give G that group topology which makes b an isomorphism of topological groups. Then G is not a Lie group since G/ ∼ = Qτ is not a Lie group. We use the projections def

pr k : R2 → R, k = 1, 2, to get morphisms pk = pr k j |G : G → R which are continuous, surjective and open; that is, they are quotient morphisms. We set Nk = ker pk . Then (q + r, r) ∈ N1 iff q + r = 0, that is N1 = {0} × Q, and (q + r, r) ∈ N2 iff r = 0 and thus N2 = Qτ × {0}. As kernels of morphisms of topological groups, both Nj are closed and G/Nk ∼ = R since both pk are quotient morphisms. Thus N1 , N2 are in N (G).

An Overview of the Definitions of a Pro-Lie Group

163

The example has some interesting aspects which illustrate various tricky features of this set-up: (i) The inclusion map j : G → R2 may be identified with the injective morphism g  → (g + N1 , g + N2 ) : G → G/N1 × G/N2 . Even though each of the quotient maps G → G/Nj is open, this inclusion map is not open onto its image G∗ = Q × R, and G does not inherit the induced topology from this map. The group G∗ is not a Lie group. (ii) The sum N1 + N2 = Qτ × Q is direct, and it has a nondiscrete topology. The quotient group (N1 + N2 )/N1 = (Qτ × Q)/({0} × Q) has basic zero neighborhoods of the form    & 1 1 ' ((U ∩ Qτ × Q) + ({0} × Q))/({0} × Q) = − n , n ∩ Q × Q /({0} × Q) # where U is a basic zero neighborhood of G of the form (U  × {0}) + (r, r) : − n1 < & $ ' r < n1 with a basic zero neighborhood of Qτ such that U  ⊆ − n1 , n1 . Thus (N1 + N2 )/N1 is isomorphic to Q with the topology induced from R. So the standard bijective morphism N2 → (N1 +N2 )/N1 is not open and thus is not an isomorphism of topological groups. This is an interesting failure of the second isomorphism theorem, because N2 ∼ = Qτ is countable, hence a σ -compact group. (iii) We note that N1 + N2 N1 + N2 G G + ⊆ G∗ ⊆ × , N1 N2 N1 N2 which we may read as Q2 ⊆ G∗ ⊆ R2 with the topology on Q2 that is induced from R2 . The group G∗ is dense, but not equal to R2 . In order to help the reader to develop a feeling for the position of the category proLieGr of pro-Lie groups within the category TopGr of all topological groups we recall that by the Completeness Theorem for Pro-Lie Groups 3.36, the category proLieGr is closed in TopGr under the formation of all limits and that (as a consequence) for each topological group G there is a unique pro-Lie group F G and a morphism ηG : G → F G with dense image such that every morphism f : G → H into a pro-Lie group factors uniquely through ηG . This formalism is completely analogous to that of the Bohr compactification of a topological group. What is the Bohr compactification AG there is the “pro-Lie-ification” F G here. Since the Bohr compactification AG as a compact group is a pro-Lie group, the Bohr compactification morphism G → AG factors through a canonical morphism F G → AG. The pro-Lie-ification also allows us to verify easily, that the category of pro-Lie groups has free objects over the category of pointed topological spaces.

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3 Pro-Lie Groups

Postscript In Definition 3.25 of this chapter we introduce a notion of pro-Lie group which is both more general and more natural than that common in the literature, and we summarize equivalent formulations of this definition towards the end of the chapter. Indeed let G be a topological group and N (G) the set of all closed normal subgroups such that G/N is a finite-dimensional Lie group. We call a topological group a pro-Lie group if G is complete and has arbitrarily small normal subgroups N such that G/N is a Lie group. We have seen in Chapter 1 that in these circumstances, G is naturally isomorphic to the projective limit limN ∈N (G) G/N and that this limit is locally compact if and only if there is a compact element N in the filter basis N (G). It is this more special situation that the literature generally refers to when it speaks of pro-Lie groups. The entire structure theory we develop in this book concentrates on pro-Lie groups and therefore on the category proLieGr of all pro-Lie groups and all continuous homomorphisms between them. So proLieGr is a full subcategory of the category TopGr of all topological groups. We have noticed that every pro-Lie group gives rise to a projective system of finitedimensional Lie groups {pMN : G/N → G/M | (M, N ) ∈ N (G) × N (G), M ⊇ N} whose limit is naturally isomorphic to G in such a way that the limit morphisms are the quotient morphisms pN : G → G/N. This is indeed a very special kind of projective limit. If an arbitrary projective system {fj k : Lk → Lj | (j, k) ∈ J × J, j ≤ k} of morphisms of topological groups between finite-dimensional Lie groups is given for def

some directed set J , then we say that G = limj ∈J Lj is a projective limit of Lie groups – implicitly assuming that we are talking about finite-dimensional real Lie groups. With this terminology at hand we can say that every pro-Lie group is a projective limit of Lie groups; in our summary of possible definitions, this is the implication A ⇒ B. The converse emerges here as a difficult question, but in the fundamental Pro-Lie Group Theorem 3.34 we show that all projective limits of Lie groups are pro-Lie groups; in the summary B ⇒ A. Thus the words “projective limit of finite-dimensional Lie groups” and “pro-Lie group” are synonyms. This is a highly nontrivial fact as we have seen in this chapter. While we proved this fact here from scratch, using the basic theory of finite-dimensional Lie groups, A. A. George Michael was able to give a very short proof that is based on the solution of Hilbert’s Fifth Problem [61]. The primary Definition A (= Definition 3.25) of pro-Lie group uses the set N (G) which is canonically attached to any topological group. Caution is needed because N (G) is not always a filter base, as Glöckner’s Example 3.40 shows, while it is in any pro-Lie group, as was shown in a paragraph preceding 3.25.

Postscript

165

The category proLieGr contains all compact groups and all locally compact abelian groups. That proLieGr is a rich environment is evident from the fact that it also contains, firstly, all finite-dimensional Lie groups and arbitrary products of these, and, secondly, the additive groups of all Lie algebras of compact groups (see [102, Proposition 9.45]) and of locally compact abelian groups (loc. cit., p. 355, Theorem 7.66 (i)); furthermore, the exponential function from the Lie algebra of a locally compact abelian group to the group itself is a morphism in proLieGr. We show that all pro-Lie groups have Lie algebras whose additive topological groups are pro-Lie groups. It is a relatively easy matter to establish that the full subcategory of TopGr of projective limits of Lie groups is closed under the formation of all limits in TopGr and is therefore a complete category in its own right. This is done in 3.8. Thus the category proLieGr of pro-Lie groups is complete. It would be a challenging task to prove this directly from the definition of pro-Lie groups; in fact we do not know how to proceed in this fashion. It is somewhat curious that a proof of the fact that every closed subgroup of a pro-Lie group is a pro-Lie group is harder than the completeness result itself. We use here the “Closed Subgroup Theorem for Projective Limits” 1.34 which we provided in Chapter 1. In the process of establishing all of this we have proved that a pro-Lie group is totally disconnected iff L(G) = {0} iff it is prodiscrete; these equivalences are plausible, but by no means obvious. Yamabe [206], [207] has shown that every locally compact group contains an open subgroup which is a pro-Lie group (see also [145], [129]). It is that portion of a locally compact group which we shall recognize as amenable to further structural analysis. The bottom line: We have arrived at the very substantial and hopefully manageable category proLieGr of topological groups with profinite-dimensional Lie algebras. It is suitable in all respects in which any category of locally compact groups is defective: – it is closed under all limits and passing to closed subgroups, and it contains all finite-dimensional Lie groups; – it permits a Bohr compactification-like reflection of the category of all topological groups into the category of pro-Lie groups proLieGr and it has a free functor Top → proLieGr; – it has an excellent – albeit in general infinite-dimensional – Lie theory; – it is closed under passing to the additive groups of the Lie algebras. And, in addition it still has the following property: – it includes all compact groups, all locally compact abelian groups, and all connected locally compact groups and thus is the true background theory for any Lie theory of locally compact groups. It is therefore feasible (at least temporarily) to introduce categories which appear to be particularly suitable envelopes of proLieGr, namely, LieAlgGenGr, the full subcategory of all topological groups having a generating Lie algebra. In 4.22 (i) below we shall prove that every pro-Lie group has a generating Lie algebra. We show that

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3 Pro-Lie Groups

proLieGr it is also contained in the category proLieAlgGr of topological groups which have pro-Lie algebras. But proLieAlgGr is complete; it does in fact contain arbitrary

products of finite-dimensional Lie groups. The following diagram gives an overview of the principal categories that we have considered.

TopGr

kkkk kkkk LieAlgGenGr

LieAlgGr

SSSS SSSS proLieAlgGr

SSSS SSS

kkk kkk proLieComGr

kkkk kkkk proLieGr

kkk kkk k k kk

CompGr

LCGr??

LieGr

SSSS SSSS

SSS SSS SSS S

proLieGr

LCAbGr

The morphisms of all categories in this diagram are continuous homomorphisms between topological groups. We need tabulate only their objects: Name of the category TopGr LieAlgGr LieAlgGenGr proLieAlgGr proLieComGr proLieGr LCGr CompGr LieGr LCAbGr

Objects of the category topological groups topological groups having a Lie algebra topological groups having a generating Lie algebra topological groups having a profinite-dimensional Lie algebra topological groups having a pro-Lie group identity component pro-Lie groups locally compact groups compact groups (finite-dimensional) Lie groups locally compact abelian groups

Complete? yes yes ??? yes yes yes no yes no no

Postscript

167

The questions marks in “LCGr??” are a warning that in this book we do not reprove the fact that locally compact groups have pro-Lie identity components and thus also have profinite-dimensional Lie algebras; see [129], [145], [206], [207]. We do know that the categories CompGr, LCAbGr and LieGr do have profinite-dimensional Lie algebras and thus are indeed subcategories of proLieAlgGr. The question marks in the line for LieAlgGenGr are there because we do not know whether the category of topological groups having generating Lie algebras is complete; it is closed under the formation of products, but we did not show that it is closed under the passage to closed subgroups. Another unsettled question that we leave behind is whether for a pro-Lie group G the component factor group G/G0 is complete, that is, is a prodiscrete group.

Chapter 4

Quotients of Pro-Lie Groups

We have proved that every projective limit of finite-dimensional Lie groups is a pro-Lie group, that the category of pro-Lie groups is complete and that every closed subgroup of a pro-Lie group is a pro-Lie group. We now address the topic of quotient groups of pro-Lie groups. Quotients arise naturally in the canonical decomposition of any morphism f : G → H between topological groups: If K is the kernel of f then there is a unique bijective morphism f  : G/K → f (G) defined by f  (gK) = f (g) so that the standard diagram commutes G ⏐ ⏐ quot G/K

f

−−−−−→

H  ⏐ ⏐incl −−−−−→ f (G), f

where quot : G → G/K is the quotient morphism and incl : f (G) → H is the inclusion embedding. Thus the first thing in understanding the structure of morphisms is understanding quotients and quotient morphisms. It belongs to the basic theory of locally compact groups to establish that quotients of locally compact groups are locally compact. The fact that quotients of finite-dimensional Lie groups are Lie groups belongs to the less elementary pieces of information of the basic theory. It is inevitable that we address the question of quotients of pro-Lie groups. Indeed, we see shortly that a quotient group of a pro-Lie group need not be a pro-Lie group. We then ask whether the category of pro-Lie groups “has quotients”. But for this question to make sense we would first have to make precise what we mean by a quotient in a category. Alas, from a category theoretical point of view, quotients are hard to put into an entirely satisfactory framework. We review this observation early in this chapter. We then show that the failure of pro-Lie groups to have complete quotients occurs already for the abelian pro-Lie groups RX as soon as X has at least continuum cardinality, and we notice that this phenomenon occurs in the supposedly familiar context of connected compact abelian groups for whose theory we refer e.g. to [102, Chapters 7 and 8]. On the positive side, we show that the Lie algebra functor on proLieGr preserves quotient morphisms; and we eventually prove a result which implies that all quotient groups of connected pro-Lie groups modulo connected closed normal subgroups are complete and therefore are pro-Lie groups. Prerequisites. No new prerequisites are demanded in this chapter.

Quotient Objects in Categories

169

Quotient Objects in Categories Quotient objects in a category arise when morphisms ϕ : A → B are decomposed in the form ϕ = με where ε : A → C is an epic and μ : C → B is some morphism such that whenever ϕ is factored in the form ϕ = ba with a : A → E and a monic b : E → B, then a has a factorisation a = a  ε. We mention in passing that in many categories where such a factorisation ϕ = μ  ε exists, μ is in fact itself a monic; these categories are said to have epic-monic factorisation. Examples are the categories of abelian groups, the category of groups, the category of rings, and their topological-algebraic counterparts. If categories have additional structure, then canonical constructions of factorisations ϕ = μ  ε are possible. Let us briefly describe how, and let us not aspire to the utmost possible generality but work on a level of generality that is still perfectly adequate for the categories we are dealing with in this book. In fact, categories of groups typically have singleton objects; in category theoretical terms they fit the following description: A null object in a category is an object 0 such that for each object A in the category there are unique morphisms 0 → A and A → 0. It is seen immediately that null objects are unique up to isomorphism. If null objects exist, then for every pair (A, B) of objects there is a morphism A → 0 → B, called a null morphism or a constant morphism, easily seen to be unique and denoted by 0BA : A → B. Categories with null objects are called pointed categories, see e.g. [102, Definition A3.6]. All categories whose objects are groups of one form or another are clearly pointed, and a constant morphism is one mapping everything to the identity element of the range group. Assume that ϕ : A → B is a morphism in a pointed category. If the equalizer (respectively, coequalizer) of ϕ and 0BA exists, then it is called the kernel κϕ : ker ϕ → A of ϕ (respectively, the cokernel qϕ : B → coker ϕ of ϕ). If a pointed category has kernels and cokernels, then for any morphism ϕ : A → B we can form the cokernel of κϕ : ker ϕ → A and denote it qϕ : A → A/ ker ϕ. Now ϕ  κϕ = 0B ker ϕ = ϕ  0A ker ϕ , and by the definition of qϕ there is a unique μϕ : A/ ker ϕ → B such that ϕ = μϕ  qϕ . If ϕ = ba is any factorisation with a : A → E, then 0B ker ϕ = ϕκϕ = baκϕ and the universal property of the kernel κb : ker b → E imply that there is a unique morphism k : ker ϕ → ker b such that κb k = aκϕ . If b is monic then κb is the null morphism and aκϕ = κb k = 0; thus, by the definition of the cokernel, there is an a  : A/ ker ϕ → E such that a = a  qf . Thus the factorisation ϕ = μϕ  qϕ has the required universal factorisation property. In such categories as the category of abelian groups, modules, groups, topological groups, etc., the inclusion morphism of the standard kernel ker ϕ into A is the categorical kernel and the standard quotient map A → A/ ker f is the categorical quotient qϕ . In these categories, the factor μϕ of a factorisation ϕ = μϕ  qϕ is automatically a monic. In general however, as we shall see in the category of pro-Lie groups, in the factorisation ϕ = μϕ qϕ itself, the morphism μϕ need not be monic. Nevertheless, the best category theoretical definition of a quotient morphism in a pointed category with kernels and cokernels is this: A morphism q : A → Q is called a quotient morphism if there is a morphism ϕ : A → B such that Q = A/ ker ϕ and q = qϕ .

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4 Quotients of Pro-Lie Groups

We should note that this formalism does have a category theoretical dual which associates with ϕ : A → B the kernel jϕ : C(ϕ) → B of the cokernel B → coker ϕ which one should duly call the coquotient of ϕ, giving rise to the canonical decomposition A ⏐ qf ⏐  A/ ker ϕ

ϕ

−−−−−→ −−−−−→ ϕ

B  ⏐j ⏐ϕ C(ϕ).

But we shall not pursue this further at this point, in particular since cokernels and coquotients in all but categories such as that of abelian groups and its variants are not what they seem at first sight. In the category proLieGr, however, the next theorem tells us what we can expect in terms of quotients and canonical decompositions of morphisms, and subsequently we shall inspect examples which show that we cannot expect to do much better. Indeed, as a consequence of the next theorem, the category proLieGr of pro-Lie groups is closed in TopGr under the formation of (group theoretical) quotients that happen to be complete. For any morphism f : G → H of pro-Lie groups with kernel K = ker f , the morphism qf defined as the composition of the (group theoretical) quotient morphism G → G/K with the embedding of G/K into its completion (G/K)N (G/K) is the cokernel of inclK : K → G and thus is the category theoretical quotient in proLieGr if H is a pro-Lie group. But we shall also see shortly, that the factorisation f = μf  qf is not epic-monic in general.

Quotient Groups of Pro-Lie Groups The Quotient Theorem for Pro-Lie Groups Theorem 4.1. (i) A quotient group of a proto-Lie group is a proto-Lie group and is isomorphic as a topological group to a dense subgroup of a pro-Lie group; in particular, every proto-Lie group has a completion. If the quotient group is complete, then it is a pro-Lie group. (ii) Let f : G → H be a morphism of topological groups from a pro-Lie group into def a complete group, and set K = ker f . Then there is a canonical decomposition: G ⏐ ⏐ q

f

−−−−−→

G/K ⏐ γG/K ⏐  (G/K)N (G/K)

H  ⏐ ⏐incl f (G)  ⏐ ⏐id

−−−−−→ f

f (G).

Quotient Groups of Pro-Lie Groups

171

In this diagram all groups are complete with the possible exception of G/K; further, G and (G/K)N (G/K) are pro-Lie groups. In particular, there is a factorisation f = μf  qf , qf = γG/K  q and μf = incl f  with an epic qf such that for any factorisation f = me of morphisms between complete groups for a monic m and some morphism e there is a factorisation of this morphism e in the form of e = e  qf . (iii) The morphism qf : G → (G/K)N (G/K) is the cokernel of inclK : K → G in the category proLieGr. (iv) (The Weak Pro-Open Mapping Theorem for Pro-Lie groups) Assume that G is a proto-Lie group such that G/N has countably many components for all N ∈ N (G). Let U be the set of open identity neighborhoods of G. If f : G → H is a surjective morphism onto a proto-Lie group, then (∀U ∈ U, M ∈ N (H )) Mf (U ) is open in H. If G has countably many components, then G/N has countably many components for all N ∈ N (G). Proof. (i) Let G be a pro-Lie group and K a closed normal subgroup. Define f : G → def

H = G/K to be the open quotient morphism. For N ∈ N (G), the set N K is a closed subgroup of G containing K, and since f is a quotient map and N K is K-saturated, the def

set N ∗ = f (N K) ⊆ H is closed and agrees with f (N ). Then N ∗ is a closed normal subgroup of H , and since f is open, we have that H /N ∗ ∼ = G/f −1 (N ∗ ) = G/N K ∼ = (G/N)/(NK/N ) is a finite-dimensional Lie group as a quotient of a finite-dimensional Lie group. Let M = {N ∗ | N ∈ N (G)}. Then M is a filter basis of closed normal subgroups of H such that all factor groups H /M, M ∈ M are Lie groups. Since N (G) converges to 1 as G is a pro-Lie group, from the continuity of f we conclude that f (N (G)) = {f (N) | N ∈ N (G)} converges to 1 in H . But since H is regular, i.e. the filter of identity neighborhoods has a basis of closed sets, M converges to 1 in H . Thus H is a proto-Lie group and by Theorems 1.29 and 1.30 we have a natural dense embedding morphism γH : H → HM into a pro-Lie group HM . By Definition 3.1, if H is complete, then, being a proto-Lie group, H is a pro-Lie group. (ii) Since H is complete by hypothesis, f (G) is the completion of f (G). There is no loss of generality in assuming H = f (G). By the standard canonical decomposition of a morphism, we have an injective morphism of topological groups F : G/K → H , F (gK) = f (g), such that f = F  q and F (G/K) = f (G). The pro-Lie group (G/K)N (G/K) is the completion of the proto-Lie group G/K according to (i). Hence F extends uniquely to a morphism f  : (G/K)N (G/K) → H , and thus f = f   γG/K  q = F  q. Finally set qf = γG/K  q and assume that f = m  e for a morphism e : G → C between complete groups and for a monomorphism m : C → H . Then K = ker f = (me)−1 (1) = e−1 (ker m) = e−1 ({1}) = ker e. Now we apply to e in place of f what we just derived for f ; hence there is a unique morphism e : (G/K)N (G/K) → C such that e = e qf . (iii) Assume that ϕ : G → X is a morphism in proLieGr such that ϕ  inclK = 0, that is, ϕ(K) = {1}. Then there is a unique morphism ψ : G/K → X given by

172

4 Quotients of Pro-Lie Groups

ψ(gK) = ϕ(g). Since X is complete as a pro-Lie group, ψ extends uniquely to the completion of G/K, i.e. there is a unique morphism ϕ  : (G/K)N (G/K) → X such that ϕ = ϕ  γG/K q = ϕ  qf . Since the category proLieGr is full in TopGr, this is what we had to show. def (iv) Let K = ker f . Then G/K is a proto- Lie group by (i) above, and f is open iff the induced bijective morphism F : G/K → H is an isomorphism. We claim that (G/K)/(N/K) ∼ = G/N has countably many components for any N/K ∈ N (G/K). Since N/K ∈ N (G/K) we know that N ∈ N (G), and thus G/N has only countably many components. It is therefore no loss of generality if we assume that f : G → H is bijective, as we shall do from now on. We note that if G has countably many components, then G/N has countably many components for each N ∈ N (G). Indeed, let ν : N → G/G0 be any surjective function. The quotient map G → (G/N )/(G/N )0 factors through G/G0 yielding a surjective function μ : G/G0 → (G/K)/(G/K)0 . Then μν : N → (G/N )/(G/N )0 is surjective, proving the claim that G/N has countably many components. def

It follows as in the proof of (i) that C = {f (N ) : N ∈ N (G)} is a filter basis of closed normal subgroups of H converging to 1. Let M ∈ N (H ) and U an open identity neighborhood of H satisfying MU = U M = U such that U/M contains no subgroups of the Lie group H /M other than the singleton one. Then we find an N ∈ N (G) such that f (N) ⊆ U ; thus f (N ) ⊆ M. Then fMN : G/N → H /M, fMN (gN) = fMN (g)M, is a surjective morphism between two locally compact groups. A connected locally compact group is algebraically generated by any compact identity neighborhood (see [102, Proposition A4.25 (iii)]); having countably many components, G/N is therefore σ -compact. Then the Open Mapping Theorem for Locally Compact Groups (see for instance [102, EA1.25]) shows that fMN is open. If pN : G → G/N and qM : H → H /M are the quotient morphisms, then qM  f = fMN  pN is open. Hence for each M ∈ N (H ) and each open identity neighborhood U of G the subset Mf (U ) is an M-invariant open identity neighborhood of H . Of course, Theorem 4.1 says in particular that any quotient of a pro-Lie group is a proto-Lie group. This may be perceived as an unsatisfactory weakness of proLie groups. We shall see shortly that we have intriguing examples showing that, for instance, the pro-Lie group RR has incomplete quotients. This will show that, sadly, this weakness cannot be remedied. We shall see in Chapter 5, Proposition 5.2, that the free abelian group Z(N) has a nondiscrete nonmetrizable prodiscrete and thus pro-Lie topology τ . If δ is the discrete topology on Z(N) , then the identity morphism f : (Z(N) , δ) → (Z(N) , τ ) is a nonopen bijective morphism between two countable pro-Lie groups. Thus a better Open Mapping Theorem than 4.1 (iv), alas, cannot be proved either. In later chapters, however, we shall see that connectivity makes all the difference, and we shall prove an Open Mapping Theorem saying that surjective morphisms between connected pro-Lie groups are open.

The Exponential Function of Compact Abelian Groups and Quotient Morphisms

173

In the proof of Theorem 4.1 we gave a self-contained proof of the factorisation property in (ii) even though it follows from (iii) via the category theoretical arguments we outlined in the paragraphs preceding 4.1. In [176, p. 196, Remark (b)] it is observed that a product of abelian complete metrizable topological groups may have incomplete quotient groups. Moreover, it is proved that every topological abelian group is a quotient of a complete topological abelian group ([176, 11.1]). Indeed, O. V. Sipacheva has proved that every topological group is a quotient of a complete topological group. (See [41, p. 100], [184, corollary to Theorem 4.1.6.].) We shall now set about showing that a product of 2ℵ0 copies of R has incomplete quotients; such examples occur in the immediate vicinity of compact connected abelian groups. Thus examples that illustrate the limitations of our Theorem 4.1 arise even in the theory of this fairly well documented class of topological groups.

The Exponential Function of Compact Abelian Groups and Quotient Morphisms This requires a detour into the theory of compact abelian groups. Namely, we shall investigate when the exponential function exp : L(G) → G of a compact connected abelian group G is open onto its image. For a detailed exposition of the exponential function of compact abelian groups we refer to [102], notably Chapters 7 and 8. We recall that L(G) = Hom(R, G) is the topological vector space of all one parameter subgroups, and that the exponential function is given by evaluation via exp X = X(1). We note that L(G) is isomorphic to a product of copies of R (see [102, Theorem 7.66 (i) and Theorem 7.30 (ii)]). For any compact abelian group G, the exponential function expG : L(G) → G (see [102, Theorem 7.66]) is a morphism of abelian topological groups. Let Ga denote the arc component of the zero element 0. Then the corestriction expG : L(G) → Ga of the exponential function to its image is a surjective morphism of topological groups. Since we deal here exclusively with abelian groups, we shall write them additively. The neutral element is zero, written 0. Since G is a compact abelian  is a discrete abelian group, and since G is connected, G  is group, its character group G torsion-free. We observe that the image of the exponential function expG : L(G) → G is Ga and that Ga = exp L(G) is dense in G. (For a proof, see [102, Theorem 8.30].) Separable abelian groups. A subgroup G of an abelian group A is said to split if there is a subgroup H of A such that A = G ⊕ H . Likewise we say that a short exact sequence j

0→G→A→B→0

(E)

is split if j (G) is a split subgroup of A. This is tantamount to saying that the equivalence class of (E) in Ext(B, G) is zero. (See e.g. [102, Lemma A1.49ff.]) A subgroup G of

174

4 Quotients of Pro-Lie Groups

an abelian group A splits automatically if G is divisible or A/G is free. (See e.g. [102, Corollary A1.36 and Proposition A1.15.]) A subgroup P of an abelian group A is called a pure subgroup if the following condition is satisfied. (i) (∀p ∈ P , a ∈ A, n ∈ N) n · a = p ⇒ (∃x ∈ P ) n · x = p. One observes directly (see also [102, PropositionA1.24]) that for a torsion-free group A, this condition is equivalent to either of the following conditions. (ii) The factor group A/G is torsion-free. (iii) (∀n ∈ N, a ∈ A) n · a ∈ G ⇒ a ∈ G. In a torsion-free abelian group A every subgroup G is contained in a unique pure subgroup [G] = {a ∈ A : (∃n ∈ N) n · a ∈ G}. (See [102, Proposition A1.25].) Moreover, the following conditions are equivalent (see [102, Proposition A1.64]): (a) Every finite rank pure subgroup of A is free. (b) Every countable subgroup of A is free. An abelian group satisfying these conditions is said to be ℵ1 -free. We are interested in a subclass of the class of ℵ1 -free groups. Proposition 4.2. Let A be a torsion-free abelian group. Then the following conditions are equivalent: (i) (∀a ∈ A) [Z · a] is free and splits. (ii) Every rank one pure subgroup is free and splits. (iii) Every finite rank pure subgroup is free and splits. Proof. Clearly, (iii) ⇒ (ii) ⇔ (i). We must prove (i) ⇒ (iii). Step 1. We first note that if A satisfies (i) then every pure subgroup B of A satisfies (i): Indeed let b ∈ B; since B is pure, the pure subgroup [Z · b] generated by b in B is pure in A, and as a subgroup of the free pure subgroup generated by b in A it is free. Since A satisfies (i), we have A = [Z · b] ⊕ K with a suitable subgroup K. By the modular law, B = [Z · b] ⊕ (K ∩ B). Setting H = K ∩ B we obtain B = [Z · b] ⊕ H as claimed. Step 2. We claim that every finite rank pure subgroup is free; in other words we show that A is ℵ1 -free. Let G be a finite rank pure subgroup of A and assume that G is a counterexample of minimal rank n. From (i) we know n > 1. Let 0 = g ∈ G. Since G satisfies (i), G = [Z · g] ⊕ H and [Z · g] is free. Since the subgroup H has smaller rank than G, it is not a counterexample and is therefore free. Hence G is free and thus cannot be a counterexample. Step 3. Now we prove that every finite rank pure subgroup G splits. By Step 2, N G = m=1 Z · em , where N ∈ N. By (i) there is a subgroup H1 of A such that A = Z · e1 ⊕ H1 and G = Z · e1 ⊕ (H1 ∩ G). Assume that H1 ⊇ H2 ⊇ · · · ⊇ Hn , n < N has been constructed in such a fashion that A = Z · e1 ⊕ · · · ⊕ Z · em ⊕ Hm

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175

and G = Z · e1 ⊕ · · · ⊕ Z · em ⊕ (Hm ∩ G), m = 1, 2, . . . , n. By the first step we may apply (i) to the pure subgroup Hn and find a subgroup Hn+1 of Hn such that Hn = Z · en+1 ⊕ Hn+1 and Hn ∩ G = Z · en+1 ∩ (Hn+1 ∩ G). This yields a descending family of subgroups Hn such that A = nm=1 Z · em ⊕ Hn and em ∈ Hn for m > n. We set H = HN . Then A = N m=1 Z · em ⊕ HN = G ⊕ H , as was to be shown. Definition 4.3. We say that an abelian group A is an S-group if it satisfies the equivalent conditions of Proposition 4.2. The S-groups have been called separable [49] which is not an advisable terminology here because we will deal with topological abelian groups for which the adjective separable refers to groups having a dense countable subset, and this is entirely different. One might have called S-groups strongly ℵ1 -free; our terminology reflects the “strongly” as well. Every free group is an S-group, since every subgroup of a free group is free, and the quotient of a free group modulo a pure subgroup is free (see e.g. [102, Proposition A1.24 (ii)]); thus every pure subgroup splits. A Whitehead group is an abelian group A such that Ext(A, Z) = {0}, that is, every extension 0→Z→G→A→0 splits. Example 4.4. The group A = ZN has the following properties: (i) A is an S-group. (ii) A is not a Whitehead group. (iii) The subgroup Z(N) of A is a countable free pure subgroup which does not split. Proof. (i), (ii) The group ZN is an ℵ1 -free group which is not a Whitehead group: see e.g. [102, Example A1.65]. We verify Condition 4.2 (ii) for A = ZN . Let P be a rank one pure subgroup of A and let k = (kn )n∈N be an element of P such that [Z · k] = P . Then the greatest common divisor of {kn : n ∈ N} is 1. The decreasing sequence gcd{k1 , k2 , . . . , kn } is eventually constant; that is there is a natural number N such that k1 , . . . , kN have the greatest common divisor 1. Then the subgroup PN generated in the finitely generated free group ZN by (k1 , . . . , kN ) is pure. By the Elementary Divisor Theorem (see e.g. [102, Theorem A1.10]) applied to ZN , after choosing a new basis, we may assume that PN = Z × {0} × · · · × {0}. It is therefore no loss of generality to assume that (kn )n∈N = (1, 0, . . . , 0, kN +1 , kN +2 , . . . ). Set G = {0} × Z × Z × · · · . Then ZN = P ⊕ G and ZN is an S-group as asserted. (iii) If m · (kn )n∈N ∈ Z(N) for some m ∈ Z then mkn = 0 for all but a finite number of n ∈ N. Then (kn )n∈N . Thus Z(N) is a pure subgroup of Z N which is obviously countable and free. The group ZN /Z(N) is a torsion-free algebraically compact group and contains a copy Zp of the p-adic integers for each prime as a direct summand. (See e.g. [58, p. 176, 42.2 and p. 169, 40.4.]) Since Zp contains countable groups which

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are not free (e.g. q1∞ Z for any prime q different from p), and since ZN is ℵ1 -free, ZN cannot contain a subgroup isomorphic to A/Z(N) . Example 4.5. There is an abelian group B with a subgroup C ∼ = Z such that B/C ∼ = ZN and that every morphism B → Z annihilates C. The group B is an ℵ1 -free group which is not an S-group. Proof. In the proof of Proposition A1.66 of [102], the following lemmas are proved: Lemma A. Let E = [0 → C → B → X → 0] be any extension of C ∼ = Z by an abelian group X. Then there is a homomorphism f : B → Z whose restriction to C is nontrivial if and only if E represents an element of finite order in Ext(X, Z). Lemma B. Ext(ZN , Z) contains 2(2

ℵ0 )

elements of infinite order.

Taken together, these lemmas yield the existence of a torsion-free group B and a cyclic subgroup C such that B/C ∼ = ZN , and that every morphism B → Z vanishes on C. The subgroup C is a subgroup of rank 1 which does not split, and since B/C is torsion-free, C is a pure subgroup ' +C &of B. Thus B is not an S-group. If P is a finite rank pure subgroup of B, then P C is a finite rank pure subgroup of B/C ∼ = A and is therefore finitely generated free; its full inverse image P  in B is a finitely generated torsion-free group and is, therefore, free. Thus P as a subgroup of a free group is free. Hence B is an ℵ1 -free group. In this area of the theory of abelian groups, ZN is a universal test example. For instance, Proposition 4.2 cannot be complemented by another equivalent condition which would say: Every countable pure subgroup splits. The example shows, in particular, that the class of S-groups is properly smaller than that of ℵ1 -free groups and is not contained in the class of Whitehead groups. Every Whitehead group is an S-group (see [49, p. 226]). Thus the class of S-groups is properly bigger than the class of Whitehead groups and thus a fortiori properly bigger than the class of free groups. ℵ1 -free groups  S-groups   Whitehead  groups  free groups Strengthening local connectivity for compact abelian groups. A locally compact  = Hom(G, T). abelian group G is completely characterized by its Pontryagin dual G (See e.g. [102, Chapters 7 and 8].) A topological group G is locally connected if and only if its identity component G0 is open in G and is locally connected; a connected locally compact abelian group G contains a unique characteristic maximal compact

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subgroup C and a subgroup V ∼ = Rn such that the morphism (v, c)  → v + c : V × C → G is an isomorphism of topological groups. (See e.g. [102, Theorem 7.57].) In discussing local connectivity of a locally compact abelian group G, it is no loss of generality to assume that G is compact and connected. A locally compact abelian  is discrete and group G is compact and connected if and only if its character group G torsion-free. (See e.g. [102, Proposition 7.5 (i), and Corollary 8.5].) Local connectivity of a compact connected abelian group is characterized as follows. Proposition 4.6. For a compact connected abelian group G, the following statements are equivalent: (i) There are arbitrarily small compact connected subgroups N such that G/N is a finite-dimensional torus group.  is the directed union of pure finitely generated free sub(ii) The character group G groups.  is ℵ1 -free. (iii) G (iv) G is locally connected. Proof. For a proof see [102, Theorem 8.36]. We compare this proposition with the following Proposition 4.7. For a compact connected abelian group G, the following statements are equivalent: (i) There are arbitrarily small compact connected subgroups N for which there is a finite-dimensional torus subgroup TN of G so that (n, t)  → n+t : N ×TN → G is an isomorphism of topological groups.  is the directed union of finitely generated free split sub(ii) The character group G groups.  is an S-group. (iii) G Proof. The equivalence of (i) and (ii) follows at once from duality.  is torsion-free. Let P be a rank one pure (ii) ⇒ (iii): By (ii), the abelian group G  By (ii) there is a finitely generated  The P = [Z · a] for some a ∈ G. subgroup of G.  free split subgroup F of G containing a. Since a direct summand is a pure subgroup we have P = [Z · a] ⊆ F . As a pure subgroup of a finitely generated torsion-free  P is a split group, P is a direct summand of F , and since F is a direct summand of G,  subgroup of G.  is the directed union of all of its finite (iii) ⇒ (ii): As a torsion-free abelian group, G rank pure subgroups P ; by (iii), every such P is split and free, and thus (ii) follows. The comparison of Propositions 4.6 and 4.7 justifies the following definition. Definition 4.8. A locally compact abelian group is said to be strongly locally connected if its identity component is open and its unique maximal compact connected subgroup satisfies the equivalent conditions of Proposition 4.7.

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In particular, a compact connected abelian group is strongly locally connected if and only if its character group is an S-group. def  N . Then G is a strongly locally connected and connected Example 4.9. Let G = Z but not arcwise connected compact abelian group. There is a compact connected, locally connected, but not strongly locally connected group H of weight 2ℵ0 containing G such that H /G is a circle group. G has a metric torus group quotient which is not a homomorphic retract.

Proof. A compact connected abelian group H is arcwise connected if and only if its  is a Whitehead group. (See e.g. [102, Theorem 8.30 (iv)].) The character group H claim thus follows by duality from Examples 4.4 and 4.5. The class of connected strongly locally connected compact abelian groups is properly larger than that of torus groups and properly smaller than that of connected locally connected compact abelian groups. The exponential function of strongly locally connected groups. We shall investigate when the exponential function expG : L(G) → G of a compact abelian group G is open onto its image in the present context. For a detailed exposition of the exponential function of compact abelian groups we refer to [102], notably Chapters 7 and 8. We need to know here that L(G) = Hom(R, G) is the topological vector space of all one parameter subgroups, i.e. continuous group morphisms X : R → G, where Hom(R, G) is given the topology of uniform convergence on compact sets. The exponential function is given by evaluation via exp X = X(1). By duality, L(G) may also be viewed  R) with the topology of pointwise convergence. We as the vector space Hom(G, note that L(G) is isomorphic to a product of copies of R (see [102, Theorem 7.66 (i) and Theorem 7.30 (ii)]). For any compact abelian group G, the exponential function expG : L(G) → G (see [102, Theorem 7.66]) is a morphism of abelian topological groups. Let Ga = im expG denote the arc component of the zero element 0. (See [102, Theorem 8.30.]) We note the exact sequence: expG  Z) → 0 0 → K(G) → L(G) −−−−−→ G → Ext(G,

(exp)

where K(G) = ker expG . The corestriction expG : L(G) → Ga of the exponential function to its image is a surjective morphism of topological groups. A surjective morphism between topological groups is open if and only if it is a quotient morphism. Thus expG is open iff the induced bijective morphism of topological groups L(G)/K(G) → Ga is an isomorphism. Proposition 4.10. Let G be a connected and strongly locally connected compact abelian group. Then expG : L(G) → Ga is open.  Then P ∈ P is Proof. Let P denote the set of pure finite rank subgroups of G.  such  a finitely generated free split subgroup of G. We select a subgroup SP ⊆ G def  = P ⊕ SP . Let NP = P ⊥ denote the annihilator of P in G and TP = S ⊥ that G P

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the annihilator of SP . By duality (n, t)  → n + t : NP × TP → G is an isomorphism of topological groups, that is, G = NP ⊕ TP algebraically and topologically.  are naturally isomorphic by the Annihilator MechaThe groups TP , G/NP and P nism (see [102, Theorem 7.64]) and thus are finite-dimensional torus groups. The morphism expNP × expTP : L(NP ) × L(TP ) → NP × TP is naturally equivalent to the exponential function of NP × TP . Let UP be the set of arcwise connected open zero-neighborhoods U of L(TP ) mapped homeomorphically onto an open zero neighborhood V of TP by expTP ; such neighborhoods U and V exist as TP is a Lie group. Then (expNP × expTP )(L(NP ) × U ) = (NP )a × V = (NP × V ) ∩ (NP × TP )a . It follows that expG (L(NP ) ⊕ U ) is an identity neighborhood of Ga in the topology induced from that of G. We claim that {L(NP ) ⊕ U : P ∈ P , U ∈ UP } is a basis for the open zero-neighborhoods of L(G), where L(NP ) is naturally considered as a cofinite-dimensional vector subspace of L(G). Once this claim is established, expG : L(G) → Ga is open and the proof of the proposition is complete.   = Since G P = colimP ∈P P by duality, we have G = limP ∈P G/NP . The functor L = Hom(R, −) preserves projective limits (see e.g. [102, Proposition 7.38 (iv)]; in fact L preserves all limits). Furthermore, L preserves quotients (see [102, Theorem 7.66 (iv)]). Hence L(G) = limP ∈P L(G)/L(NP ). Let rP : L(G) → L(G)/L(NP ) denote the quotient morphism, qP : L(G)/L(NP ) → L(TP ) the natural isomorphism, and pP = qP ◦ rP : L(G) = L(NP ) ⊕ L(TP ) → L(TP ) the projection. For any zero neighborhood W of L(G), by the Fundamental Theorem on Projective Limits 1.27 (i), there is a P ∈ P and a zero neighborhood U ∈ UP in L(TP ) such that pP−1 (U ) ⊆ W . Since pP−1 (U ) = L(NP ) ⊕ U , the claim is proved. We now get the result which shows that Theorem 4.1 cannot be improved. The product RR is a pro-Lie group (see 3.4 (ii) and 3.28), and it is of the simplest kind while not being locally compact. def

Corollary 4.11. (i) The uncountable product V = RR has a closed totally disconnected algebraically free subgroup K of countable rank such that the quotient V /K is incomplete and its completion G is a compact connected and strongly locally connected abelian group of continuum weight. (ii) Let q : V → V /K denote the quotient map and γV /K : V /K → G be the completion map. There is a morphism f : V → C into a compact hence complete group whose kernel is K and which has the property that the factorisation map f  : G → C determined uniquely by f = f   γV /K  q is not injective.  N be the compact abelian group of 4.4. The rank of ZN Proof. (i): Let G = Z agrees with the cardinal of ZN and that is the cardinal 2N of the continuum. Then N L(G) = Hom(R, G) = Hom(ZN , R) ∼ = RR . Thus we take for V the ad= R2 ∼ ditive group of L(G) and K = K(G) and know that ε : V /K → Ga is an isomorphism of topological groups by Proposition 4.10. The completion of Ga is G, and w(G) = card ZN = 2ℵ0 . The kernel K(G) is algebraically isomorphic to Hom(ZN , Z) (see [102, Theorem 7.66 (ii)]). But Hom(ZN , Z) ∼ = Z(N) (see [49, p. 61, Corollary 2.5]). Thus K is free of countable rank.

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(ii): In [102, Example A1.65] one finds the construction of a character χ : ZN → T of order 2 (i.e., 2 · χ = 0 in additive notation) which does not factor in the form ϕ

p

ZN −−→ R −−→ T,

p(r) = r + Z.

Then, as an element of G = Hom(ZN , T), the character χ is not in the image Ga of Hom(ZN , p) = expG : Hom(ZN , R) = L(G) → G. Set Z = {0, χ }. Then Z is a closed subgroup of G such that Ga ∩ Z = {0}. Put C = G/Z and let f  : G → C be the quotient morphism whose kernel is Z. The restriction F : Ga → C is injective. Let f : V → C be defined by f (X) = F (expG X). By (i) the corestriction q : V = L(G) → Ga of the exponential function is a quotient morphism, and F = f  q is the canonical epic-monic factorisation of F . Since G is isomorphic to the completion G of Ga and the inclusion Ga → G is the completion morphism γGa : Ga → G, the assertion follows. The significance of 4.11 (ii) is as follows: In the category of all complete topological abelian groups, the completion of a quotient plays the role of a quotient in the category as it has the expected universal properties; nevertheless, it will in general fail to have familiar properties as 4.11 (ii) illustrates. The possible incompleteness of quotients plays a somewhat disturbing role in the general theory of pro-Lie groups. The simplest nontrivial projective limits of finite-dimensional Lie groups are the products RX . The product RN is metrizable and complete, hence every quotient is complete. Corollary 4.11 shows that the “next largest product”, RR already has incomplete quotients, and it is remarkable that there are such quotients whose completion is compact. While we are utilizing the exponential function of compact abelian groups for the purpose of exhibiting incomplete quotients, we record additional information that is relevant in this context. Proposition 4.12. Assume that the corestriction expG : L(G) → Ga of the exponential function of a compact connected abelian group is a quotient morphism. Then (i) Ga has arbitrarily small open arcwise connected identity neighborhoods in the topology induced from that of G. (ii) G is locally connected.  is ℵ1 -free. (iii) G Proof. (i) A quotient morphism is open. But L(G) is a locally convex topological vector space and thus has arbitrarily small arcwise connected neighborhoods of zero which are mapped onto open identity neighborhoods of Ga by expG . (ii) Let W be an identity neighborhood of G. Then there is an identity neighborhood V such that V 2 ⊆ W . By (i), Ga has an open arcwise connected identity neighborhood U satisfying U ⊆ V . Then the closure U of U in G is contained in V ⊆ V V ⊆ W . There is an open set UG in G such that U = UG ∩ Ga , and since Ga is dense in G we have U = UG . Also, since U is arcwise connected, U is connected

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and is an identity neighborhood in G. Thus U is a connected identity neighborhood in G which is contained in W . Thus G is locally connected. (iii) This follows from Proposition 4.6 and (ii) above. Thus the exponential function of a compact connected abelian group can be open onto its image only if the group is locally connected. Proposition 4.13. If the arc component Ga of the zero element of a compact connected abelian group G is locally arcwise connected, then the corestriction expG : L(G) → Ga of the exponential function is open. Proof. On the group G there is a filter basis N (G) converging to 1 and consisting of closed compact subgroups N such that G/N is a finite-dimensional torus. By [102, Theorem 7.66 (iv)], there are arbitrarily small identity neighborhoods of G of the form N ⊕ V where V = expG U for an open n-cell neighborhood U in a finite-dimensional vector subspace F of L(G) such that (n, X)  → n + expG X : N × U → N ⊕ V is a homeomorphism. Now Na ⊕ V = (N ⊕ V )a is the arc component of 0 in Ga ∩ (N ⊕ V ). Since Ga is locally arcwise connected, arc components of open sets of Ga are open in Ga , and thus Na ⊕ V is open in Ga . Therefore Ga has arbitrarily small identity neighborhoods of the form Na ⊕ V . However, these are of the form Na ⊕ V = expN L(N ) ⊕ expG U = expG (L(N ) ⊕ U ) = expG (L(N ) ⊕ U ). Now we know that G = limN ∈N (G) G/N and just as in the proof of 3.1 we conclude that therefore L(G) = limN ∈N (G) L(G)/L(N ) and that L(G) has arbitrarily small neighborhoods of the form L(N )⊕U . This proves that expG is an open morphism. In fact one can show, that strong local connectedness is both necessary and sufficient for the exponential function of a compact abelian group to be surjective onto its image. Indeed the following theorem holds. Theorem 4.14 (Characterisation Theorem for Strong Local Connectivity of Compact Connected Abelian Groups). For a compact connected abelian group G and its zero arc-component Ga , the following conditions are equivalent: (i) (ii) (iii) (iv)

G is strongly locally connected. The exponential function expG : L(G) → G is open onto its image. Ga is locally arcwise connected.  is an S-group, that is, every finite rank pure subgroup of G  is free and is a G direct summand.

Proof. See [106, Theorem 5.1]. We have presented the implications (iv) ⇔ (i) ⇒ (ii) ⇒ (iii) in Propositions 4.7, 4.10, and 4.12. The implication (ii) ⇒ (i) is harder to prove.

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Remarks 4.15. (i) If a topological group G contains a closed central subgroup N such that N ∼ = RR , then G has an incomplete quotient group. (ii) Let {Aj : j ∈ J } be a family of abelian topological groups such that def  card J ≥ 2ℵ0 and each Aj contains a copy of R. Then G = j ∈J Aj has an incomplete quotient group. (iii) Let {Ej : j ∈ J } be a family of Hausdorff topological vector spaces such that def  card J ≥ 2ℵ0 . The E = j ∈J Ej has an incomplete quotient group. (v) If a topological group has a quotient group which has an incomplete quotient group, then it itself has an incomplete quotient group. (vi) Let {Gj : j ∈ J } be a family of topological groups such that card J ≥ 2ℵ0 and each Gj contains closed normal subgroups Mj ⊆ Nj such that Nj /Mj ∼ = R and [G, Nj ] ⊆ Mj (where [A, B] denotes the subgroup generated by the commutators def  aba −1 b−1 , a ∈ A, b ∈ B). Then G = j ∈J Gj has an incomplete quotient group. Exercise E4.1. Verify the assertions in Remarks 4.15 [Hint. (i) From the example in Corollary 4.11 above we find a closed, totally disconnected subgroup D of N such that N/D is incomplete. Since N is central, D is normal in G and then G/D contains N/D, an incomplete group, as a closed subgroup. The remaining assertions are straightforward.] The example in Corollary 4.11 (i) and the examples in Exercise E4.1 show very clearly that Theorem 4.1 (i) cannot be improved to read that quotients of pro-Lie groups are pro-Lie groups; most of our examples are in fact connected. In [34] it is shown that the class of abelian topological groups which are isomorphic to a product C × RX × ZY where C is a compact group and X and Y are countable sets is closed under the passage to closed subgroups and to quotient groups. This points out the fact that quotient groups of RX are complete if X is countable. In this sense the example of RR is minimal in the class of weakly complete topological vector spaces, if one momentarily accepts the Continuum Hypothesis. In [116] it is proved that a closed connected subgroup of RX is a closed vector subspace; consequently it is a direct summand algebraically and topologically (see [102, Theorem 7.30 (iv)]; in particular the quotient is a weakly complete topological vector space. More generally, we shall see shortly sufficient conditions on a pro-Lie group G and a closed normal subgroup N for G/N to be a pro-Lie group. (See Theorem 4.28 below).

The One Parameter Subgroup Lifting Theorem The lifting of one parameter subgroups deals with the following situation: Assume that f : G → H is a quotient morphism and Y ∈ L(H ); under which circumstances is there an X ∈ L(G) such that L(f )(X) = Y ?

The One Parameter Subgroup Lifting Theorem

Lemma 4.16. Assume that

183

ϕ

P ⏐ −−→ R ⏐ ⏐ ⏐ π Y G −−→ H

(1)

f

def

is a pullback of topological groups. Set K = ker ϕ. Then the following conditions are equivalent: (i) (ii) (ii ) (iii) (iv)

K is a semidirect factor and ϕ is surjective. ϕ is a retraction. ϕ|P0 : P0 → R is a retraction, where P0 is the identity component of P . There is an X ∈ L(G) such that L(f )(X) = Y . There is a subgroup R of P such that KR = P and K ∩ R = {1}, and further that ϕ|R : R → R is open.

These conditions imply: (v) There is a closed subgroup R of P such that KR = P and K ∩ R = {1}. Proof. (i) ⇔ (ii): The equivalence of (i) and (ii) is a standard exercise in topological group theory (see e.g. E1.5). (ii) ⇒ (ii ): If a morphism σ : R → P satisfies ϕ  σ = idR , then σ (R) ⊆ P0 as R is connected, and thus its corestriction σ : R → P0 satisfies ϕ  σ = idR . (ii ) ⇒ (ii): Conversely, if σ : R → P0 satisfies ϕ  σ = idR , then its coextension σ : R → P satisfies ϕ  σ = idR . (ii) ⇒ (iii): If X : R → P is a one parameter subgroup satisfying ϕ  X  = idR def

then X = π  X : R → G is a one parameter subgroup of G such that L(f )(X) = f  X = f  π  X = Y  ϕ  X  = Y  idR = Y . (iii) ⇒ (ii): Assume Y = L(f )(X) = f X. Then for all r ∈ R we have f (X(r)) = Y (r). Now the explicit form of the pullback is P = {(g, r) ∈ G × R | f (g) = Y (r)} and ϕ(g, r) = r (see e.g. Theorem 1.5). Hence (X(r), r) ∈ P for all r ∈ R and if we set X (r) = (X(r), r), then X  : R → P is a morphism satisfying ϕ(X  (r)) = r for all r. (i) ⇒ (iv) ⇒ (v) is trivial. (iv) ⇒ (ii): The morphism ϕ|R : R → R is continuous and open. Thus ϕ(R) is an open subgroup of R and therefore equals R. So ϕ|R is surjective, and since K ∩ R = {0} it is also injective. Hence it is an isomorphism of topological groups and thus is invertible; the coextension σ : R → P of (ϕ|R)−1 : R → R satisfies ϕ  σ = idR . Lemma 4.17. If f in the pullback (1) is surjective, then ϕ is surjective. If f is open, then ϕ is open. If f is a quotient morphism so is ϕ. Proof. Surjectivity: if r ∈ R then, since f is surjective, there is a g ∈ G such that f (g) = Y (r).

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Openness: The filter of identity neighborhoods of P has a basis of open sets of the form W = P ∩ (U × I ), where U is an open identity neighborhood of G and I an open interval around 0 in R. Then ϕ(W ) = {r ∈ I | (∃g ∈ U ) f (g) = Y (r)} = I ∩ Y −1 (f (U )). Since f is an open map, f (U ) is an open subset of H and thus by the continuity of Y , the set ϕ(W ) is open. Quotients: This assertion follows from the combination of the preceding two. Lemma 4.18. In the pullback (1), assume that G is a pro-Lie group. Then P is a pro-Lie group. Proof. By Theorem 1.5 (a) we have P = {(g, r) ∈ G × R : f (g) = Y (r)} in the category TopGr of topological groups. Since G and R are pro-Lie groups, the product G× R is a pro-Lie group since by Theorem 3.3 the category proLieGr is closed under the formation of products. Since the groups G, R, and H are all Hausdorff, the subgroup P is closed. Hence by Theorem 3.35 it follows that P is a pro-Lie group. We now are ready for a proof of the lifting of one parameter subgroups. This is not easy because in the absence of countability assumptions, this requires the Axiom of Choice, and the absence of compactness in the present situation forces us to rely on completeness and the convergence of Cauchy filters. The proof will require from the reader a certain facility handling “multivalued morphisms” as a special type of binary relations; but most of what is required will be self-explanatory in the proof. Lemma 4.19 (The One Parameter Subgroup Lifting Lemma). Let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Then every one parameter subgroup Y : R → H lifts to one of G, that is, there is a one parameter subgroup σ of G such that Y = f  σ . Proof. By Lemmas 4.16, 4.17, and 4.18, we may assume that H = R and we have to show that f is a retraction. Let K = ker f . Since {N ∗ = f (N) | N ∈ N (G)} converges to 0 in R, and since there are no subgroups in ]−1, 1[ other than {0} there is an N ∈ N (G) such that f (N) = N ∗ = {0}, and thus N ⊆ K. Then for all N ∈ N (G), N ⊇ N , the morphism f induces a quotient morphism fN : G/N → R, fN (gN ) = f (g), and fN (gN ) = 0 iff f (g) = 0 iff g ∈ K, that is, ker fN = K/N . If we let pN : K → K/N and qN : G → G/N denote the quotient morphisms, then we have a commutative diagram 1 →

N ⏐ ⏐ incl 1 → K ⏐ ⏐ pN  1 → K/N

idN

−−−→

N ⏐ ⏐ incl f incl −−−→ G −−→ R ⏐ ⏐ → 0 ⏐q ⏐id N  R fN incl −−−→ G/N −−→ R → 0

(2)

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185

with exact rows and columns. Due to the fact that the exponential map of a Lie group is a local homeomorphism at 0, an open morphism ψ : L1 → L2 between Lie groups induces an open and therefore surjective morphism L(ψ) between their Lie algebras: L(ψ)

L(L ⏐ 1 ) −−−→ L(L ⏐ 2) ⏐expL expL1 ⏐   2 L1 −−−→ L2 . ψ

Hence there is a morphism σN : R → G/N such that fN  σN = idR . The binary −1  σN : R → G satisfies the following conditions: relation  = qN def

(i) (0) = N and every (r) ⊆ G is a coset modulo N. (ii) The graph of  is a closed subgroup of R × G. (iii) We have a commutative diagram of binary relations of which all but  are functions: idR R −−−→ R ⏐ ⏐ ⏐ ⏐id   R f G −−−→ R (3) ⏐ ⏐ ⏐ ⏐id qN   R G/N −−→ R. fN

A binary relation  : R → G satisfying (i), (ii) and (iii) will be called a multivalued morphism associated with N . The set S of all multivalued morphisms  : R → G associated with some N ∈ N (G) is partially ordered under containment ⊆. By Zorn’s Lemma we find a maximal filter F ⊆ S. It is our goal to show def

that M = {(0) |  ∈ F } is cofinal in N (G). Assuming that this is proved, we note that for each r ∈ R and  ∈ F the subset (r) is a coset N x = xN with N = (0) ∈ N (G), and thus (r)(r)−1 = N x(N x)−1 = N; since M converges to 1, we conclude that {(r) |  ∈ F } is a Cauchy filter basis. Since G is complete, it converges to an element σ (r) ∈ G, giving us a function σ : R → G. As each (r), being a coset modulo N = (0) ∈ M, is closed, we have σ (r) ∈ (r) for all  ∈ F . Consequently, since (3) is commutative for each  ∈ F for N = (0) we have the following commutative diagram for all N ∈ M: R ⏐ ⏐ σ G ⏐ ⏐ qN  G/N

idR

−−−→ f

−−−→ −−→ fN

R ⏐ ⏐id  R R ⏐ ⏐id  R R.

(4)

The upper rectangle shows that f  σ = idR , and the fact that each qN   : R → G is a morphism of topological groups shows that qN  σ : R → G/N is continuous.

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4 Quotients of Pro-Lie Groups

Theorem 2.1 (i) shows that G has arbitrarily small open identity neighborhoods U satisfying U N = U for some N ∈ M. Then if V is a zero neighborhood of R such that −1 (U/N ) = U . This shows that σ is continuous. qN (σ (V )) ⊆ U/N , then σ (V ) ∈ qN Hence σ is the required coretraction for f . Thus the remainder of the proof will show that M is cofinal in N (G). Suppose that this is not the case. Then there exists an N ∈ N (G), N ⊇ N such that M ⊆ N for all M ∈ M ⊆ N (G). Let us temporarily fix M; then M ∩ N ∈ N (G), and thus def

G† = G/(M ∩ N) is a Lie group: G

MNG GG ww GG w GG ww w w GG w w N M GG ww GG w GG ww GG ww G ww M ∩N

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

G† .

This shows that for fixed M everything takes place in the Lie group G† in which def

def

M † = M/(M ∩ N) and N † = N/(M ∩ N) are closed normal Lie subgroups with M † ∩ N † = {1}. Thus μ : M † → G/N, μ(m(M ∩ N )) = mN is a morphism of Lie groups mapping M † bijectively onto MN/N and inducing an isomorphism of Lie algebras L(M † ) → L(MN/N ) ⊆ L(G/N ). Now M † is a closed normal subgroup of the Lie group G† and thus M † /M0† is a discrete normal subgroup of the Lie group G† /M0† . We let M ‡ be the open subgroup of M † containing (M † )0 and being such that M ‡ /M0† = (M † /M0† ) ∩ (G† )0 /M0† . Hence M ‡ /M0† is a discrete normal subgroup of a connected Lie group. Consequently it is finitely generated and thus countable. Therefore M ‡ has countably many components and so μ(M0† ) is an analytic subgroup Man ⊆ G/N agreeing with (MN/N )0 and having Lie algebra L(Man ) = L(MN/N ) = L((MN/N )0 ). (See [102, Theorem 5.52ff.]) Accordingly, {L(MN/N ) | M ∈ M} is a filter basis of finite-dimensional vector subspaces of L(G/N ). Hence there is a smallest element m = L(M# N/N ) in it such that for all M ≤ M# in M we have L(MN/N) = m. Let us abbreviate q(M# ∩N ) : G → G/(M# ∩ N ) by q # : G → G# , further f(M# ∩N) : G# → R by f # , and M# /(M# ∩ N ) by M # . Since L(μ) : L(M # ) → L(M # N/N ) = m is an isomorphism we have ,, q#

M(M # ∩ N) M# ∩ N

- = q # (M0# ) for M ⊆ M# in M. 0

(5)

The One Parameter Subgroup Lifting Theorem

187

There is a  # ∈ F such that M # =  # (0). Then for all  ∈ F contained in  # , the subgroup q # ((0)) of the Lie group G# is contained in q # ( # (0)), satisfies q # ((0)) = M0# , and (f #  q #  )(R) = R. Thus for all r ∈ R we have q # ( # (r)) = q # ((r)) since the right side is contained in the left and both are cosets modulo M # . In the Lie group G# we have the configuration G

M# NG GG v v GG vv v GG vv GG v v M# H N HH ww w HH ww HH ww HH ww M# ∩ N

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

G# .

−1 # Let σ # = σM# ∩N : G# → R be defined by σ = qM #   # . Then  # = qM #  σ and we have a commutative diagram of binary relations def

idR

−−−→

R ⏐ ⏐ #  G ⏐ qM # ⏐ 

f

−−−→

G/M #

−−−→ fM #

R ⏐ ⏐id  R R ⏐ ⏐id  R

(3# )

R.

def

We conclude that S = q # ( # (R)) =  # (R)/(M # ∩ N ) is a closed subgroup of G# whose Lie algebra L(S) cannot be contained in K # = K/(M # ∩ N ) = ker f # . From dim G# /K # = 1 we conclude L(G# ) = L(K # ) + L(S)

and L(S) = L(S) ∩ L(K # ) + R · X

for a suitable element X ∈ L(G# ) satisfying f # (expG# X) = 1. Setting τ : R → S, τ (r) = expG# r · X we obtain a coretraction for f # : G# → R. The binary relation def

 = (q # )−1  τ : R → G is a member of S. Moreover, for all  ∈ F we have q # ()(r) ⊇ τ (r) for all r ∈ R. Hence  ∩  is a member of S. Now the maximality of F shows that  ∈ F . But this implies that M # ∩ N = (0) ∈ M and that is a contradiction to our supposition allowing us a choice of an N such that M ∩ N = M for all M ∈ M. This contradiction finally completes the proof. There are some subtleties here which we should point out. Following Theorem 5.52 in [102] we have seen that the additive group h of a Banach space mapped surjectively onto an abelian Lie group G (which itself is quotient of a Banach space modulo a

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4 Quotients of Pro-Lie Groups

discrete subgroup) such that G has a one parameter subgroup which does not lift to h. This cannot happen if the domain is separable, but it does happen in the category of not necessarily finite-dimensional Lie groups. While being surjective, the morphism in question is not open and the Open Mapping Theorem fails. On the other hand, let G be a free abelian topological group in the sense of Graev (see [71]) generated by the space |R| underlying the topological abelian (additive) group R of real numbers. (For details on free topological groups see for instance [150]; see also [130].) Since the generating space |R| is arcwise connected and contains the identity, G itself is arcwise connected. Further, G is a kω group; hence it is a complete topological group ([150, pp. 379, 380]). Accordingly, G is not metrizable, because if it were it would be a σ -compact Baire space group and thus would be locally compact – which it is not. (See also [151].) Now we use the universal property of G to extend the identity map |R| → R to a morphism of topological groups q : G → R. If we consider |R| as embedded into G (which we may) then the restriction ρ of q to |R| is the identity map. Set K = ker q and define m : |R| × K → G by m(r, k) = rk; then m has an inverse given by m−1 (g) = (ρ −1 q(g), (ρ −1 q(g))−1 g). So m is a homeomorphism such that q m = ρ pr 1 . Thus, topologically, q is equivalent to a projection of a product onto one of the factors and is therefore a quotient map. However, algebraically G is a free group generated by the elements of |R| and thus does not contain any divisible elements, and so L(G) = {0}. Hence there is no morphism X : R → G such that q  X = idR . The quotient morphism q does not lift and L(q) : L(G) → L(R) = R is not surjective. We have seen that the functor L preserves all limits and thus, in particular, all kernels (since ker f for a morphism f of topological groups is nothing but the equalizer of f and the constant morphism). We shall say that a functor F : A → B between categories of topological groups is strictly exact if it preserves kernels and quotients. A morphism e : N → G of topological groups is said to be a strict morphism if its corestriction N → im e is an open morphism, that is, the bijective morphism N/ ker e → im e induced by it is an isomorphism of topological groups. Also recall from Theorem 4.1, that a quotient H of a pro-Lie group G is a proto-Lie group and that it has a completion H which is a pro-Lie group. Now, as a corollary of the One Parameter Subgroup Lifting Lemma we obtain the following theorem. The Strict Exactness Theorem for L Theorem 4.20. Let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Then the following conclusions hold. (i0 ) H is a proto-Lie group and the canonical embedding γH : H → HN (H ) =

lim

N ∈N (H )

H /N

according to 1.29 and 1.30 induces an isomorphism L(γH ) : L(H ) → L(HN (H ) )

The One Parameter Subgroup Lifting Theorem

189

of pro-Lie algebras. In particular L(H ) is a pro-Lie algebra. (i) The pro-Lie algebra morphism L(f ) : L(G) → L(H ) is a quotient morphism. In particular, the functor L : proLieGr → proLieAlg is strictly exact. (ii) If f

e

N −−→ G −−→ H is an exact sequence of morphisms of pro-Lie groups with a strict morphism e, then L(f )

L(e)

L(N ) −−−−→ L(G) −−−−→ L(H ) is an exact sequence of pro-Lie algebras. Proof. (i0 ) From the Quotient Theorem of Pro-Lie Groups 4.1 we recall that HN (H ) = limN∈N (H ) H /N is the completion H of the proto-Lie group H . Let νN : H → H /N denote the limit morphism and qN : H → H /N be the quotient morphism for N ∈ N (H ). From Theorem 1.29 we recall that νN  γH = qN for all N ∈ N (H ). The functor L preserves limits, and so for each N ∈ N (H ) we have a commutative diagram L(γH )

L(H −−−−→ L(H ⏐ ) ⏐ ) ⏐ ⏐ qN  L(ν) L(H /N ) −−−−→ L(H /N ),

(∗)

idL(H /N)

where the L(νN ) are the limit maps. The composition f

qN

qN  f : G −−→ H −−→ H /N is a quotient morphism of pro-Lie groups. The One Parameter Subgroup Lifting Lemma 4.19 tells us that it induces a surjective morphism L(qN  f ) : L(G) → L(H /N). Surjective morphisms of weakly complete topological vector spaces are quotient maps (see Appendix 2, Theorem A2.12 (b)). Since L is a functor, we have L(qN  f ) = L(qN )  L(f ); since L(qN )  L(f ) is surjective, L(qN ) : L(H ) → L(H /N) is surjective. Thus in the diagram (∗), the maps L(νN )  L(γH ) are all quotient maps, and this implies by Theorem 1.20 (i) that L(γH ) has a dense image. Also, L(γH ) has a zero kernel, since L preserves kernels and γH is an embedding. From Theorem 1.30 it follows that L(γH ) is an embedding. Therefore the composition L(f )

L(γH )

L(γH )  L(f ) : L(G) −−−−→ L(H ) −−−−→ L(H ) is a morphism of pro-Lie algebras with a dense image. From Appendix 2, Theorem A2.12 (a) we know that every morphism of weakly complete topological vector spaces with dense image splits and thus is surjective. Since the underlying vector spaces of L(G) and L(H ) are weakly complete by 3.8 and 3.12, the morphism L(γH )L(f ) is

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4 Quotients of Pro-Lie Groups

surjective. Therefore, the embedding L(γH ) is surjective and thus is an isomorphism. This is what we had to show. (i) We observed that L preserves kernels because kernels are limits. In (i0 ) we saw that L(H ) is a pro-Lie algebra, and the One Parameter Subgroup Lifting Lemma 4.19 shows that L(f ) is surjective, hence a quotient since surjective morphisms of weakly complete topological vector spaces are quotients by Theorem A2.12(b). Therefore, L preserves quotients, and so L is strictly exact. This completes the proof of (i). (ii) Let e

f

N −−→ G −−→ H be morphisms of pro-Lie groups such that im e = ker f and e is open onto its image. Set K = ker f , and decompose e as κ  ε where ε : N → K is the corestriction of e to its image and κ : K → G is the inclusion morphism. Then L(e) = L(κ)  L(ε) and L(κ) : L(K) → L(G) is the kernel of L(f ) since L preserves kernels, and L(ε) : L(N) → L(K) is surjective since L preserves quotients. Thus im L(e) = L(K) = ker L(f ).

It is remarkable that we do not require that H is a pro-Lie group; yet we obtain that L(H ) is a pro-Lie algebra. Such a situation can indeed arise as is shown by 4.9 and 4.10 where we have a quotient morphism from RR onto an incomplete group with a totally disconnected kernel inducing on the Lie algebra level the identity map of RR . The identity morphism Rd → R from the discrete group of real numbers to the group of real numbers with its natural topology is a bijective morphism of Lie groups inducing the zero morphism L(Rd ) = {0} → R ∼ = L(R). Thus one cannot drop the assumption in the theorem that e be a strict morphism. Sometimes one has an Open Mapping Theorem available which gives us this assumption for free – for instance if ker f is a Baire space and N is locally compact and σ -compact. In fact, we shall prove an Open Mapping Theorem for almost connected pro-Lie groups later in the book. In that context we shall show elementarily the existence of a connected proto-Lie group 2 ℵ0 H whose Lie algebra is algebraically isomorphic to Rℵ0 , while L(H ) ∼ = R2 . Thus there is no chance of proving (i0 ) of Theorem 4.20 without the assumption that H is in fact a quotient of a pro-Lie group. Corollary 4.21. (i) If N is a closed normal subgroup of a pro-Lie group G, then the quotient morphism q : G → G/N induces a map L(q) : L(G) → L(G/N ) which is a quotient morphism with kernel L(N ). Accordingly there is a natural isomorphism X + L(N) → L(f )(X) : L(G)/L(N ) → L(G/N ). (ii) Let G be a pro-Lie group. Then {L(N ) | N ∈ N (G)} converges to zero and is cofinal in the filter  (L(G)) of all ideals i such that L(G)/i is finite-dimensional. Furthermore, L(G) is the projective limit limN ∈N (G) L(G)/L(N ) of a projective system of bonding morphisms and limit maps all of which are quotient morphisms, and

The One Parameter Subgroup Lifting Theorem

191

there is a commutative diagram L(γG ) G ∼ L(G) −−−−→ L(GN (G) ) = L(limN ∈N ⏐ ⏐ (G) N ) = limN ∈N (G) ⏐ ⏐ expG  L(limN∈N (G) expG/N ) G −−−−→ GN (G) = limN ∈N (G) G/N.

L(G) L(N )

γG

Proof. Assertion (i) is an immediate consequence of the Strict Exactness Theorem 4.20. (ii) We know that L preserves limits. Thus L(γG ) : L(G) → L(GN (G) ) is an isomorphism. By (i) above, L(G/N ) ∼ = L(G)/L(N ) and thus L(G) ∼ =

lim

N ∈N (G)

L(G)/L(N ).

Thus by 1.27 (ii), the filter basis {L(N ) | N ∈ N (G)} of the kernels of the limit maps converges to 0 and the projective system of the L(G)/L(N ) has the natural quotient morphisms as bonding maps; by 1.27 (ii) it follows that the limit maps are quotient morphisms as well. It then follows that this filter basis is cofinal in  (L(G)). (Compare 1.40.) Recall from Definition 2.21 that E(G) denotes the subgroup expG L(G) algebraically generated by expG L(G). Corollary 4.22. (i) For a pro-Lie group G, the subgroup E(G) = expG L(G) is dense in G0 , i.e. E(G) = expG L(G) equals G0 . In particular, a connected nonsingleton pro-Lie group has nontrivial one parameter subgroups. Moreover, if h is a closed proper subalgebra of L(G), then expG h = E(G). (ii) A morphism f : G → H of pro-Lie groups induces a surjective (hence quotient) morphism L(f ) : L(G) → L(H ) in each of the following cases: (a) f is open. (b) f is surjective and G is almost connected. (iii) Assume that a morphism f : G → H of pro-Lie groups induces a surjective morphism L(f ) : L(G) → L(H ); by (ii) above this is the case, in particular, if f is open or if f is surjective and G is almost connected. Then the induced morphism E(f ) : E(G) → E(H ) is surjective, that is E(H ) = f (E(G)). As a consequence, H0 = f (G0 ). If H is a Lie group, then H0 ⊆ f (G). If H is in addition connected, then f is surjective. (iv) A morphism f : G → H of pro-Lie groups into a finite-dimensional Lie group inducing a surjective morphism L(f ) is locally trivial, that is, there is an open identity neighborhood V of H and a continuous map s : V → G such that f (s(h)) = h for all h ∈ V . In particular, for N = ker f , ϕ : f −1 (V ) → N × V ,

ϕ(g) = (gs(f (g))−1 , f (g))

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4 Quotients of Pro-Lie Groups

is a homeomorphism with inverse ψ given by ψ(n, h) = ns(h). (vi) Let G be a pro-Lie group and assume that for all N from a basis of N (G) the quotient G/N is connected. Then G is connected. Proof. (i) First we show that a nonsingleton connected pro-Lie group has nontrivial one parameter subgroups. Let G be a nonsingleton connected pro-Lie group. There is a g ∈ G, g = 1. Since lim N (G) = 1 there is an N ∈ N (G) such that g ∈ / N. Then G/N is a nonsingleton connected Lie group. Thus L(G/N ) = {0}. Then L(G) = {0} by 4.21 (i). Next we let G be an arbitrary pro-Lie group. The closed subgroup E(G) = expG L(G) is fully characteristic, hence normal. By the One Parameter Subgroup Lifting Lemma 4.19, every one parameter subgroup of G/E(G) lifts to one in G which is contained in E(G) by the definition of E(G). Hence L(G/E(G)) = {0}. Thus G/E(G) is totally disconnected by what we just proved, and thus G0 ⊆ E(G) ⊆ G0 . def

For a proof of the last assertion of (i), let us write g = L(G) and let h be closed subalgebra of g such that h = g. Then there is a 0-neighborhood U of g such that h + U = g. By Corollary 4.21 (ii) we find a closed normal subgroup N of G such that def

n = L(N) ⊆ U . Thus h + n = g. Since dim g/n < ∞ and all vector subspaces of a finite-dimensional real vector space are closed we know that h+n is a closed subalgebra of g. If we can prove that expG (h + n) = E(G) then certainly expG h = E(G). Thus by replacing h by h + n and renaming the subalgebra we shall assume without loss of generality that n ⊆ h. By Corollary 4.21 (i) we have a commutative diagram {0} →

n ⏐ ⏐ id {0} → ⏐ n expN ⏐  {1} → N

−−−→

incl

quot

j

q

h −−−→ h/n → {0} ⏐ ⏐ ⏐ ⏐ incl  L(j ) L(q) −−−→ ⏐ g −−−→ L(G/N ⏐ ) → {0} ⏐exp expG ⏐   G/N −−−→ G −−−→ G/N → {1},

and we may naturally identify L(G/N ) with g/n; under this identification, expG/N (h/n) gets identified with q(expG (h)) = (expG (h))N/N. Now expG hN/N = (expG (h))N/N = expG/N (h/n). By the same token, E(G)N/N = expG gN/N = expG/N (g/n) = (G/N )0 since G/N is a Lie group. Since h/n = g/n the analytic subgroup expG/N (h/n) of (G/N )0 is proper. Thus expG hN = E(G)N and since expG h ⊆ E(G), this must mean expG h = E(G). (ii) (a) Let f : G → H be an open morphism of topological groups. Then f (G) is an open, hence closed, subgroup of H (see e.g. [102, Proposition A4.25 (ii)]) and thus a pro-Lie group by Corollary 4.8. The open and surjective corestriction G → f (G) (inducing an isomorphism of topological groups G/ ker f → f (G)) is a quotient morphism between pro-Lie groups and thus induces a quotient morphism

The One Parameter Subgroup Lifting Theorem

193

L(f ) : L(G) → L(f (G)) by the Strict Exactness Theorem 4.20. Since f (G) is open in H , the inclusion j : f (G) → H induces an isomorphism L(j ) : L(f (G)) → L(H ) of topological Lie algebras. Thus L(f ) : L(G) → L(H ) is a quotient morphism. (b) By Theorem A2.12 (b) of Appendix 2, a surjective morphism of weakly complete topological vector spaces is a quotient morphism; so we have to show that L(f ) is surjective, that is, that every one parameter subgroup of H can be lifted to one of G. By Lemmas 4.17 and 4.18, it is no restriction to assume that H = R = L(H ). Assume that X ∈ L(G) is such that L(f )(X) = 0. Then L(f )(R · X) = R = L(H ). It therefore suffices to show the existence of an X ∈ L(G) which is not annihilated by L(f ). However, suppose that L(f )(L(G)) = {0}. Then f (E(G)) = {1H }. By the continuity of f it follows that f (E(G)) = {1H }. By (i) above, E(G) = G0 . Since G is almost connected, G/G0 is compact. Since f (G0 ) = {1H }, there is a morphism F : G/G0 → H = R, F (gG0 ) = f (g). Then f (G) = F (G/G0 ) is a compact subgroup of H = R and therefore is {1H }, and this contradicts the surjectivity of f . (iii) By assumption, L(H ) = L(f )(L(G)), and thus expH L(H ) = expH L(f )(L(G)) = f (expG L(G)). Consequently E(H ) = expH L(H ) = f (expG L(G)) = f expG L(G) = f (E(G)).

Thus H0 = E(H ) = f (E(G)) ⊆ f (G0 ) ⊆ H0 = H0 , and this shows f (G0 ) = H0 . Now assume that H is a connected Lie group. Then expH L(H ) is a neighborhood of 1 and thus E(H ) is an open and hence also closed subgroup of H and thus H0 ⊆ E(H ). Since E(H ) is connected, E(H ) = H0 . Now H = E(H ) = f (E(H )) ⊆ f (G), and thus H0 ⊆ f (G). If H = H0 , then f is surjective. (iv) The morphism L(f ) : L(G) → L(H ) is a surjective morphism of weakly complete topological vector spaces and therefore splits by Theorem A2.12 (a), that is, there is a morphism of weakly complete topological vector spaces σ : L(H ) → L(G) such that L(f )  σ = idL(H ) . Since H is a finite-dimensional Lie group, there is an open zero neighborhood U of L(H ) and an open identity neighborhood V of H such that expH |U : U → V is a homeomorphism. Define s : V → G by s = expG  σ  (expH |U )−1 . Let h ∈ V . Then f (s(v)) = (f  expG σ )((expH |U )−1 (h)) = (expH L(f )  σ )((expH |U )−1 (h)) = expH ((expH |U )−1 (h)) = h, as asserted. The remainder is an easy exercise by verifying that ψ  ϕ and ϕ  ψ are the respective identity functions. (v) Let qN : G → G/N denote the quotient morphism. By (ii) we have G/N = E(G/N) = qN (E(G)). Thus G = E(G)N for all N ∈ N (G) and so G0 = E(G) = E(G) = G. There is a connected commutative infinite-dimensional Banach Lie group G whose exponential function maps a proper closed hyperplane h in its Lie algebra g surjectively onto the group (see [102, Chapter 5, paragraph following Theorem 5.52]); this illustrates that 4.22 (i) is not exactly a trivial matter.

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4 Quotients of Pro-Lie Groups

Let H = Tp be the p-adic solenoid (see Example 1.20 (ii)), let G = L(H ) = R, and let f : G → H be the exponential function expH : L(H ) → H . Then L(f ) : L(G) → L(H ),

L(G) = R

is an isomorphism, but f is far from surjective. In fact, im f = Ha is the identity  arc component of H and as abelian groups, π0 (H ) = H /Ha and Ext p1∞ · Z, Z are isomorphic groups of continuum cardinality. (See Chapter 8 in [102].) The relation H0 = f (G0 ) for a quotient morphism f such as in 4.22 (iii) cannot be improved as the example of the following quotient morphism of locally compact abelian groups shows: Let G = R × Zp for the group of p-adic integers Zp , let H = G/{(n, −n) | n ∈ Z} ∼ = Tp and let f be the corresponding quotient morphism. Note that H is compact and connected. (See [102, Exercise E1.11 following Definition 1.30]. We consider Z as a subgroup of Tp as well.) Then G0 = R × {0}, and f (G0 ) = Ha = H = H0 . However, we want to point out, that the proof of 4.22 (iii) does show more than what was asserted. Observe that for any topological group G we can write E(G) = expG L(G) and E(G) = E(G). If G is a proto-Lie group, then its completion is a pro-Lie group G∗ = GN (G) and we have g ⊆ g∗ (writing G ⊆ G∗ ). If equality g = g∗ holds then expG g = expG∗ g∗  is dense in G∗ and is contained in G, and so is dense in G. Therefore: Remark 4.22 (iii)∗ . Assume that a morphism f : G → H of topological groups induces a surjective function L(f ) : L(G) → L(H ). Then the induced morphism E(f ) : E(G) → E(H ) is surjective, that is E(H ) = f (E(G)). If G0 = E(G) and H0 = E(H ), then H0 = f (G0 ). In particular, if G is a pro-Lie group and H is a proto-Lie group whose pro-Lie algebra coincides with the pro-Lie algebra of its completion, then H0 = f (G0 ). The vanishing of the Lie algebra means the absence of nontrivial one parameter subgroups. This property should be discussed here. Just for the record we repeat at this point what we showed in Proposition 3.30. In the light of the Pro-Lie Group Theorem 3.34 that result can be slightly reformulated. Corollary 4.23 = 3.30. For a pro-Lie group G the following statements are equivalent: (a) (b) (c) (d)

L(G) = {0}. G is totally disconnected. G is zero-dimensional. G is prodiscrete.



This generalizes the classical result that a totally disconnected finite-dimensional Lie group is discrete.

Sufficient Conditions for Quotients to be Complete

195

Sufficient Conditions for Quotients to be Complete In the structure theory of locally compact groups, the concept of “almost connected groups” is quite helpful, but it applies perfectly well to arbitrary topological groups.We introduced it in our preface and in Definition 1(ii) in the overview chapter, and we reiterated it many times since. For easy reference we recall: Definition 4.24 = 1(ii). A topological group G is called almost connected if G/G0 is compact. Clearly, all compact groups and all connected topological groups are almost connected. If G is almost connected and N is a closed normal subgroup of G, then G0 N /N ⊆ (G/N)0 , whence (G/N )/(G/N )0 is a quotient group of (G/N )/(G0 N/N) ∼ = G/G0 N ∼ = (G/G0 )/(G0 N /G0 ), which is in turn a quotient group of the compact group G/G0 . Thus quotients of almost connected groups are almost connected. For the next steps we observe, that the filter basis N (G) that we use to define pro-Lie groups may be “thinned out” to a cofinal filter basis M(G) if G is almost connected. We say that M and N in N (G) are close if M ∩ N is open in M and N . This relation is reflexive and symmetric. We claim that it is transitive: Indeed, if N1 ∩ N2 is open in Nj , j = 1, 2 and N2 ∩ N3 is open in Nj , j = 2, 3 then N1 ∩ N2 ∩ N3 is open in N2 and then in N1 ∩ N2 and N2 ∩ N3 ; therefore the triple intersection is also open in N1 and N3 whence N1 ∩ N3 is open in N1 and N3 . Therefore closeness is an equivalence relation, and we write M ∼ N if M and N are close. Now we define def

M(G) = {N ∈ N (G) | (M ⊆ N and M ∼ N ) ⇒ |N/M| < ∞}.

(6)

Lemma 4.25. (i) If G is an almost connected pro-Lie group, then M(G) as defined in (6) is cofinal in N (G). In particular, M(G) is a filter basis. (ii) If M(G) is cofinal, then γ : G → GM(G) = limM∈M(G) G/M is an isomorphism. Proof. (i) Let N ∈ N (G) and M ∼ N in N (G) such that N ⊇ M. The quotient def

G/M → G/N has the discrete kernel D = N/M and so induces a covering map of the identity components (G/M)0 → (G/N )0 whose kernel is K = (N/M) ∩ (G/M)0 = (N ∩G0 M)/M. Since G is almost connected, the Lie group G/M is almost connected, and thus (G/M)/(G/M)0 ∼ = G/G0 M is finite; consequently D/K is finite. The group K is central (see e.g. [102, A4.27]), in particular abelian, and it is finitely generated. (See [102, Proposition 5.75], applied to a maximal connected abelian subgroup H : all of these contain the center (see [86, p. 189], Theorem 1.2, or [85, p. 280, Satz III.7.11]).)

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Let tor K denote the finite torsion subgroup of K. D = N/M    K    tor K   {1}

 D/K finite 

K/ tor K ∼ = Zn(M)

 finite

The torsion-free rank n(M) of K/ tor K is a natural number. We claim that n(N ) is the maximal rank of free abelian subgroups of D: Indeed let F be a free abelian subgroup of D. Then F K/K ∼ = F /(F ∩K) is finite, hence F ∩K is a free abelian group such that rank(F ∩ K) = rank F (see e.g. [102, Theorem A1.10]). But rank(F ∩ K) ≤ n(M), and this establishes the claim. Let GN denote the simply connected covering group of (G/N )0 . Then for each M ∼ N, M ⊆ N there is a covering morphism GN → (G/M)0 . The set {n(M) | M ⊆ N, M ∼ N} is bounded by the torsion-free rank of π1 (GN ). Let us pick an M ⊆ N in the ∼-equivalence class of N with maximal n(M). Now let M  ⊆ M be a member in the ∼-class of M. Then M  ∼ N and M  ⊆ M, and thus n(M  ) = n(M) by the maximality of n(M). We claim that |M/M  | is finite. Suppose that this is not the case. Then in the sequence of coverings (G/M  )0 → (G/M)0 → (G/N )0 the kernel of the first one has a positive torsion-free rank n , and the kernel of the second has torsion-free rank n(M). Hence the torsion-free rank n(M  ) of the composition is n + n(M) > n(M), contrary to the maximality of n(M). This contradiction proves our claim that |M/M  | is finite. Hence M ∈ M(G) and M ⊆ N and this shows that M(G) is cofinal in N (G). (ii) By the Cofinality Lemma 1.21 we have lim

M∈M(G)

G/M =

lim

N ∈N (G)

G/N = GN (G) .

Since G is pro-Lie we have G ∼ = GN (G) . For almost connected pro-Lie groups, we shall see an alternative approach to M(G) in 9.45. We shall now introduce a topology which we call the Z-topology and which is reminiscent of the Zariski topology on algebraic varieties and, in particular, on algebraic groups. This topology will be defined on almost connected pro-Lie groups. While it is a T1 -topology (singletons are closed sets) it gives rise to one of those rare occasions where we choose to use topologies which are not Hausdorff. When the topology is restricted to a Lie group which is an algebraic group, it will in general be coarser than the Zariski topology. Recall that a subbasis for the set of closed sets C of a topology O(X) on a set X is a set B ⊆ C such that every closed set is the intersection of sets each of which is a finite

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union of members of B. We say that the topology O(X) is generated by B. Every set of subsets of a set X can serve as a subbasis for a topology. Recall that a collection of subsets of a set is said to satisfy the finite intersection property if every finite subcollection has a nonempty intersection, that is, iff it generates a filter. Let X be a topological space. By the Alexander Subbasis Theorem (see e.g. [131, p. 139, Theorem 6]), the topology O(X) generated by a subbasis B for the set of closed sets of X is compact iff every set of sets from B satisfying the finite intersection property has a nonempty intersection. Note in passing that a proof of the Alexander Subbasis Theorem requires the Axiom of Choice. Definition 4.26 (The Z-topology on a pro-Lie group). Let G be a pro-Lie group and def

set B(G) = {gN | g ∈ G, N ∈ M(G)}. The topology Z(G) on G generated by B(G) as a subbasis for the set of closed sets is called the Z-topology. We shall often abbreviate Z(G) by Z. Note that we are not saying that G with the Z-topology is a topological group. Proposition 4.27. The Z-topology Z on a pro-Lie group G has the following properties: (i) All left and right translations are homeomorphisms in the Z-topology. (ii) Inversion x  → x −1 : G → G is a homeomorphism in the Z-topology.  (iii) If M(G) = {1} then Z is a T1 -topology. This is the case if G is almost connected. (iv) If G is almost connected, then Z is a compact T1 -topology. Proof. (i), (ii) It suffices to observe that the subbasis B(G) is bijectively mapped onto itself by left and right translations (since x(gN ) = (xg)N respectively, (gN )x = (gx)N) and by inversion (since (gN )−1 = N −1 g −1 = g −1 N ). (iii) By (i) it suffices to show that {1} is closed, i.e., that for each g = 1 there is a closed set containing 1 but not g. Thus let g = 1. Since M(G) = {1} there is an N ∈ M(G) such that g ∈ / N . Then N is the required closed set. (iv) We assume that G is an almost connected pro-Lie group and show that the Z-topology is compact. We proceed in steps. Step (a). Assume that G is a Lie group, in which case {1} ∈ M(G). Now let F = {gj Nj | j ∈ J } be a filter basis of subbasic sets with suitable Nj ∈ M(G). We like to show that it has a nonempty intersection and then apply the Alexander Subbasis Theorem to prove that G is compact for the Z-topology. Define F0 = {gj (Nj )0 | j ∈ J }. Since the collection of finite dimensional vector spaces {L(Nj ) | j ∈ J } satisfies the descending chain condition, there is a j0 ∈ J such that for all j ≥ j0 we have L(Nj ) = L(Nj0 ) and hence (Nj )0 = (Nj0 )0 . Thus there is no harm in assuming that the identity components of the members of F all agree with some M ∈ M(G). But then we pass to the quotient G/M. To simplify notation we then assume that M is a singleton and that all members of F are discrete normal subgroups of the Lie group G. Now we use our hypothesis that G is almost connected in its group topology. This property is preserved by the reductions we made. Hence G/G0 is finite since in a Lie

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group the identity component is open. Then there is a g ∈ G such that gG0 ∩ F = Ø for all F ∈ F . We may consider the filter basis g −1F instead  and assume in the end that {G0 ∩ F : F ∈ F } is a filter basis F  . Since F  ⊆ F . Therefore there is no harm in assuming F  = F . Thus we are dealing with a connected Lie group G and a filter basis F of cosets modulo discrete normal subgroups D ∈ M(G). Since {1} ∈ M(G) and {1} is open in each D ∈ M(G), D ∼ {1} for all D. By the definition of M(G), this means that D is finite. Thus the filter basis F consists of finite sets and so has a nonempty intersection. An application of the Alexander Subbasis Theorem completes the proof of the compactness of G in this case. Step (b). Assume that G is a strict projective limit of its Lie group quotients G/N , def  N ∈ M(G) and assume that H = N ∈M(G) G/M is compact in its Z-topology. We claim that then G is compact in its Z-topology. For a proof consider M, N ∈ M(G), M ⊇ N, let SMN denote the subgroup of G/M×G/N of all pairs (gM, hN ) with hN ⊆ gM. Then hM = hN. We claim that (G/M ×G/N )/SMN = (G/M ×{N })SMN /SMN is isomorphic to G/M. Indeed G/M × G/N is an almost connected Lie group, and thus is σ -compact and local compact, and G/M is a locally compact. Then by the Open Mapping Theorem for Locally Compact Groups (see for instance [79, p. 42, Theorem 5.29]) applies and  establishes the claim. Hence SMN ∈ M(G/M × G/N). Now let PMN ⊆ H = P ∈M(G) G/P be  the subgroup of all (gP P )P ∈N (G) such that gN ∈ gM M. Then PMN ∼ = SMN × P ∈M(G)\{M,N } G/P and thus H /PMN ∼ = G/M and therefore PMN ∈ M(H ). In particular PMN is Z-closed in H . Hence limP ∈M(G) G/P = M,N ∈N (G),N ⊆M PMN is a Z-closed subgroup of H . Since H is Z-compact by assumption, it follows that G ∼ = limN ∈M(G) G/N is Z-compact. After Steps (a) and (b) the proof of Claim (iv) will be finished if we prove the next and final step:  Step (c). G is a product j ∈J Gj of almost connected Lie groups Gj . We claim that G is compact in the Z-topology. Observation (i). If M ∈ M(G), then there is a unique largest cofinite subset EM of J such that j ∈EM Gj ⊆ M, where we have identified the partial product in an unmistakable fashion with a subgroup of G. Proof of Observation (i). Since G/M is a Lie group and thus has no small subgroups, we find an open neighborhood U of M such that U M = U and every subgroup of G contained in U is contained in M. Let V be a basic identity neighborhood for the product topology of G such that V ⊆ U . Then there is a finite  subset EM ⊆ J and identity neighborhoods Wj in Gj for j ∈ EM such that V = j ∈J Vj for " Wj if j ∈ EM , Vj = Gj if j ∈ J \ EM .    are two Then j ∈J \EM Gj ⊆ V ⊆ U and therefore j ∈EM Gj ⊆ M. If EM and EM such sets, then       Gj = Gj Gj ⊆ M.  j ∈J \EM ∩EM

j ∈J \EM

 j ∈J \EM

Sufficient Conditions for Quotients to be Complete

199 def

Thus  we may assume that EM was selected to be the smallest set such that P (M) = j ∈J \EM Gj , {1} × P (M) ⊆ M.  Now let N(M) be the projection of M onto j ∈EM Gj along P (M) ⊆ M. Then M = N(M) × P (M) where N(M) is a closed normal and almost connected normal subgroup of G (incidentally not containing any of the factorsGj , j ∈ EM ) such that M = N(M) × P (M). For a subset I ⊆ J let us write GI = j ∈I Gj , and let Fin(J ) denote the set of all finite subsets of J . Then we have Observation (ii). The set M(G) is the set of all M = N × GJ \E ,

E ∈ Fin(J ),

N ∈ M(GE ).

In particular, each M ∈ M(G) contains a member of M(G) of the form {1}×GJ \E , E ∈ Fin(J ).  Observation (iii). In the Z-topology, G = j ∈J Gj is compact for any infinite family of almost connected Lie groups Gj . In order to show that G is compact in the Ztopology, by the Alexander Subbasis Theorem it is sufficient to show that a filter basis B of subbasic closed sets, that is, cosets B = g(B)M(B) ∈ B,

g(B) ∈ G, M(B) ∈ M(G),

has a nonempty intersection. Notice that M(B) = B −1 B is uniquely determined by B and B ⊇ C implies M(B) ⊇ M(C), while the representative g(B) is unique only modulo M(B). If g ∈ B∩C for B, C ∈ B, then B = gM(B) and C = gM(C), whence B ∩ C = g(M(B) ∩ M(C)); in general M(G) is not closed under finite intersections. As we have seen in Observation (ii), for each B ∈ B, the normal subgroup M(B) of G is of the form M(B)EB × GJ \EB for a suitable minimal finite set EB ∈ Fin(J ) and a suitable subgroup M(B)EB ∈ M(GEB ). We write g(B) as g(B)EB × gJ \EB ∈ GEB × GJ \EB . Then we have B = g(B)EB M(B)EB × GJ \EB .

(∗)

The set of all filter bases B  containing B and having cosets g(B  )M(B  )B  as members is inductive with respect to containment ⊆. Hence it contains maximal ones by Zorn’s  Lemma. The intersection of any of these maximal filter bases is contained in B. It is therefore no restriction of generality if, in order to simplify notation, we assume that B itself is maximal. Clearly B is infinite. Let E ∈ Fin(J ); we consider the projection fE : G → GE and inspect the filterbasis fE (B) of cosets modulo normal subgroups in the almost connected Lie group GE . Equation (∗) permits us the comparison B = g(B)EB M(B)EB × GE\EB × GJ \(E∪EB ) , ker fE = {1E∩EB } × GEB \E × {1E\EB } × GJ \(E∪EB ) ,

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with obvious identifications of the partial products of the Gj . Accordingly we see that fE (B) = fE(E∪EB ) (g(B)EB M(B)EB × GE\EB ). Note that g(B)EB M(B)EB × GE\EB = (g(B)EB × {1E\EB })·(M(B)EB × GE\EB ) is a coset modulo the closed normal subgroup M(B)EB × GE\EB . However, M(B)EB ∈ M(GEB ), and thus the component (M(B)EB )0 is open in M(B)EB and therefore M(B)EB ∼ (M(B)EB )0 ; hence M(B)EB has finitely many components and therefore M(B)EB × GE\EB has finitely many components. The image fE(E∪EB ) ((M(B)EB × GE\EB )0 ) is an analytic subgroup of the Lie group GE and thus fE (M(B)) = fE(E∪EB ) (M(B)EB × GE\EB ) is a finite extension of an analytic subgroup of GE . In view of the fact that the set of finite extensions of analytic subgroups in an almost connected Lie group satisfies the finite descending chain condition, we have seen that (†) for each E ∈ Fin(J ) the filterbasis fE (B) contains a smallest element AE , that is, there is a C ∈ B such that for all D ∈ B AE = fE (C) ⊆ fE (D). We now claim that (‡) for each B ∈ B there is a unique element hB ∈ GEB such that def

B = {hB } × GJ \EB ∈ B and B ⊆ B. For a proof of this claim let B ∈ B. Then by (∗) we have B = g(B)EB M(B)EB × GJ \EB . By (†) the filter basis fEB (B) has a minimal element AEB ⊆ g(B)EB M(B)EB . Let a ∈ AE . Then g(B)EB M(B)EB = aM(B)EB and a ∈ fEB (C) for all C ∈ B. Hence, if for C ∈ B we set Ca = fE−1 (a) ∩ C, then B def

(∀C ∈ B) Ca = Ø,

and Ca−1 Ca = GJ \EB ∩ M(C) ⊇ GJ \(EB ∪EC ) ∈ M(G).

In particular, since fEB |GEB is (up to natural identification) the identity of GEB , (a) ∩ B = {a} × GJ \EB . If C ∈ B, then Ba ∩ C = fE−1 (a) ∩ B ∩ C. we have Ba = fE−1 B B Since B is a filterbasis, there is a D ∈ B such that D ⊆ B ∩ C. Then Ø = Da = fE−1 (a) ∩ D ⊆ fE−1 (a) ∩ B ∩ C = Ba ∩ C. Thus {Ba } ∪ B is contained in a filter basis B B  B of cosets modulo closed normal subgroups of G which are members of M(G). By the maximality of B we conclude B  = B. This means B ⊇ Ba ∈ B, and if we set

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201

hB = a and B = Ba , then this is the assertion (‡). For a proof of the uniqueness of hB , assume that B contains also {hB } × GJ \EB ∈ B. Since B is a filterbasis, {hB } × GJ \EB ∩ {hB } × GJ \EB = Ø. This entails hB = hB and thus secures uniqueness. From (∗) we obtain for each pair C ⊆ B in B the comparison B = g(B)EB M(B)EB × GEC \EB × GJ \EC , × GJ \EC , C = g(C)EC M(C)EC

(∗∗) (∗∗∗)

Accordingly, we have the comparison B = {hB } × GEC \EB × GJ \EC , × GJ \EC , C = {hC }

(++) (+++)

From (++) and (+++) we have (∀B, C ∈ B) B ⊇ C ⇒ fEB EC (hC ) = hB .

(#)

Next we show that the updirected family {EB : B ∈ B} is cofinal in Fin(J ), that is, for each E ∈ Fin(J ) there is a B ∈ B such that E ⊆ EB . Since E is finite and {EB : B ∈ B} is up-directed, this is equivalent to saying that each j ∈ J is contained  in some EB , that is B∈B EB = J . By way of contradiction, suppose that this is not the case. Then there is a j ∈ J such that j ∈ / EB for all B. Then (∗) takes the following form: B = gEB MEB × Gj × GJ \(EB ∪{j }) . Then the filter basis of all sets B  = gEB MEB × {1j } × GJ \(EB ∪{j }) can be enlarged to contain B, and that contradicts the maximality of B. Thus {EB : B ∈ B} is cofinal in Fin(J ) as asserted.  Then condition (#) implies that there is a family  h = (hj )j ∈J ∈ G = j ∈J Gj such that hEB = (h j )j ∈EB ∈ G EB and thus h ∈ B∈B B. Since B ⊇ B for all B ∈ B we conclude h ∈ B. Thus B = Ø which is what we had to show to complete Observation (iii) and thereby Step c. In Step (c) we have shown that arbitrary products of families of almost connected pro-Lie groups are Z-compact. In Step (b) we have shown that if this is the case, then for any almost connected pro-Lie group G, the arbitrary strict projective limits G∼ = GM(G) = limN ∈M(G) G/M is Z-compact. This completes the proof of Claim (iv) of the proposition. The proof of assertion (iv) is delicate in so far as we have to keep clearly in focus that we want to prove compactness for the Z-topology. While it is true that an arbitrary

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product of almost  connected pro-Lie groups Gj is compact in the Tychonoff product topology of  j (Gj , Z(Gj )) this contributes nothing to the fact, proven in (iv), that  ( j Gj , Z( j Gj )) is compact, as this topology of this space in general is finer than the product topology. Exercise E4.2. Verify the details of the following examples. (i) On the groups G = Z and G = R the Z-topology is the cofinite topology, that is the topology whose closed sets are all finite sets and the set G. But Z fails to be compact in the topology having as a subbasis for its closed sets the collection {gN : g ∈ G, N ∈ N (Z)}. $ # [Hint. First observe that M(G) = {0}, Z . Regarding the second assertion, the filter basis {1 + p + p2 + · · · + p n−1 + pn Z | n ∈ N} consists of cosets of groups in N (Z) and has empty intersection in Z.] (ii) Let Rc be R with the cofinite topology. The product topology on R2c has a basis of open sets of the form R2 \ L where L is a finite union of straight lines each of which is either horizontal or vertical. Thus the diagonal of R2 is dense in R2c . It is, however, closed in R2 for the Z-topology of R2 . The diagonal is the graph of the identity morphism R → R. Thus graphs of morphisms continuous in the ordinary topology are not closed in the product space for the Z-topologies on the range and the domain in general. With this preparation we can prove the first part of the following theorem giving one set of sufficient conditions for a quotient group of a pro-Lie group to be a pro-Lie group. We complement this result by alternative sufficient conditions which are easily obtained by citing relevant literature, and by one condition whose sufficiency we shall establish much later in the book in Chapter 9. The Quotient Theorem for Pro-Lie Groups Revisited Theorem 4.28. Assume that G is a pro-Lie group and K is a closed normal subgroup. Then G/K is a pro-Lie group if at least one of the following conditions is satisfied: (i) (ii) (iii) (iv)

K is almost connected and G/G0 is complete. K satisfies the First Axiom of Countability. K is locally compact. G is almost connected and K is the kernel of a morphism whose image is a pro-Lie group.

Proof. (i) We accomplish the proof in two parts. In the first part we assume that G/G0 def

is compact, that is, that G is also almost connected. By Theorem 4.1, H = G/K is a proto-Lie group. Let f : G → H be the quotient morphism and let C be a Cauchy filter of H . We have to show that C converges. Since G is almost connected, M(G) is cofinal in N (G) by 4.6. For each N ∈ M(G) let N ∗ = f (N) and let pN ∗ : H → H /N ∗ be the quotient morphism. Then the image pN ∗ (C) in the Lie group H /N ∗ is a Cauchy filter and thus has a limit hN . In

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203

fact, (hN )N∈M(G) ∈ H{N ∗ |N ∈M(G)} = limN ∈M(G) H /N ∗ ∼ = HN (H ) ; indeed C has to def

converge to a point in the completion of H . Now let FN = (pN ∗ f )−1 (hN ). Then {FN | N ∈ N (G)} is a filter basis consisting of cosets of closed normal subgroups KN of G where K = ker f . We claim that KN ∈ M(G). Now KN ∈ N (G). Let M ⊆ KN in N (G) such that M ∼ KN, that is, M is open in KN. Then M ∩ K is open in K. Since K is almost connected, KM/M ∼ = K/(M ∩ K) is finite. We must show that KN/M is finite; for this purpose it is now no loss of generality, after replacing M by KM if necessary, to assume that K ⊆ M. Then M is open in MN, whence MN is closed and thus KN = MN . Now MN/M ∼ = N/(M ∩ N ) is discrete, that is M ∩ N is open in N whence M ∩ N ∼ N . Now N ∈ M(G) implies that KN /M = MN/M ∼ Since = N/(M ∩ N) is finite. Thus KN ∈ M(G) as claimed.  G is compact in the Z-topology by Proposition 4.8 (iv), there is a g ∈ N ∈N (G) FN . Then pN ∗ (f (g)) = hN for all N ∈ N (G) from which we deduce that f (g) = lim C. Thus every Cauchy filter in H converges and so H is complete. By (i), H is a proto-Lie group, and thus by the definition of pro-Lie groups (3.25) we have shown that H is a pro-Lie group. In the second part of the proof we assume that G/G0 is complete. As a quotient of K/K0 , the factor group K/(K ∩ G0 ) ∼ = (K/K0 )/((K ∩ G0 )/K0 ) is compact. Moreover, in the factor group G/(K ∩ G0 ), the subgroup G0 /(K ∩ G0 ) is closed. Hence the product (G0 K/(K ∩ G0 )) · (K/(K ∩ G0 )) = G0 K/(K ∩ G0 ) is closed in G/(K ∩ G0 ) and therefore G0 K is closed in G and so is a pro-Lie group by the Closed Subgroup Theorem 3.35. The factor group G0 K/G0 is a continuous homomorphic image of the compact group K/(K ∩ G0 ) under the morphism k(K ∩ G0 )  → kG0 and consequently is a compact group. Therefore G0 K is an almost connected proLie group. Then by Part 1 of the proof, G0 K/K is a pro-Lie group. Further G/G0 is complete by assumption. Since K/(K ∩ G0 ) is compact, its homomorphic image G0 K/K is compact. Hence G/G0 K ∼ = (G/G0 )/(G0 K/G0 ) is complete by (iii) below. Now the factor group G/K has the complete normal subgroup G0 K/K such that (G/K)/(G0 K/K) ∼ = G/G0 K is complete, and it is therefore complete by [176, p. 225, 12.3]. Thus the proof of (i) is complete. (ii) and (iii): In view of [176, p. 242, Lemma 13.13], both of these are special cases of the sufficient condition given in [176, p. 206, Theorem 11.18]. (iv) will be established later in Chapter 9 in Lemma 9.57. Of course, if G itself satisfies the First Axiom of Countability, then (ii) follows. But there are simpler alternative arguments in this case. Recall that a topological group is metrizable if and only if it satisfies the First Axiom of Countability. (See for instance [102, Theorem A4.16].) Quotients of complete metrizable topological groups are complete because Cauchy sequences can be lifted. (See [26, Chap. 9, §3, no 1, Proposition 4].) The assertion that G/N is a pro-Lie group then follows from 4.1 (i). Naturally, pro-Lie groups often fail to be metrizable. Dierolf and Roelcke explain in [176] that there is a common cover for the two conditions of metrizability and local compactness; metrizable and locally compact spaces are almost metrizable, and it is true that for a complete topological group G and

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an almost metrizable closed normal subgroup N the factor group G/N is complete. For the concept of almost metrizability we refer to [176, p. 241, Definition 13.11]. We mention that it follows from the work of Wigner [201] that there is yet another sufficient condition for the conclusion of Theorem 4.28 to hold: (ii∗ ) K is isomorphic to an arbitrary product of groups each of which satisfies the First Axiom of Countability. This condition trivially implies (ii) and is implied by (ii∗∗ ) K is isomorphic to an arbitrary product of Lie groups. We shall see in Chapter 8 that a simply connected reductive Lie group is a product of simply connected simple Lie groups and copies of R (see Theorem 8.14 and Corollary 8.15). (ii∗∗ ) is implied by (ii∗∗∗ ) K is a simply connected reductive pro-Lie group. As we already noted above in comments subsequent to the example in Corollary 4.11, in general, the quotient of a complete topological group is not necessarily complete; in the case of additive groups of topological vector spaces this was observed at an early stage ([132], [75]). Grosso modo, the completeness of quotient groups of a complete topological group G is a delicate matter, and this is why, in order to find sufficient conditions appropriate for the category proLieGr, somewhat protracted arguments were needed in the proof that Condition (i) of Theorem 4.28 is sufficient for completeness. The Axiom of Choice is buried in the proof of Z-compactness, e.g. in the Alexander Subbasis Theorem. The arguments we cited from the book of Dierolf and Roelcke are all but trivial. Only the case that G is assumed to be metrizable in 4.28 (ii) is simpler, but this is also the most restrictive condition. The following question is in a sense a converse to that which is answered by the Closed Subgroup Theorem 1.34 and the Quotient Theorem 4.1: Let G be a topological group and N a closed normal subgroup of G such that both N and G/N are pro-Lie groups. Under what circumstances is G a pro-Lie group? Among the following three examples, the first two illustrate that this fails to be the case under the rather simple circumstances of semidirect products. The third example however, showing that every pro-Lie group has a tangent bundle, illustrates that often semidirect products do work well also for pro-Lie groups. Examples 4.29. (i) Let L be a nontrivial compact Lie group and let Z act automordef

phically on P = LZ by the shift. Set G = P  Z. Then N = P × {0} is a compact normal pro-Lie subgroup, and G/N is discrete, and thus a Lie group. But G is not a pro-Lie group.

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(ii) Let D be a discrete group, and A a compact nontrivial group of automorphisms of D. Then the compact group AN acts automorphically on the discrete group V = D (N) via (an )n∈N · (dn )n∈N = (an dn )n∈N . Set G = V  AN . Then N = V × {1} is a discrete normal subgroup, thus in particular a normal Lie subgroup. The factor group G/N ∼ = AN is compact and therefore a pro-Lie group. But G is not a pro-Lie group. def (iii) For each pro-Lie group G, the semidirect product T (G) = L(G) Ad G is a well-defined pro-Lie group. Proof. Exercise E4.3. Exercise E4.3. Prove the assertions made in the description of the Examples 4.29 (i) and (ii). [Hint. (i) G has arbitrarily small identity neighborhoods of the form, U = LN\F × {1} for some finite set F ⊆ Z (where we identify LA in an obvious fashion with a subgroup of ZN for any subset A of Z). Assume that F = Ø and x = (xn )n∈Z ∈ LN\F are such that xn = 1. If xn−m ∈ F then a shift of (x, 1) is outside U . Hence U does not contain any nondegenerate normal subgroups of G. (ii) G has arbitrarily small identity neighborhoods U = {1} × AN\F with a finite subset F of N. If a = (an )n∈N ∈ AN with an = 1, then there is a d ∈ D such that / {1} × A, and thus if an (d) = d. Then (d, 1)(1, an )(d −1 , 1) = (d · an (d)−1 , an ) ∈ v = (dj )j ∈N with dj = 1 for j = n, then (v, 1)(1, a)(v −1 , 1) ∈ / {1} × AN . So U does not contain any nondegenerate normal subgroup. (iii) Any identity neighborhood of T (G) contains one of the form U ×V where U is a zero neighborhood of L(G) and V is an identity neighborhood of G. In 4.21 (ii) we see that there is a closed ideal j of L(G) such that j ⊆ U and dim L(G)/j < ∞. Likewise there is a normal subgroup of G such that N ⊆ V and G/N is a Lie group. Then j×N is a normal subgroup of T (G) such that j×N ⊆ U ×V and T (G)/(j×N ) ∼ = L(G)/jG/N is a Lie group. Thus T (G) is a pro-Lie group, as asserted.] In the proof of (iii), the reader will notice very quickly that j and N may be chosen in such a way that j = L(N ) so that in fact, setting M = L(N ) × N, by what we shall see in Corollary 4.22, T (G)/M ∼ = L(G/N ) AdG/N G/N . Now T (G/N ) = L(G/N) AdG/N G/N is the tangent bundle of the Lie group G/N . Thus T (G) is approximated by the tangent bundles of the Lie group quotients G/N of G. We record some basic facts on the natural relationship between pro-Lie groups and proto-Lie groups. Exercise E4.4. The category of proto-Lie groups and morphisms of topological groups between them is denoted by protoLie. For a topological group G we write L(G) = GN (G) = limN ∈N (G) G/N. Recall that we have a morphism γG : G → L(G) (see 1.29, 1.39) Prove the following facts: (i) For any topological group G, the topological group L(G) is a pro-Lie group. (ii) γG is an embedding if and only if G is a proto-Lie group and an isomorphism if and only if G is a pro-Lie group.

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(iii) For each morphism f : G → H of proto-Lie groups into a pro-Lie group H there is a unique morphism f  : L(G) → H such that f = f   γG . protoLie γG

proLieGr

G ⏐ ⏐ ∀f 

−−→

L(G) ⏐ ⏐  f

L(G) ⏐ ⏐  ∃!f

H

−−→

H

H.

idH

(iv) The assignment G  → L(G) on proto-Lie groups extends to a functor L : protoLie → proLieGr which is left adjoint to the inclusion functor proLieGr → protoLie. (See e.g. [102, Appendix 2, Theorem A3.28ff.].) [Hint. (i) We have L(G) = limN ∈N (G) G/N where G/N is a Lie group for each N by the definition of N (G). In particular, L(G) is a complete topological group. The limit maps νN : L(G) → G/N are quotient morphisms inducing isomorphisms  | N ∈ N (G)}  → G/N (with N  = ker νN ) by 1.29 (iii). Then by 1.27 (ii) {N L(G)/N  is is a filter basis of closed normal subgroups of L(G) converging to 1 and L(G)/N a Lie group for all N ∈ N (G). Now Proposition 3.27 shows that L(G) is a pro-Lie group. (ii) This follows from 3.26. (iii) Assume now that f : G → H is a morphism of proto-Lie groups. Since H is a pro-Lie group, it is complete. Since G is a proto-Lie group, it has a completion by 3.26 and γG : G → L(G) is the completion morphism. Hence there is a unique extension of f to a morphism f  : L(G) → H . (iv) This is now a consequence of (iii) and Theorem A3.28 of [102]. On a larger scale, the inclusion functor from the category pro-Lie groups proLieGr into the category TopGr of topological groups has a left adjoint L : TopGr → proLieGr given by the construction of Exercise E4.5.; in Chapter 2 this functor was derived from more category theoretical principles in 2.25 (iii). There is a full subcategory of the category protoLie which arises naturally from the fact that a quotient group of a pro-Lie group is a proto-Lie group. The following exercise comments on this category. Exercise E4.5. Let Q denote the full subcategory of TopGr of all topological groups which are quotient groups of pro-Lie groups. Prove the following facts about this category: (i) Q is a (full) subcategory of protoLie. (ii) Q is closed in TopGr under the formation of products and retracts. (iii) Q is closed under the formation of closed subgroups. (iv) Q is closed under the formation of quotient groups. (v) The restriction of the functor P : protoLie → proLieGr to Q is left adjoint to the inclusion functor.

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[Hint. (i) is a consequence of the Quotient Theorem for Pro-Lie Groups 4.1 (ii): Show that a product of quotient morphisms is a quotient morphism and that the composition of quotient morphisms is a quotient morphism; use these facts to show that Q is closed in TopGr under the formation of products and retracts. (iii) If H /N is a closed subgroup of G/N, then H is a closed subgroup of G and if G is a pro-Lie group, then H is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Assertions (iv) and (v) are straightforward.] Exercise E4.6. Recall that we say that a topological group has no small normal subgroups if there is an identity neighborhood not containing any subgroups except the singleton one. Let G be a pro-Lie group, N a closed normal subgroup, and qN : G → G/N the associated quotient morphism. Then we have, on the level of sets, an injective function Hom(qN , H ) : Hom(G/N, H ) → Hom(G, H ), Hom(qN , H )(f ) = f  qN . If G is a pro-Lie group and N ⊆ M in N (G), giving an associated quotient morphism qMN : G/N → G/M, then we have an associated injective function Hom(qMN , H ) : Hom(G/M, H ) → Hom(G/N, H ),

M ⊆ N, M, N ∈ N (G).

This is an injective system of sets and functions, and we obtain an injective limit colimN∈N (G) Hom(G/N, H ) for which the maps Hom(qN , H ) : Hom(G/N, H ) → Hom(G, H ) provide an injective function iN (G) : colim Hom(G/N, H ) → Hom(G, H ). N ∈N (G)

We shall identify colimN ∈N (G) Hom(G/N, H ) with a subset of Hom(G, H ), which means identifying a morphism f : G/N → H with its canonical composition f qN ∈ Hom(G, H ). Prove the following statement: Proposition A. A proto-Lie group is a Lie group if and only if it has no small subgroups. [Hint. Let U be an identity neighborhood of G which contains no nondegenerate subgroups. Since G is a proto-Lie group, there is an N ∈ N (G) that is contained in U and thus satisfies N = {1}.] Proposition B. If G is a pro-Lie group, and H is a group without small subgroups, then colim Hom(G/N, H ) = Hom(G, H ). N ∈N (G)

[Hint. Let f : G → H be a member of Hom(G, H ) and let U be an identity neighborhood of G which does not contain a nondegenerate subgroup of H . Then f −1 (U ) is an identity neighborhood of G which, according to lim N (G) = 1 contains some N ∈ N (G). Then f (N) is a subgroup of H contained in U and thus is {1}, and thus f factors in the form f = f   qN . Therefore f ∈ Hom(G/N ) after our identification of f and f  .]

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Quotients and Quotient Maps between Pro-Lie Groups In Theorem 4.1 (iv) we proved the next best thing to an Open Mapping Theorem without having made a special definition. Let us do this here. Definition 4.30. (i) A morphism f : G → H of topological groups is called relatively open if the set {f (U ) | U is an identity neighborhood of G} is a basis for the filter of identity neighborhoods of H . (ii) A morphism f : G → H from a topological group to a pro-Lie group will be called pro-open, if for each M ∈ N (H ) the function g  → f (g)M : G → H /M is open. An open morphism is a relatively open morphism, and if the range is a pro-Lie group, it is pro-open. Let G be the additive group of real numbers endowed with the following topology: consider on Q the topology induced by the natural topology of R and let the open identity neighborhoods of Q form a basis for the identity neighborhoods of R. This is a metrizable group topology on R. If H = R with its natural topology, and if f : G → H is the identity map then f is a surjective morphism of topological groups which is relatively open but fails to be open. If H = Tp is the p-adic solenoid p

p

p

lim{T ←− T ←− T ←− · · · } then H is a pro-Lie group. Let Zp be the subgroup . / 1 p 1 p 1 p lim Z/Z ←− 2 Z/Z ←− 3 Z/Z ←− · · · . p p p This group and the group Zp of p-adic integers are isomorphic. Let D be the group Zp with its discrete topology and set G = D × R. Then G is an abelian finite-dimensional Lie group (hence a pro-Lie group) with uncountably many components. Define f : G → H by f ((zn + Z)n∈N , r) = (zn + p −n r + Z)n∈N . Then f is a surjective morphism of topological groups whose kernel is isomorphic to Z (see [102, Example E1.11 following Definition 1.30]). Let πn : Tp → T be the n-th projection. Then 3 4 2 ε ε (πn f )({0}×] − ε, ε[) = − n , n + Z Z ⊆ T, p p and thus f is pro-open. But it fails to be relatively open, let alone open. In a remark following the proof of Theorem 4.1 we mentioned a nondiscrete proLie group topology τ on the countable group Z(N) making the identity map from the discrete group Z(N) to (Z(N) , τ ) a bijective morphism between two countable pro-Lie groups that fails to be open.

Quotients and Quotient Maps between Pro-Lie Groups

209

Lemma 4.31. (The Pro-Open Mapping Theorem for Pro-Lie Groups) (i) Let G be a pro-Lie group with arbitrarily small elements N ∈ N (G) such that the Lie group G/N has countably many components. This is the case if G has only countably many components. If f : G → H is a surjective morphism of topological groups onto a pro-Lie group H , then f is pro-open. (ii) If f : G → H is a pro-open morphism of pro-Lie groups such that f (N ) ∈ N (H ) for all N ∈ N (), then f is relatively open. (iii) If, in addition to the hypotheses of (i) above, f is locally closed (in the sense that for any identity neighborhood W of G there is an identity neighborhood U contained in W such that f (U ) is closed), then f is open. Proof. Exercise E4.7. Exercise E4.7. Verify Lemma 4.31. [Hint. (i) This was in effect proved in Theorem 4.1 (iv). (ii) We have to show that for arbitrarily small identity neighborhoods U of G the closure of the image f (U ) is an identity neighborhood of H . Since G is a pro-Lie group, by Theorem 1.27 (i), we may assume that we are given an open identity neighborhood U of G for which there is an N ∈ N (G) such that U N = N U = U . Then by (i), f (U )N ∗ /N ∗ is an identity neighborhood of H /N ∗ and thus f (U )N ∗ = f (U )f (N ) ⊆ f (U )f (N) = f (U N) = f (U ) is an identity neighborhood of H . (iii) is an immediate consequence of (i) and (ii).] This is all we shall say here in the line of the Open Mapping Theorem. Notice that a morphism is locally closed if it is a closed map; this is the case for instance if it is a proper map. Definition 4.32. A topological group P is called procountable, if the filter basis of all open subgroups N such that P /N is countable converges to the identity.  An arbitrary product P = j ∈J Dj of any family of countable discrete groups Dj is procountable. Indeed, a complete topological group is procountable iff it is isomorphic to a closed subgroup of a product of a family of countable discrete groups. Lemma 4.33. Let P be a procountable group and f : P → H a morphism of topological groups into a connected pro-Lie group G. Then f (P ) is totally disconnected. def

Proof. Let A = f (P )0 . We have to show that A is singleton. From the Closed Subgroup Theorem 3.35 we know that A is a connected abelian pro-Lie group and that f −1 (A) is a procountable subgroup of P . For the purposes of the proof we may assume that P = f −1 (A), that is, that A = f (P ). Suppose now that A is nonsingleton. Then we have a quotient morphism q : A → L where L is a nonsingleton connected Lie def

group. Let j : A → A be the inclusion morphism. Then F = q  j  f : P → L is morphism into a Lie group with a dense connected image F (P ). Now let U be a zero neighborhood in L such that {0} is the only subgroup contained in U . Since P is procountable, there is an open normal subgroup N contained in F −1 (U ) such that

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4 Quotients of Pro-Lie Groups

P /N is countable. Then F (P ) = q(A) is a homomorphic image of P /N and is, therefore, a dense countable connected subgroup of L. But any countable subspace of a connected metrizable space is totally disconnected (Exercise E4.9 below). Hence q(A) is singleton. So L is singleton, contrary to what we have supposed. Exercise E4.8. A countable subset of a metrizable space is totally disconnected. [Hint. Let (X, d) be a metric space and C a countable subset. We may assume that X has at least two different points. Let c ∈ C and define f : X → R+ = [0, ∞[ by f (x) = d(c, x). Then f is continuous, f (X) ⊆ R+ is connected and is therefore a nondegenerate interval Ic containing 0 = f (c). The set f (C) is countable; hence Ic \f (C) is uncountable and clusters toward 0. There is a sequence (rn )n∈N in Ic \f (C) such that rn < n1 . Then the open balls of radius rn in C form a basis of open-closed neighborhoods of c in C.] We note that, more generally, a connected space on which the continuous real valued functions separate the points, has to have a cardinality that is at least that of the continuum. There are countable connected Hausdorff spaces. (See for instance [52, pp. 352, 353].)

Postscript There are two substantial results in this chapter. The first is that the functor L preserves quotient morphisms. More precisely, let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Let Y ∈ L(H ) be a one parameter group of H . Then there is a one parameter group X ∈ L(G) such that f  X = L(f )(X) = Y , that is, the morphism L(f ) : L(G) → L(H ) induced on the Lie algebra level is surjective. In particular, if f is a quotient morphism between pro-Lie groups then L(f ) is a quotient morphism of pro-Lie algebras. Accordingly, the functor L : proLieGr → proLieAlg not only preserves all limits, but it also preserves quotient morphisms. This is a nontrivial fact which, not surprisingly, has substantial consequences. Its proof requires the Axiom of Choice. It is important to ask whether a quotient of a pro-Lie group is a pro-Lie group. Unfortunately, the answer is in the negative, because quotient groups of complete topological groups are not necessarily complete and may not even have a completion. Indeed Sipacheva [184] showed that every topological group is a quotient group of a complete topological group. However, the quotient of a pro-Lie group is always a proto-Lie group, has a completion, and this completion is a pro-Lie group. Moreover, if f : G → H is a morphism of pro-Lie groups, and if q : G → Q is the natural dense morphism from G to the pro-Lie group completion of G/ ker f , then there is a unique morphism of pro-Lie groups f  : Q → H such that f = f  q. This indeed looks like a good universal property for a quotient object Q even though q is not surjective as we expect a quotient morphism to be. Unfortunately, f  may not be injective either. Perhaps the failure of such a category as the category of pro-Lie groups to behave neatly

Postscript

211

with respect to passing to quotient objects makes it plausible that there may be no easy category theoretical approach to quotients on the level of objects and arrows. Surely the simplest connected pro-Lie groups are the products of reals RX . The quotients of these are complete if X is countable, and we show in this chapter that RX has incomplete quotients as soon as card X ≥ 2ℵ0 . Surprisingly, this all happens in a context which we think we are familiar with, to wit, the context of compact connected abelian groups and their exponential function. So, in contrast with quotients of locally compact groups which are always locally compact, and in contrast with quotients of classical Lie groups which are always Lie groups, quotients of pro-Lie groups often fail to be pro-Lie groups – and this happens already for the additive groups of weakly complete topological vector spaces. In such circumstances it is important to have sufficient conditions under which quotients of pro-Lie groups are complete. Classically, first countable complete topological groups have complete quotients. We demonstrate a result which pertains to connected and indeed more generally to almost connected pro-Lie groups. Here we have used the terminology that a topological group G is almost connected if the factor group G/G0 modulo the identity component is compact. Indeed we show that the quotient group G/N of a pro-Lie group G modulo an almost connected closed normal subgroup N is a pro-Lie group again provided that the protodiscrete factor group G/G0 is complete. (see Lemma 3.31). The proof of this fact is not superficial. It uses the Axiom of Choice as well. We do not know an example of a pro-Lie group G such that G/G0 fails to be complete. The literature on topological groups such as [176] provides us with the information that a factor group G/N of a pro-Lie group G is a pro-Lie group whenever N is locally compact. In Chapter 9 in Theorem 9.60 we shall prove a fairly good Open Mapping Theorem for pro-Lie groups which applies whenever the domain is almost connected. We mentioned this in Theorem 4.28. In the meantime we include a weaker result showing that a surjective morphism from a pro-Lie group G with countably many components onto a pro-Lie group is “pro-open” where the pro-openness of a morphism between pro-Lie groups is a substitute for openness.

Chapter 5

Abelian Pro-Lie Groups

We saw in [102] that compact abelian groups are not simply an informative, important, and interesting special case of more general compact groups, but rather an integral part of the structure of these groups. Even at the superficial level we know that the closure of any cyclic group or of any one parameter subgroup in a locally compact group is either isomorphic to Z, respectively, to R, or else is a compact abelian subgroup. A similar situation persists in the case of pro-Lie groups. We know that in any pro-Lie group the closure of any cyclic subgroup or any one parameter subgroup is an abelian pro-Lie group. Further every locally compact abelian group is an abelian pro-Lie group. A topological group is called a weakly complete vector group if it is isomorphic to the additive group of a weakly complete topological vector space, and therefore is isomorphic to RJ for some set J . (See Corollary A2.9 in Appendix 2.) We shall see that the structure of connected abelian pro-Lie groups reduces completely to that of weakly complete vector groups and compact connected abelian groups for which we have a rather exhaustive structure and Lie theory (as shown, for instance, in [102, Chapters 7 and 8]). As the structure theory of pro-Lie groups unfolds we shall have to recall results which we prove in this chapter. There are certain aspects of the duality theory of locally compact abelian groups which extend to abelian pro-Lie groups although this larger class does not have a perfect duality theory outside the domain of connected groups. We shall discuss duality theory as well. Prerequisites. We need the general theory of pro-Lie groups up to and including Chapter 4 and elements of the duality theory of topological abelian groups such as it is, for instance, presented in [102, Chapter 7]. In particular, the theory of weakly complete topological vector spaces and their duality theory is assumed to be known (see Appendix 2).

Examples of Abelian Pro-Lie Groups We begin by offering some orientation of the class of abelian pro-Lie groups by presenting a list of examples. Let us firstly recall (see for instance [199]) the following basic types of locally compact nondiscrete fields: (a) The field R of real numbers. (b) The field Qp of p-adic rationals for some prime p. (c) The field GF(p)[[X]] of Laurent series in one variable with the exponent valuation over the field with p elements.

Examples of Abelian Pro-Lie Groups

213

All other nondiscrete locally compact fields are finite extensions of these; in cases (a) and (b) the characteristic is 0 and in case (c) it is finite. Of course, every field F with the discrete topology is a locally compact field.  ∞ 1  Let Z(p∞ ) = n=1 p n · Z /Z denote the Prüfer group of all elements of p-power order in T = R/Z. We consider on the group Z of integers, the group Q of rationals, and the Prüfer group Z(p∞ ) as endowed with their discrete topologies. Examples 5.1. Let J be an arbitrary infinite set. The following examples are abelian pro-Lie groups. (i) All locally compact abelian groups. (ii) All products of locally compact abelian groups, specifically: (a) the groups RJ ; (b) the groups (Qp )J ; (c) the groups QJ for the additive group of rational numbers Q with its discrete topology; (d) the groups ZJ for the group Z of integers with its discrete topology; (e) the groups Z(p∞ )J . An infinite product of noncompact locally compact groups is not locally compact, so, for infinite J , none of the groups in (ii)(a)–(ii)(e) is locally compact, but if J is countable, they are Polish (that is, completely metrizable and second countable). A countable product of discrete infinite countable sets in the product topology is homeomorphic to the space R \ Q of the irrational numbers in the topology induced by R (see [26, Chap. IX, §6, Exercise 7]). So QN , ZN , and Z(p∞ )N are abelian prodiscrete groups on the Polish space of irrational numbers. These elementary examples show, that the category of abelian pro-Lie groups is considerably larger than that of locally compact groups. The groups in (ii)(a)–(ii)(c) are divisible and torsion-free, and the groups in (ii)(e) are divisible and have a dense torsion group. Recall from the Closed Subgroup Theorem for Pro-Lie Groups 3.35 that every closed subgroup of a pro-Lie group is a pro-Lie group, and from the Completeness Theorem for Pro-Lie Groups 3.36 that the product of any family of pro-Lie groups is a pro-Lie group. It is then clear that from even simple examples a wealth of examples can be easily constructed. There is a less obvious but very instructive example which we present separately. This example is based on the example which we discussed in Chapter 4 right after the Quotient Theorem for Pro-Lie Groups 4.1: see Proposition 4.2 and the discussion leading up to this proposition and all the information presented subsequently through the Characterisation Theorem for Strong Local Connectivity of Compact Connected Abelian Groups 4.14. This discourse is due to Poguntke and the authors as described in [106]. However, for the present example we need only what we shall specify explicitly here.

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5 Abelian Pro-Lie Groups

If J is any set and j ∈ J , then δj : J → R is defined by " 1 if x = j , δj (x) = 0 otherwise. For any subset S ⊆ RJ let S denote the subgroup algebraically generated by S as is usual. Proposition 5.2. The free abelian group Z(N) has a nondiscrete topology making it into a prodiscrete abelian group F in such a fashion that the following conditions are satisfied: (i) There is an injection j : F → W mapping F isomorphically (algebraically and topologically) onto a closed subgroup of the weakly complete vector group W of topological dimension 2ℵ0 (that is, W ∼ = RR ) such that W/F is an incomplete group whose completion is a compact connected and locally connected group, and that the R linear span span F is dense in W . def

(ii) The subset B = {δn : n ∈ N} satisfies F = B and j (B) is unbounded in W . (iii) If K ⊆ F is any compact subset, then there is a finite subset M ⊆ N such that k ∈ K ⊆ Z(N) implies that the support supp(k) = {m ∈ N : k(m) = 0} is contained in M. In particular, every compact subset of F is contained in a finite rank subgroup of F . Proof. (i) Let G be the character group of the discrete group ZN . Then G is a compact, connected and locally connected but not arcwise connected group. Let Ga denote the arc component of the identity element in G. It was proved in [106] that the corestriction of the exponential function expG : L(G) → Ga is a quotient map. Let W = L(G) ∼ = ℵ0 Hom(ZN , R) ∼ = R2 and F = ker expG ⊆ L(G) = W . The exponential function of (locally) compact abelian groups is extensively discussed in [102, Proposition 7.36ff.]; in [102, Theorem 7.66], the kernel of the exponential function is denoted by K(G). Now F as a closed subgroup of a pro-Lie group is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups. It is totally disconnected (see [102, Theorem 7.66 (ii)]), and so by Lemma 3.31 F is a prodiscrete group. By [102, Proposition 7.35 (v)(d)], the linear span span F = span K(G) is dense in L(G) = W . Since Ga does not contain any copy of a cofinite-dimensional closed vector subspace of L(G), the function expG : L(G) → Ga cannot induce a local isomorphism and thus its kernel F = K(G) is not discrete. Furthermore, from [102, Theorem 7.66 (ii)] we observe that F ∼ = Hom(ZN , Z), algebraically, and from [49, p. 53 Corollary 15 and p.560, Corollary 24], we get that  : Z(N) → Hom(ZN , Z), ((pm )m∈N )((zm )m∈N ) = m∈N pm zm is an isomorphism of abelian groups and that, accordingly, F is a free group algebraically generated by (B).

Weil’s Lemma

215

(ii) We now prove that (B) ⊆ Hom(ZN , Z) ⊆ Hom(ZN , R) = W is unbounded. Note that (δn ) : ZN → Z is simply the evaluation evn : ZN → Z, evn (f ) = f (n). Since ZN is considered with the discrete topology, the topology on W ∼ = Hom(ZN , Z) N is that of pointwise convergence, that is, the topology induced from ZZ . Let s : N → Z be an arbitrary element of ZN . Then pr s : Hom(ZN , Z) → Z is given by pr s (ϕ) = ϕ(s) and thus pr s ((B)) = {evn (s) : n ∈ N} = {s(n) : n ∈ N} = im s. Therefore the projection of B into the s-component of Hom(ZN , Z) ⊆ Hom(ZN , R) ∼ = W is bounded if and only if s is bounded. Since there are unbounded elements s ∈ ZN , the set j (B) is unbounded in W . (iii) Let K ⊆ Z(N) be a compact subset and suppose that it fails to satisfy the claim. Then there is a sequence of elements kn ∈ K such that Mn = max supp(kn ) is a strictly increasing sequence. Now we define a function s : N → Z recursively as follows: Set s(1) = 0 and s(m) = 0 for m ∈ / {Mn : n ∈ N }. Assume that s(m) has been defined 5 m for 1 ≤ m ≤ Mn in such a way that σm = M j =1 km (j )s(j ) ≥ m. Now solve the inequality def

n + 1 ≤ σn+1 =

Mn 6

kn+1 (j )s(j ) + kn+1 (Mn+1 )s(Mn+1 )

j =1

#5 $ ). The s-projection pr ((K)) = k(n)s(n) : k ∈ K ⊇ {σn = for s(M n+1 s n∈ N 5 k (m)s(m) : m ∈ J } contains arbitrarily large elements σ ≥ n and thus cannot n m∈N n be bounded, in contradiction to the compactness of K. We realize that (ii) is implied by (iii); we preferred to give separate proofs for better elucidation of this remarkable example. The group F cannot be metrizable, because as a complete abelian group its underlying space would be a Baire space and thus as a countable topological group would have to be discrete. It is therefore noteworthy that there are pro-Lie groups whose underlying space is not a Baire space. As a countable group, it is trivially σ -compact, that is the countable union of compact sets. We observe in conclusion of our brief discussion of examples of abelian pro-Lie groups that we have seen prodiscrete abelian groups on the space R \ Q of irrationals and on a countable nondiscrete space.

Weil’s Lemma In the domain of locally compact groups, Weil’s Lemma says: Let g be an element of a locally compact group and g the subgroup generated by it. Then one (and only one) of the two following cases occurs (i) n  → g n : Z → g is an isomorphism of topological groups. (ii) g is compact.

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Exercise E5.1. Prove Weil’s Lemma. [Hint. We prove minutely more: Let E = R or E = Z and consider a morphism X : E → G. There is no loss in assuming that G = X(E), that is, that G is abelian and X has a dense image and so X is an epimorphism of abelian topological groups. Let K = ker X. The closed subgroup K of E is either cyclic or R. Thus E/K is singleton, a circle, or finite, or isomorphic to E. Since X factors through the quotient map E → E/K with a morphism X : E/K → im X, E ⏐ ⏐ quot E/K

X

−−→ −−→

A  ⏐ ⏐incl im X,

X

E/K and thus im X is compact in the first two cases. In these cases G is compact and we are in Case (ii) since im X is dense in G. We may and shall assume henceforth that X is injective and thus is a monic of topological abelian groups. There are elementary proofs of Weil’s Lemma in the literature, see for instance [102, Proposition 7.43]. A proof using the duality of locally compact groups and a : G →E  is a monic and epic of locally bit of Lie theory is as follows. The dual X compact abelian groups and thus is injective and has dense image. Since R has no  has no nonsingleton compact subgroups and thus, nonsingleton compact subgroups, G in particular, has no small subgroups. Hence it is of the form Rp × Tq × D for a  is an injective torsion-free discrete group D. Then p + q ≤ 1 (considering that X p+q  linear map from R → R). Case p + q = 0: Then A is discrete and so A is  maps A 0 ∼ compact. Case p = 1: Here the injective map X = R isomorphically onto ∼  E = R, and D = 0 follows. Case q = 1: Similarly, A = E = T and thus A ∼ = Z.] Weil’s Lemma for Pro-Lie Groups Theorem 5.3. Let E be either R or Z and X : E → G a morphism of topological groups into a pro-Lie group. Then one (and only one) of the two following cases occurs: (i) r  → X(r) : E → X(E) is an isomorphism of topological groups. (ii) X(E) is compact. def

Proof. The closed subgroup A = X(E) is an abelian pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Thus we may assume that G is abelian and the image of X is dense. Now let N ∈ N (G) and pN : G → G/N the quotient map. Then G/N is an abelian Lie group for a morphism pN X : E → G/N with dense image. By Weil’s Lemma for locally compact groups, either pN  X is an isomorphism of topological groups or else G/N is compact. If M ⊇ N in N (G) and pN  X is an isomorphism, then pM  X and the bonding map G/M → G/N are isomorphisms as well. Thus there are two mutually exclusive cases:

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217

(A) (∀N ∈ N (G)) pN  X is an isomorphism of topological groups and all bonding maps G/M → G/N are isomorphisms. (B) There is a cofinal subset Nc (G) ⊆ N (G) such that G/N is compact. In Case (A), the limit G ∼ = limN ∈N (G) G/N is isomorphic to E. In Case (B) let N ∈ N (G) and let ↑N = {P ∈ N (G) : P ⊆ N }. Then for all P ∈ N (G) the quotient G/P is compact. By the Cofinality Lemma 1.21 (ii) we know G∼ = limP ∈↑N G/P . Since all G/P are compact for P ∈ ↑N , the group G is compact in this case, and the theorem is proved. Definition 5.4. (i) Let G be a topological group. Then comp(L)(G) denotes the set of all X ∈ L(G) such that expG R · X is compact. A one parameter subgroup X ∈ comp(L)(G) is called a relatively compact one parameter subgroup. Furthermore, comp(G) denotes the set {x ∈ G : x is compact}. An element x ∈ comp(G) is called a relatively compact element of G. (ii) A topological abelian group G is said to be elementwise compact if G = comp(G). It is said to be compact-free if comp(G) = {0}. In any topological abelian group G, the set comp(G) of relatively compact elements is a subgroup, since g, h ∈ comp(G) implies that gh is contained in the compact subgroup gh. A discrete abelian group is elementwise compact if and only if it is a torsion group; it is compact-free if and only if it is torsion-free. If f : G → H is a morphism of abelian pro-Lie groups, then f (comp(G)) ⊆ comp(f (G)) ⊆ comp(H ); accordingly f (comp(G)) ⊆ comp(H ). The quotient morphism f : Z → Z/2Z shows that in general f (comp(G)) = comp(H ) even for quotient morphisms. Theorem 5.5. Let G be an abelian pro-Lie group. Then (i) comp(G) is a closed subgroup of G and therefore an elementwise compact proLie group in its own right. (ii) comp(G) ∼ = limN ∈N (G) comp(G/N ). (iii) For N ∈ N (G), let CN ⊆ G be the closed subgroup of G containing N for  which CN /N = comp(G/N ). Then comp(G) = N ∈N (G) CN and comp(G) = limN∈N (G) CN /N (iv) The factor group G/ comp(G) is compact-free.  Proof. (i) Since G ∼ = limN ∈N (G) G/N  ⊆ N ∈N (G) G/N we may assume that G is a closed subgroup of a product P = j ∈J Lj of abelian Lie groups Lj . Any abelian Lie group is of the form L = Rp × Tq × D for a discrete group D (see for instance [102, Corollary 7.58 (iii)]). Then comp(L) = {0}× Tq ×tor D for the torsion subgroup tor D of D. Hence comp(L) is closed. Thus pr j (comp(G)) ⊆ pr j (comp(G)) ⊆ comp(Lj ) as comp(Lj ) is closed. Now let g = (gj )j ∈J ∈ comp(G) ⊆ P . Then gj ∈ comp(Lj ) def  and thus gj  is compact and g ∈ K = j ∈J gj . As a product of compact groups, K is compact and thus g ⊆ K is compact and thus g ∈ comp(G). Hence comp(G) is closed and thus a pro-Lie group by the Closed Subgroup Theorem 3.35.

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(ii) The assignment G → comp(G) defines a functor from the category of abelian pro-Lie groups to itself. If G is a pointwise compact abelian pro-Lie group and H is any abelian pro-Lie group, then any morphism of topological groups f : G → H factors through comp(H ), that is, is of the form f = inclcomp(H ),H f  with a unique morphism f  : H → comp(H ). That is, comp is a right adjoint to the inclusion functor of the full subcategory of pointwise compact abelian pro-Lie groups in the category AbproLieGr of abelian pro-Lie groups. See for instance [102, Proposition A3.36]. Therefore it preserves limits (see for instance [102, Theorem A3.52]). Since comp preserves limits, it preserves, in particular, projective limits. (iii) follows from the Closed Subgroup Theorem for Projective Limits 1.34 (v). (iv) Let g comp(G) be nonzero in G/ comp(G) that is, g ∈ / comp(G). Then by / comp(G/N ) and then (ii) there is an N ∈ N (G) such that g ∈ / CN . Then gN ∈ (gN) comp(G/N ) is not relatively compact in (G/N )/ comp(G/N ). Since the quotient morphism G → G/N maps comp(G) into comp(G/N ), there is an induced quotient morphism F : G/ comp(G) → (G/N )/ comp(G/N ) given by F (g comp(G)) = (g/N) comp(G/N ). As this element is not relatively compact, the element g comp(G) cannot be relatively compact in G/ comp(G). Thus G/ comp(G) is compact-free. The examples 5.1 (ii)(b) and (e) are elementwise compact abelian pro-Lie groups which are not locally compact since J is infinite. The examples 5.1 (ii)(a), (c), and (d), and the example in Proposition 5.2 are compact-free abelian pro-Lie groups. In the first part of the following definition we recall when a topological group G with identity component G0 is called almost connected (see Definitions 1 and 4.24). Definition 5.6. Let G be a topological group. (i) G is said to be almost connected if G/G0 is compact. (ii) G is said to be compactly generated if there is a compact subset K of G such that G = K. (iii) G is said to be compactly topologically generated if there is a compact subset K of G such that G = K. Lemma 5.7. Let f : G → H be a morphism of topological groups and assume, firstly, that G is almost connected and, secondly, that f (G) is dense in H . Then H is almost connected. In particular, each quotient group of an almost connected group is almost connected. Proof. Since f (G0 ) is connected and contains the identity, f (G0 ) ⊆ H0 and therefore the morphism ϕ : G/G0 → H /H0 , ϕ(gG0 ) = f (g)H0 is well-defined. By assumption G/G0 is compact. So the continuous image ϕ(G/G0 ) is compact and thus, since H /H0 is Hausdorff, is closed in H /H0 . Since f (G) is dense in H , it follows that ϕ(G/G0 ) is dense in H /H0 . Therefore, H /H0 = ϕ(G/G0 ), and so H /H0 is compact. Proposition 5.8. Assume that G is an abelian pro-Lie group satisfying at least one of the two conditions (i) G is almost connected;

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(ii) G is topologically compactly generated. Then comp(G) is compact and therefore is the unique largest compact subgroup of G. Proof. We may assume that G = comp(G) and must show that G is compact. Then by Theorem 5.5 (ii) we have G = limN ∈N (G) comp(G/N ); it therefore suffices to verify that comp(G/N ) is compact for all N. Let N ∈ N (G). The Lie group G/N is isomorphic to L = Rp × Tq × D with a discrete group D (compare the proof of Theorem 5.5 (i)), and thus comp(L) = {0} × Tq × tor(D) for the torsion group tor(D) of D. Thus we have to verify that tor(D) is finite. In case (i), G/N ∼ = L is an almost connected Lie group by Lemma 5.7 and thus D is itself finite. In case (ii), G/N ∼ = L has a dense subgroup generated by a compact set, and this is then true for the discrete factor D. Then D is finitely generated and thus is isomorphic to the direct sum of a finite group and a finitely generated free group (see for instance [102, Theorem A1.11]). Thus tor(D) is finite in this case as well. Lemma 5.9. Assume that G is an abelian pro-Lie group. Then L(comp(G)) = comp(L)(G)), and there is a closed vector subspace W such that (X, Y )  → X + Y : W × comp(L)(G)) → W ⊕ comp(L)(G) = L(G) is an isomorphism of weakly complete topological vector spaces. Proof. By Definition 5.4, a one parameter subgroup X : R → G is in comp(L)(G) iff X(R) is compact iff X(R) ∈ comp(G). Thus comp(L)(G) = L(comp(G)). Since comp(G) is a closed subgroup of G by Theorem 5.5 (i), L(comp(G)) is a closed vector subspace of L(G). Then it has an algebraic and topological vector space complement, by Theorem A2.11 (i) of Appendix 2.

Vector Group Splitting Theorems Lemma 5.10. Let G be an abelian pro-Lie group, and assume that comp(G) is compact. Then the following conclusions hold. (i) There is a weakly complete vector group W and a compact abelian group C which is a product of circle groups such that G may be considered as a closed subgroup of W × C such that G ∩ ({0} × C) = comp(G). (ii) G/ comp(G) is embedded as a closed subgroup into the weakly complete vector group W . Proof. (i) For each N ∈ N (G) we have an embedding iN : G/N → W (N )×C(N ) for a finite-dimensional vector group W (N) and a finite-dimensional torus C(N ). Hence

220 G∼ =

5 Abelian Pro-Lie Groups

lim

N∈N (G)

G/N ⊆





i=

N∈N (G) iN

G/N −−−−−−−→

N ∈N (G)



W (N ) × C(N ) = W × C

N ∈N (G)

 for a  weakly complete vector group W = N ∈N (G) W (N ) and a compact group C∼ = N∈N (G) C(N). Since i is an embedding we may write G ⊆ W × C and assume that G is closed. Then comp(G) ⊆ comp(W × C) = {0} × C. But conversely, as every element in {0} × C is relatively compact, we have G ∩ ({0} × C) ⊆ comp G. (ii) By (i) above we may assume G ⊆ W × C for a weakly complete vector group W and torus C and {0} × C is the maximal compact subgroup of W × C. Hence comp(G) ⊆ {0} × C, and since C is compact, comp(G) ⊆ {0} × C. Let p : P → W be the projection of W × C onto W with kernel {0} × C. Then p is a proper and hence closed morphism of topological groups; therefore p|G : G → p(G) is a quotient morphism onto a closed subgroup of W . Since ker(p|G) = comp(G) we have G/ comp(G) ∼ = p(G) and the lemma follows. We remark in passing that an abelian Lie group, being isomorphic to a group × Tq × D for some discrete group (see for instance [102, 1 Exercise E5.18], or Corollary 7.58), is isomorphic to a subgroup of a connected abelian Lie group if and only if it is algebraically generated by a compact subset. If a topological abelian group is isomorphic to the additive topological group of a weakly complete topological vector space, it is called a weakly complete vector group. Rp

Lemma 5.11. Let G be an abelian pro-Lie group such that comp(G) = {1}. Then expG : L(G) → G0 is an isomorphism of topological groups. Proof. It is no loss of generality to assume that G = G0 , and we shall do that from now on. If X ∈ ker expG , that is X(1) = 0, then expG R.X = X(R) is a homomorphic image of R/Z and is therefore compact. Hence X(R) ⊆ comp(G) = {1} and thus X = 0. So expG is injective. Let N ∈ N (G). Since G is connected, G/N is a connected abelian Lie group, and thus the hypotheses of Lemma 5.10 are satisfied. Thus by Lemma 5.9 we may and will now assume that G is a closed subgroup of a weakly complete vector group V and we let i : G → V denote the inclusion map. We may identify V with L(V ) and expV with idV . There is a commutative diagram L(i)

L(G) −−→ ⏐ expG ⏐  G −−→ i

V ⏐ ⏐id  V V.

The morphism L(i) implements an isomorphism of L(G) onto a closed vector subspace of V . This implies that the corestriction expG : L(G) → expG L(G) is an isomorphism of topological groups, and thus expG L(G) is a closed vector subspace of V via i. By Corollary 4.22, exp L(G) is dense in G = G0 . Since expG L(G) is closed, we have G = expG L(G), and expG : L(G) → G is an isomorphism.

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Lemma 5.12 (Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups). Let G be an almost connected abelian pro-Lie group. Then there is a closed subgroup V of G such that expG |L(V ) : L(V ) → V is an isomorphism of topological groups and that (X, g)  → (expG X) + g : L(V ) × comp(G) → V ⊕ comp(G) = G is an isomorphism of topological groups. In particular, every connected abelian pro-Lie group is isomorphic to RJ × C for some set J and some compact connected abelian group C. Proof. By Theorem 5.5 (i), comp(G) is a closed subgroup. def

Let q : G → H = G/ comp(G) be the quotient morphism. By the Strict Exactness Theorem for L 4.20 we have a commutative diagram of strict exact sequences incl

L(q)

incl

q

0 → comp(L)(G) −−−→ L(G) −−−→ L(H ⏐ ⏐ ⏐ ) →0 ⏐ ⏐ ⏐exp expG | comp(L)(G) expG  H 0→ comp(G) −−−→ G −−−→ H → 0. By Lemma 5.10, the morphism L(q) splits, that is, there is a morphism of weakly complete topological vector spaces σ : L(H ) → L(G) such that L(q)  σ = idL(H ) . By 5.5 (iv) we have comp(H ) = {1}. Now we use the fact that comp(G) is compact by Proposition 5.8 to conclude that H is complete and thus is a pro-Lie group by Theorem 4.28 (iii). Since G is connected, H is connected. Then Lemma 5.11 implies that expH : L(H ) → H is an isomorphism. We define s : H → G by s = expG  σ  −1 −1 −1 exp−1 H . Then qs = qexpG  σ expH = expH  L(q) σ  expH = expH  expH = idH . Thus q splits. Now set V = s(H ) and let μ : V × comp(G) → G be defined by μ(v, g) = v + g and ν : G → V × G by ν(g) = (g − s(q(g)), s(q(g))). Then μ and ν are inverses of each other, and this completes the proof of the lemma. By applying the Vector Group Splitting Lemma 5.12 to the identity component G0 of an abelian pro-Lie group we see at once that: In any abelian pro-Lie group G, the identity component has a unique largest compact subgroup comp(G0 ) and G0 is isomorphic to a direct product of G0 / comp(G0 ) and comp(G0 ), where G0 / comp(G0 ) is isomorphic to the additive group of the weakly complete topological vector space L(G)/ comp(L)(G) and comp(G). Thus connected abelian pro-Lie groups are already reduced to weakly complete vector groups (Example 5.1 (ii)(a)) and compact connected groups. Definition 5.13. Let G be an abelian pro-Lie group. Let V be any closed subgroup of G such that (i) V is isomorphic to the additive topological group of a weakly complete topological vector space,

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(ii) (v, c)  → v +c : V ×comp(G0 ) → G0 is an isomorphism of topological groups. Then V is called a vector group complement. A vector group complement in a connected abelian pro-Lie group is not unique, but all vector group complements are isomorphic to G/ comp(G). We shall in fact want to know whether we can find a complement that is invariant under the action of a compact group of automorphisms of G. We therefore inspect the issue a bit more closely. Since all of this happens in the identity component of an abelian pro-Lie group, we shall not restrict the generality if we assume that G is in fact connected. Assume that  is a topological group and G is a pro-Lie group. Assume that (ω, g)  → ω · g :  × G → G is an automorphic action, that is, each g  → ω · g is an automorphism of topological groups and 1 · g = g, ω1 ω2 · g = ω1 · (ω2 · g) for all g ∈ G, and ω1 , ω2 ∈ . Since L is a functor, we have an action (ω, X)  → ω · X of  def on g = L(G) such that exp(ω · X) = ω · (exp X). By this definition if X : R → G is a one-parameter subgroup of G, then ω.X : R → G is defined by (ω · X)(r) = ω · X(r). If (ω, g)  → ω · g :  × G → G is a continuous function then we claim that (ω, X)  → ω · X :  × g → g is also continuous: Indeed, recall that on g = Hom(R, G) we consider the topology of uniform convergence on compact subsets and let C ≥ 0, ω ∈ , X ∈ g, and U an identity neighborhood of G. Let V be an identity neighborhood of G such that ω · V ⊆ U and consider a Y ∈ g such that (∀r ∈ [−C, C]) Then

X(r)−1 Y (r) ∈ V .

(∗)

(ω · X)(r)−1 (ω · Y )(r) = ω(X(r)−1 Y (r)) ⊆ ω · V ⊆ U

for all r ∈ [−C, C]. By the continuity of ϕ :  × R → G, ϕ(ρ, r) = (ω · X)(r)−1 (ρω · Y )(r), for each r ∈ [−C, C], there is an identity neighborhood r of  and an open neighborhood Wr of r ∈ R such that ϕ(r × Wr ) ⊆ U . Since [C, C] is compact, we find  def  r1 , . . . , rn ∈ [−C.C] such that [−C, C] ⊆ nm=1 Wrm . Let  = nm=1 rm . Then ϕ( × [−C, C]) ⊆ U . That is, for all ω ∈ ω and all Y satisfying (∗) we have (ω · X)(r)−1 (ω · Y )(r) ∈ V . And this proves our claim. We shall say that (ω, g)  → ω · g :  × G → G is a continuous action iff it is an action and a continuous function.

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If G is a topological abelian group and C a closed subgroup, then there is a complementary subgroup V of G such that G = V ⊕ C algebraically and topologically iff the morphism p : G → G, p(v + c) = c satisfies p 2 = p and C = im p; conversely, if such a function is given then ker p is a complementary subgroup for C. Proposition 5.14. (i) Let V be any vector space complement of a connected abelian pro-Lie group G and let us write G = V × C with C = comp(G). For any morphism of topological groups f : V → C, the subgroup graph(f ) = {(v, f (v)) : v ∈ V } is also a vector group complement. All vector group complements are obtained in this way. If f : V → C is given, then αf : G → G, αf (v, c) = (v, c + f (v)) is an automorphism of G mapping V ×{0} to graph(f ), while Pf : G → G, Pf (v, c) = (0, c−f (v)) is an idempotent endomorphism of G whose kernel is a vector space complement and whose image and fixed point set is C. (ii) Let  be a compact group and assume that there is a continuous automorphic action  × G → G on the connected abelian pro-Lie group G. Then there is a vector space complement in G that is invariant under . Proof. (i) Exercise E5.2 below. (ii) Since C is a characteristic subgroup, it is invariant under the action of , and it follows that L(C) is invariant in g = L(G). The vector subspace L(C) has a vector space complement in the weakly complete vector space g. (See Chapter 7, Theorem 7.7 (iv).) Thus we have an idempotent endomorphism of topological vector spaces q : g → g, im q = L(C). Since  is compact and acts continuously and linearly on the locally convex complete vector space g, the integral 7 p= ωqω−1 dω : g → g ω∈

exists and is an idempotent equivariant endomorphism of topological vector spaces, that is, it satisfies p(ω · X) = ω · p(X) and im p = L(C). (See [102, Example 3.37 (ii).]) def

Then W = ker p is an  invariant vector subspace of g such that g = W ⊕ L(C) algebraically and topologically. So, using the terminology of (i) above, writing G = V × C and identifying L(G) with L(V ) × L(C), we can now invoke (i) for L(G) and write W = {(X, Y + λ(X)) : X ∈ L(V ), Y ∈ L(C)} for a suitable linear morphism ϕ : L(V ) → L(C). The exponential function expV : L(V ) → V is an isomorphism. We define f : V → C by f (expV X) = expC ϕ(X) and get a vector space complement expG W = {(v, f (v)) : v ∈ V } ⊆ V × C, and by (i) again, this an algebraic and topological complement for C, that is, G = expG W ⊕ C. Moreover, if ω ∈ , then ω · (X, ϕ(X)) = (X  , ϕ(X )) for some X , whence ω · (expV X, f (expV X)) = ω · expG (X, ϕ(X)) = expG ω(X, ϕ(X)) = expG (X , ϕ(X )) = (expV X , f (expV X )) ∈ Vϕ . Thus expG W is -invariant.

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5 Abelian Pro-Lie Groups

Exercise E5.2. Prove Remark 5.14 (i). [Hint. The function α : G0 → G0 , α(v, c) = (v, c + f (v)) is an automorphism of topological groups mapping V × {0} onto graph(f ). If W ⊆ G0 = V × C is a vector group complement, then since W is a vector group complement there is a projection pr W : G0 → W with kernel {0} × C. Let pr C : G0 → C denote the projection with kernel V × {0}. Define f : V → C by pr C (pr W (v, 0) − (v, 0)). Then f is a morphism of topological groups; if w = (v, c) ∈ W then w = pr W (v, 0) and w − (v, 0) = (0, c) whence c = f (v). The assertions on αf and Pf are straightforward.] We now work towards removing the hypothesis of almost connectivity from the Vector Group Splitting Lemma 5.12. Lemma 5.15. Let W be a weakly complete topological vector space and G a proto-Lie group. Assume that f : W → G is a bijective morphism of topological abelian groups. Then f is an isomorphism of topological groups.  denote the completion of G. Then G  is a connected abelian pro-Lie group; Proof. Let G by theVector Group Splitting Lemma 5.12, it is therefore of the form V ⊕C algebraically and topologically for a weakly complete topological vector subgroup V and a compact  → V denote the projection onto V along C. The function subgroup C. Let pr V : G def

ϕ = pr V f : W → V is a dense morphism of weakly complete vector groups and is therefore a quotient morphism of weakly complete topological vector spaces by Theorem A2.12 (a) of Appendix 2. Thus there are closed vector subspaces V1 and V2 of W such that W = V1 ⊕ V2 algebraically and topologically such that V2 = ker ϕ and ϕ  i : V1 → V , where i : V1 → W is the inclusion, is an isomorphism of weakly  σ = f i(ϕi)−1 complete topological vector spaces. Now the morphism σ : V → G, −1 −1 satisfies pr V σ (v) = pr V f  i  (ϕ  i) = ϕ  i  (ϕ  i) = idV . This means that  = σ (V ) ⊕ C, σ (V ) = f (V1 ). In order to simplify notation, after replacing V by G def

σ (V ), if necessary, we may actually assume that V = f (V1 ). Then D = f (V2 ) ⊆ C, and we have G = V × D with a dense subgroup D of C. We recall that G is a protoLie group; then D ∼ = G/V is a connected proto-Lie group and a dense subgroup of a compact group. Let N ∈ N (D); then D/N is a Lie group and a dense subgroup of C/N . A Lie group is complete, and thus D/N is closed in C/N, that is, D/N = C/N for all N ∈ N (D).  Hence G is complete and f : W → V × C Therefore D = C and thus G = G. is bijective. We can write W = W1 × W2 such that f = f1 × f2 where f1 is an isomorphism from W1 onto V and f2 : W2 → C is a bijective morphism of topological abelian groups from a weakly complete vector group W2 onto a compact group C. In particular, every point in C is on a one parameter subgroup, that is exp : L(C) → C is surjective. But C, as a bijective image of a real vector space, is torsion-free and divisible.  is divisible and torsion-free and so is a rational vector space, that is, a direct Thus C . If C = Q J for some sum of copies of Q, and therefore C is a power of copies of Q

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225

)J as L preserves limits, and expC may set J , then L(C) may be identified with L(Q J ∼ ) → Q  is not surjective, because the be identified with (expQ  ) . But expQ  : R = L(Q  nontrivial compact homomorphic images of R are isomorphic to R/Z ∼ = T and T ∼ =Q  . Thus J = Ø and C = {0}. This means that G = V and since  T ∼ =Q∼ = Z ∼ =Q W → V is an isomorphism of topological groups. f ∼ = Note that Lemma 5.15 is a very special example of an Open Mapping Theorem (see 9.60 and its proof). Lemma 5.16. Assume that G is an abelian topological group with closed subgroups G1 and H such that G1 is either (a) a weakly complete topological vector subgroup, or (b) a compact subgroup. Assume further that G1 + H = G, that G1 ∩ H = {0}, and that G/H is a proto-Lie group. Then μ : G1 × H → G, μ(v, h) = v + h is an isomorphism of topological groups. Proof. Clearly μ is a bijective morphism of topological groups. We must show that its inverse is continuous. The morphism β : G1 → G/H , β(v) = v + H is a continuous bijection. In Case (a), by Lemma 5.15 β is open. In Case (b) it is a homeomorphism since G1 is compact. That is, β −1 : G/G0 → G0 is continuous in both cases. Let def

q : G → G/G0 be the quotient map. Then α = β  q : G → G0 is a morphism of topological groups, and μ−1 (g) = (α(g), g − α(g)) is likewise a morphism of topological groups. Notice that by the Quotient Theorem for Pro-Lie Groups 4.1, the quotient G/H is a proto-Lie group if G is a pro-Lie group. Proposition 5.17. Assume that G is an abelian proto-Lie group and that G1 is a closed connected subgroup which is a finite-dimensional Lie group. Then there is a closed subgroup H such that the morphism (v, h)  → v + h : G1 × H → G is an isomorphism of topological groups. Proof. We claim that it is no loss of generality to assume that G is a pro-Lie group.  be the completion of G. Since the subgroup G1 is a finite-dimensionIndeed, let G al Lie group, it is locally compact. Locally compact subgroups are closed (see for instance [102, Corollary A4.24.]). Thus G1 is also a closed subgroup of the pro If we can show that there is a closed subgroup H  of G  such that Lie group G.  → G  is an isomorphism of topological groups, then (v,  h) → v +  h : G1 × H def  we get a subgroup of G such that (v, h)  → g + h : G1 × H → G setting H = G ∩ H is an isomorphism of topological groups. This proves our claim; from here on we shall assume that G is a pro-Lie group.

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The subgroup G1 , being a finite-dimensional Lie group, has no small subgroups. Hence there is a zero neighborhood U such that {0} is the only subgroup contained in G1 ∩ U . Now let N ∈ N (G) be contained in U . Then N ∩ G1 ⊆ U ∩ G1 = {0}. Recall that G/N is a Lie group and (G1 + N )/N is isomorphic to G1 /(G1 ∩ N ) ∼ = G1 by the Closed Subgroup Theorem for Projective Limits 1.34 (iv). So (G1 + N )/N is isomorphic to Rp ⊕ Tq and is a closed subgroup of G/N which is isomorphic to Rm ⊕ Tn ⊕ D for a discrete subgroup D. It is therefore a direct summand algebraically and topologically, that is, there is a closed subgroup H of G containing N such that H /N is a complementary summand for (G1 +N )/N . Thus G1 +H = G and G1 ∩H ⊆ def

G1 ∩ N = {0}. Then if C = comp(G1 ) = {0}, it follows from Lemma 5.16 (a), that (v, h) → v + h : G1 × H → G is an isomorphism of topological groups. Therefore, in the general case, G/C ∼ = G1 /C × (H + C)/C and thus there is a vector subgroup V of G such that G ∼ = V × (C + H ); it remains to be observed that C + H ∼ = C × H. But that is Lemma 5.16 (b). Next we need a lemma on weakly complete topological vector spaces. Recall that an affine subspace A of a vector space W is a subset of the form A = g + V for some vector subspace V . The affine subspace is linear iff g ∈ V . Lemma 5.18. Let W be a weakly complete topological vector space and F a filter basis of closed affine subspaces. (a) Assume that F is a closed vector subspace of W and all A ∈ F are linear. Then F+



F =



F + H.

(∗)

If, in addition, dim E/F < ∞, then   F+ F = (F + H ).

(∗∗)

H ∈F

H ∈F

(b) Assume that all A ∈ F are linear. Then the following conditions are equivalent: (i)  lim F = 0. (ii) F = {0}.  (c) F = Ø. (d) Assume that F ⊆ N (W ). Then the following conditions are equivalent: (i) lim F = 0. (ii) F is a basis of N (W ). Proof. See Theorems A2.13 and A2.14 of Appendix 2. In terms of a terminology that has been used for situations like this, (c) can be expressed in the following form: Weakly complete topological vector spaces are linearly compact.

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We shall prove next that every weakly complete vector subgroup in an abelian pro-Lie group splits. Another way of expressing this in a category theoretical fashion is this: Weakly complete vector groups and torus groups are relative injectives for the class of embeddings in the category of abelian pro-Lie groups. For a full understanding recall that for a class E of monomorphisms, an object I in a category is called a relative injective for E , if for any monic ι : A → B from E and any morphism f : A → I there is a morphism F : B → I such that f  ι = f . idI

I −−→  ⏐ f⏐ A −−→ ι

I  ⏐ ⏐F B.

(See for instance [102, paragraph following 9.77].) By definition, an object is an injective, if it is a relative injective for the class of all monomorphisms. Exercise E5.3. A pushout in a category is a pullback in the opposite category (see Definition 1.1 (iii)). Show Proposition A. Let I be an object in a category and consider the following two statements: (i) I is a relative injective for E . (ii) Any monic ι : I → B from E is a coretraction, that is there is an ε : B → I such that ε  ι = idI . Then (i) ⇒ (ii) and if the category has pushouts pushing forward E -monics, then the two conditions are equivalent. This is the case in the category AbTopGr of abelian topological groups and in the category AbproLieGr of abelian pro-Lie groups for the class E of closed embeddings. Proposition B. Any product of relative injectives is a relative injective. [Hint. Proof of A: For a proof of (i) ⇒ (ii) simply take A = I and f = idI in the definition. For a proof of the reverse implication let ι : A → B be a monomorphism from E and f : A → I any morphism into an object I satisfying (ii). Form the pushout ϕ

I −−→  ⏐ f⏐ A −−→ ι

P  ⏐β ⏐ B.

(∗)

Since pushouts push forward E -monics and ι is an E -monic, the morphism ϕ is an E -monic as well. Hence by (ii) there is an ε : P → I such that ε  ϕ = idI . Set F = εβ. Then F  ι = εβι = εϕf = idI f = f . ϕ

ε

ι

idB

I −−→ P   −−→ ⏐ ⏐β f⏐ ⏐ A −−→ B −−→

I  ⏐ ⏐F B.

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Finally, we recall from Theorem 1.5 (a), that the pushout (∗) is formed by creating first the coproduct P  of I and B and then the coequalizer e : P  → P of the two morphisms copr I f and copr B ι (assuming that coproducts and coequalizers exist). In the category AbTopGr of abelian topological groups P  = I × B and copr I (x) = def

(x, 0), copr B (b) = (0, b). The coequalizer e is the cokernel of δ = copr I f − copr B ι, that is the quotient morphism e : I × B → (I × B)/Q with Q = δ(A) = {(f (a), −a) : a ∈ A}, where we have assumed A = A ⊆ B and ι to be the inclusion map. Notice that Q is closed in I × B: If (x, b) = limj (f (aj ), −aj ) for a net (aj )j ∈J in A, then b ∈ A = A and x = limj f (aj ) = f (−b) since f is continuous. (Indeed, Q is in essence the graph of f .) Now ϕ : I → (I × B)/Q is given by ϕ(x) = (x, 0) + Q. Then x ∈ ker ϕ iff (x, 0) ∈ Q, that is, (x, 0) = (f (a), −a) for some a ∈ A. Then a = 0 and as a consequence x = f (a) = f (0) = 0. Thus ϕ maps I bijectively onto ((I × {0}) + Q)/Q = (I × A)/Q. Since Q is closed in B, the group I × A is closed in I × A. We claim that the morphism ψ : I → (I × A)/Q, ψ(x) = (x, 0) + Q is open. So let U be a symmetric identity neighborhood of I . Find a symmetric identity neighborhood V of I such that V + V ⊆ U . Now we let W = f −1 (V ); this is a symmetric identity neighborhood of A since f is continuous. Now we show that ψ(U ) contains (V × W ) + Q/Q and this will show that ψ(U ) is an identity neighborhood of (I × A)/Q and thus prove the asserted openness of ψ. Let (v, w) ∈ V × W . We claim that there are elements u ∈ U and a ∈ A such that (u, 0) + (f (a), −a) = (v, w). Indeed, we let a = −w. Then u − f (w) = v, implies u = v + f (w) ∈ V + V ∈ U and this proves the claim. In the category AbproLieGr everything remains the same except that in order to form the pushout in the category AbproLieGr we have to form the completion P of (I × B)/Q, because (I × B)/Q is a proto-Lie group but not necessarily a pro-Lie group (see Chapter 4, the Quotient Theorem for Pro-Lie Groups 4.1). Thus ϕ : I → P is the composition of the closed embedding morphism γ : I → (I × B)/Q and the dense embedding morphism η : (I × B)/Q) → P . But I is a pro-Lie group and is therefore complete, and since ϕ(I ) = η(γ (I )) is an isomorphic copy of I it is a complete and therefore closed subgroup of B. Hence ϕ is a closed embedding in this case as well. Proof of B. Let {Ij : j ∈ J } be a family of relative injectives and ι : A → B a mono from E . Assume that f : A → j ∈J Ij is any morphism. Fix j since Ij is an E injective, there is an Fj : B → Ij such that Fj  ι = pr j f . Then the universal  property of the product (see Definition 1.4 (i)) give us a morphism F : B → j ∈J Ij such that pr j F = Fj . Then pr j F ι = Fj  ι = pr j f and this implies F  ι = f by the uniqueness in the universal property of the product.] Theorem 5.19. Assume that G is an abelian pro-Lie group with a closed subgroup G1 and assume that there are sets I and J such that G1 ∼ = RI × TJ . Then G1 is a

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homomorphic retract of G, that is, a direct summand algebraically and topologically. So, G ∼ = G1 × G/G1 . Proof. By Exercise E5.3, Proposition A, applied to the category AbproLieGr of abelian pro-Lie groups, we are claiming that G1 is a relative injective for the class of closed homomorphic embeddings in AbTopGr. By Exercise E5.3, Proposition B above, it suffices to prove this for each of the cases (i) G1 = R, and (ii) G1 = T. For each of these cases Proposition 5.17 applies and proves the claim. It may be a useful exercise to present an independent and category theory free proof of the preceding theorem in the case that G1 ∼ = RI is a weakly complete topological vector space. It proceeds as follows: Let S be the set of all closed subgroups S of G satisfying the following conditions: (i) S ∩ G1 is a vector group. (ii) (∀g ∈ G) S ∩ (g + G1 ) = Ø. We claim that (S, ⊇) is an inductive poset. For a proof of the claim let T be a chain def  in S and set T = T . We  have to show that T satisfies (i) and (ii). (i) We note T ∩ G1 = S∈T S ∩ G1 , and all S ∩ G1 are closed vector groups; hence their intersection is a closed vector group. (ii) Let g ∈ G; we must show that T ∩ (g + G1 ) = Ø. Now for each S ∈ T we find an sS ∈ S ∩ (g + G1 ) by (ii). Then g + G1 = sS + G1 . We claim that (∗) S ∩ (g + G1 ) = sS + (S ∩ G1 ). Indeed if s ∈ S ∩(g+G1 ), then s ∈ g+G1 = sS +G1 and thus s −sS ∈ S ∩G1 and thus s ∈ sS +(S∩G1 ). Conversely, if s ∈ S∩G1 , then sS +s ∈ S∩(sS +G1 ) = S∩(g+G1 ). By (i) we know that S ∩ G1 is a (closed) vector subgroup VS of G1 . Thus from (∗) we obtain (∗∗) Ø = (S − g) ∩ G1 = sS − g + VS , where sG − g ∈ G1 . Now the family {sS − g + VS : S ∈ T } is a filter basis of closed affine subspaces of the vector group G1 . By Lemma 5.18 (c), there is a  t∈ (sS − g + VS ) ∈ G1 , S∈T



 and thus t + g ∈ S∈T (sS + VS ) = S∈T S ∩ (g + G1 ) = T ∩ (g + G1 ). This completes the proof that (S, ⊇) is inductive. Using Zorn’s Lemma, let H be a minimal member of S. We claim that H ∩G1 = {0}. Suppose that the claim were false. def

Then H1 = H ∩ G1 is a nonzero weakly complete vector group. Let N be a vector subgroup of H1 , such that H1 /N is a finite-dimensional vector group (for instance, one isomorphic to R). Now G/N as a quotient of a pro-Lie group, is a proto-Lie group by Theorem 4.1. Then by Proposition 5.17, there is a closed subgroup S of H containing N such that (H1 /N ) + (S/N ) = H /N, and (H1 /N ) ∩ (S/N ) = {N }. Thus

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H1 ∩ S = N is a vector subgroup and G1 + S = G1 + H = G and so the subgroup S of H /N satisfies (i) and (ii) above. The minimality of H then entails S = H and thus N = H1 ∩ S = H1 and that is a contradiction to the choice of N. This proves our claim that there is a closed subgroup H of G such that G = G1 + H and G1 ∩ H = {0}. Thus the function (v, h)  → v + h : G1 × H → G is a bijective morphism of topological groups by Lemma 5.16. Recall from the Vector Group Splitting Lemma 5.12 and Definition 5.13 that every abelian pro-Lie group has a vector group complement. Vector Group Splitting Theorem for Abelian Pro-Lie Groups Theorem 5.20. Let G be an abelian pro-Lie group and V a vector group complement. Then there is a closed subgroup H such that: (i) (v, h)  → v + h : V × H → G is an isomorphism of topological groups. (ii) H0 is compact and equals comp G0 and comp(H ) = comp(G); in particular, comp(G) ⊆ H . (iii) H /H0 ∼ = G/G0 , and this group is prodiscrete. (iv) G/ comp(G) ∼ = V × S for some prodiscrete abelian group S without nontrivial compact subgroups. (v) G has a characteristic closed subgroup G1 = G0 + comp(G) which is isomorphic to V × comp(H ) such that G/G1 is prodiscrete without nontrivial compact subgroups. (vi) The exponential function expG of G = V ⊕ H decomposes as expG = expV ⊕ expH where expV : L(V ) → V is an isomorphism of weakly complete vector groups and expH = expcomp(G0 ) : L((comp(G0 )) → comp(G0 ) is the exponential function of the unique largest compact connected subgroup; here L(comp(G0 )) = comp(L)(G) is the set of relatively compact one parameter subgroups of G. (vii) The arc component Ga of G is V ⊕Ha = V ⊕comp(G0 )a = im L(G). Moreover, if h is a closed vector subspace of L(G) such that exp h = Ga , then h = L(G). Proof. (i) By Theorem 5.19, H exists such that (i) is satisfied. (ii) Let us write G = V × H . Then G0 = V × H0 . Since V × {0} is a vector group complement, G = (V × {0}) ⊕ comp(G0 ) algebraically and topologically. Then the projection of G0 onto H0 along V maps the compact subgroup comp(G0 ) onto H0 . Thus H0 is compact. So {0} × H0 ⊆ comp(G0 ), and since G0 = V × H0 , the factor group comp(G0 )/({0} × H0 ) is isomorphic to a subgroup of V . Since V as a vector group has no nontrivial compact subgroup, comp(G) = {0} × H0 follows. (iii) Retaining the convention G = V × H after (i), we have G0 = V × H0 . Then G/G0 =

V × H ∼ (V × H )/(V × {0}) ∼ = = H /H0 . V × H0 (V × H0 )/(V × {0})

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By the Closed Subgroup Theorem 3.35, H is a pro-Lie group. Since H0 is compact, H /H0 is complete by Theorem 4.28 (iii). Thus by Lemma 3.31 H /H0 is a prodiscrete group (iv) Again we write G = V × H and have comp(G) = {0} × comp(H ). Thus G/ comp(G) =

V ×H ∼ = V × (H / comp(H )). {0} × comp(H )

By Lemma 5.5 (iii), H / comp(H ) is compact-free. By (iii) above, H /H0 is prodiscrete. As a quotient of the prodiscrete group H /H0 , the quotient H / comp(H ) ∼ = (H /H0 )/(comp(H )/H0 ) is a protodiscrete group by Proposition 3.30 (b). (v) comp(G) is a closed characteristic subgroup and by (iv) the factor group G/ comp(G) decomposes into a direct product V ×S in which V ×{0} is the connected component and thus is characteristic. The kernel G1 of the composition of the quotient morphism G → G/ comp(G) and the projection G/ comp(G) → S is a closed characteristic subgroup equal to G0 + comp(G) and G/G1 ∼ = S. Applying (i) to G1 we get G1 ∼ = V × comp(G). (vi) follows immediately from (i) and Definition 5.4 (i) in view of the fact that for any topological vector space V the exponential function expV : L(V ) → V , expG X = X(1), is an isomorphism of topological vector spaces as all one parameter subgroups are of the form X = r  → r · v for a unique vector v = vX . See also Lemma 5.15. (vii) As G = V ⊕ H is a direct product decomposition we have Ga = Va ⊕ Ha . But V , as the additive topological group of a topological vector space is arcwise connected, and Ha = (H0 )a = comp(G0 )a . By [102, Theorem 8.30 (ii)], we have comp(G0 )a = L(comp(G0 )) = L(H ). Thus from (vi) we get Ga = expG L(G). Now let h be a closed subalgebra of L(G). If h = L(G), then expG h = E(G) = Ga by Corollary 4.21 (i). In the context of conclusions (iv) and (v) we recall from 5.10 (ii) that every prodiscrete group without compact subgroup embeds into a weakly complete vector group. But Proposition 5.2 shows that such embeddings may have bizarre properties. For locally compact abelian groups, 5.20 (i) yields a core result of their structure theory; it is presented practically in every source book on locally compact abelian groups (see for instance [102, Theorem 7.57]). The examples in 5.1 (ii)(b)–(e) illustrate certain limitations of this main result. The examples in 5.1 (ii)(b) and (ii)(e) show what prodiscrete pointwise compact groups may look like; neither has a compact open subgroup and therefore both fail to be locally compact. The examples in (ii)(b) are torsion-free and divisible, the examples in (ii)(e) have a dense proper torsion subgroup and are divisible. The examples in 5.1 (ii)(c) and (d) are compact-free; those in (ii)(c) are divisible, those in (ii)(d) have no nondegenerate divisible subgroups. Thus unlike in the locally compact case, we cannot expect that inside the factor H , the subgroup comp(H ) is open, or, equivalently, that the factor group G/G0 has an open compact subgroup.

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It is easy to mix these examples. There are compact abelian groups in which the component does not split (see [102, Example 8.11]). Theorem 5.20 completely elucidates the structure of the identity component G0 , it largely clarifies the structure of G1 (although comp(G) is best understood in the locally compact case), and it reduces the more subtle problems on G to the compactfree prodiscrete factor group G/G1 . One should recall the example in Proposition 5.2 which typically might occur as a prodiscrete factor group. Corollary 5.21. Let G be an abelian pro-Lie group and V a vector group complement. Then the following statements are equivalent: (i) (ii) (iii) (iv)

G/G0 is locally compact. G/V is locally compact. comp(G/V ) is locally compact and open in G/V . There is a locally compact subgroup H of G containing comp(G) as an open subgroup such that such that (v, h)  → v + h : V × H → G is an isomorphism of topological groups.

Proof. (i) ⇔ (ii): We have G/G0 ∼ = (G/V )/(G0 /V ) and G0 /V is compact by Lemma 5.12. The quotient of a locally compact group is locally compact, and the extension of a locally compact group by a locally compact group is locally compact. (iv) ⇒ (iii): Since comp(G) = comp(H ) this is clear. (iii) ⇒ (ii): Trivial. (ii) ⇒ (iv): The locally compact abelian group G/V has a compact identity component (G/V )0 = comp(G/V )0 and thus has a compact open subgroup C. Let K be the full inverse image of C in G. Then K is an almost connected open subgroup of G to which the Vector Group Splitting Theorem applies. Thus K is the direct product of V and the unique maximal compact subgroup comp(K) of K. Then (G/ comp(K))0 = K/ comp(K) ∼ = V , this is an open divisible subgroup of G/ comp(K). But then G/ comp(K) is the direct sum of (K/ comp(K))0 and a discrete group H / comp(K) with a closed subgroup H of G containing comp(K) as an open subgroup. In particular, comp(K) ⊆ comp(H ) and comp(H ) is open in H . From (G/ comp(K)) ∼ = (K/ comp(K)) × (H / comp(K)) and K∼ = V × comp(K) we derive G∼ = V × H. As we have comp(G) = comp(H ), the implication (ii) ⇒ (iv) is proved.

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Compactly Generated Abelian Pro-Lie Groups The Vector Group Splitting Theorem tells us that each abelian pro-Lie group G is built up in a lucid fashion from a weakly complete vector group and a more special abelian pro-Lie group H which is turn is an extension of the characteristic closed subgroup comp(G) = comp(H ) by a prodiscrete compact-free factor group H / comp(H ) ∼ = G/(G0 + comp(G)). The examples 5.1 (ii)(b) and (ii)(e) (and the groups easily manufactured from these by passing to products, subgroups and quotients) indicate that we are not to expect very explicit information on comp(G) without further hypotheses, and a similar statement holds for prodiscrete compact-free groups (see 5.1 (ii)(c), (ii)(d) and 5.2). A topological space is called a Polish space if it is completely metrizable and second countable. It is said to be σ -compact, if it is a countable union of compact subspaces. It is said to be separable if it has a dense countable subset.  Countable products of Polish spaces are Polish. For instance, any product n∈N Ln of a countable sequence of second countable Lie groups is a Polish pro-Lie group; this applies in particular to RN or ZN . A mixture of topological and algebraic properties of topological groups is exemplified by the concepts introduced in 5.6, to which we return presently. Remark 5.22. (i) Every almost connected locally compact group is compactly generated. (ii) Every compactly generated topological group is σ -compact. (iii) A topological group whose underlying space is a Baire space and which is σ -compact is a locally compact topological group. (iv) A σ -compact Polish group is locally compact. (v) A compactly generated Baire group is locally compact. Proof. (i) Let K be a compact neighborhood of the identity. Then K is an open subgroup which has finite index in G. Let F be any finite set which meets each coset modulo K. Then K ∪ F is a compact generating set of G. def

(ii) If K is a compact generating set of G, C = KK −1 is a compact generating then ∞ −1 set satisfying C = C; then G = C = n=1 C n . (iii) A Baire space cannot be the union of a countable set of nowhere dense closed subsets. A topological group containing a compact set with nonempty interior is locally compact. (iv) By the Baire Category Theorem (see [26, Chapter 9, §5, no 3, Théorème 1]), every Polish space is a Baire space. (v) is clear from the preceding. The following remarks are straightforward from the definitions, from Proposition 5.8, and the Vector Group Splitting Theorem 5.20. Remark 5.23. Let G ∼ = V × H be an abelian pro-Lie group with a vector group complement V and H as in Theorem 5.20. Then the following statements are equivalent:

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(i) G is Polish iff both V and H are Polish. (ii) G is σ -compact iff V, comp(G), and H / comp(H ) are σ -compact. (iii) G is compactly generated iff V and H / comp(H ) are compactly generated and comp(G) is compact. (iv) G is separable iff V and H are separable. These simple remarks lend some urgency to a more detailed understanding of the situation of weakly complete topological vector spaces; we shall turn to this topic in the next section. Remark 5.24. For a discrete abelian group, the following statements are equivalent: (i) (ii) (iii) (iv)

G is finitely generated free. G is isomorphic to a closed additive subgroup of Rn for some natural number n. G is isomorphic to a closed additive subgroup of RJ for some set J . G is isomorphic to a closed additive subgroup of a weakly complete topological vector space.

Proof. For the equivalence of (i) and (ii) see for instance [102, Theorem A1.12 (i)]. Trivially (ii) ⇒ (iii) ⇒ (iv). Assume (iv), that is, that G is a closed discrete subgroup of a weakly complete vector group W . Since G is discrete, there is an identity neighborhood U1 of W such that W ∩ U1 = {0}. Let U be an open identity neighborhood of W such that U + U + U + U ⊆ U1 . Since lim N (W ) = 0 there is a V ∈ N (W ) such that V ⊆ U and thus U + V ⊆ U + U . By replacing U by U + V where necessary we assume that U + V = U and U + U ⊆ U1 . If u ∈ (G − U ) ∩ U then u = g − u for some 0 = g ∈ G and u ∈ U ; thus g = u + u ∈ G ∩ U + U ⊆ G ∩ U1 = {0}. Thus V is the complement of (G \ {0}) − U in G + V . So G ∼ = (G + V )/V is a discrete hence closed subgroup of the finite-dimensional vector space W/V . This result provides an alternative proof of the fact that the prodiscrete free abelian group F of infinite rank in 5.2 cannot be discrete. Proposition 5.25. Let G be an abelian compact-free pro-Lie group. (i) If N (G) has a basis M of subgroups such that G/M is compactly generated, then G is isomorphic to a closed subgroup of a product RI × ZJ and hence also of a weakly complete topological vector space RK . (ii) If G is compactly generated, then (i) applies. (iii) The group G is compactly generated and Polish iff it is isomorphic to Rm × Zn iff it is locally compact. Proof. (i) We assume that for N ∈ M we have G/N = VN ⊕ FN ⊕ tor G/N where VN is a finite-dimensional vector group, FN is finitely generated free and tor G/N is the finite torsion  group of G/N. It follows that G∼may  be identified with a closed subgroup of P = N∈M G/N = V ×F ×C where V = N ∈M VN is a weakly complete vector   ∼ ∼ group, F = N ∈M FN and C = N ∈M tor G/N . Then comp(P ) = {0} × {0} × C

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235

and G ∩ comp(P ) = {0} since G is compact-free. The projection P → V × F is a proper, hence closed morphism, with kernel comp(P ), mapping G onto a closed subgroup of F which is isomorphic to G/(G ∩ comp(P )) ∼ = G. Since V is a product of copies of R and F is a product of copies of Z, assertion (i) is proved. (ii) If G is compactly generated and N ∈ N (G) then N is a closed subgroup such that G/N is compactly generated, hence is of the form specified in the proof of (i). (iii) If G is Polish and compactly generated, then it is locally compact by 5.22 (i),(iv) and thus, being compact-free, is isomorphic to Rm × Zn . Corollary 5.26. Any compactly generated, compact-free prodiscrete group is isomorphic to a closed subgroup of a group ZJ . If it is not of finite rank, then it is not isomorphic to a subgroup of ZN . This situation is illustrated by the example in Proposition 5.2. In the proof of Proposition 5.8 it was only needed that the Lie group quotients of the abelian Lie group G in question were compactly generated. This together with Proposition 5.25 yields at once: Corollary 5.27. If G is an abelian pro-Lie group whose Lie group quotients are compactly generated, then comp(G) is compact and G/ comp(G) is embeddable in a weakly complete vector group. This applies, in particular, to all compactly generated abelian pro-Lie groups.

Weakly Complete Topological Vector Spaces Revisited We begin with an observation showing that the idea of topologically compactly generated pro-Lie groups may not be very restrictive. Remark 5.28. A weakly complete vector group is topologically compactly generated. A group of the form ZJ for any set J is topologically compactly generated. Proof. See Remark A2.15 of Appendix 2. Lemma 5.29. For a weakly complete topological vector space W , the following statements are equivalent: (A) (B) (C) (D)

W W W W

is σ -compact. is locally compact. is finite-dimensional. is compactly generated.

Proof. See Proposition A2.17 of Appendix 2. ∼ RJ by CorolLet W be a weakly complete topological vector space. Then W = lary A2.9 of Appendix 2. The cardinal card J is called the topological dimension of W . (See [103].)

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Lemma 5.30. For a weakly complete topological vector space W , the following statements are equivalent: (i) W ∼ = RJ with card J ≤ ℵ0 . (ii) W is locally compact or is isomorphic to RN . (iii) W is finite-dimensional or is isomorphic to RN . (iv) W is second countable. (v) W is first countable. (vi) W is Polish. Proof. See Appendix 2, Proposition A2.18. Lemma 5.31. For a weakly complete topological vector space W , the following statements are equivalent: (a) W is separable. (b) W contains a dense vector subspace of countable linear dimension over R. (c) W is isomorphic as a topological vector space to RJ with card J ≤ 2ℵ0 . They are implied by the equivalent statements of Lemma 5.30. Proof. See Proposition A2.19 of Appendix 2. For the following theorem we recall the definition of the characteristic closed subgroup G1 = G0 + comp(G) ∼ = V × comp(G) of an abelian pro-Lie group in Vector Group Splitting Theorem 5.20 (v) for a vector group complement V (see Definition 5.13 (ii)). Theorem 5.32 (The Compact Generation Theorem for Abelian Pro-Lie Groups). (i) For a compactly generated abelian pro-Lie group G the characteristic closed subgroup comp(G) is compact and the characteristic closed subgroup G1 is locally compact. (ii) In particular, every vector group complement V is isomorphic to a euclidean group Rm for some m ∈ N0 = {0, 1, 2, . . . }. (iii) The factor group G/G1 is a compactly generated prodiscrete group without compact subgroups. If G/G1 is Polish, then G is locally compact and G∼ = Rm × comp(G) × Zn . (iv) Assume that G is a pro-Lie group containing a finitely generated abelian dense subgroup. Then comp(G) is compact and G ∼ = Rm comp(G) × Zn , m, n ∈ N. In particular, G is locally compact. (v) A finitely generated abelian pro-Lie group is discrete. Proof. By Theorem 5.20, G ∼ = V × H such that H0 is compact. The factors V and H are compactly generated as homomorphic images of G. By Lemma 5.29, V ∼ = Rn for some nonnegative integer n. By Proposition 5.8, comp(H ) is compact and by 5.20, comp(G) = comp(H ) and H0 ⊆ comp(H ). Thus G1 ∼ = V × comp(G) is locally compact. Also, H / comp(H ) is totally disconnected, and by Theorem 4.28 (iii) this

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quotient is a pro-Lie group and hence is prodiscrete by Proposition 4.23. If the factor group G/G1 is Polish, then it is finitely generated free by 5.25 (iii), and the remainder of (iii) follows. For a proof of (iv) notice first that G is obviously abelian, containing a dense finitely generated abelian subgroup. All quotient groups of G satisfy the hypothesis of (iv), notably each vector group complement of G has a dense finitely generated subgroup and therefore has a dense finite-dimensional vector group; finite-dimensional vector groups are locally compact and therefore complete. Thus each vector group complement is isomorphic to Rm for some m ∈ N. def Also G/G1 contains a dense finitely generated subgroup. Set F = G/G1 ; then comp(F ) = {0}, and by 5.10 (ii), F may be considered as a closed subgroup of a weakly complete vector group V . Then R ⊗ F is algebraically isomorphic with the R-linear span E of F . So dimR E is finite. Every finite-dimensional topological vector space over R is locally compact and is isomorphic to Rp . Any closed free subgroup of Rp , however, is discrete (see for instance [102], Appendix 1, the Closed Subgroups and Quotients Theorem A1.12). This shows that G/G1 is discrete and thus is finitely generated free. Since free discrete factor groups are always direct summands, it now follows from the Vector Group Splitting Theorem 5.20 that G ∼ = Rm × comp(G) × Zn . So comp(G) contains a finitely generated dense subgroup g1  ⊕ · · · ⊕ gk . (For the structure of finitely generated abelian groups see for instance [102], Appendix 1, the Fundamental Theorem of Finitely Generated Abelian Groups A1.11.) By Weil’s Lemma for Pro-Lie Groups 5.3, for each j = 1, . . . , k, either gj ∈ comp(G) or def gj  ∼ = Z. Hence gj  is compact for all j and then K = g1  + · · · + gk  is compact, hence contained in comp(G). On the other hand, K being closed and containing g1 , . . . , gk , contains comp(G). Thus K = comp(G) and this shows that comp(G) is compact. This completes the proof of (iv). Conclusion (v) follows now from (iv): If the group G is finitely generated, then it is countable. A countable locally compact group is discrete by the Baire Category Theorem (or by the existence of Haar measure). Hence a finitely generated locally compact group is discrete. The observation 5.32 (iv) is (in an obvious sense) a generalization of Weil’s Lemma for Pro-Lie Groups 5.3. It is not known whether a compactly generated abelian prodiscrete compact-free group is finitely generated free. By Proposition 5.2, the free abelian group of countably many generators has a nondiscrete nonmetric pro-Lie group topology and fails to be compactly generated.

The Duality Theory of Abelian Pro-Lie Groups The structure theory results we discussed permit us to derive results on the duality of abelian pro-Lie groups. Recall that this class contains the class of all locally compact abelian groups properly.

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 = Hom(G, T) denote its dual with For any topological abelian group G we let G the compact open topology. (See e.g. [102, Chapter 7].) There is a natural morphism   given by ηG (g)(χ ) = χ(g) which may or may not of abelian groups ηG : G → G be continuous; information regarding this issue is to be found for instance in [102], notably in Theorem 7.7. We shall call a topological abelian group semireflexive if   is bijective and reflexive if ηG is an isomorphism of topological groups; ηG : G → G in the latter case G is also said to have duality (see [102, Definition 7.8]). There is an example of a nondiscrete but prodiscrete abelian torsion group due to Banaszczyk [5, p. 159, Example 17.11], which is semireflexive but not reflexive (see also Chapter 14, Example Ac.2. Therefore we know that the category of abelian pro-Lie groups is not self-dual under Pontryagin duality. Recall the following observations. Proposition 5.33. The dual of a weakly complete vector group V is naturally isomorphic to the additive group of its vector space dual V  endowed with the finest locally convex vector space topology. The dual of the additive group of a real vector space E with the finest locally convex vector space topology is naturally isomorphic to the additive group of its vector space dual E  with the weak ∗-topology, that is the topology of pointwise convergence. Both a weakly complete vector group and the additive group of a real vector space given its finest locally convex topology are reflexive. Proof. See Appendix 2, Theorem A2.8.  is naturally Proposition 5.34. Let A and B be topological abelian groups. Then A×B   isomorphic to A × B, and if A and B are reflexive, then A × B is reflexive. Proof. Exercise . Exercise E5.4. Prove Proposition 5.34. [Hint. The proof is straightforward.] Proposition 5.35. Let G be an abelian pro-Lie group and let V be a vector group complement. Then G is reflexive, respectively, semireflexive iff G/V is reflexive, re is isomorphic to a product E × A spectively, semireflexive. The character group G where E is the additive group of a real vector space with its finest locally convex topology and A is the character group of an abelian pro-Lie group whose identity component is compact. Proof. By the Vector Group Splitting Theorem 5.20, G is isomorphic to V × H for a weakly complete vector group V and a pro-Lie group H with compact identity component H0 . The vector group V is reflexive by Proposition 5.33 and its dual is the additive group of a vector space with its finest locally convex vector space topology. The assertion now follows from Proposition 5.34.

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This result reduces the issue of duality of abelian pro-Lie groups to the class of abelian pro-Lie groups whose identity component is compact. Keep in mind for the following corollary that every connected topological group is almost connected. Thus it applies to all connected abelian pro-Lie groups. Theorem 5.36. Every almost connected abelian pro-Lie group is reflexive, and its character group is a direct sum of the additive topological group of a real vector space endowed with the finest locally convex topology and a discrete abelian group. Pontryagin duality establishes a contravariant functorial bijection between the categories of almost connected abelian pro-Lie groups and the full subcategory of the category of topological abelian groups containing all direct sums of vector groups with the finest locally convex topology and discrete abelian groups. Proof. By the Vector Group Splitting Theorem 5.20 according to which an abelian pro-Lie group is almost connected if and only if it is a direct product of a weakly complete vector group V and a compact group C. By Proposition 5.33, V is reflexive and its Pontryagin dual is a vector group obtained as the additive topological group of a real vector space which is given its finest locally convex topology. By the Pontryagin duality between the categories of compact abelian groups and discrete abelian groups (see for instance [102], notably Chapters 7 and 8), C is reflexive and its dual is a discrete abelian group. Then the preceding Proposition 5.35 proves the first assertion. The remaining duality statement is then automatic. Exercise E5.5. Prove the following result. Proposition. If G is an abelian pro-Lie group then the characters of G separate the   ηG (g)(χ ) = χ(g) is injective. points, and the canonical morphism ηG : G → G, [Hint. If g = 0, find N ∈ N (G) not containing g. Let q : G → G/N be the quotient morphism. On the abelian Lie group G/N find a character χ not annihilating q(g) = 0. Then χ  q is a character of G not annihilating g. The kernel of ηG consists of all g which are annihilated by all characters and this is singleton.] The following lemmas deal with not necessarily abelian pro-Lie groups. Lemma 5.37. Assume that G is a pro-Lie group. Then G0 is the intersection of open subgroups. Proof. By Proposition 3.31, G/G0 is protodiscrete. Hence the filter basis N (G/G0 ) consists of open normal subgroups and converges to the identity. If g ∈ G \ G0 , then gG0 = G0 and so there is a normal subgroup N of G such that N/G0 ∈ N (G/G0 ) not containing gG0 . But then N is an open normal subgroup not containing g. A pair (G, H ) consisting of a topological group G and a subgroup is said to have enough compact sets if for each compact subset K of G/H there is a compact subset C of G such that (CH )/H ⊇ K. (See [102, Definition 7.12 (ii)].)

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Lemma 5.38. Let G be a topological group and N a closed normal subgroup. (i) Assume that the quotient morphism G → G/N has local cross sections. Then for every N ∈ N (G), the pair (G, N ) has enough compact sets. (ii) If N ∈ N (G), then G → G/N has local cross sections provided G is a pro-Lie group or G is protodiscrete. Proof. (i) Assume that the quotient morphism qN : G → G/N has local cross sections. Then there is an open identity neighborhood U satisfying U N = U and a continuous function σ : U/N → G such that (qN |U )  σ : U/N → U/N is the identity. Let V be a compact identity neighborhood in the Lie group G/N which is contained in U/N and let V ◦ denote its interior.  Now let K be a compact subset of G/N . Then K ⊆ k∈K kV ◦ . Thus by the compactness of K we find elements k1 , . . . , kn ∈ K such that K ⊆ k1 V ∪ · · · ∪ kn V . −1 We pick elements gj ∈ qN (kj ), j = 1, . . . , n and set C = g1 σ (V ) ∪ · · · ∪ gn σ (V ). Then C is compact as a finite union of compact sets, and qN (C) =

n  j =1

qN (gj )qN σ (V ) =

n 

kj V ⊇ K.

j =1

(ii) If N ∈ N (G) and G is a pro-Lie group then qN has local cross sections by Theorem 4.22 (iv). If G is protodiscrete, and N ∈ N (G), then G/N is discrete; so local (and even global cross) sections exist trivially. Lemma 5.39. Let G be an abelian proto-Lie group and let N ∈ N (G). Assume that the pair (G, N) has enough compact sets. Then:  → N ⊥ , λG,N (χ )(g) = χ(g+N ), is an isomorphism (i) The morphism λG,N : G/N of topological groups. (ii) N ⊥ is a compactly generated locally compact abelian group. (iii) The group N ⊥ is compact if and only if N is open. Proof. (i) If the pair (G, N ) has enough compact subsets, the claim that λG,N is an isomorphism of topological groups follows at once from [102, Lemma 7.17 (i)]. (ii) Since G/N is an abelian Lie group, it is isomorphic to Rp × Tq × D for a  is discrete group D (see for instance [102, Corollary 7.58 (iii)]). Thus N ⊥ ∼ = G/N p q   isomorphic to R × Z × D, where D is a compact abelian group and every compact  for a discrete abelian group. Now Rp × D  is an almost abelian group occurs as D connected locally compact abelian group, and every almost connected locally compact abelian group is of this form by the Vector Group Splitting Lemma. (iii) follows from the fact that G/N is discrete iff N is open and the facts that the character group of a discrete group is compact and vice versa.  its charProposition 5.40. Assume that G is an abelian proto-Lie group and A = G acter group. Then:

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(i) A is the directed union of the family of locally compact subgroups N ⊥ , N ∈ N (G), each of which is algebraically generated by a compact subset. (ii) If G is protodiscrete then A = comp(A). Proof. (i) Let U be a zero-neighborhood of T in which every subgroup is singleton. If χ : G → T is a character of an abelian pro-Lie group G, then all subgroups contained in χ −1 (U ) are annihilated. Since lim N (G) = 0, there is an N that  ∈ N (G) such ⊥ = χ (N ) ⊆ U and thus χ (N) = {0}, that is, χ ∈ N ⊥ . Thus G N ∈N (G) N . Since N (G) = {N : N ∈ N (G)} is a filter basis, {N ⊥ : N ∈ N (G)} is an upwards directed family of closed subgroups of A. By Lemma 5.39, every N ⊥ is a locally compact and compactly generated subgroup. (ii)A pro-Lie group G is totally disconnected iff it is prodiscrete by Proposition 4.23. If this condition is satisfied, then N (G) is a basis for the filter of identity neighborhoods. Thus by 5.39 (iii), every N ⊥ is compact. Hence by (i), we get (ii). def

The character group A = (ZN ) of a power of copies of Z (see Example 5.1 (ii)(b)) is a sum of copies of the circle group T with the compact open topology; it is an arcwise connected topological abelian group. From Theorem 5.20 we know connected abelian pro-Lie groups are isomorphic to V × C for a weakly complete vector group V and a compact connected group C, and thus we can say without even detailing the topology on A that this group is not a pro-Lie group. It does, however, satisfy A = comp(A).  its dual. Then Proposition 5.41. Let G be an abelian pro-Lie group and A = G G⊥ = comp(A). 0 0 is Proof. (i) By 3.31, G/G0 is protodiscrete, and thus by Proposition 5.40 (ii), G/G ⊥  the union of its compact subgroups. Now λ : G/G0 → G0 , λ((χ )(g)) = χ(g + G0 ) is a bijective morphism of topological abelian groups. Since a bijective morphism of topological groups maps each compact subgroup isomorphically onto its image, G⊥ 0 is ⊆ comp(A). a union of compact subgroups. Thus G⊥ 0 ⊥ ⊥  (ii) κ : A/G⊥ 0 → G0 , κ(χ +G0 ) = χ (h) maps A/G0 continuously and bijectively   onto A|G0 ⊆ G0 . But by Theorem 5.36 we know that G0 ∼ = E ×D where E is a vector space with its finest locally convex vector space topology, and D, as the character group of a compact connected abelian group is a discrete and torsion-free abelian group (see 0 ) is singleton. Therefore A|G0 and thus A/G⊥ [102, Corollary 8.5]). Hence comp(G 0 do not contain any nondegenerate compact subgroups. It follows that comp(A) ⊆ G⊥ 0. . Therefore comp(A) = G⊥ 0

The Toral Homomorphic Images of an Abelian Pro-Lie Group From the duality of compact abelian groups we cite some results which we shall immediately apply to abelian pro-Lie groups. Let C denote a compact abelian group.

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 contains a free subgroup of maximal rank such Then the (discrete) character group C  that C/F is a torsion group. The subgroup F is not unique. Indeed, one way of get → Q ⊗ C,  f (x) = 1 ⊗ x, ting F is as follows. We consider the morphism f : C where the tensor product is taken over the ring of integers. Then ker f is exactly  of C,  and the tensor product Q ⊗ C  is a rational vector the torsion subgroup tor C   contains a space which is spanned over Q by f (C). Thus by Zorn’s Lemma, f (C)   Q-basis B, whose Z-span in Q ⊗ C is a free subgroup B whose rank is dimQ (Q ⊗ C), def  Let A = f −1 (B) be the full inverse image of this the torsion-free rank of C. ∼ free abelian group. Then A/ tor C = B is a free abelian group. Since free groups  ⊕ F . Then are projective, there is a free subgroup F of A such that A = tor C f |F : F → B is an isomorphism. The factor group A/F is torsion as is the group ∼   is torsion.   which is a torsion group. Thus C/F C/A ⊆ (Q ⊗ C)/B = f (C)/B def  The character group T = F is a torus, that is, a product of circles, and the dual of the   gives a surjective morphism f: C  → T by duality, inclusion morphism j : F → C  ∼ and C = C. Hence we have a surjective morphism of compact groups q : C → T such  R) → Hom(T, R) is an isomorphism, since C → Q⊗R that Hom( q , T) : Hom(C,  and T → Q ⊗ T induce isomorphisms  R) → Hom(C,  R), Hom(Q ⊗ C, Hom(Q ⊗ T , R) → Hom(T , R). Since L(C) = Hom(R, C), respectively, L(T ) = Hom(R, T ), is isomorphic to  R), respectively, Hom(T , R), we conclude that L(q) : L(C) → L(T ) is an Hom(C, def

isomorphism. This is also seen from the fact that D = ker q is the character group of  , a torsion group and is, therefore, a totally disconnected compact abelian group. C/F Thus the exact sequence q 0 → D → C −−→ T → 0 gives rise to an exact sequence L(q)

0 → L(D) = {0} → L(C) −−−→ L(T ) → 0. For further details see [102, Chapter 8, Definition 8.12–8.20]. In particular, it is shown in [102, Corollary 8.18], that for a connected compact abelian group G, the subgroups D may be chosen arbitrarily small. These considerations motivate Definition 5.42. Let G be an abelian topological group. A closed subgroup D of G will be called a cotorus subgroup if G/D is a torus and the quotient morphism q : G → G/D induces an isomorphism L(q) : L(G) → L(C/D) of Lie algebras. Any group isomorphic to G/D for a cotorus subgroup D of G will be called a toral homomorphic image of G We noted in the discussion preceding this definitions that, in a compact abelian group, cotorus subgroups always exist, and that they are zero-dimensional. We will now show that an abelian pro-Lie group G always has cotorus subgroups, and indeed arbitrarily small ones if G is connected.

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Proposition 5.43. An abelian pro-Lie group G always has at least one cotorus subgroup D, and D is prodiscrete. If G is connected, then D may be chosen arbitrarily small. Proof. By Theorem 5.20, G has a unique largest compact connected subgroup comp G0 = (comp G)0 . Let D1 be a cotorus subgroup of comp G0 . Set G1 = G/D1 . Then (G1 )0 ∼ = R I × comp(G1 )0 ∼ = RI × TJ for suitable sets I and J , I ∪J ∼ and L(G) = R . Now we apply Theorem 5.19 and deduce that G1 is the direct product of (G1 )0 and G1 /(G1 )0 ∼ = G/G0 . Thus we obtain a quotient morphism q : G → RI × TJ × G/G0 . Now we set D = q −1 (ZI × {0} × G/G0 ). Then G ∼ R I × TJ × G/G0 ∼ I = T × TJ . = I D Z × {0} × G/G0 Moreover D/D1 ∼ = ZI × (G/G0 ). That is, we have an exact sequence 0 → D1 → G → ZI × (G/G0 ) → 0, yielding an exact sequence 0 → L(D1 ) → L(G) → L(Z)I × L(G/G0 ) → 0 by Theorem 4.20 and the limit hence product preservation of L. Now L(D1 ) = {0} and L(G/G0 ) = {0} by Proposition 3.30. It follows that L(D) = {0} and that therefore, in view of 4.20, L(G) → L(G/D) is an isomorphism. So D is a cotorus subgroup of G. If G is connected, we may take D = D1 and since D1 may be taken arbitrarily small by [102, Corollary 8.18], we are finished. Corollary 5.44. If G is an abelian pro-Lie group then there is a quotient morphism q : G → TK for some set K such that L(q) is an isomorphism and there is a commutative diagram L(q) L(G) −−−→ R⏐K ⏐ ⏐exp K expG ⏐   T G −−−→ TK . q

Proof. This is just a reformulation of Proposition 5.43. Later, in Chapter 6 and Chapter 8 we shall see that the additive group of L(G)  the simply connected universal group of G. In [102, Chapter 9] it is is written as G, discussed that the fundamental group π1 (C) of a compact group is naturally isomorphic  Z). Thus there are many reflexive topological abelian groups G (definition to Hom(C,  R) agrees with in this chapter in the discussion preceding 5.33) for which Hom(G,  = Hom(C,  R) ∼ C = Rdim C where dim C is the cardinal representing the torsion for which  Among these there are those like G = C  or (Q ⊗ C) free rank of C. (dim C) ∼  π1 (G) = {0} while the torus homomorphic image T = F = (Z ) ∼ = Tdim C has

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the largest possible fundamental group π1 (T ) = Hom(T, Z) ∼ = Hom(Z(dim C) , Z) = dim C dim C ∼ Hom(Z, Z) . =Z To some extent this can be generalized to an arbitrary (not necessarily commutative) pro-Lie group G by considering cotorus subgroups D of the center Z(G). Proposition 5.45. Let G be a pro-Lie group and D a cotorus subgroup of its center. def Then there is a quotient map q : G → H = G/D such that L(q) : L(G) → L(H ) is an isomorphism and Z(H )0 is a torus. Moreover, if G is connected, Z(H ) itself is connected and is a torus, and D may be chosen arbitrarily small. Proof. By Theorem 4.20 we have an exact sequence L(q)

0 → L(D) → L(G) −−−→ L(H ) → 0. Since D is prodiscrete by Proposition 5.43, we conclude L(D) = {0} (see 4.23). Since bijective morphisms between weakly complete topological vector spaces are isomorphisms, we infer that L(q) is an isomorphism. Now let G be connected; consider the center Z(H ) of H and its full inverse image def

A = q −1 (Z(H )). Since Z(H ) is the center of H , we have comm(G × A) ⊆ D. Now comm(G × {x}) is connected for each x and contains 1. But D is totally disconnected, and thus comm(G × A) = {1}, that is, A ⊆ Z(G). Therefore Z(H ) = Z(G)/D = Z(H )0 since D is a cotorus subgroup of Z(G). Then, by Proposition 5.43, D may be chosen arbitrarily small. The main purpose of the cotorus subgroups, however, is that they permit us to prove for pro-Lie groups the existence of good substitutes for universal covering morphisms of Lie groups. The Resolution Theorem of Abelian Pro-Lie Groups Theorem 5.46. Let G be an abelian pro-Lie group and write G = V × H with a vector group complement V according to the Vector Group Splitting Theorem for Abelian ProLie Groups 5.20. Let D be a cotorus subgroup of H . Then D is a prodiscrete group and the morphism δ : L(G) × D → G,

δ(X, x) = (expG X)x,

is a quotient morphism inducing an isomorphism of Lie algebras and having the prodiscrete kernel {(X, exp −x) : exp X ∈ D} ∼ = exp−1 D ∼ = exp−1 (D ∩ comp(G0 )). Proof. Since L(G) = L(V ) × L(H ) and expV : L(V ) → V is an isomorphism, the assertion on G it true if it is true on H . Thus without loss of generality we now assume that V = {0}. Then G0 = (comp G)0 is compact and G = G0 D with a compact def

intersection  = D ∩ G0 , so that the morphism ϕ : G0 × D → G,

ϕ(g0 , d) = g0 d

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is a quotient morphism with kernel {(−d, d) : d ∈ } ∼ = . The compact subgroup  is a cotorus subgroup of G0 . So the morphism μ : L(G) ×  → G0 , μ(X, x) = (exp X)x, is a quotient morphism by the Resolution Theorem for Compact Groups ([102, Theorem 8.20]). So we conclude that ν : L(G) ×  × D → G,

ν(X, d1 , d) = (expG X)d1 d

which essentially is the composition ϕ  ν of two quotient morphisms is itself a quotient morphism. Since  is compact, the map κ :  × D → D,

κ(d1 , d) = d1 d

is a quotient morphism. So idL(G) ×κ : L(G) ×  × D → L(G) × D is a quotient map. Now ν = δ  (idL(G) ×κ) and under these circumstances, δ is a quotient morphism. So by Theorem 4.20, L(δ) is a quotient morphism. The assertion about ker δ is straightforward; so ker δ is totally disconnected, and thus L(ker δ) = {0} by 4.23. Hence L(δ) is an isomorphism. A quotient morphism with a prodiscrete kernel is the next best thing we can have to a covering morphism. We have two quotient morphisms with a totally disconnected kernel at work, permitting us to formulate the following “Sandwich Theorem”. The Sandwich Theorem for Abelian Pro-Lie Groups Corollary 5.47. Let G and D be as in Theorem 5.46 and write  = D ∩ comp(G)0 . Then there are quotient maps with totally disconnected kernels δ

ρ

L(G) × D −−→G−−→

G0 D ×  

such that G0 / ∼ = RI × TJ for suitable sets I and J where L(G) ∼ = RI ∪J and that the composition D G0 × ρ  δ : L(G) × D →   is the obvious map, namely, the product of the exponential function L(G) → G0 / and the quotient morphism D → D/. Proof. The information on δ was established in Theorem 5.46, and the information on ρ in Proposition 5.43 and its proof.

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Postscript By contrast with the category of locally compact abelian groups, the larger category AbproLieGr of abelian pro-Lie groups is closed in the category of all abelian topological groups and continuous group morphisms under the formation of arbitrary limits and the passing to closed subgroups. Each locally compact abelian group is a proLie group and thus arbitrary products of locally compact abelian groups and arbitrary closed subgroups of such products are abelian pro-Lie groups. Therefore, the category AbproLieGr of abelian pro-Lie groups is rather large. In the first section we recorded some individual examples so that the reader can form a first impression of the type of topological abelian groups we face. This addresses in particular those readers whose intuition is trained by studying locally compact abelian groups which are the daily bread of classical harmonic analysis and whose structure was exposed in [102], Chapters 7 and 8. The most remarkable of the prodiscrete examples presented here is the example of the free abelian group on countably infinitely many generators which supports a nonmetric totally disconnected pro-Lie group structure with bizarre properties (Proposition 5.1). One of the most prominent examples of connected abelian pro-Lie groups is the additive topological group of the dual vector space of an arbitrary real vector space, where the dual E  = HomR (E, R) of a real vector space is given the topology of pointwise convergence; this topology is also called the weak ∗-topology. It makes the dual into a complete topological vector space which is called a weakly complete topological vector space. Therefore its additive topological group is called a weakly complete vector group, and whenever it occurs as a subgroup of a topological group, it is called a weakly complete vector subgroup. Weakly complete vector groups have a perfect duality theory as is discussed in Appendix 2. The character group and the vector spaces dual of a topological vector space are naturally isomorphic. The compact open topology on the dual of a weakly complete vector space is the finest locally convex topology on the vector space dual. Since the character group of a weakly complete vector group is a vector space and thus is a direct sum of copies of R, every weakly complete vector group is isomorphic to a vector group of the form RJ for some set J . Thus a weakly complete vector group is locally compact if and only if it is finite-dimensional. The transfinite topological dimension of RJ as discussed in [103] is card J . Therefore the topological dimension of a weakly complete vector group agrees with the linear dimension of its dual. Pontryagin duality establishes a dual equivalence between the category of compact abelian groups and the category of (discrete) abelian groups. Therefore an abelian pro-Lie group is a direct product of a weakly complete vector group and a compact abelian group if and only if its dual is a direct sum of a real vector space given its finest locally convex topology, and some discrete abelian group. The main result of this chapter is that every abelian pro-Lie group G is isomorphic to the direct product W ×H of a weakly complete vector subgroup W of G and an abelian pro-Lie group H whose identity component H0 is compact and is the intersection of

Postscript

247

open subgroups. In particular, a connected abelian pro-Lie group is the direct product of a vector group and a compact group. Thus for such abelian pro-Lie groups the structure theory is completely reduced to the theory of weakly complete topological vector spaces on the one hand and to the familiar compact group situation on the other hand. From this result we derive that a necessary and sufficient condition for a connected abelian pro-Lie group to be locally compact is that it is algebraically generated by a compact set. In a somewhat loose way one could describe the situation by saying that abelian pro-Lie groups are very well understood if they are connected (or even not far from being connected) while the area of totally disconnected, that is, prodiscrete abelian groups, is a wide open field. We have observed that almost connected abelian pro-Lie groups are reflexive and have a lucid duality theory which is completely known due to the known vector space duality of weakly complete topological vector spaces and the duality of compact abelian groups. The Lie theory of abelian pro-Lie groups is likewise lucid: Since L(G) = L(G0 ) for any topological abelian group, it deals with the connected part of G only, and if G0 is an abelian pro-Lie group then the Vector Group Splitting Theorem says that G0 is algebraically and topologically the direct sum G0 = V ⊕ comp(G) of a weakly complete vector subgroup complement V and the unique largest compact subgroup comp(G); hence L(G) = L(V ) ⊕ L(comp(G), where expV : L(V ) → V is an isomorphism of a weakly complete vector group and where the exponential function expcomp(G) : L(comp(G)) → comp(G) is very well understood from [102], Chapters 7 and 8. From the theory of the exponential function of compact groups we know that, except for the case of a torus group G, the exponential function expG is rarely if ever surjective. In [102, Chapter 8, Theorem 8.20] it was shown in the so called “Resolution Theorem for Compact Abelian Groups” that this defect could be compensated for by finding a suitable totally disconnected closed subgroup D and a quotient morphism δ : L(G) × D → G such that L(δ) is an isomorphism of Lie algebras. It is noteworthy, that the Resolution Theorem remains intact for arbitrary abelian pro-Lie groups. In the end we have a sandwich theorem of the following nature: There is a totally disconnected subgroup D intersecting G0 in a compact totally disconnected subgroup , and there are quotient morphisms G D ×   inducing isomorphisms on the Lie algebra level such that the composition is the product of the exponential function δ

ρ

L(G) × D −−→ G −−→

expG

−−−→ L(G) ⏐ ⏐ ∼ = RI ∪J

G0  ⏐ ⏐∼ =

−−−→ RI × TJ quot

248

5 Abelian Pro-Lie Groups

and the quotient map D → D/. These results highlight once more the role of weakly complete topological vector spaces and compact groups play in the structure theory of abelian pro-Lie groups. We shall see more resolution theorems in the nonabelian environment in Chapter 12.

Chapter 6

Lie’s Third Fundamental Theorem

The concept of a prosimply connected pro-Lie group is introduced. It coincides with the purely topological concept of a simply connected topological group when the proLie group is in fact a Lie group. It is proved that for every pro-Lie algebra there is a prosimply connected pro-Lie group whose Lie algebra it is, and this prosimply connected pro-Lie group is unique up to isomorphism. In fact, all of this is done functorially and the universal properties arising in this way are studied. Prerequisites. We need the general theory of pro-Lie groups such as presented in Chapters 3 and 4. We also require from the reader a certain familiarity with basic category theory; the concept of an adjoint functor which was used extensively in Chapter 2 will be used essentially here as well.

Lie’s Third Fundamental Theorem for Pro-Lie Groups Consider a pro-Lie group G and the filter basis N (G) of closed normal subgroups N such that G/N is a Lie group. Then G ∈ N (G) and the singleton G/G is a simply connected Lie group. Let us define def

N S(G) = {N ∈ N (G) : G/N is simply connected}. Recall from Chapter 1, Definition 1.12ff. that a subset C of a directed set is cofinal if for each d ∈ D there is a c ∈ C such that d ≤ c. Definition 6.1. A pro-Lie group G is called prosimply connected if N S(G) is cofinal in N (G). We notice that for a prosimply connected pro-Lie group G, the set N S(G) is a filter basis and that G = limN ∈N S(G) G/N by the Cofinality Lemma 1.13 (or 1.21). From Corollary 4.22 (iii) it follows that a prosimply connected pro-Lie group is connected. A pro-Lie group G is a (finite-dimensional) Lie group if and only if {1} ∈ N (G). Accordingly, a Lie group is prosimply connected if and only if it is simply connected. For the simply connected Lie group R, the filter basis N (R) consists of R itself and all cyclic subgroups including {0}. For {0}, R = N ∈ N (R) the quotient R/N is a circle group and is therefore not simply connected. Therefore it would not have been a good idea, in Definition 6.1 to postulate that G/N is simply connected for all N ∈ N (G). Simple connectivity may be defined in several reasonable but nonequivalent ways, including firstly the way it was done in [102, Definition A2.6]. According to this

250

6 Lie’s Third Fundamental Theorem

definition, a space X is called simply connected if it has the following universal property: For any covering map p : E → B between topological spaces, any point e0 ∈ E and any continuous function f : X → B with p(e0 ) = f (x0 ) for some x0 ∈ X there is a continuous map f: X → E such that p  f = f and f(x0 ) = e0 . Let us say that a space is loopwise simply connected if it is connected and π1 (X) = 0. (See e.g. [102, Definition 8.60.]) Examples 6.2. (i) Any product of simply connected finite-dimensional Lie groups is simply connected in any sense and is also a prosimply connected pro-Lie group. (ii) Let G be a compact connected abelian group; then G is a pro-Lie group (see [102, Corollary 2.43]). Every Lie group quotient G/N , N ∈ N (G) is a torus. Thus G is never prosimply connected.  Z). Thus if G is the 1-dimensional By [102, Theorem 8.62], π1 (G) ∼ = Hom(G, ∼ , then G solenoid Q = Q and π1 (G) = {0}, that is, G is loopwise simply connected. However, no compact connected abelian group is simply connected (see [102, Theorem 9.29]). While the definition of prosimple connectedness of a pro-Lie group G does not agree with any of the traditional concepts of simple connectivity, we shall show in Corollary 8.15 in a later chapter that it does agree with the concept of simple connectivity expressed in terms of the lifting property described in the paragraph preceding 6.2 above (see e.g. [102, Definition A2.6]), and the one expressed by the vanishing of the fundamental group π1 (G). Proposition 6.3. Let G be a pro-Lie group. (a) Assume N ∈ N (G). Consider the following two statements: (i) N ∈ N S(G). (ii) N is connected. Then (i) ⇒ (ii), and if G is prosimply connected, both statements are equivalent. (b) Assume that N (G) contains a cofinal subset C of connected subgroups N of G. Then G is locally connected. (c) Any prosimply connected pro-Lie group is locally connected. Proof. We begin by proving (a). (i) ⇒ (ii): Let N ∈ N S(G); then G/N is a simply connected Lie group. Since the identity component N0 of N is characteristic and closed, it is normal in G, and G/N0 is a pro-Lie group by the Quotient Theorem for Pro-Lie Groups Revisited 4.28 (i). Now N/N0 is a pro-Lie group as a closed subgroup of G/N0 by the Closed Subgroup Theorem 3.35. Since N/N0 is a totally disconnected normal subgroup of a connected group, it is central by Lemma 12.55. Thus N/N0 is a totally disconnected abelian proLie group and so is prodiscrete by Proposition 4.23. Hence there is an open subgroup M of N such that M/N0 , being central in G/N0 , is normal in G/N0 , whence M is normal in G. The quotient morphism q : G/M → G/N has the discrete kernel N/M and therefore is a covering morphism. Since G/N is simply connected, q is an

Lie’s Third Fundamental Theorem for Pro-Lie Groups

251

isomorphism, that is M = N. Therefore, the prodiscrete group N/N0 has no open normal subgroups and thus is singleton. Hence N = N0 which is what we had to show. (ii) ⇒ (i): Assume that G is prosimply connected and that N is connected. Since N S(G) is cofinal in N (G), there is an M such that M ⊆ N. Thus there is a quotient morphism q : G/M → G/N. Now assume that ϕ : (G/N ) → G/N is the universal covering morphism of connected Lie groups. Since G/M is simply connected, by the lifting property, there is a morphism f : G/M → (G/N ) such that ϕ  f = q. Let qM : G → G/M denote the quotient morphism. Since ϕ is a covering morphism and f is the lifting of a quotient morphism, f and thus f  qM are quotient morphism. Also (G/N), being the universal covering of the Lie group G/N , is a Lie group. Thus def N1 = ker(f  qN ) ∈ N (G) and N1 ⊆ N. Also, N/N1 ∼ = ker ϕ, and the kernel ker ϕ of the universal covering of G/N is discrete. Hence N1 is open in the connected group N, whence N1 = N. Thus ϕ is an isomorphism and so G/N is a simply connected Lie group, that is, N ∈ N S(G). (b) Assume the existence of C as stated in (b). We have to show that G has arbitrarily small connected identity neighborhoods. Indeed G has arbitrarily small identity neighborhoods V containing an N ∈ C such that V N = V and that V /N is an open cell neighborhood of the identity in the Lie group G/N . Let W be any nonempty open closed subset of V . Since N is connected, for all w ∈ W we get wN ⊆ W . Hence W N = W . Then W/N is nonempty open closed in the open cell V /N and thus W = V . So V is connected, and (b) is proved. (c) is now an immediate consequence of (a) and (b). If G = SO(3)N , then the members of N (G) are precisely the cofinite partial products of G, and all of these are connected. But N S(G) = {G}. One notices directly, that G is locally connected. At any rate, for any prosimply connected pro-Lie group G, by Proposition 6.3, all members of N S(G) are connected. The occurrence of prosimply connected pro-Lie groups is not a rare event. Indeed, Theorem 6.4 below asserts the existence of a plethora of prosimply connected pro-Lie groups. Subsequent results will place the construction into a more functorial frame. But first we recall a basic result of Lie group theory utilizing the universal property of simple connectivity: Lemma L. Let S be a simply connected Lie group and L a Lie group, and assume that there is a morphism α : L(S) → L(L). Then there is a unique morphism of Lie groups a : S → L such that L(a) = α. The following diagram commutes: α

L(S) −−→ L(L) ⏐ ⏐ ⏐exp expS ⏐   L S −−→ L.

(SC)

a

Proof (Indication). The proof is standard Lie theory: Let B be a convex zero neighborhood of L(S) such that expS induces a homeomorphism expS |B : B → V onto an identity neighborhood of S. Then expL  α  (expS |B)−1 : B → L is a local morphism

252

6 Lie’s Third Fundamental Theorem

which by the simple connectivity of S extends uniquely to a morphism a : S → L such that the diagram (SC) commutes (see e.g. [102, Corollary A2.26]; note that U is assumed to be connected there). This completes the indication of proof of the lemma. Theorem 6.4 (Lie’s Third Fundamental Theorem for Pro-Lie Algebras). Let g be a pro-Lie algebra. Then there is a prosimply connected pro-Lie group (g) for which there is an isomorphism ηg : g → L((g)). Proof. For each i ∈  (g) there is (up to isomorphism) a unique simply connected Lie group Gi such that there is an isomorphism ϕi : g/i → L(Gi ). Since all Gi are simply connected, for i ⊇ j in  (g) by Lemma L, the quotient map quotij : g/j → g/i induces a unique surjective morphism fij : Gj → Gi such that the following diagram commutes: quotij

←−−−−− g/j g/i ⏐ ⏐ ⏐ ⏐ϕ ϕi   j L(G −−−− L(G ⏐ i ) ←− ⏐ j) L(fij ) ⏐expG expG ⏐  i j Gi ←−−−−− Gj . fij

Now {fij : Gj → Gi | (i, j) ∈  (g) ×  (g), j ⊇ i} def

is a projective system of finite-dimensional Lie groups. Let (g) = limi∈ (g) Gi be its limit and let fi : (g) → Gi be the limit maps. We claim that (g) is a prodef

Lie group. Indeed the filter basis N S(G) = {ker fi | i ∈  (g)} converges to 1 by Theorem 1.27 (ii), and (g) is complete as the limit of complete groups, whence γGN S(G) : (g) → GN S(G) is an isomorphism by 1.30 and 1.33, inverting the natural isomorphism η : GN S(G) → (g) of 1.27 (ii). For each i ∈  (g), the following diagram commutes: η / (g) GN S(G) MMM MMM MMMfi νker fi fi MMM MMM M &   / Gj , (g)/ ker fi  fi

where η = γG−1 . The filter basis N S(G) is cofinal in N ((g)). The fi are quotient N S(G) morphisms by 1.27 (iii). Therefore each fi : (g)/ ker fi → Gi is an isomorphism of finite-dimensional Lie groups. Thus all quotients (g)/ ker fi are simply connected. Now GN is a pro-Lie group by 1.29 and 1.40. Thus (g) is a pro-Lie group as claimed. By Definition 6.1 and what we showed above, it is simply connected. Since L is a continuous functor by 2.25 (ii) we have an isomorphism α : L((g)) → limi∈ (g) L(Gi ) and thus we obtain an isomorphism ηg as a composition

Lie’s Third Fundamental Theorem for Pro-Lie Groups γg

253

α −1

limi∈ (g) ϕi

g −−→ limi∈ (g) g/i −−−−−−→ limi∈ (g) L(Gi ) −−→ L(limi∈ (g) Gi ) = L((g)). This completes the proof. Exercise E6.1. Prove the following assertion: Let g be an abelian pro-Lie algebra, that is, a weakly complete topological vector space; then the additive group (g) of g is a simply connected pro-Lie group such that L((g)) = Hom(R, (g)) ∼ = g. This theorem has a functorial enhancement of considerable elegance. We have to refer to the concept of adjoint functors. (See e.g. [102, Appendix 2, Definition A3.29].) Lie’s Third Fundamental Theorem, Functorial Version Theorem 6.5. The construction  of Theorem 6.4 assigning to a pro-Lie algebra a prosimply connected pro-Lie group extends to a functor  : proLieAlg → proLieGr which is left adjoint to the Lie algebra functor L : proLieGr → proLieAlg. Proof. By TheoremA3.28 of [102] we have to verify the universal property best pursued by the following diagram. proLieAlg ηg

proLieGr

g ⏐ ⏐ ∀f 

−−−−→

L((g)) ⏐ ⏐  L(f )

L(H )

−−−−→

L(H )

idL(H )

(g) ⏐ ⏐  ∃!f

()

H

For each morphism of topological Lie algebras f : g → L(H ) for a pro-Lie group H we have to find a unique morphism of topological groups f  : (g) → H such that L(f  )  ηg = f . Thus assume we are given f : g → L(H ). For each N ∈ N (H ) let qN : H → H /N be the quotient map. Then incl

L(qN )

0 → L(N ) −−−−→ L(H ) −−−−→ L(H /N ) is exact since L preserves kernels (see e.g. 3.22). Since dim L(H /N ) < ∞ we have def

L(N) ∈  (L)(H ). Then i(N ) = f −1 (L(N )) ∈  (g) and f induces a morphism of topological Lie algebras fN : g/i(N ) → L(G/N ). By definition, (g) = limi∈ (g) Gi for a projective system of simply connected Lie groups Gi such that there is an isomorphism ϕi : g/i → L(Gi ). Since Gi(N ) is simply connected there is a unique morphism of topological groups gN : Gi(N ) → H /N such that the following diagram is commutative: −1 ϕi(N) fN L(G⏐i(N ) ) −−−−→ g/i(N ) −−−−→ L(H⏐/N ) ⏐ ⏐exp expG  H /N i(N)  Gi(N ) −−−−→ Gi(N ) −−−−→ H /N; idGi(N)

gN

254

6 Lie’s Third Fundamental Theorem

−1 consequently, L(gN ) = ϕi(N )  fN . Let

pi : αi : βi :

(g) −→ Gi , limj∈ (g) g/j −→ g/i, limj∈ (g) L(Gj ) −→ L(Gi )

be the limit morphisms. Then we have, in particular, a commutative diagram in which lim is short for limi∈ (g) : λg

limi ϕi

L(qN )

id

g −−−−→ lim −−−−→ lim L(G g ⏐ ⏐i ⏐ i ) ←−−− ⏐ ⏐β αi(N) ⏐ =   i(N) id g g − − − − → − − − − → L(G i(N ⏐ ⏐) ⏐i(N ) ) ←−−− ϕi(N) quot ⏐ ⏐ ⏐L(g ) f fN   i(N) id L(H ) −−−−→ L(H /N ) −−−−→ L(H /N ) ←−−− α

L((g)) ⏐ ⏐L(p (N ))  i L(G⏐i(N ) ) ⏐L(g )  i(N) L(H /N ).

def

For each N ∈ N (H ) we set fN = gN pi(N ) : (g) → H /N. The map α −1 (limi ϕi )λg from the top left corner of the diagram to the top right corner is ηg by the definition of ηg in the proof of 6.4. Thus the outside of the diagram yields L(fN )  ηg = L(qn )  f.

(1)

If N1 ⊇ N2 in N (H ) we straightforwardly see that fN 2 = qN1 N2 fN 1 with the obvious quotient morphism qN1 N2 : H /N2 → H /N1 . Then the universal property of the limit H = limN∈N (H ) H /N attaches to the cone {fN : L((g)) → L(H /N ) | N ∈ N (H )} a unique morphism f  : (g) → H such that qN  f  = fN for all N ∈ N (H ) and thus L(qN )  L(f  ) = L(fN ). (2) So from (1) and (2) we have L(qN )L(f  )ηg = L(qN )f for all N ∈ N (H ). Since L is limit preserving, the L(qN ) form a limit cone and thus separate the points. Therefore L(f  )  ηg = f. The question of the uniqueness of f  remains. Assume also that f  : (g) → H satisfies L(f  )  ηg = f = L(f  )  ηg . Since ηg is an epic (being even an isomorphism) this means L(f  ) = L(f  ). Now consider the two superimposed commutative diagrams L(f  )

L((g)) ⏐ ⏐ exp(g) ⏐  (g)

−−−−→ −−−−→  L(f ) f

−−−−→ −−−− →  f

L(H ⏐ ) ⏐ ⏐expH  H

and conclude f   exp(g) = f   exp(g) . Since the exponential function of a pro-Lie group is generating, we conclude f  = f  , and this completes the proof.

Lie’s Third Fundamental Theorem for Pro-Lie Groups

255

From purely category theoretical considerations, we obtain the following version of the universal properties of the adjoint functors L and  (see e.g. [102, Appendix 3, Proposition A3.36]). There is a natural morphism πG : (L(G)) → G of pro-Lie groups with the following universal property: Given a pro-Lie group G and any morphism f : (h) → G for some pro-Lie algebra h, there is a unique morphism f  : h → L(G) of pro-Lie algebras such that f = πG  (f  ). proLieAlg

proLieGr πG

L(G)  ⏐ ∃!f  ⏐

(L(G))  ⏐  ⏐(f )

−−−−→

G  ⏐∀f ⏐

h

(h)

−−−−→

(h).

id(h)

(⊥)

However, with the information from Chapter 4 that L preserves quotient morphisms it serves a useful purpose if we elaborate on this situation in the concrete case of pro-Lie groups. The Theorem on the Reflection into the Category of Prosimply Connected Pro-Lie Groups Theorem 6.6. (i) Let G be a pro-Lie group. Then there is a prosimply connected  and a pro-open morphism pro-Lie group G  −−−→ G πG : G which induces an isomorphism of pro-Lie algebras ∼ =

 −−−→ L(G). L(πG ) : L(G)  extends to a functor proLieGr → proLieGr onto the (ii) The assignment G  → G full subcategory proSimpConLieGr of prosimply connected pro-Lie groups, which is right adjoint to the inclusion functor proSimpConLieGr → proLieGr. In particular,  → G is a natural transformation with a prodiscrete kernel P (G) def = ker πG . πG : G (iii) The following statements (a) and (b) are equivalent:  → G is an isomorphism. (a) πG : G (b) G is prosimply connected. (iv) The image of πG is dense in G0 . (v) Let f : G → H be a morphism of connected pro-Lie groups such that L(f ) is an isomorphism and H is prosimply connected. Then f is an isomorphism.

256

6 Lie’s Third Fundamental Theorem

(vi) By a slight abuse of notation denote the corestriction of the functor  to the category of prosimply connected pro-Lie groups by  : proLieAlg → proSimpConLieGr and the restriction of the functor L to the category of prosimply connected pro-Lie groups by L : proSimpConLieGr → proLieAlg. Then the two functors  : proLieAlg → proSimpConLieGr,

L : proSimpConLieGr → proLieAlg

implement an equivalence of categories. (vii) If f : G → H is a bijective morphism of connected pro-Lie groups, then  → H  is L(f ) : L(G) → L(H ) is an isomorphism of pro-Lie algebras and f: G an isomorphism of prosimply connected pro-Lie algebras. In particular, a bijective morphism of prosimply connected pro-Lie groups is an isomorphism.  = (g) be the pro-Lie Proof. (i) Set g = L(G); then g is a pro-Lie algebra. We let G  group constructed in Theorem 6.1; then there is an isomorphism ηg : L(G) → L(G).  We construct πG : G → G. Let M ∈ N (G) and note that the quotient morphism qM : G → G/M induces an exact sequence L(qM )

incl

0 → L(M) −−−−→ L(G) = g −−−−→ L(G/M) → 0 ∼ limi∈ (g) g/i and L(qM ) induces is exact by Corollary 4.16. Hence L(M) ∈  (g) = †  = limi∈ (g) Gi for a family of an isomorphism qM : g/L(M) → L(G/M). Recall G simply connected finite-dimensional Lie groups Gi with an isomorphism ϕi : g/i → L(Gi ). In particular, there is an isomorphism ϕL(M) : g/L(M) → L(GL(M) ). Then † −1 qM  ϕL(M) : L(GL(M) ) → L(G/M) is an isomorphism. Hence the two Lie groups GL(M) and G/M are locally isomorphic (see e.g. [102, Theorem 5.42]; this theorem does not depend on the linearity of the Lie groups involved); thus there is an open connected identity neighborhood U of GL(M) , an open identity neighborhood V of G/M, and a homeomorphism μ0 : U → V satisfying μ0 (uu ) = μ0 (u)μ0 (u ) for u, u , uu ∈ U . Since GL(M) is a finite-dimensional simply connected Lie group, there is a unique morphism μM : GL(M) → G/M extending μ0 and implementing a local isomorphism (see e.g. [102, Corollary A2.26], as in the proof of Lemma L used in the proof of Theorem 6.4); the following diagram is commutative: L(G ⏐ i) ⏐ expG L(M)  GL(M)

L(μM )

−−−−→ L(G/M) ⏐ ⏐exp  G/M −−−−→ G/M, μM

 → Gi be a limit morphism. Then for each and L(μM ) =  Let pi : G  → G/N , αM = μM  pL(M) . The cone M ∈ N (G) we have a morphism αM : G property qM1 M2  αM2 = αM1 for M1 ⊇ M2 in N (G) is straightforwardly verified. Since G is a pro-Lie group, λG : G → limN ∈N (G) G/N is an isomorphism, and by  → G such that the universal property of the limit, there is a unique morphism πG : G qM  πG = αM , i.e., that the following diagram commutes for all M ∈ N (G): † qM

−1 ϕL(M) .

Lie’s Third Fundamental Theorem for Pro-Lie Groups

 G ⏐ ⏐ pM  GL(M)

πG

−−−−→ −−−−→ μM

G ⏐ ⏐q M G/M.

257

(3)

Then πG is pro-open and for each M there is a commutative diagram L(πG )

 −−−−→ L(⏐G) ⏐ L(pM ) L(GL(M) ) −−−−→ L(μM )

L(G) ⏐ ⏐L(q )  M L(G/M).

(4)

Recall that L(μM ) is an isomorphism of Lie algebras. The functor L preserves limits. Thus L(γG )

g = L(G) −−−−→

lim

M∈N (G)

† −1 limM∈N (G) (qM )

L(G/M) −−−−−−−−−−→

lim

M∈N (G)

L(G)/L(M)

is an isomorphism. This implies that the filter basis {L(M) ∈  (g) | M ∈ N (G)} converges to 0 in g. From Proposition 1.40 it follows that it is cofinal in  (g) and that,  may be identified with limM∈N (G) GL(M) by the Cofinality Lemma 1.21, therefore G in such a way that the pL(M) are the limit maps. As a consequence, in diagram (4), the vertical maps are the respective limit maps, and since the L(μM ) are isomorphisms it follows that L(πG ) is an isomorphism. This concludes the construction of πG and finishes the proof of (i).  = (L(G)) with the back adjunction πG (see e.g. [102, We have reconstructed G Definition A3.37]) in (⊥).  extends to a right adjoint functor to the inclusion (ii) In order to prove that G  → G  functor and that πG : G → G is a natural transformation, it again suffices to verify the universal property in a fashion analogous to that in the proof of 6.5: proSimpConLieGr

 G  ⏐  f ⏐ H

proLieGr πG  −−− G −→  ⏐  ⏐f

H

−−−−→ idH

G  ⏐f ⏐ H

We have to show that for each morphism of pro-Lie groups f : H → G from a simply  of topological connected pro-Lie group H there is a unique morphism f  : H → G groups such that π  f  = f . So let f : H → G be given and let M ∈ N (G). Since L(μM ) is an isomorphism, μM : GL(M) → G/M implements a covering morphism of the identity component (G/M)0 . Since H is simply connected, by Lemma L, qM f

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6 Lie’s Third Fundamental Theorem

has a unique lifting νM : H → GL(M) such that H ⏐ ⏐ f G

νM

−−−−→ GL(M) ⏐ ⏐μ  M −−−−→ G/M

(5)

qM

is commutative for all M ∈ N (G). We recall that {L(M) | M ∈ N (G)} is cofinal in  (g) and that therefore  = lim GL(M) G M∈N (G)

up to a natural identification, where the limit maps are denoted pM . If M1 ⊆ M2 then straightforwardly pM1 M2 νM2 = νM1 . Thus by the universal property of the limit, there  such that pM f  = νM for all M ∈ N (G). Then is a unique morphism f  : H → G    qM πG f = μM pM f = μM fM = qM f for all M ∈ N (G). Since the qM separate points, πG f  = f . Thus we have the existence of f  ; it remains to show that it is unique. If also πG f  = f = πG f  , then κ : H → ker πG , κ(h) = f  (h)(f  (h)−1 , is a continuous function from a connected space to ker πG . But L(ker πG ) = ker L(πG ) = {0} since L(πG ) is an isomorphism. Hence ker πG is totally disconnected, equivalently, prodiscrete by 3.30 (iv). Thus κ is constant and this proves f  = f  and completes the proof of (ii).  is prosimply connected. Let us prove (b) ⇒ (a): (iii) Trivially, (a) ⇒ (b) since G  such that We apply (i) with H = G and f = idG . This gives us a ν : G → G   being the πG ν idG . Thus G is a retract of G in proLieGr, the semidirect factor of G totally disconnected group ker πG . As a topological direct factor of a connected space, it must be singleton. Thus πG is a monic retraction and therefore is an isomorphism. (See [102, Remark A3.13 (ii)].)  → L(G) is an isomorphism. Then (iv) From (i) we know that L(πG ) : L(G) 0 ) by 4.22 (ii). Since G 0 = G  as G  is prosimply connected, the assertion G0 = πG (G is proved. The following commutative diagram may be helpful to visualize the situation: L(πG )

 −−−−→ L(G) L(⏐G) ⏐ ⏐ ⏐exp expG˜   G  G. G −−−−→ πG

(v) Let g = L(G); since H is prosimply connected we may write H = (h) and identify h and L((h)) via the isomorphism ηh in () in the proof of 6.5. Then by the universal property expressed in (), there is a unique morphism f ∗ : H = (h) → G such that L(f ∗ ) = L(f )−1 . Now L(f ∗  f ) = L(f ∗ )  L(f ) = idL(G) = L(idG ), and by the uniqueness in the universal property, we have f ∗  f = idG . Similarly, f  f ∗ = idH . Hence f ∗ = f −1 and f is an isomorphism. (vi) Let g be a pro-Lie algebra. Then ηG : g → L((g)) is an isomorphism for each pro-Lie algebra g by 6.4, and πG : (L(G)) → G is an isomorphism for each

Lie’s Third Fundamental Theorem for Pro-Lie Groups

259

prosimply connected pro-Lie group G by (iii) above. These statements together prove Claim (vi). (vii) By Theorem 2.25, the functor L preserves kernels; hence the injectivity of f implies the injectivity of L(f ). By the connectivity of G and by Theorem 4.22 (ii)(b), the morphism L(f ) is a surjective continuous and open morphism of pro-Lie algebras and is therefore an isomorphism. As does any functor, the functor  : proLieAlg → →H  proSimpConLieGr preserves isomorphisms. Since = L we obtain that f: G is an isomorphism. The last assertion then follows from this and (iii) above. We shall call the kernel P (G) of G the Poincaré group of G. In the case that G is a Lie group, this notation agrees with the classical one and thus correctly extends this  → G as the universal morphism of G; in the concept. We also shall refer to πG : G  will be called case of Lie groups this is the universal covering morphism. The group G the universal group. (Cf. Theorem 8.21 below.)  is isomorphic to the additive group of L(G) If G is a compact abelian group, then G and πG is equivalent to expG : L(G) → G which is a morphism of topological groups  and G will be quite different in almost all aspects; in this case. Therefore, in general, G this is a fact one knows from Chapter 8 in [102], on compact abelian groups. We note that 6.6 (iv) says that the categories proLieAlg of pro-Lie algebras and continuous Lie algebra morphisms and the full subcategory of the category of topological groups and continuous group homomorphisms consisting of prosimply connected proLie groups are faithful images of each other; whatever is true in one of them, grosso modo, holds in the other as well, and the two functors  and L achieve the translation. Since  : proLieAlg → proLieGr is a left adjoint, it preserves colimits, it preserves cokernels. Before we exploit this useful fact, let us quickly review the concept of a cokernel. In a pointed category (that is, a category with a terminal and initial object and thus with zero morphisms; see e.g. [102, Appendix 3]) the kernel of a morphism f : X → Y is morphism e : K → X which is the equalizer of f and the zero morphism X → Y (in the category of groups, for instance, the constant morphism). As an equalizer, a kernel K is a limit. Dually, the cokernel of a morphism f : X → Y is a morphism c : Y → C which is the coequalizer of f and the zero morphism. In the category of Hausdorff topological groups, the cokernel c is the quotient morphism c : Y → Y /N where N is the smallest closed normal subgroup containing f (X). In fact, whenever ϕ : Y → Z is a morphism of topological groups such that ϕ f : X → Z is the constant morphism, then f (X) ⊆ ker ϕ and thus N ⊆ ker ϕ. So ϕ factors uniquely through the quotient morphism c : Y → Y /N. The Strict Exactness Theorem for  Theorem 6.7. Assume that

e

f

n −−−→ g −−−→ h is a strict exact sequence of morphisms of pro-Lie algebras; that is, assume that im e = ker f . Then (f ) (e) (n) −−−−→ (g) −−−−→ (h)

260

6 Lie’s Third Fundamental Theorem

is a strict exact sequence of morphisms of pro-Lie groups, that is, the image of (e) : (n) → (g) is the kernel of (f ) : (g) → (h) and this kernel is prosimply connected. Proof. The morphism f factors through g/ ker f and ker f is also the kernel of the quotient morphism g → g/ ker f . We may therefore replace f by the quotient morphism g → g/ ker f and will therefore assume, without losing generality, that f is itself a quotient morphism. It is then the cokernel of e and since  preserves cokernels, (q) is the cokernel of (e) in proLieGr. We let K = ker (f ). Then the quotient morphism p : (g) → (g)/K is the cokernel of (e) in TopGr. Let Q ⊇ (g)/K be the completion and j : (g)/K → Q the inclusion. Then the coextension j  p : (g) → Q is the cokernel of (e) in proLieGr. By the uniqueness of cokernels, there is an isomorphism ι : (h) → Q such that ι  (f ) = j  p: (f ) (g) −−−−→ ((h) ⏐ ⏐ ⏐ ⏐ι p (∗)  (g)/K −−−−→ Q. j

By the Strict Exactness Theorem for L 4.20, the map L(j ) : L((g)/K) → L(Q) is an isomorphism. Trivially, L(ι) : L((g)) → L(Q) is an isomorphism. Since L preserves kernels, we have L(K) = ker L((f )) ⊆ L((g)). In the commutative diagram e

f

L((e))

L((f ))

n −−−−→ g −−−−→ h ⏐ ⏐ ⏐ ⏐ ⏐ ηh ηg ⏐ ηn    L((n)) −−−−→ L((g)) −−−−→ L((h)) the vertical arrows are isomorphisms by 6.5 and thus, if we identify pro-Lie algebras def

via the natural automorphism η, we see that k = L(K) equals e(n). This means that the corestriction of (e) to K is the unique morphism e : (n) → K such that L(e ) : n = L((n)) → k = e(n) is the corestriction of e to k. The situation is illustrated by the following commutative diagram L(e )

−−−−→ ⏐k n ⏐ exp(n) ⏐ expK ⏐   (n) −−−− → K  e

−−−−→

L(i)

f

i

(f )

g −−−−→ ⏐ ⏐exp  (g) −−−−→ (g) −−−−→

h ⏐ ⏐exp  (h) (h),

where i : K → (g) denotes the inclusion morphism. Thus if we identify L((g)/K) with g/k (via 4.20), and η with the identity then L(p) identifies with the quotient map g → g/k and we obtain from (∗) upon applying the functor L the commuting diagram f

g −−−−→ ⏐ ⏐ quot g/k −−−−→ L(j )

h ⏐ ⏐L(ι)  L(Q).

Lie’s Third Fundamental Theorem for Pro-Lie Groups

261

We claim that K is connected, that is K0 = K. Let j ∗ : (g)/K0 → Q∗ denote completion. Then L(j ∗ ) : L((g)/K0 ) = g/k → L(Q∗ ) is an isomorphism and the surjective morphism s : (g)/K0 → (g)/K, s(xK0 ) = xK induces a unique extension s ∗ : Q∗ → Q which in turn induces an isomorphism L(s) : L(Q∗ ) → L(Q). But Q∗ is a connected pro-Lie group since (g)/N0 is connected, and Q is prosimply connected since Q ∼ = (h). As a consequence s ∗ is an isomorphism of topological groups by 6.6 (v). Since the morphism s is a restriction of s ∗ , it is injective. As it is also surjective it is bijective. But this means K = K0 as asserted. Now we apply the Closed Subgroup Theorem for Projective Limits 1.34 for (g) in place of G, for N S((g)) in place of N and K in place of H . Then N ∼ = limM∈N S((g)) K/(K ∩ M), where K/(K ∩ M) ∼ = KM/M. Now for each M ∈ N S((g)), the quotient group K/(K ∩ M) is a quotient of a connected pro-Lie group, and is therefore a connected proto-Lie group by 4.1; on the other hand, KM/M is a subgroup of the Lie group (g)/M and therefore has an identity neighborhood, in which the singleton subgroup is the only subgroup. Thus KM/M ∼ = K/(K ∩ M) is a normal Lie subgroup of the simply connected Lie group (g)/M and is therefore simply connected (see e.g. [17, Chap. 3, §6, no 6, Proposition 14] or [85, p. 224, III.3.17]). So K is a projective limit of simply connected Lie groups and is therefore prosimply connected by 6.1. Hence πK : (k) → K is an isomorphism of topological groups by 6.6 (iii). If k : k → g is the inclusion, then e = k  L(e ) and we note a commutative diagram (L(e ))

(i)

e

incl

−−−−→ (k) −−−−→ (n) ⏐ ⏐ ⏐ ⏐ π K0  id (n) −−−− → K0 −−−−→ 

(g) ⏐ ⏐ id (g).

This means that im (e) = K. Thus im (e) = ker (f ) and this is exactly the claim of the theorem. Theorem on the Quotient Preservation in Lie’s Third Theorem Corollary 6.8. (i) Assume that f : g → h is a surjective morphism of pro-Lie algebras. Then the morphism of topological groups (f ) : (g) → (h) is a quotient morphism. In particular, it is surjective. Moreover, ker (f ) is connected. (ii) Assume that G is a pro-Lie group and N a closed normal subgroup. Let F : G → def : G →H  H = G/N be the quotient morphism and j : N → G the inclusion. Then F is a quotient morphism of prosimply connected pro-Lie groups whose kernel is the  in G,  that is, there is a strict exact sequence image of N j˜

 F

−−−−→G −−−−→H  → 1. 1→N If H is a pro-Lie group, then there is a commutative diagram whose rows are strict exact sequences

262

6 Lie’s Third Fundamental Theorem ˜



j

F

j F   −−− →1 1→N −→ H ⏐ −−−−→ G ⏐ ⏐ ⏐ ⏐ ⏐ pN  pG  pH  1 → N −−−−→ G −−−−→ H → 1.

(iii) Let G a prosimply connected pro-Lie group and N a closed normal subgroup. Then the following two statements are equivalent: (a) G/N is prosimply connected. (b) N is prosimply connected. (c) N is connected. (iv) Assume that F : G → H is a surjective morphism of pro-Lie groups and that : G →H  is a quotient morphism. G is connected. Then F Proof. (i) The surjectivity of f implies the bijectivity of the induced map f  : g/n → h. In the category of weakly complete topological vector spaces, however, a bijective morphism, as the adjoint of a bijective morphism of vector spaces, is an isomorphism. (Compare this with [102, Theorem 7.30 (iv)], cited below as Theorem 7.7 (iv).) Hence f : g → h is a quotient morphism and if n denotes the kernel of f , then f is in fact the cokernel of the inclusion morphism e : n → g. Applying  to the exact sequence f

e

{0} → n −−−→ g −−−→ h → {0}, by the Strict Exactness Theorem for , 6.7, we get an exact sequence of morphisms of pro-Lie groups (f )

(e)

const

{1} → (n) −−−−→ (g) −−−−→ (h) −−−−→ {1}. In particular im (e) = ker (f ), which is a closed connected normal subgroup N of def

G = (g), and im (g) = ker const = (h). Thus (f ) is a surjective morphism of connected pro-Lie groups with connected kernel. It is also the cokernel of (e) in proLieGr since  preserves cokernels. In TopGr, the quotient morphism q : G → G/N is the cokernel of (e). By Theorem 4.1 the factor group G/N is a proto-Lie group and its completion Q ⊇ G/N is a pro-Lie group. Let j : G/N → Q be the inclusion. Then j  q : G → Q is the cokernel of (e) in proLieGr. By the uniqueness of cokernels, there is an isomorphism ι : (h) → Q such that G ⏐ ⏐ q G/N

(f )

−−−−→ (h) ⏐ ⏐ι  −−−−→ Q j

commutes. Since (f ) is surjective and j is injective, it follows that j is bijective. That is, G/N = Q is complete and therefore is a pro-Lie group. Then (f ) and q are equivalent, and thus (f ) is in fact a quotient morphism.

Lie’s Third Fundamental Theorem for Pro-Lie Groups

263

(ii) By the Strict Exactness Theorem 4.20 for L, we have a strict exact sequence L(F )

incl

0 → n −−−−→ g −−−−→ h → 0 with a quotient morphism L(F ) and with n = ker L(F ) ∼ = L(N ) by the kernel preservation of L. Now by the Strict Exactness Theorem 6.7 for  we obtain a strict exact sequence (incl)

L(F )

1 → (n) −−−−→ (g) −−−−→ (h) → 1, where L(F ) is a quotient morphism by (i) above. Now we recall from Theorem 6.6 and its proof that  · =   L and that πG :  g → G is that natural morphism inducing an isomorphism on the Lie algebra level. Thus (ii) follows. (iii) We let F : G → H be the quotient morphisms as in (ii). The group G is prosimply connected iff pG is an isomorphism by Theorem 6.6 (iii). If, in the commuting diagram in (ii), two of the three vertical morphisms pN , pG , and pH are isomorphisms, so is the third one. Hence by 6.6 (iii) again, (a) and (b) are equivalent.  with G by the isomorphism Trivially (b) implies (c). Finally assume (c). We identify G  pG and consider both N and N as subgroups of G via the inclusions j˜ and j . Then pN  ⊆ N . (Note that this is consistent with F = pH  F  is an inclusion map, that is N     and N = ker F ⊆ ker(pH  F ) = ker F = N.) Since L(pN ) : L(N ) → L(N ) is an  = N0 . Our assumption, however, isomorphism, from Corollary 4.22 we conclude N is N0 = N, and so pN is the identity map, whence (b) holds. (iv) By the connectivity of G and by Theorem 4.22 (ii)(b), the morphism L(F ) is a  = (L(F )) is a quotient quotient morphism of pro-Lie algebras; then by (i) above, F morphism of pro-Lie groups. Theorem on the Preservation of Embeddings in Lie’s Third Theorem Corollary 6.9. (i) If j is any closed ideal of a pro-Lie algebra g, then the inclusion i : j → g induces an embedding (i) : (j) → (g). That is, (j) may be considered as a closed normal subgroup of (g). (ii) If N is the kernel of a morphism F : G → H of pro-Lie groups and j : N → G →G  is an embedding so that N  may be identified the inclusion morphism, then j˜ : N  with the kernel of F . i

q

Proof. (i) The exact sequence 0 → j −−→ g −−→ g/j → 0 by the Exactness Theorem 6.7 gives an exact sequence i

(q)

{1} → (j) −−−→ (g) −−−→ (g/j) → {1}. If K = ker (q), then K is a closed normal subgroup and thus a pro-Lie group by the Closed Subgroup Theorem 3.35. Since L preserves kernels, L(K) = j. By the Exactness Theorem 6.7, the corestriction ε : (j) → K of (j ) is a bijective morphism inducing an isomorphism L(ε) : ((j) → L(K) = j) which becomes the identity when L((j)) is identified with j via ηj . It follows from the uniqueness in

264

6 Lie’s Third Fundamental Theorem

the universal property of  (see 6.6) that ε is none other than the canonical morphism  → K. In 6.8, however, we saw that K was prosimply connected. πK : (L(K)) = K But then by 6.6 (iii), ε = πK is an isomorphism. But this means that (i) is an embedding algebraically and topologically, as asserted. (ii) Since the functor L preserves kernels, the Lie algebra n = L(N ) is (up to natural isomorphism) the kernel of L(F ) : L(G) → L(H ). Now we apply (i) above to n in place of j and recall  · =   L again to obtain the assertion.

Semidirect Products We discussed in Chapter 1 the concept of a semidirect product of topological groups in Exercise E1.5. The concept of a semidirect product in the category of topological groups is very fruitful, both traditionally in classical Lie group theory and in the theory of pro-Lie groups – as we still have to see. Let us review the concept with particular emphasis on the functorial aspects of it. Remember that a self map f of a set is called idempotent if f 2 = f . Also recall that for topological groups N and H and a morphism α : H → Aut(N ) such that (h, n)  → α(h)(n) : H × N → N is continuous, the product N × H becomes a topological group N α H with the multiplication (n, h)(n , h ) = (nα(h)(n ), hh ), called the semidirect product of N by H (with respect to α). If n and h are topological Lie algebras and δ : h → Der(n) is a morphism of Lie algebras such that (Y, X)  → δ(Y )(X) : h × n → n is continuous, then n × h becomes a Lie algebra n ⊕δ h with respect to componentwise scalar multiplication and addition and the bracket multiplication [(X, Y ), (X , Y  )] = (δ(Y )(X  ) − δ(Y  )(X), [Y, Y  ]), called the semidirect sum of n with h (with respect to δ). Exercise E6.2. Verify that the bracket multiplication of the semidirect sum is antisymmetric and satisfies the Jacobi identity (see Definition A1.1 (ii), p. 624). Proposition 6.10. (a) Let G be a topological group and N a closed normal subgroup. Then the following conditions are equivalent: (i) There is an idempotent endomorphism ϕ of G such that ker ϕ = N . (ii) There are morphisms q : G → K and σ : K → G such that q  σ = idK and ker q = N. (iii) There is a closed subgroup H such that the function (n, h)  → nh : N × H → G is an isomorphism of the semidirect product N I H of N by H with respect to the morphism I defined by inner automorphisms I (h)(n) = hnh−1 onto G. If these conditions are satisfied, H = ϕ(G) = σ (K).

Semidirect Products

265

(b) Let g be a topological Lie algebra and n a closed ideal. Then the following conditions are equivalent: (i) There is an idempotent endomorphism  of g such that ker  = n. (ii) There are morphisms p : g → k and ρ : k → g such that p  ρ = idk and ker p = n. (iii) There is a closed subalgebra h such that the function (X, Y )  → X+Y : n×h → g is an isomorphism of the semidirect sum n ⊕ad h of n by h with respect to the morphism ad defined by the adjoint representation ad(Y )(X) = [Y, X] onto g. If these conditions are satisfied, h = (g) = ρ(k). Proof. We prove part (a): (i) ⇒ (ii): We let K = ϕ(G) and let q be the corestriction of ϕ to its image. Define σ : K → G be the inclusion map. Then ker q = ker ϕ = N, and if k ∈ K, then k = ϕ(g) for some g ∈ G and (q σ )(k) = q(ϕ(g)) = ϕ 2 (g) = ϕ(g) = k. (ii) ⇒ (iii): Define H = σ (K). Let μ : N I H → G be defined by μ(n, h) = nh. Then μ is easily seen to be a morphism of topological groups. The function g  → (gσ (q(g)), q(g)) is readily verified to be the inverse map μ−1 of μ. (iii) ⇒ (i): Assume that μ : N I H → G is an isomorphism of topological groups, μ(n, h) = nh. Let i : H → G be the inclusion morphism and pr H : N I H → H the projection onto H . Define ϕ : G → G by ϕ = i  pr 2 μ−1 . Assume g = nh. Then μ−1 (g) = (n, h), and μ−1 (h) = (1, h), whence ϕ 2 (g) = i pr 2 μ−1 i pr 2 μ−1 (g) = h = i pr 2 μ−1 (g) = ϕ(g). Part (b) is proved in a completely analogous fashion and is left as an exercise. Exercise E6.3. Spell out the details of the proof of Part (b) of Proposition 6.10. Theorem 6.11 (Preservation of Semidirect Products). Let g be the semidirect sum of def the closed ideal n by the closed subalgebra h. Then G = (g) is the semidirect product of a closed normal subgroup N naturally isomorphic to (n) and a subgroup H isomorphic to (h). Proof. By Proposition 6.10 (b), the hypothesis is equivalent to the existence of an idempotent endomorphism  of g with kernel n and image h. We apply the functor  def

and obtain and idempotent endomorphism ϕ = () : G → G. Let i : n → g denote the inclusion morphism. The exact sequence i



0 → n −−→ g −−→ g gives rise to an exact sequence (i)

(ϕ)

1 → (n)−−−−→(g)−−−−→(g) by the Exactness Theorem for  6.7, and (j ) is an embedding by Corollary 6.6. Set N = (j)((n)). Then (n) ∼ = N = ker () = ker ϕ. It now follows from

266

6 Lie’s Third Fundamental Theorem def

Proposition 6.10 that G is the semidirect product of N by H = ϕ(G). It remains to identify im ϕ. def

Now let p : g → h = (g) be the corestriction of  to its image. Then p is a quotient morphism inducing an isomorphism g/n → h. Then the exact sequence p

g −−→ h → 0 induces an exact sequence (p)

G = (g) −−−−→ (h) → 0 def

by the Exactness Theorem for  6.7, and q = (p) : G → (h) is a quotient morphism by Corollary 6.8. If we let ρ : h → g denote the inclusion, we have pρ = idh , whence def

def

q  σ = id(h) for σ = (ρ). Then H = σ ((h)) is isomorphic to (h). Also  = ρ  p and thus ϕ = () = (ρ)  (p) = σ  q whence im ϕ = im σ = H .

Postscript From classical Lie group theory we know that if G is a finite-dimensional Lie group, then there exists a unique (up to isomorphism) simply connected finite-dimensional Lie  such that their Lie algebras L(G) and L(G)  are isomorphic; further, there is group G,  onto G implementing a local isomorphism. an open continuous homomorphism of G When one moves out of the world of finite-dimensional Lie groups or rather out of the world of manifolds, the first task is to be clear about what “simply connected” should mean. For a connected topological space X with a base point x0 several definitions of simple connectivity are in use. For instance, • X is called simply connected if it has the following universal property: For any covering map p : E → B between topological spaces, any point e0 ∈ E and any continuous function f : X → B with p(e0 ) = f (x0 ) for some x0 ∈ X there is a continuous map f: X → E such that p  f = f and f(x0 ) = e0 . f

X ⏐ −−−−→ ⏐ idX 

E ⏐ ⏐p 

X −−−−→

B.

f

Alternatively, • X is called simply connected if π1 (X, x0 ) = {0}, that is if every loop attached to x0 is contractible.

Postscript

267

No nondegenerate connected compact abelian group is simply connected with respect to the first definition. Any connected compact abelian group G such that  Z) = {0} (such as e.g. G = Q ) is simply connected with respect to the second Hom(G, definition. For arcwise connected, locally arcwise connected, and locally arcwise simply connected spaces, in particular for manifolds, the two definitions agree (see e.g. [102, Proposition A2.10]). In [102], we opted for the first definition because of its universal properties which are practically built into the definition. Taking these complications into consideration, we ask what should be the right concept of simple connectivity for pro-Lie groups? We say that • a pro-Lie group G is prosimply connected if N (G) contains a cofinal subset N S(G) such that G/N is a simply connected Lie group for each N ∈ N S(G). This definition takes the characteristic property G ∼ = limN ∈N (G) G/N into account. But is it useful for the structure theory of pro-Lie groups? With the aid of the structure theory of pro-Lie groups that we will develop in later chapters, we will show that a pro-Lie group G is prosimply connected if and only it is simply connected, and that it satisfies π1 (G) = {0} whenever it is prosimply connected. That is a reassuring fact. But recall that no compact connected abelian group is prosimply connected. That may make us a bit nervous. However, within the full category of pro-Lie groups in the category of topological groups, the full subcategory of prosimply connected pro-Lie groups has precisely the right universal properties. Firstly, for every profinite Lie algebra g there is a unique prosimply connected pro-Lie group G (up to isomorphism) such that L(G) = g. (Lie’s Third Fundamental Theorem). Thus there is essentially a “bijection” between the possible Lie algebras of pro-Lie groups and the prosimply connected specimens in proLieGr. If g is abelian, then G is the additive group of g; this is reasonable and has no analog in such categories as the category of locally compact groups. Secondly, for each connected pro-Lie group G there is a (functorially attached)  and a pro-open natural morphism πG : G →G prosimply connected pro-Lie group G  inducing an isomorphism L(πG ) : L(G) → L(G) of Lie algebras, and every morphism  of G into a prosimply connected pro-Lie group uniquely lifts to a morphism from G, factoring through πG . All of this very satisfactorily generalizes familiar facts from classical Lie group theory. Yet none of these facts is exactly trivial. For a compact  is naturally isomorphic to the additive group of L(G) and πG : G →G abelian group, G identifies with expG : L(G) → G. With our theorems, among other things we are saying: the category proLieAlg of profinite Lie algebras has “a precise copy”, namely, proSimpConLieGr, the full subcategory of the category proLieGr of pro-Lie groups containing precisely the prosimply

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connected pro-Lie groups. Moreover, the category of connected Lie groups allows a left-reflection into this category, that is a left adjoint functor which behaves like a retraction onto proSimpConLieGr. We have seen that Lie’s Third Fundamental Theorem in reality complements the Lie algebra functor to an adjoint situation. This is of course true in the finite-dimensional situation, but as a rule, less emphasis is placed on this fact in elementary Lie theory. Among other reasons, the most compelling one is that in the finite-dimensional case we are not dealing with complete categories. Adjoint situations are most appropriate where limits are present, because right adjoint functors (such as L in our case) preserve all limits. The functor  : proLieAlg → proLieGr that is left adjoint to the Lie algebra functor L : proLieGr → proLieAlg has excellent exactness properties, as we have seen in Theorem 6.7 and its two corollaries. This is a parallel to the Exactness Theorem for L 4.20 in Chapter 4. The theorems on the preservation of quotients 6.8 and of embeddings 6.9 are particularly relevant since quotients are somewhat problematic as we have seen in Chapter 4, and since we cannot have an Open Mapping Theorem for pro-Lie groups in general. The general covering theory for topological groups introduced by Berestovskii and Plaut fits very well with our theory for pro-Lie groups (see [6], [7], [8]). For a  connected pro-Lie group G, the Berestovskii–Plaut-cover agrees with G.

Chapter 7

Profinite-Dimensional Modules and Lie Algebras

In the previous chapters we recognized the significance of the category of topological groups with profinite-dimensional Lie algebras. This category can be understood only if one first comprehends the nature of pro-Lie algebras, that is, profinite-dimensional Lie algebras themselves. In this chapter we develop the structure theory of pro-Lie algebras by utilizing the duality between the category of weakly complete topological vector spaces and the category of vector spaces we review completely in Appendix 2. The use of this duality is made possible through representation theory or, equivalently, module theory of a fixed Lie algebra L. In fact we shall consider profinite-dimensional modules which emulate the definition of pro-Lie algebras in such a fashion that the adjoint module of a pro-Lie algebra is exactly a profinite-dimensional module. While pro-Lie algebras themselves do not dualize in an obvious way, profinite-dimensional modules do, and we shall make ample use of this fact. Prerequisites. In this chapter we shall make use of the duality theory of weakly complete topological vector spaces as it is presented in Appendix 2. We shall collect the essential features in Theorem 7.7. The representation theory of Lie algebras used here is simple. But we do resort to the basic structure theory of finite-dimensional Lie algebras presented in source books that are widely used such as [16]. In this chapter we do not use any information from Chapters 3, 4 and 5.

Modules over a Lie Algebra We begin with some elementary facts on the linear algebra of modules over a Lie algebra. Definition 7.1. (i) Let L be a Lie algebra and E a vector space. Then E is an L-module if there is a bilinear map (x, v)  → x · v : L × E → E

satisfying

[x, y] · v = x · (y · v) − y · (x · v)

for all x, y ∈ L and v ∈ E. A function f : E1 → E2 between L-modules is said to be a morphism of L-modules if it is linear and satisfies (∀x ∈ L, v ∈ E1 )

f (x · v) = x · f (v).

(1)

(ii) If L is a Lie algebra and V is a topological vector space, then V is said to be a continuous L-module if the underlying vector space is an L-module in the sense of (i) above and the functions v  → x · v : V → V are continuous for all x ∈ L.

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(iii) If L is a topological Lie algebra, then a topological vector space V is said to be a topological L-module if it is a continuous L-module and the maps x  → x · v : L → V are continuous for all v ∈ V ; in other words, if (x, v)  → x · v : L × V → V is continuous in each variable separately. Let us consider some prominent examples. Example 7.2. For any vector space E, let gl(E) denote the Lie algebra of all endomorphisms of E with the Lie bracket [ϕ, ψ] = ϕψ − ψϕ. If L is any Lie algebra, let |L| denote the underlying vector space. If we define ad x : L → L by (ad x)(y) = [x, y], then the Jacobi identity shows that ad : L → gl(|L|) is a morphism of Lie algebras, called the adjoint representation. If for x ∈ L and v ∈ |L| we set x · v = (ad x)v, then |L| becomes an L-module called the adjoint module. If L is a topological Lie algebra then the underlying topological vector space |L| becomes a topological L-module, again called the adjoint module. In particular, if g is a pro-Lie algebra and |g| is the underlying weakly complete topological vector space, then |g| is a topological g-module via the adjoint operation. If E is an L-module, for x ∈ L we define xE : E → E by xE (v) = x · v. Then x  → xE : L → gl(E) is a morphism of Lie algebras. Conversely, if π : L → gl(E) is a representation, i.e. a morphism of Lie algebras, then E becomes an L-module via x · v = π(x)(v). Thus the set of representations of L on V is in bijective correspondence with the L-module structures on E. If V is a topological vector space and an L-module, then each xV is an endomorphism of the topological vector space V , and if gl(V ) denotes the Lie algebra of all endomorphisms of the topological vector space V with the Lie bracket [ϕ, ψ] = ϕψ − ψϕ, then x  → xV : L → gl(V ) is a representation. Conversely, every representation of L into gl(V ) yields on the topological vector space V the structure of an L-module. If L is a topological Lie algebra and V a topological L-module, then x  → xV : L → gl(V ) is a morphism of topological Lie algebras if gl(V ) is given the topology of pointwise convergence, sometimes called the strong operator topology on V . We shall now make use of the duality of topological vector spaces. If V is a topological vector space, its topological dual is the vector space V  = Hom(V , R) of all continuous linear functionals. In accordance with this definition, V  is a vector space without additional structure, but it may be equipped with many relevant vector space topologies; we shall consider two of them here. The coarser of the two is the weak *-topology, also called the topology of pointwise convergence, that is the topology induced from the product RV . If V  is equipped with this topology, we denote it by V  . The second topology we consider is the topology of uniform convergence on compact subsets of V also called the compact open topology. (See [102, Proposition 7.1ff].) If the topological dual V  is endowed with the compact open topology, we denote  have the same underlying vector . The topological vector spaces V  and V it by V . For a space V  , but the topology of V  may be properly coarser than that of V  is defined to be TopGr(G, T) = Hom(G, T) topological group G the character group G with the compact open topology; of course, this works well only when G is abelian. If V

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271

is a topological vector space, then the natural morphism Hom(V , R) → Hom(V , R/Z) is an isomorphism of topological vector spaces when both hom-groups are given the compact open topology (see [102, Proposition 7.5 (iii)]). This justifies our notation of  for V  with the compact open topology. V Now let L be a Lie-algebra and V a topological vector space. If ω ∈ V  , then ω  xV : V → R is a continuous linear functional. Thus if we define x · ω by x · ω = −ω  xV ∈ V  we have (∀x ∈ L, v ∈ V )

(x · ω)(v) = −ω(x · v).

(2)

Lemma 7.3. (i) Let L be a Lie algebra and V a topological vector space and L→V , module. With respect to the bilinear map (x, ω) → x · ω = −ω  xV : L × V  the topological vector space V is an L-module. (ii) If L is a topological Lie algebra and V a topological L-module, then V  is a topological L-module. Proof. (i) First we compute ([x, y] · ω)(v) = −ω([x, y] · v) = −ω(x · (y · v)) + ω(y · (x · v)) = −y · ω(x · v) + x · ω(y · v) = x · (y · ω)(v) − y · (x · ω)(v). Thus V  is an L-module. Now we observe the required continuity property, namely, that xV  , defined by xV  (ω) = x · ω = −ω  xV , is continuous for all x ∈ L. But that is immediate from the continuity of all xV and ω. (ii) Let ω ∈ V  and v ∈ V . All functions x  → x · v : L → V are continuous. Hence all functions x  → (x · ω)(v) = −ω(x · v) : L → R are continuous and that means that the linear function x  → x · ω : L → V  is continuous with respect to the topology of pointwise convergence on V  . Lemma 7.4. (i) Let L be a Lie algebra and V a topological vector space and L-mod→V , the ule. With respect to the bilinear map (x, ω)  → x · ω = −ω  xV : L × V  is an L-module. topological vector space V (ii) If L is a topological Lie algebra and V a topological L-module such that  is a topological L-module. (x, v)  → x · v : L × V → V is continuous, then V Proof. (i) After 7.1 (ii) we have to verify only that for all x ∈ L the linear maps →V  are continuous. For a proof we assume that a compact subset C of V and xV : V a zero neighborhood U of R are given and thus determine a basic zero neighborhood W (C, U ) = {ω ∈ V  | ω(C) ⊆ U }. Since xV is continuous, −x · C = −xV (C) is compact, and every functional ω satisfying ω(−x · C) ⊆ U also satisfies ω(−x · C) = (x · ω)(C) ⊆ U . Thus xV maps W (−x · C, U ) into W (C, U ) →V , (ii) For a proof of the continuity of the functions x → x · ω = −ω  xV : V we again assume that a compact subset C of V and an open zero neighborhood U of R are given. We must find a zero neighborhood W of L such that x ∈ W implies (x · ω)(C) ⊆ U i.e., ω(−x · C) ⊆ U . Now ω−1 (U ) is an open identity neighborhood

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of V . If the function (x, v) → x · v : L × V → V is assumed to be continuous, then an open symmetric zero neighborhood W of L exists such that W · C ⊆ ω−1 (U ), and this proves the assertion. Definition 7.5. 7.5. The L-module V  is called the dual module of V . Example 7.6. Let L be a topological Lie algebra and let |L| denote the underlying topological vector space. Define L to be the topological dual |L| of |L| in the weak *-topology. Then the dual module of the adjoint module |L| on L is a topological  x)(ω) = L-module which is called the coadjoint module, written Lcoad . If we set (ad   x · ω = ω  ad x, then ad : L → gl(L) is a continuous morphism of Lie algebras if one endows gl(|L|) with the topology of pointwise convergence.  denotes the topological dual of |L| with the compact open topology, then L  is If L a topological L-module with respect to the coadjoint action.

Duality of Modules We consider two categories of topological vector spaces which are dual to each other under the passage to the topological dual with the compact open topology. (See [102, Proposition 7.1ff].) Namely, we consider firstly the category Vect of vector spaces and linear maps between them, and secondly, the category WCVect of weakly complete topological vector spaces and continuous linear maps between them. Any real vector space can be considered as a topological vector space with respect to the finest locally convex vector space topology (see Appendix 2, Proposition A2.3 and the passage that precedes it), and any linear map between vector spaces is automatically continuous with respect to this topology on the domain and the range space. An abelian topological group A is called reflexive if the natural evaluation morphism  η: A →  A into the double dual is an isomorphism of topological groups. (See [102, Definition 7.8ff].) We denote by RefTopGr the full subcategory of TopGr whose objects are all reflexive abelian topological groups. Note that a morphism f : V → W of topological groups between the additive groups of topological vector spaces is always linear. To see this note that if r = m n with m, n ∈ N, then n · f (r · v) = f (m · v) = m · f (v), whence f (r · v) = r · f (v), and this relation extends by continuity to all r ∈ R. the set of all ω ∈ V  If S is a subset of a topological vector space V , then S ⊥ denotes   ⊥ such that ω(S) = {0}. If S is a subset of V , then S = ω∈S ker ω. This will not cause confusion if we always keep track of the vector space in which S is located. The vector subspace S ⊥ is called the annihilator of S. For easy reference, we review the duality of vector spaces and weakly complete topological vector spaces.

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Theorem 7.7 (Duality of Real Vector Spaces). Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then   is an isomorphism of topological vector spaces. (i) E is reflexive; i.e. ηE : E → E  and E  is a weakly complete Thus E belongs to RefTopGr. Moreover, E  = E, topological vector space.   is an isomorphism of topological vector spaces, (ii) V is reflexive; i.e. ηV : V → V  has the finest locally convex topology, and thus V belongs to RefTopGr. The dual V   ∼   and V = V = V . (iii) Vect and WCVect are subcategories of RefTopGr, and the contravariant functor  · : RefTopGr → RefTopGr exchanges the categories Vect and WCVect. (iv) Every closed vector subspace V1 of V is algebraically and topologically a direct summand; that is there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces. Every surjective morphism of weakly complete topological vector spaces f : V → W splits; that is, there is a morphism σ : W → V such that f  σ = idW . (v) For every closed vector subspace H of E, the relation H ⊥⊥ = H ∼ = (E  /H ⊥ )  ⊥ ⊥  holds and E /H is isomorphic to H . The map F  → F is an antiisomorphism of the complete lattice of vector subspaces of E onto the lattice of closed vector subspaces of E  . Proof. See Appendix 2, notably Theorems A2.8, A2.11, and A2.12. A subset M of an L-module V is called a submodule if L · M ⊆ M. If M is a closed submodule then V /M is an L-module with respect to the bilinear map (x, v + M)  → x ∗ (v + M) = x · v + M. Definition 7.8. Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. (i) V is called a profinite-dimensional L-module if it is complete as a topological vector space and the filter basis M of closed submodules M ⊆ V such that dim V /M < ∞ converges to 0. (ii) E is called a locally finite-dimensional L-module if for each finite subset S of E there is a finite-dimensional submodule F of E containing S. We specifically draw the reader’s attention to the facts that according to our definitions, a profinite-dimensional L-module is a topological vector space with a module structure, while a locally finite-dimensional L-module is merely a vector space with an L-module structure. It is possible at all times to consider it as a topological vector space with a module structure by equipping it with the finest locally convex topology, but doing that is not likely to give us additional insights. Thus a locally finite-dimensional module is a purely algebraic object even if L is a topological Lie algebra. Let g be a finitely generated topological Lie algebra on an infinite-dimensional vector space. Then the underlying vector space |g| with the adjoint module structure cannot be locally finite-dimensional, since a finite set of generators of g is not contained

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in a finite-dimensional subalgebra, let alone an ideal, and thus it is not contained in a finite-dimensional submodule. A simple example is as follows: Let s = so(3) and ej ∈ s, j ∈ Z(3) = Z/3Z, a basis of s such ' that [e & j , ej +1 ] = ej +2 for j = 0, 1, 2 mod 3. Then [s, e0 ] contains e1 and e2 and s, [s, e0 ] = s. Now set g = sN and x = (xn )n∈N with xn = e0 for all n. In ' &N the adjoint module we have g·x = [s, e0 ]N and g·(g·x) = s, [s, e0 ] = sN = g. Thus the adjoint module is singly generated algebraically and yields a profinite-dimensional g-module whose underlying vector space does not yield a locally finite-dimensional g-module, since {x} is not contained in a finite-dimensional submodule. Proposition 7.9. For an L-module on a topological vector space V , the following statements are equivalent: (i) V is a profinite-dimensional L-module.  = limM∈M V /M (see 1.29) is an isomorphism. (ii) The morphism γV : V → V Every profinite-dimensional L-module is weakly complete. Proof. The assertion follows from Theorems 1.29 and 1.30. Proposition 7.10. For an L-module E the following conditions are equivalent: (i) (ii) (iii) (iv)

E is locally finite-dimensional. Each element of E is contained in a finite-dimensional submodule. E is the union of its finite-dimensional submodules. E is a sum of a family of finite-dimensional submodules.

Proof. Exercise E7.1. Exercise E7.1. Prove Proposition 7.10. [Hint. (i) ⇒ (ii) ⇒ (iii) ⇒ (i) and (iii) ⇒ (iv) are trivial. If (iv) holds, then E = 5 . Let F denote the family j ∈J Vj for a family of finite-dimensional submodules Vj of all finite sums of the submodules Vj . Then (iv) says E = F and this implies (iii).] The Duality Theorem for Profinite-Dimensional and Locally Finite-Dimensional L-Modules Theorem 7.11. Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. Then the following conclusions hold.  = E  is weakly complete. (ia) The underlying vector space of the dual module E   of E  is E (up to natural isomorphism). If E is a locally finiteThe dual module E  is a profinite-dimensional L-module. dimensional L-module, then E (ib) If L is a topological Lie algebra and E is a topological L-module then the dual  = E  is a topological L-module. module E   has the finest locally convex topology. Its dual (V ) = V  is (ii) The dual module V  V (up to natural isomorphism). If V is a profinite-dimensional L-module, then V is a locally finite-dimensional L-module.

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(iii) The functor  : Vect → WCVect maps the category of locally finite-dimensional L-modules to the category of profinite-dimensional topological L-modules. The functor  : WCVect → Vect maps the category of profinite-dimensional topological L-modules to the category of locally finite-dimensional L-modules. (iv) Every closed L-submodule V1 of V is a direct topological vector space summand; that is, there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces. Every surjective morphism of profinite-dimensional topological L-modules f : V → W splits in terms of topological vector spaces; i.e. there is a morphism of topological vector spaces σ : W → V such that f  σ = idW . (v) A vector subspace H of E is an L-submodule iff its annihilator H ⊥ in E  is an L-submodule of E  . A closed vector subspace K of V is an L-submodule of V iff its annihilator in V  is an L-submodule of V  . For every L-submodule H of E, the . The map relation H ⊥⊥ = H ∼ = (E  /H ⊥ ) holds and E  /H ⊥ is isomorphic to H F  → F ⊥ is an antiisomorphism of the complete lattice of L-submodules of E onto the lattice of closed L-submodules of E  , and F ⊥⊥ = F for each submodule F of E and each closed submodule F of V .  and E  is a weakly complete topological vector Proof. (i) By 7.7 (i) and (iii), E  = E  space. By 7.3, E is an L-module, and if L is a topological Lie algebra and E is a  ∼ topological L-module, then E  is a topological L-module as well. The relation E =E follows from 7.7 (i). Now assume that E is a locally finite-dimensional L-module.  is complete and that the filter basis of closed vector Since we know that E  = E  subspaces N with dim E/N < ∞ converges to 0, in order to verify condition 7.8 (i)  we have to show that every closed vector subspace N of E  such that E/N  is for V = E  finite-dimensional contains an L-submodule M such that E/M is finite-dimensional. By the Annihilator Mechanism 7.7 (v), the annihilator N ⊥ of N in E is finite-dimensional. Since E is a locally finite-dimensional L-module, there is a finite-dimensional L-module F of E containing N ⊥ . Then we let M be the annihilator F ⊥ of F in E  .  we have   ⊥∼ Since annihilators of submodules are submodules and E/M = E/F =F  dim E/M < ∞. Thus the proof of (i) is complete.  has the finest locally convex topology and by (i) above and 7.7 (i) (ii) By 7.7 (iii), V  ) = V ∼ and (ii), we have (V = V . Now we assume that V is a profinite-dimensional L-module. We claim that V  is a locally finite-dimensional L-module. So let S be a finite subset of V  . Let S ⊥ be the annihilator of this set in V . Then by 7.7 (v), the vector space V /S ⊥ ∼ = (span S) is finite-dimensional. Since V is a profinite-dimensional L-module, there is a submodule M such that M ⊆ S ⊥ and that dim V /M < ∞. def ∼ Then F = M ⊥ , the annihilator of M in V  is a submodule and since F = V /M we ⊥ ⊥⊥ know that it is finite-dimensional. Also S ⊆ M implies S ⊆ S ⊆ M⊥ = F . Statement (iii) is now a consequence of (i) and (ii), and conditions (iv) follows directly from 7.7 (iv). (v) Let H be a vector subspace of E. Then x · H ⊆ H for x ∈ L iff ω(H ) = {0} implies x · ω(H ) = −ω(x · H ) = {0} for all ω ∈ E  since H = H ⊥⊥ by 7.7 (v). Thus x · H ⊆ H iff x · H ⊥ ⊆ H ⊥ . For a closed vector subspace K of V the same argument

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applies since for a closed vector subspace K of V one has V = V ⊥⊥ by 7.7 (v). The remainder of (v) is an immediate consequence of 7.7 (v). Corollary 7.12. Let L be a topological Lie algebra and V a weakly complete topological L-module. Then the following statements are equivalent: (i) V is a profinite-dimensional L-module. (ii) V  is a locally finite-dimensional L-module. Proof. Since E = V  with the finest locally convex topology and V are duals of each other by 7.7, the corollary is an immediate consequence of Theorem 7.11. The theory of profinite-dimensional modules is primarily developed to deal with profinite-dimensional Lie algebras. This is why the following example is crucial. Exercise E7.2. Prove the following proposition: Let g be a weakly complete topological Lie algebra. Then the following statements are equivalent. (i) g is a pro-Lie algebra. (ii) The coadjoint module gcoad of g is locally finite-dimensional. [Hint. With g = L and V = |g| (the underlying topological vector space), this is but a special case of Corollary 7.12.] As we progress further into duality theory we have reached the point to recall that categorical concepts such as limits and their various manifestations such as products, pullbacks, equalizers projective limits come in pairs, as each concept has its dual concept. Technically, if D : J → C is a diagram its colimit is the limit of the diagram D : J → C op in the opposite category. However, in order to have things self-contained in this book as much as possible, we explicitly formulate the definitions of a colimit and colimit cone. Definition 7.13. Let J and C be categories. A diagram D : J → C is said to have a colimit colim D ∈ ob C if there is a cone λ : D → const(colim D), called the colimit cone1 such that for each cone α : D → const A there is a unique morphism α  : colim D → A such that α = const(α  )  λ. Exercise E7.3. Formulate the definitions of coproduct, coequalizer, pushout, and injective limit. [Hint. Dualize the concepts introduced in 1.4.] of vector Lemma 7.14. (i) Let E be a vector space and {Fj | j ∈ J } a directed family  subspaces of E; i.e., J is a directed set and j ≤ k implies Jj ⊆ Jk . Then E = j ∈J Fj and E = colimj ∈J Fj are equivalent statements. 1 The

“cones” in this definition are “upside down” and should be called “cocones”. However, there are limits to the grammatical and etymological pliability of language. For instance, it is not true that a coconut is the dual of a nut. In fact, it is the bidual of a nut as the conut is its dual. For further references on philological analysis see the footnote in [102] regarding the terminology of Definitions 9.30.

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(ii) Let V be a profinite-dimensional L-module and M any filter basis of submodules M such that dim V /M < ∞ for all M ∈ M and that M converges to 0. Then the dual module V  is the union of the directed set of finite-dimensional submodules M ⊥ , M ∈ M. Proof. Exercise E7.4. Exercise E7.4. Verify Lemma 7.14. [Hint. (i) E = colimj ∈J Fj means that for every family of linear maps fj : Fj → F into a vector space F such that j ≤ k implies fk |Fj = fj there is a unique linear map f : E → F such that f |Fj = fj . Prove that this implies and is implied by E = colimj ∈J Fj . (ii) follows from (i) and the Duality Theorem 7.11.]

Semisimple and Reductive Modules Definition 7.15. (i) An L-module E is said to be simple if {0} and E = {0} are its only submodules. (ii) An L-module E is called semisimple if every submodule is a direct module summand. (iii) Let V be a profinite-dimensional topological vector space and an L-module. Then the module is said to be reductive if its dual module is semisimple. (iv) A pro-Lie algebra g is called reductive if its adjoint module is reductive. Note that a simple abelian module is isomorphic to R with the zero module operation x · r = 0 for all x ∈ L and r ∈ R. Recall that the direct sum  j ∈J Ej of a family Ej , j ∈ J of vector spaces is the set of all families (xj )j ∈J ∈ j ∈J Ej of elements xj ∈ Ej vanishing for all but a finite number of exceptions.5If each Ej is a vector subspace of a vector space E, then we shall say that the sum j ∈J Ej ⊆ E is direct if the morphism 6 8 6 xj : Ej → Ej (xj )j ∈J  → j ∈J

j ∈J

j ∈J

is an isomorphism of vector spaces. The Structure Theorem of Semisimple Locally Finite-Dimensional Modules Theorem 7.16. Let L be a Lie algebra and E a locally finite-dimensional L-module. Then the following statements are equivalent: (i) (i ) (ii) (iii)

E is a semisimple L-module. Every submodule is a direct module summand. Every finite-dimensional submodule of E is semisimple. E is the union of finite-dimensional semisimple submodules.

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(iv) E is (isomorphic to) a direct sum simple submodules Ej .

j ∈J

Ej of a family of finite-dimensional

Proof. (i) ⇔ (i ): Definition 7.15(ii). (i) ⇒ (ii): Let F be a finite-dimensional submodule of E and F1 a submodule of F . We must show that F1 has a complementary module summand in F . Since E is semisimple, there is a submodule E2 of E such that E = F1 ⊕ E2 . Set F2 = F ∩ E2 . Then F = F1 ⊕ F2 is readily checked (“Law of Modularity”). (ii) ⇒ (iii): Since E is a locally finite-dimensional module, every element is contained in a finite-dimensional submodule F . Hence this implication is trivial. (iii) ⇒ (iv): We consider sets S of finite-dimensional simple 5 submodules of E such that the morphism ε : F → E sending (x ) to F F ∈S F ∈S F ∈S xF ∈ E is injective. The set F of all sets S is partially ordered under inclusion. We claim that F is inductive. def  S and assert that For a proof let S be a totally ordered subset of F . We set5S = S ∈ F . Indeed we claim that the morphism (xF )F ∈S  → 5 x F ∈S F : F ∈S F → E is injective. Indeed let (xF )F ∈S be in its kernel. Then F ∈S xF = 0. The set T = {F ∈ S : xF = 0} is finite, hence is contained in some Sx ∈ S since S is the union of all S in S. But by definition of F the function 6 8 yF : F →E (yF )F ∈Sx  → F ∈Sx

F ∈Sx

is injective. Now (xF )F ∈ Sx is in the kernel of this map, and this implies xF = 0 for all F ∈ Sx . But xF = 0 for F ∈ S \ T and thus (xF )F ∈S is zero. This proves the claim and shows the assertion that S ∈ F 5. Hence F is indeed inductive. Now let S be a maximal element in F and set E0 = F ∈S F . If E0 = E we are finished. Suppose that E0 = E. Then by (ii) there is a finite-dimensional semisimple submodule W of E which is not contained in E0 . The submodule E0 ∩ W has a direct complement E1 in W , and thus E0 + E1 is a direct sum. Since E is locally finite-dimensional, so is E1 and thus there is a finite-dimensional nonzero submodule S of minimal dimension. Clearly S is a simple module. Since S ⊆ E1 , the sum E0 + S is direct. Hence the family S ∪ {S} is a member of F and is properly larger than S. This is a contradiction to the choice of S. Hence E0 = {0} and the proof is5 complete. (iv) ⇒ (i): We assume that E is a direct sum j ∈J Ej of submodules. Let F be a submodule of E. We must find a module complement of F5in E. For a fixed j ∈ J we either have Ej ⊆ F or Ej ∩ F = {0}. Let F1 = j ∈J, Ej ⊆F Ej and 5 E . Then E = F ⊕ F , and by the modular law we have F2 = 1 2 j ∈J, Ej ∩F ={0} j F = F1 ⊕ (F ∩ F2 ). Thus if we find a module complement of F ∩ F2 in F2 we are done. Thus in order to simplify notation, we might just as well 5 assume that F ∩Ej = {0} for all j ∈ J . The set  of all subsets I ⊆ J such that F ∩ 5j ∈I Ej = {0} is seen to be inductive again due to the fact that for every element x = j ∈J xj only finitely 5 many xj are nonzero. By Zorn’s Lemma let us pick a maximal subset I and set F = j ∈I Ej . Then clearly F ∩ F =5 {0}. We claim that E = F + F . Suppose that this is not the case. Then there is an x = j ∈J xj ∈ E \ (F + F ). In particular, there must be an i ∈ J \ I

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such that xi ∈ / F + F . We may say x = xi ∈ Ei . It follows that (F + F5) ∩ Ei = {0} which in turn entails F ∩ (F + Ei ) = {0}5because otherwise 0 = y = j ∈I ∪{i} yj ∈ F ∩ (F + Ej ) would imply yi = y − j ∈I yj ∈ Ei ∩ (F + F ) = {0} and thus 0 = y ∈ F ∩ F which is not possible. But now I ∪ {i} ∈  contradicting the assumed maximality of I . This contradiction proves the claim and finishes the proof. Exercise E7.5. Prove the following result. Proposition. In the class of L-modules, the following statements hold: (i) A submodule of a semisimple module is semisimple. (ii) A quotient of a semisimple module is semisimple. (iii) A direct sum of locally finite-dimensional semisimple modules is semisimple. [Hint. (i) Let F be a submodule of a semisimple module E and assume that F1 is a submodule of F . Since E is semisimple, there is a submodule E2 of E such that def

E = F1 ⊕ E2 . Define F2 = F ∩ E2 . Show that F = F1 ⊕ F2 . (ii) Let f : E → F be a surjective module morphism. Since E is semisimple, we find a submodule F1 of E such that E = ker f ⊕ F1 . Show that f |F1 : F1 → F is an isomorphism of modules and invoke (i) above to show that F1 is semisimple. (iii) By Theorem 7.16 every summand is a direct sum of simple submodules and thus the direct sum is a direct sum of simple submodules. Hence by Theorem 7.16 again the direct sum is semisimple.] Corollary 7.17. Assume that E is a locally finite-dimensional L-module. Let Ess be the union of all finite-dimensional semisimple submodules. Then: (i) Ess is the unique largest semisimple submodule. (ii) Ess is a direct sum of finite-dimensional simple modules. Proof. (i) By Theorem 7.16, the sum of two finite-dimensional semisimple submodules is a finite-dimensional semisimple submodule. Hence the set of all finite-dimensional semisimple submodules is directed. Therefore its union is a submodule. By Theorem 7.16 again it is semisimple, and it is obviously the largest semisimple submodule. (ii) This is an immediate consequence of (i) and Theorem 7.16. Theorem 7.18. (a) Let V be a profinite-dimensional L-module for a Lie algebra L. Then the following statements are equivalent: (i) V is reductive. (i ) Every closed submodule is a direct module summand algebraically and topologically. (ii) Every finite-dimensional quotient module of V is reductive. (iii) V is the projective limit of finite-dimensional reductive module quotients. (iv) V is isomorphic to a product of finite-dimensional simple modules. (b) Every profinite-dimensional L-module has a unique smallest submodule V ss such that V /V ss is reductive.

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Proof. Via duality 7.11, (a) follows from 7.16 and (b) from 7.17. In any L-module E we have y · (x · v) = yx · v − xy · v + x · (y · v) − x · v + x · v = [x, y] · v + x · (y · v − v) + x · v; hence the linear span of all x · v as (x, v) ranges through L × E is a submodule Eeff . Definition 7.19. For an L-module E define E0 = {v ∈ E | (∀x ∈ L) x · v = 0}; this set is clearly a submodule, called the maximal zero submodule. The submodule Eeff is called the effective submodule. In a locally finite-dimensional L-module E the submodule Ess is called the semisimple radical. Clearly, in E0 , every vector subspace is a submodule. If E is a locally finite-dimensional L-module, then E0 is just a vector space and thus is isomorphic to a direct sum R(I ) for some set I ; if V is a profinite-dimensional L-module, then V0 is just a weakly complete topological vector space and thus is isomorphic to RI for some set I (Corollary A2.9). In a locally finite-dimensional L-module, E0 ⊆ Ess . Proposition 7.20. (i) Assume that E is a simple L-module. Then E agrees with E0 if and only if it is zero or one-dimensional; if this is not the case, then E agrees with Eeff . In the latter case, if 0 = v ∈ E, then vL = E. (ii) 5If E is a semisimple locally finite-dimensional L-module, then E is the direct sum j ∈J Ej of a family of simple submodules Ej of E, and 6 E0 = Ej , (3) j ∈J dim Ej =1

Eeff =

6

Ej .

(4)

j ∈J dim Ej >1

In particular, E = E0 ⊕ Eeff .

(5)

Proof. (i) Let E be a simple L-module. Then E0 is either {0} or E. In the latter case, every vector subspace being a submodule, its dimension is either 0 or 1. In the former case, E = Eeff . Let 0 = v ∈ E. Then L · v is a submodule. Since E0 = {0} it is nonzero. Hence L · v = E 5 (ii)5 From 7.16 we deduce the direct 5 sum representation E = j ∈J Ej . Define F0 = j ∈J, dim Ej =1 Ej , and F1 = j ∈J, dim Ej >1 Ej . Let 0 = vj ∈ Ej . Then " {0} if dim Ej = 1, L · xj = Ej if dim Ej > 1 by 5 (i) above and thus F0 ⊆ E0 , F1 ⊆ Eeff . Now let v ∈ E0 . Then 0 = x · v = j ∈J x · vj for all x ∈ L, and thus x · vj = 0 for all x ∈ L and j ∈ J . Hence vj ∈ E0 for all j and thus v ∈ F0 . Therefore F0 = E0 .

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But Eeff = (F0 ∩ Eeff ) ⊕ F1 . Now let 5 F0 ⊕ F1 = E and F1 ⊆ Eeff imply5 v = j ∈J,dim Ej =1 vj ∈ F0 ∩ Eeff . Then v = nm=1 xm · wm with xm ∈ L, wm ∈ E. 5 Now wm = j ∈J wmj with wmj ∈ Ej , and v=

6

xm · wmj =

m=1,...,n j ∈J

6

xm m=1,...,n j ∈J dim Ej >1

· wmj ∈ F1 .

But then v ∈ F0 ∩ F1 = {0}, and therefore v = 0. It follows that F0 ∩ Eeff = {0}, and thus that F1 = Eeff . Proposition 7.21. Let V be a profinite-dimensional L-module and V  its dual module.  and (V )⊥ = V  . Then (V0 )⊥ = Veff eff 0 Proof. Consider an ω ∈ V  ; then ω ∈ (Veff )⊥ iff ω(x · v) = 0, for all x ∈ L and v ∈ V iff x · ω = 0 for all x ∈ L iff ω ∈ V0 . Thus V0 = (Veff )⊥ . Similarly  )⊥ by duality. Applying ⊥, in view of the Duality Theorem 7.11 (v) we get V0 = (Veff ⊥  )⊥⊥ = V  = V  since every vector subspace of V  is closed. V0 = (Veff eff eff The Structure Theorem of Reductive Profinite-Dimensional L-Modules Theorem 7.22. Let V be a reductive profinite-dimensional L-module. Then  V is isomorphic to a product V0 × Veff , and V0 ∼ = RI for some set I and Veff ∼ = j ∈J Vj where each Vj is a simple submodule of V such that 1 < dim Vj < ∞. Proof. Since the module V is reductive and profinite-dimensional, its dual V  is semisimple and locally finite-dimensional. Thus by Proposition 7.20 we have V  =  . Then Theorem 7.11 (v) and Proposition 7.21 imply V = (V  )⊥ ⊕ (V  )⊥ = V0 ⊕ Veff eff 0 ⊥ = V  /V  ∼ V  , and by V0 ⊕ (V0 )⊥⊥ = V0 ⊕ Veff . The dual of Veff is V  /Veff 0 = eff 7.20 (ii), this module is a direct sum of simple modules with finite dimension greater 1. Thus Veff by duality is a product of simple modules of finite dimension greater than 1.

Reductive Pro-Lie Algebras Let us apply some of these results to pro-Lie algebras. def

Lemma 7.23. Let g be a pro-Lie algebra and i a closed ideal such that h = g/i is simple. Then dim h < ∞ and the annihilator i⊥ in the coadjoint module gcoad of g is a finite-dimensional simple submodule and is isomorphic to the coadjoint module of h.

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Proof. We consider the adjoint module of g and its dual, the coadjoint module. The vector space  h is isomorphic to i⊥ (see [102, Theorem 7.30 (v)]) and thus i⊥ is isomorphic to the coadjoint module hcoad of h which is a simple Lie algebra and therefore has a simple adjoint module, whence the coadjoint module i⊥ is simple. Proposition 7.24. Let g be a pro-Lie algebra and  (g) the filter basis of closed ideals i such that dim g/i  < ∞. Further let gcoad = Hom(|g|, R) denote the coadjoint module. Then gcoad = i∈ (g) i⊥ , and i⊥ is a finite-dimensional module which is the coadjoint module of the finite-dimensional Lie algebra g/i. Proof. From the Duality Theorem 7.11 we know that (i⊥ ) ∼ = g/i and (g/i) = i⊥ ⊥ for i ∈  =  (g). Since  is a filter basis, {i | i ∈  } is a directed family of finitedimensional vector subspaces of  E. As g = limi∈ g/i we have E = colimi∈ i⊥ , and by 7.15 this means exactly E = i∈ i⊥ . Definition 7.25. (a) Let g be a topological Lie algebra. Then its center z(g) is the set {x ∈ g | (∀y ∈ g) [x, y] = 0}. The span of all elements [x, y], x, y ∈ L is a subalgebra of L, called the commutator subalgebra [g, g]. The notation g = [g, g] is also used. The closure [g, g] is called the closed commutator algebra. If [g, g] = g, then g is called perfect. (b) A pro-Lie algebra g is called reductive if its adjoint module gad is a reductive g-module. It is called semisimple if it is reductive and its center z(g) is zero. In terms of the adjoint module gad of g, the center is the maximal zero submodule (gad )0 , the commutator subalgebra [g, g] is the effective submodule (gad )eff . In particular, the center and the commutator algebra are ideals. We shall see presently that the commutator subalgebra of a reductive pro-Lie algebra is closed. The following lemma uses a piece of information on finite-dimensional simple Lie algebras which we provide in Appendix 3. Indeed by Corollary A3.3 we know that Lemma BR. In any finite-dimensional semisimple real Lie algebra, each element is the sum of at most two brackets. This allows us to prove the following lemma. Lemma 7.26. Let {s j | j ∈ J } be any family of finite-dimensional real semisimple Lie algebras. Then s = j ∈J sj is its own algebraic commutator algebra s = [s, s], that is, the linear span of all elements [X, Y ] for X, Y ∈ s. In particular, every semisimple pro-Lie algebra is perfect. Proof. For subsets A and B of a Lie algebra g we define brack (A, B) = n

n 6

 [Xj , Yj ] | Xj ∈ A, Yj ∈ B, j = 1, . . . , n .

j =1

By Lemma BR above we have sj = brack 2 (sj , sj ). Since  brack n (s, s) = brack n (sj , sj ) for n ∈ N j ∈J

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we note in particular that brack 2 (s, s) =

 j ∈J

brack 2 (sj , sj ) =



sj = s.

j ∈J

Clearly, brack 2 (g, g) ⊆ [s, s]. Thus s = [s, s] follows, and this says that s is perfect. The Structure Theorem of Reductive Pro-Lie Algebras Theorem 7.27. (a) For a pro-Lie algebra g the following conditions are equivalent. (i) g is reductive. (i ) Every closed ideal of g is an ideal direct summand algebraically and topologically. (ii) g is the product of a family of finite-dimensional simple or one-dimensional ideals of g. (b) Let g be a reductive pro-Lie algebra. Then the commutator algebra [g, g] is closed and is a product of finite simple real Lie algebras. Further g ∼ = z(g) ⊕ [g, g] algebraically and topologically, and z(g) ∼ = RI for some set I . (c) Every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. Proof. (a) The Lie algebra g is reductive iff the adjoint module gad is reductive, and by Theorem 7.18, this is the case iff gad is a product of finite-dimensional simple gmodules. Now a vector subspace of g is a submodule of the adjoint module gad iff it is an ideal of g. This remark completes the proof of the equivalence of (i), (i ) and (ii).  By (a) above g ∼ = RI × j ∈J sj with a family of simple finite-dimensional real Lie algebras sj . By the preceding Lemma 7.26, the second factor s satisfies [s, s] = s, and the first factor is abelian. The assertion follows. (c) This is a consequence of Theorem 7.18 (b) applied to the adjoint module of g. We shall call ncored (g) the coreductive radical. We shall say more about ncored (g) later in Theorem 7.66 and Theorem 7.67. From Definition 7.25 (b) and Theorem 7.27 (b) we get at once the following conclusions. Corollary 7.28. Any semisimple pro-Lie algebra is perfect. Structure of Semisimple Pro-Lie Algebras Corollary 7.29. For a pro-Lie algebra, the following statements are equivalent. (I) g is semisimple. (II) g is the product of a family of finite-dimensional simple ideals of g.

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Lemma 7.30. Assume that E is a locally finite-dimensional L-module and S is a semisimple submodule such that S0 = {0} and E/S is a zero module. Let K denote the set {x ∈ L : (∀v ∈ E) x · v = 0}. Then S = Eeff and if L/K is semisimple, then E is semisimple. In particular, (6) E = S ⊕ E0 . Proof. By Proposition 7.20, S = Seff , and therefore Seff ⊆ Eeff . But Eeff /S ⊆ (E/S)eff = {0} implies Eeff ⊆ S, and so S = Eeff . It remains to show that E is semisimple. Let F be a finite-dimensional submodule. By Theorem 7.16 it suffices to show that F is semisimple. Now S ∩ F is contained in a def

finite sum D = Ej1 ⊕ · · · ⊕ Ejn of simple submodules of dimension > 1; by replacing F by F +D we may assume that F ∩S = D. Then F /(F ∩S) ∼ = (F +S)/S ⊆ (E/S)0 . We are reduced to the finite-dimensional case and may assume that dim E < ∞. Let F1 be a submodule of E containing S such that dim F1 /S = 1; if we can show that F1 is semisimple, then F1 = S ⊕ R · x1 with an element x1 such that L · x1 = {0}. Since this works for arbitrary one-dimensional submodules of E/S we will obtain E = S ⊕ R · x1 ⊕ · · · ⊕ R · xk , k = dim E/S, which will finish the proof. Hence we may as well assume that dim E/S = 1. We may assume that K = {0} because E is an L/K-module in the obvious way, and submodules and quotient modules are the same. But then, as we assume that L is semisimple, the assertion follows from [16, §6, no 3, Définition 3]. Proposition 7.31 (Levi–Mal’cev Theorem for Reductive Pro-Lie Algebras). Let g be a pro-Lie algebra such that g/z(g) is semisimple. Then g is reductive and g is an ideal direct sum, algebraically and topologically, g = z(g) ⊕ [g, g].

(7)

Proof. We apply Lemma 7.30 with L = g, E = gcoad , S = z(g)⊥ , K = z(g). Then, by duality, we have the module isomorphisms S ∼ = (g/z(g)) and E/S ∼ = z(g) . By hypothesis, g/z(g) is semisimple and thus, by Definitions 7.15 and 7.27 (i), S is a semisimple module. Also, since the g-submodule z(g) of the adjoint module g is the zero module, its dual E/S is a zero module. The module L/K = g/z(g) is semisimple. Thus Lemma 7.30 applies and shows that E is semisimple and this means by Definition 7.15 (iv), g is reductive. So 7.28 implies g = z(g) ⊕ [g, g]. Now [g, g] ∼ = g/z(g) is a semisimple pro-Lie algebra, and then g is a reductive one. Thus (7) follows from Theorem 7.27.

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Transfinitely Solvable Lie Algebras The familiar concept of solvability of a finite-dimensional Lie algebra requires some thought in the absence of dimensional restrictions. We are dealing with topological Lie algebras, and thus there are added complications due to the possible definition of “topological solvability” which again is not apparent in the situation of finite-dimensional Lie algebras. Similar comments apply to nilpotency which we shall treat in a way that is parallel to that of solvability. Definition 7.32. Let g be a Lie algebra. Set g(0) = g and define sequences of ideals g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g(α) is defined for α < β.  (i) If β is a limit ordinal, set g(β) = α<β g(α) . (ii) If β = α + 1, set g(β) = [g(α) , g(α) ]. For cardinality reasons, there is a smallest ordinal γ such that g(γ +1) = g(γ ) . Set = g(γ ) . Let ω denote the first infinite ordinal. Then g is said to be transfinitely solvable, if g(∞) = {0}. If g is transfinitely solvable and γ ≤ ω, then g is called countably solvable. If γ is finite and g(γ ) = {0}, then g is called solvable.

g(∞)

Recall that, if γ = 0, that is, [g, g] = g, then g is called perfect. In a sense, transfinitely solvable Lie algebras and perfect ones are opposite types. Staying with solvable Lie algebras for a moment we look at a variation of our theme. Definition 7.33. Let g be a Lie algebra. A sequence of ideals (jα )α≤ρ indexed by an initial section of ordinals is said to be an abelian sequence of ideals if it is descending and the following conditions hold. Let β < ρ.  (i) If β is a limit ordinal, then jβ = α<β jβ . (ii) If β = α + 1, then jβ /jα is abelian. The sequence is said to be a nilsequence of ideals if it is descending and condition (i) above and the following condition are satisfied: (ii ) If β = α + 1, then jα /jβ is central in g/jβ . A sequence (jα )α≤ρ of ideals is said to be terminating if it is descending and jρ = {0}. The sequence (g(α) )α≤γ is an abelian sequence of ideals; it is that abelian sequence of ideals which descends fastest. Indeed we have Lemma 7.34. Let (jα )α≤ρ be an abelian sequence of ideals of g. Then (∀α ≤ ρ) g(α) ⊆ jα .

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Proof. We prove the assertion by transfinite induction.  Let β ≤ ρand assume that g (α) ⊆ jα for α < β. If β is a limit ordinal, then g(β) = α<β g(α) ⊆ α<β jα = jβ . If, however, β = α +1, then g(α) ⊆ jα and there is homomorphism ϕ : g(α) → jα /jβ given by ϕ(x) = x + jβ . Since the range of ϕ is abelian, g(β) = [g(α) , g(α) ] ⊆ ker ϕ = jβ . This completes the induction. Clearly g is transfinitely solvable if (g(α) )α≤ρ is terminating. If so, then γ ≤ ρ. Proposition 7.35. For a Lie algebra g, the following statements are equivalent: (i) g is transfinitely solvable. (ii) g has a terminating abelian sequence of ideals. Proof. Since clearly (i) ⇒ (ii) we must prove that (ii) implies (i). Let (jα )α≤ρ be a terminating abelian sequence of ideals. By Lemma 7.34 we have g(ρ) ⊆ jρ = {0}, and thus the sequence (g(α) )α≤ρ is terminating, and g is therefore transfinitely solvable. We next turn to topological Lie algebras. Here it is the closed commutator subalgebras which are more relevant because they give rise to Hausdorff quotient algebras. Definition 7.36. Let g be a topological Lie subalgebra of a topological Lie algebra h. (For instance, h = g.) Set g((0)) = g and define sequences of ideals g((α)) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g((α)) is defined for α < β.  (i) If β is a limit ordinal, set g((β)) = α<β g((α)) . (ii) If β = α + 1, set g((β)) = [g((α)) , g((α)) ]. For cardinality reasons, there is a smallest ordinal γ such that g((γ +1)) = g((γ )) . Set g((∞)) = g((γ )) . Let ω denote the first infinite ordinal. Then g is said to be transfinitely topologically solvable, if g(∞) = {0}. If g is transfinitely topologically solvable and γ ≤ ω, then g is called countably topologically solvable. If γ is finite and g((γ )) = {0}, then g is called topologically solvable. If γ = 0, that is, [g, g] = g, then g is said to be topologically perfect. The definitions of various concepts of solvability have parallel definitions of corresponding concepts of nilpotency. First we deal with the purely algebraic concepts. Definition 7.37. Let g be a Lie algebra. Set g[0] = g and define sequences of ideals g[α] ⊇ g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g[α] ⊆ g(α) is defined for α < β.   (i) If β is a limit ordinal, set g[β] = α<β g[α] ⊇ α<β g(α) . (ii) If β = α + 1, set g[β] = [g, g[α] ] ⊇ [g[α] , g[α] ] = g(β) .

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For cardinality reasons, there is a smallest ordinal δ such that g[δ+1] = g[δ] . Set = g[δ] . Then g is said to be transfinitely nilpotent, if g[∞] = {0}. If g is transfinitely nilpotent and δ ≤ ω, then g is called countably nilpotent. If δ is finite and g[δ] = {0}, then g is called nilpotent.

g[∞]

Because g(α) ⊆ g[α] , any transfinitely nilpotent Lie algebra is transfinitely solvable. Every free Lie algebra L(X) over a set X (see e.g. [17, Chapitre 2, §2]) is countably nilpotent (and thus transfinitely nilpotent) (see [17, Chapitre 2, §2, Proposition 7]). Next let us turn to the topological variants of nilpotency. Definition 7.38. Let g be a topological Lie subalgebra of a topological Lie algebra h. (For instance, h = g.) Set g[[0]] = g and define sequences of ideals g[[α]] ⊇ g((α)) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g[[α]] ⊇ g((α)) is defined for α < β.   (i) If β is a limit ordinal, set g[[β]] = α<β g[[α]] ⊇ α<β g((α)) . (ii) If β = α + 1, set g[[β]] = [g, g[[α]] ] ⊇ [g[[α]] , g[[α]] ] = g((β)) . For cardinality reasons, there is a smallest ordinal δ such that g[[δ+1]] = g[[δ]] . Set g[[∞]] = g[[δ]] . Then g is said to be transfinitely topologically nilpotent, if g[∞] = {0}. Because of g((α)) ⊆ g[[α]] , any transfinitely topologically nilpotent Lie algebra is transfinitely topologically solvable. Definition 7.39. If g is transfinitely topologically nilpotent and δ ≤ ω, then g is called countably topologically nilpotent. If δ is finite and g[[δ]] = {0}, then g is called topologically nilpotent. Of course there are relationships between all of these concepts. The following lemma is the first tool. Lemma 7.40. Let g be a topological Lie algebra and let a and b vector subspaces of g. Then (i) [a, b] = [a, b], & ' & ' (ii) [g, g], [g, g] = [g, g], [g, g] , & ' & ' (iii) g, [g, g] = g, [g, g] , (iv) g(n) = g((n)) for all n ∈ N, (v) g[n] = g[[n]] for all n ∈ N. Assume that g is a subalgebra of a topological Lie algebra h; then (vi) g((α)) = g((α)) for all α, (vii) g[[α]] = g[[α]] for all α.

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Proof. (ii) and (iii) are consequences of (i), and (iv) and (v) follow by finite induction from (i). By definition we have g((0)) = g = g = g((0)) , and similarly g[[0]] = g[[0]] . Now (i) implies g(1)) = g((1)) and then g((α)) = g((α)) from there on. Similarly g[[α]] = g[[α]] . Thus (i) implies (vi) and (vii). So we have to establish (i). Obviously the left side of (i) is contained in the right side. The function f : g × g → g given by f (a, b) = [a, b] is continuous since g is a topological Lie algebra. Then f (a × b) = f (a × b) ⊆ f (a × b), and this set is contained in the closed span [a, b] of f (a × b). Thus the closed span [a, b] is contained in [a, b]. In the following proposition, for a topological Lie algebra g we again let γ be the first ordinal such that g(γ +1) = g(γ ) and γ the first ordinal such that g((γ +1)) = g((γ )) . Proposition 7.41. (i) A transfinitely topologically solvable topological Lie algebra is transfinitely solvable and γ ≤ γ . (ii) A solvable Lie algebra is topologically solvable. (iii) A finite-dimensional Lie algebra is solvable if and only if it is topologically solvable. (iv) A subalgebra g of a topological Lie algebra h is topologically solvable, respectively, nilpotent if and only if g is topologically solvable, respectively, nilpotent. (v) A subalgebra of a transfinitely solvable Lie algebra is transfinitely solvable. A closed subalgebra of a transfinitely topologically solvable Lie algebra is transfinitely topologically solvable. Proof. (i) Let g be transfinitely topologically solvable. The sequence (g((α)) )α≤γ is a terminating abelian sequence of ideals. Thus g is transfinitely solvable by Proposition 7.35. (ii) By Lemma 7.40 (iv), the two sequences (g(n) )n∈N and (g((n)) )n∈N , agree. Thus (n) if g = {0} for some natural number n then g((n)) = {0}. (iii) If the Lie algebra g is finite-dimensional, then it satisfies the descending chain condition and thus the sequence of the g(n) must become stationary after a finite number of steps. So (ii) applies. Assertion (iv) is an immediate consequence of 7.40 (vi) and (vii). (v) Let a be a subalgebra of a transfinitely solvable Lie algebra g. Then we claim a(α) ⊆ g(α) . This is certainly true for α = 0; assume that it is true for α < β. If (α) , g(α) ] = g(β) . Assume now that β = α + 1 then a(β) = a(α+1) = [a(α) , a(α) ] ⊆ [g (α) (β) β is a limit ordinal. Then a = α<β a ⊆ α<β g(α) = g(β) . This proves the claim. Actually, this proposition shows among other things that the concept of a “topologically solvable topological Lie algebra” is really superfluous because it agrees with that of a “solvable topological Lie algebra”. In the transfinite situation it is the limit ordinals that cause the trouble because taking infinite intersections and forming closures do not commute.

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However, we are interested in pro-Lie algebras, a special type of topological Lie algebras. We shall see shortly, that the situation is quite satisfactory for pro-Lie algebras. But let us first observe that we encounter a third reasonable concept of solvability for pro-Lie algebras. The topological case is treated similarly. Definition 7.42. A pro-Lie algebra g is called prosolvable if every finite-dimensional quotient algebra is solvable. It is called pronilpotent if every finite-dimensional quotient algebra is nilpotent. We need a bit of information on weakly complete topological vector spaces. Lemma 7.43. Let E be a weakly complete topological vector space and F a closed subspace. Further let F be a filter basis of closed vector subspaces. Then   F+ F = F + H. (∗) H ∈F

If, in addition, dim E/F < ∞, then   F+ F = (F + H ).

(∗∗)

H ∈F

Proof. See Appendix 2, Theorem A2.13. In lattice theoretic terminology: The lattice of vector subspaces of V is meet continuous and the lattice of closed vector subspaces of E is join continuous. This entails the following lemma. Lemma 7.44. Assume that g is a pro-Lie algebra. (i) For each closed ideal j of g, (g/j)((α)) = g((α)) + j/j.

(∗)

(ii) In particular, every continuous homomorphic image of a transfinitely topologically solvable pro-Lie algebra is transfinitely topologically solvable. (iii) If j is a closed ideal such that dim g/j < ∞, then for every limit ordinal β there is an αj < β such that for all α the relation αj ≤ α < β implies g((β)) + j = g(α) + j.

(∗∗)

(iv) γ ≤ ω, that is g((ω+1)) = [g((ω)) , g((ω)) ] = g((ω)) . (v) For every pro-Lie algebra g, the characteristic ideal g((ω)) is topologically perfect and for each j ∈  (g) there is a natural number nj such that for all natural numbers n satisfying nj ≤ n we have g((ω)) + j = g(n) + j and [g((ω)) , g((ω)) ] + j = g((ω)) + j.

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Proof. (i) We prove (∗) by transfinite induction. Assume the assertion to be proved for  all ordinals α < β. If β = α + 1, then (g/j)((β)) = (g/j)((α)) = (g((α)) + j)/j) = 

(g((α)) ) + j)/j = (g((α)) + j)/j = g((α+1)) + j/j = g((β)) + j/j. This completes the induction in this case. Now assume that β is a limit ordinal. Then (g/j)((β)) =     ((α)) = ((α)) + j/j = ( ((α)) + j)/j = j + ((α)) /j = α<β (g/j) α<β g α<β g α<β g g((β)) + j /j by the definitions and by the preceding Lemma 7.43. This completes the induction. Thus (∗) is proved. (ii) If g is transfinitely topologically solvable, we have g((γ )) = {0} and thus (g/j)((γ )) = g((γ )) + j/j = {0}. This shows that g/j is transfinitely topologically solvable. Now every surjective morphism of weakly complete topological vector spaces splits (see Appendix 2, Theorem A2.12 (a)) and thus is a quotient morphism. Hence any continuous homomorphic image of g is isomorphic to g/j for some closed ideal j. (iii) Let j be a closed ideal of g such that g/j is finite-dimensional. Then the descending chain (g(α) + j)α<β has a smallest element, say g(αj ) + j for a smallest ordinal αj < β. Then (#) g(αj ) + j = g(α) + j for all α with αj ≤ α < β. Since g/j is finite-dimensional, (†) all vector subspaces containing j are closed by the definition of the quotient topology of g/j, since all vector subspaces of a finitedimensional (Hausdorff topological) vector space are closed. We claim that g((α)) ⊆ g(α) + j for all ordinals α. For α = 0 this is trivially true. Let us assume that this claim holds for α, we shall show that it holds for β = α +1. Indeed, g((β)) = [g((α)) , g((α)) ] ⊆ [g(α) + j, g(α) + j] ⊆ [g(α) , g(α) ] + j = g(β) + j = g(β) + j by (†) above. So induction works for the step from α to α + 1. Now assume that β is a limit ordinal and that the claim is true for all α < β.Then by the definition of g((α)) and   ((β)) ((α)) by transfinite induction in 7.39 we have g = α<β g ⊆ α  <β (g(α ) + j) = g(α) + j = g((α)) + j for all α satisfying αj ≤ α < β by induction and  hypothesis  by (#). Thus for all these α we obtain g((β)) + j ⊆ g((α)) + j = α  <β g((α )) + j =   j + α  <β g((α )) = j + g((β)) by 7.43 and the inductive definition of g((β)) . Thus equality holds throughout and this is the assertion (∗∗) for all α satisfying αj ≤ α < β (iv) We compute [g((β)) , g((β)) ] + j = [g((β)) + j, g((β)) + j] + j = [g(αj ) + j, (α j g ) + j] + j = [g(αj ) , g(αj ) ] + j = g(αj +1) + j = g(αj ) + j = g((β)) + j = g((β)) + j for any limit ordinal β and any ideal j ∈  (g). For fixed β we form the intersection over all j ∈  (g) on both sides and find g((β)) on the right side since this is a closed subset of g, and [g((β)) , g((β)) ] = g((β+1)) on the left. Thus g((β+1)) = g((β)) for all limit ordinals β. In particular, this applies to β = ω. Statement (v) just summarizes (iii) and (iv).

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Proposition 7.45. Let g be a pro-Lie algebra. Then the following assertions are equivalent: (i) g is countably topologically solvable. (ii) g is transfinitely topologically solvable. (iii) g is prosolvable. These conditions imply the following ones: (iv) g is countably solvable. (v) g is transfinitely solvable. (vi) g does not contain a finite-dimensional simple Lie algebra. Proof. The implication (i) ⇒ (ii) is trivial. (ii) ⇒ (iii): Let j ∈  (g). Then by the preceding Lemma 7.44 (ii), if (i) is satisfied, then the homomorphic image g/j is transfinitely solvable, and since dim g/j < ∞, it is solvable. Hence (iii) is satisfied. (iii) ⇒ (i): Let j ∈  (g). Then since g/j is solvable by (iii), there is a natural (n) ((ω)) ⊆ j. Thus number m such that g ⊆ j for all n ≥ m. Then 7.44 (iv) implies g ((ω)) g ⊆  (g) = {0}. This proves (i). Thus (i), (ii), and (iii) are equivalent. (i) ⇒ (iv): We apply 7.41 (i) with γ being finite or ω and find that γ is finite or ω, that is, g is countably solvable. The implication (iv) ⇒ (v) is trivial and the implication (v) ⇒ (vi) follows from 7.41 (v). This proposition is still preliminary; indeed we will show in Theorem 7.53 that (v) implies (i) and thus all five conditions are equivalent.

The Radical and Levi–Mal’cev: Existence Definition 7.46. If a topological Lie algebra g has a unique largest transfinitely topologically solvable ideal then it is called its radical or in order to be very specific, solvable radical and is denoted by r(g). Classical results of finite-dimensional Lie algebras show that every finite-dimensional real Lie algebra g has a radical; the quotient algebra g/r(g) is semisimple, and every ideal of this quotient algebra is semisimple. (See any source of finite-dimensional Lie algebras such as [16, p. 72, Théorème 1, and p. 75, Corollaire 1]; or [122].) From 7.41 (iv) it follows that the radical is always closed. Lemma 7.47. Let i be any closed ideal of a topological Lie algebra g, and assume that g/r(g) is a product of simple finite-dimensional Lie algebras. Then r(g/i) = r(g) + i/i. If g is finite-dimensional, then g/r(g) is semisimple and thus the hypothesis on g/r(g) is satisfied.

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Proof. The right side is a solvable ideal of g/i and thus contained in the left by the definition of r(g/i). We define an ideal a of g as the full inverse image of r(g/i) in r(g); thus r(g/i) = a/i. Then r(g) + i ⊆ a and we have to show that a = r(g) + i. Firstly, a is closed by Lemma 3.17 (b). As an ideal of the quotient algebra g/r(g), the quotient a/r(g) is a product of simple finite-dimensional Lie algebras. The algebra a/r(g) + i is solvable as a homomorphic image of a/i = r(g/i) and it is a product of simple finite-dimensional algebras as a homomorphic image of a/r(g). Hence it is zero. Thus a = r(g) + i. Theorem 7.48 (Theorem on the Existence of the Solvable Radical). Every pro-Lie algebra g has a radical r(g); the radical is closed, and the factor algebra g/r(g) is semisimple, that is, is a product of finite-dimensional simple Lie algebras. A pro-Lie algebra g is semisimple if and only if its radical r(g) vanishes. Proof. For each j ∈  (g), the finite-dimensional Lie algebra g/j has a radical r(g/j) whose full inverse image in g we denote by rj (g), so that rj (g)/j ∼ = r(g/j). If i ⊇ j in  (g), then there is a natural quotient morphism g/j → g/i, mapping r(g/j) = rj (g)/j onto r(g/i) = ri (g)/i). In particular, the system {r(g/j) → r(g/i) | (i, j) ∈  (g) ×  (g), i ⊇ j} is a projective system of finite-dimensional solvable Lie algebras. If x ∈ rj (g) then x + j ∈ r(g/j) and so x + i ∈ r(g/i), that is, x ∈ ri (g). Therefore rj ⊆ ri (g). Thus {rj : j ∈  (g)} is a filter basis of cofinite-dimensional closed ideals of g. Let r(g) denote the intersection of all rj (g), j ∈ g; we shall show that r(g) is the radical of g. Obviously, r(g) is a closed ideal as the intersection of a filter basis of closed ideals. Claim (a). For each j ∈  (g), we have rj (g) = r(g) + j. Clearly, the right side is contained in the left. But also r(g) + j is closed as the full inverse image of the vector subspace (r(g) + j)/j of the finite-dimensional vector space g/j all   of whose vector subspaces are closed. Moreover, r(g) + j = j + i∈ (g) ri (g) = (i∈ (g) (ri (g) + j) by Lemma 7.43(∗∗). Now ri (g) + j = rj (g) for all i ∈  (g) satisfying i ⊆ j by 7.47. Hence r(g) + j = rj (g), which is Claim (a). Claim (b). The filter basis {rj (g)/r(g) : j ∈  (g)} converges to zero in g/r(g). Let U be a zero neighborhood of g. We may assume that there is a j ∈  (g) such that U + j = U and U/j is a zero neighborhood of g/j in which {j} is the only subgroup, that is, j is the only subgroup containing j which is contained in U . (See 1.27.) Then (U + r(g))/r(g) is a basic zero neighborhood of g/r(g) containing (j + r(g))/r(g) which is equal to rj (g)/r(g) by Claim (a). Thus for every zero neighborhood of g/r(g) there is a j ∈  (g) such that rj (g)/r(g) is contained in it. This establishes Claim (b). Claim (c). g/r(g) ∼ = limj∈ (g) g/rj (g). This follows from claim (b) via Theorem 1.33. Claim (d). r(g) ∼ = limj∈ (g) r(g/j). In view of Claim (a) we notice that r(g/j) = rj (g)/j = (r(g) + j)/j ∼ = r(g)/(r(g) ∩ j). But the filter basis {r(g) ∩ j : j ∈  (g)} converges to zero in r(g) since  (g) converges to zero in g. Therefore r(g) ∼ =

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limj∈ (g) r(g)/(r(g) ∩ j) by Theorem 1.33. This establishes Claim (d). (See also Theorem 1.34.) Claim (e). g/r(g) is a product of finite-dimensional simple Lie algebras, and r(g) is prosolvable. In view of Theorem 7.18 and Definition 7.25 (b), the semisimplicity of the algebra g/r(g) follows from Claim (c) and the fact that g/rj (g) ∼ = (g/j)/r(g/j) is semisimple. Then the first part of the claim is a consequence of Corollary 7.29. The second assertion follows from Claim (d) and Definition 7.42. Claim (f). r(g) is the radical. Let a be a transfinitely topologically solvable ideal. Then for each j ∈  (g), by 7.44 (ii) the homomorphic image a + j/j is a solvable ideal of the finite-dimensional Lie algebra g/j and is therefore contained in r(g/j) = rj (g)/j, that is a ⊆ rj (g) for all j ∈  (g) and thus a ⊆ j∈ (g) rj (g) = r(g). By Claim (e) this shows that r(g) is the unique largest transfinitely topologically solvable ideal. In order to prove the last assertion of the theorem, we note from the preceding results, that r(g) = {0} implies that g is semisimple. Conversely, if g is semisimple, then every finite-dimensional homomorphic image is semisimple and thus has a zero radical; since g is a pro-Lie algebra and all homomorphic images of r(g) are transfinitely topologically solvable by 7.44 (ii), we conclude r(g) = {0}. Definition 7.49. If g is a topological Lie algebra and r(g) its radical, a subalgebra s of g is called a Levi summand if g is the vector space direct sum g = r(g) ⊕ s and (r, s)  → r + s : r(g) × s → g is an isomorphism of topological vector space. If x is an element of a finite-dimensional Lie algebra g (over a field with characteristic 0) for which the vector space ad x of g is nilpotent, say 5N endomorphism 1 n is called a special inner au(ad x)N+1 = 0, then α = ead x = · (ad x) n=0 n! tomorphism of g. For the following result we refer to [16, p. 89, Théorème 5]. Theorem 7.50 (The Levi–Mal’cev Theorem for Finite Dimensional Lie Algebras). Every finite-dimensional Lie algebra (over a field of characteristic 0) has a Levi summand. If s1 and s2 are Levi summands, then there is a special inner automorphism α such that s2 = α(s1 ). A simplified version of this theorem for pro-Lie algebras is the following: Proposition 7.51. Let g be a pro-Lie algebra for which the radical is the center. Then g is reductive and therefore is the product of z(g) and the product of a family of simple finite-dimensional real Lie algebras. Proof. The proof is immediate from Theorem 7.28 and Proposition 7.34. For a proof of a general Levi–Mal’cev Theorem for pro-Lie algebras we have to do some extra work. Moreover, the existence of Levi summands is one matter and the conjugacy another. We first deal with the existence.

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Levi–Mal’cev Theorem for Pro-Lie Algebras: Existence Theorem 7.52. Every pro-Lie algebra has Levi summands. Specifically: (i) A pro-Lie algebra g is the semidirect sum r(g) ⊕ s of the radical and a closed semisimple subalgebra s. (ii) The radical r(g) is countably topologically solvable, and any Levi summand is the cartesian product of a family of finite-dimensional simple Lie algebras. (iii) If h is a closed subalgebra of g such that g = r(g) + h then h contains a Levi summand s of g. Proof. (i) For i ∈  (g) we call a subalgebra s of g an i-complement, if i ⊆ s and s/i is a Levi summand for g/i. By Theorem 7.50, i-complements exist for each i. (Note: If g is semisimple, then g is an i-complement for all i ∈  (g).) Let gcoad be the coadjoint module of g. We let X · ω denote the element given by (X · ω)(Y ) = −ω[X, Y ]. Assume that V is a vector subspace of gcoad and consider the following conditions: (a) r(g)⊥ ∩ V = {0}, (b) X · V ⊆ V for all X ∈ V ⊥ . def

If these conditions are satisfied, then s = V ⊥ is a subalgebra of g such that r(g) + s = g; indeed if X, Y ∈ g, let ω ∈ V , and then ω([X, Y ]) = −(X · ω)(Y ) = 0 by (b). Thus [X, Y ] is in the annihilator of V , i.e. is contained in s. The second assertion is simply the dual of (a). Conversely, if V is the annihilator of a subalgebra s of g satisfying g = r(g) + s, then (a) and (b) are satisfied as is readily verified. Let us provisionally call a vector subspace of gcoad involutive if it satisfies (a) and (b). The set of involutive vector subspaces is at once seen to be inductive. By Zorn’s Lemma we find a maximal involutive vector subspace M. If we can show that def

gcoad = r(s)⊥ + M, then duality will show that s = M ⊥ is a Levi summand for g. We now let s = M ⊥ denote the annihilator in g of a maximal involutive vector subspace M of gcoad . Then s is a subalgebra of g which is minimal with respect to the properties (a∗ ) g = r(g) + s. (b∗ ) s is a closed subalgebra. Because g/r(g) ∼ = s/(s ∩ r(g)), the quotient algebra s/(s ∩ r(g)) is a semisimple pro-Lie algebra by 7.48. Since s ∩ r(g) is a transfinitely topologically solvable ideal of s it is the radical r(s) of s. We are finished when we can show that r(s) = {0}. Suppose that this is not the case. Then r(s) = {0}. In order to show a contradiction we may factor [r(s), r(s)] and thus assume that r(s) is abelian. Since s is a pro-Lie algebra, there is an i ∈  (s) such that r(s/i) = (r(s) + i)/i ∼ = r(s)/(r(s) ∩ i) is nonzero. We may factor the ideal i and thus assume that r(s) is finite-dimensional, that is, is isomorphic to Rn .

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We consider the representation π : s → gl(r(s)) induced by the adjoint representation. Its kernel ker π is the centralizer c of r(s) in s, it is an ideal, contains r(s), and s/c is mapped faithfully into the finite-dimensional Lie algebra gl(r(g)) and is, therefore, finite-dimensional. The factor algebra s/r(s), is a product of finite-dimensional simple Lie algebras by 7.48; hence the ideal c/r(s) is a cofinite-dimensional partial product. Thus we have s/r(s) = t/r(s) ⊕ c/r(s) where t/r(s) is a finite direct sum of finitedimensional simple Lie algebras, and where c/r(s) is a complementary semisimple summand. By Theorem 7.50 we have t = r(s) + s1 with a Levi summand s1 for t. But c has r(s) as its center and c/r(s) is semisimple. Now we apply 7.51 and conclude that c is a reductive pro-Lie algebra and thus is the ideal direct sum r(s) ⊕ s2 with a def Levi summand s2 for c. We set s∗ = s1 ⊕ s2 ∼ = s/r(s). Then s∗ is a Levi summand for g. Now g = r(s) + s∗ and s∗ ⊆ s. By the minimality of s we conclude s∗ = s and thus r(s) = {0}, contrary to our supposition. This contradiction proves that there is a closed subalgebra s of g such that g = r(g) + s and r(g) ∩ s = {0}. By Theorem 7.7, the algebraically direct sum r(g) ⊕ s is also topologically a direct sum. (ii) From Proposition 7.45 we know that r(g) is countably topologically solvable and from Theorem 7.28 we know that s is a product of finite-dimensional simple ideals of s. (iii) Finally assume that g = r(g) + h for a closed subalgebra h. Then the map x + (h ∩ r(g))  → x + r(g) : h/(h ∩ r(g)) → g/r(g) is an isomorphism of pro-Lie algebras by Lemma 3.17. Thus h ∩ r(g) is a prosolvable ideal of h modulo which h is isomorphic to the semisimple pro-Lie algebra g/r(g). Hence it is the radical r(h) of h. We now apply the theorem to h and find a Levi summand s of h so that h = r(h) ⊕ s. Then g = r(g) + h = r(g) + ((r(g) ∩ h) ⊕ s) = r(g) + s and s ∼ = h/r(h) ∼ = g/r(g). Thus s is a Levi summand of g as well. This completes the proof of our theorem. The Equivalence Theorem for Solvability of Pro-Lie Algebras Theorem 7.53. Let g be a pro-Lie algebra. Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

g is countably topologically solvable. g is transfinitely topologically solvable. g is prosolvable. g is countably solvable. g is transfinitely solvable. g does not contain a finite-dimensional simple Lie algebra.

Proof. By Proposition 7.45 we know that each of the conditions implies the next. Assume (vi) is true. Then from the Levi–Mal’cev Existence Theorem 7.52 we know that each Mal’cev summand has to be zero, that is that g = r(g). Then g is countably topologically solvable.

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We are concerned with the Lie theory of pro-Lie groups, and thus, on the side of the Lie algebras, with the class of pro-Lie algebras. The preceding Theorem 7.53 tells us that for the class of pro-Lie algebras, the many variations of the concept of solvability for topological Lie algebras coalesce and indeed agree with the classical concept of solvability or, at the utmost, with the purely algebraic concept of countable solvability. In particular, from hindsight, higher ordinals than ω do not appear in the context of solvability of pro-Lie groups. It appears remarkable that we were not able to prove directly that transfinite solvability or even ω-solvability implies any of the seemingly stronger conditions. Proposition 7.54. If j is any closed ideal in a pro-Lie algebra g then r(g/j) = (r(g) + j)/j. Proof. By Lemma 3.17 (b), (r(g) + j)/j is a closed solvable ideal of g/j, hence is contained in r(g/j). Further, (g/j)/((r(g) + j)/j) ∼ = g/(r(g) + j) ∼ = (g/r(g))/(r(g) + j)/r(g) is semisimple by 7.48 and thus is a product of simple finite-dimensional Lie algebras. Thus Lemma 7.47 applies and proves (r(g) + j)/j = r(g/j ).

Transfinitely Nilpotent Lie Algebras This takes care of solvability. What do we know about nilpotency? First we record an analog of 7.38. Let us say that a transfinite sequence of ideals jα of a Lie algebra g is a  descending nil-sequence of ideals, if for a limit ordinal β we have jβ = α<β jα and for β = α + 1 we have [g, jα ] ⊆ jβ . Then the sequence (g[α] )α≤ρ is that nil-sequence which descends most quickly. We say that a nil-sequence of ideals jα terminates if there is an ordinal κ such that jκ = {0}. Exercise E7.6. Prove the following companion result of Proposition 7.35. For a Lie algebra g, the following statements are equivalent: (i) g is transfinitely nilpotent. (ii) g has a terminating nil-sequence of ideals. [Hint. Emulate the proof of Proposition 7.38.] We recall the definition of the ordinals δ and δ from Definition 7.38. Proposition 7.55. (i) A transfinitely topologically nilpotent topological Lie algebra is transfinitely nilpotent and δ ≤ δ. (ii) A nilpotent Lie algebra is topologically nilpotent.

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(iii) A finite-dimensional Lie algebra is nilpotent if and only if it is topologically nilpotent. (iv) A subalgebra g of a topological Lie algebra h is topologically nilpotent, respectively, nilpotent if and only if g is topologically nilpotent, respectively, nilpotent. (v) A subalgebra of a transfinitely nilpotent Lie algebra is transfinitely nilpotent. A closed subalgebra of a transfinitely topologically nilpotent Lie algebra is transfinitely topologically nilpotent. Proof. Exercise. Exercise E7.7. Prove Proposition 7.55. [Hint. Emulate the proof of Proposition 7.41.] Lemma 7.56. Let g be a pro-Lie algebra. (i) For each closed ideal j of g, (g/j)[[α]] = g[[α]] + j/j.

(†)

(ii) In particular, every continuous homomorphic image of a transfinitely topologically nilpotent pro-Lie algebra is transfinitely topologically nilpotent. (iii) If j is a closed ideal such that dim g/j < ∞, then for every limit ordinal β there is an αj < β such that for all α the relation αj ≤ α < β implies g[[β]] + j = g[α] + j. (iv) γ ≤ ω, that is

(∗∗)

g[[ω+1]] = [g, g[[ω]] ] = g[[ω]] .

(v) For every pro-Lie algebra g, the characteristic ideal g[[ω]] satisfies [g, g[[ω]] ] = and for each j ∈  (g) there is a natural number nj such that for all natural numbers n satisfying nj ≤ n we have g[[ω]] ,

g[[ω]] + j = g[n] + j and [g, g[[ω]] ] + j = g[[ω]] + j. Proof. Exercise. Exercise E7.8. Prove Lemma 7.56. [Hint. Emulate the proof of Lemma 7.44.] The Equivalence Theorem for Nilpotency of Pro-Lie Algebras Theorem 7.57. Let g be a pro-Lie Algebra. Then the following assertions are equivalent: (i) g is countably topologically nilpotent. (ii) g is transfinitely topologically nilpotent. (iii) g is pronilpotent.

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(iv) For every pair x, y of elements in g the vector space endomorphism ad x satisfies limn (ad x)n y = 0. (v) g is countably nilpotent. (vi) g is transfinitely nilpotent. Proof. The proof of the equivalence of (i), (ii), and (iii) is an exercise as are the implications (i) ⇒ (v) ⇒ (vi). Next we shall prove that (iii) implies (iv). Assume that g is pronilpotent, let x, y ∈ g, and let j ∈  (g). Then g/j is nilpotent. Hence ad(x + j) is nilpotent on g/j. Hence there is an n such that (ad x)n (g) ⊆ j. Thus (ad x)n y ∈ j. Since lim  (g) = 0 this shows that limn (ad x)n y = 0. Now assume (iv). Let j ∈  (g). Then g/j is a finite-dimensional Lie algebra in which for every pair of elements x + j and y + j we have limn ad(x + j )n (y + j) = 0. We claim that the element ad(x + j) is nilpotent. Suppose this is not the case, then def

ad x has a nonzero eigenvalue λ. Then x1 = |λ|−1 x gives rise to a vector space endomorphism ad(x1 + j) of g/j has an eigenvalue of absolute value 1. Now there is a one- or two-dimensional ad x1 + j-invariant subspace V on which ad x1 induces an orthogonal transformation with respect to a suitable inner product on V . If y ∈ / j, then n the relation limn ad(x1 + j) (y + j) = 0 yields a contradiction to the orthogonality of ad(x1 + j)|V . This contradiction proves the claim that ad(x + j) is nilpotent on g/j for all x. But this shows that g/j is a nilpotent Lie algebra which establishes (iii). Thus we know that (i) through (iv) are equivalent. Finally, we shall assume (vi), that is, that g is transfinitely nilpotent, and prove (iv), that is (∀x, y ∈ g) lim(ad x)n y = 0. n

Since we assume that g is transfinitely g[δ] = {0}. Let us fix an arbitrary x abbreviation. Define inductively " Dgα gβ =  α<β gα

nilpotent there exists an ordinal δ such that ∈ g and set D = ad x for the purpose of if β = α + 1, if β is a limit ordinal.

Intuitively, we think of gα as the image of g under the hypothetical transfinite iterate “D α ”. By transfinite induction it is easy to see that gα ⊆ g[α] . As a consequence, gδ = {0}. Since every closed vector subspace of g is a weakly complete topological vector space the image under D of any closed vector subspace of g is closed (see Appendix 2, Theorem A2.12 (b)). Hence it follows by transfinite induction from the definition of gα above that all of the ideals gα are closed. Now let j ∈  (g). By Lemma 7.43(∗∗) we have   gα = (j + gα ). (†) (∀β) j + α<β

α<β

:  We set  g = g/j; since D = ad x and j is an ideal, we have Dj ⊆ j. If D g → g is

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 + j) = [x, y] + j, then D  = adg/j (x + j), and we define defined by D(y " gα if β = α + 1, D  gβ =  gα if β is a limit ordinal. α<β  Then we claim that (∀α)  gα = (gα + j)/j.

(‡)

Indeed assume that this is the case for α < β; then there are two cases. If β = α + 1, gα = D(g  α + j)/j by the induction hypothesis, and this last term equals then  gβ = D  and gβ . If, however, ([x, gα ] + j)/j = (Dgα + j)/j =(gβ + j)/j bythe definitions of D gα = α<β (gα + j)/j by induction hypothesis. β is a limit ordinal, then  gβ = α<β     Then the last term equals (g + j))/j = j + α<β gα )/j = (j + gβ )/j by (†) α<β α and by the definition of gβ . Thus claim (‡) is proved by transfinite induction. Since the Lie algebra  g is finite-dimensional, it satisfies the descending chain condition for vector subspaces. Hence there is a (smallest) natural number N such that  gN =  gN +1 ,  gN . It follows by transfinite induction that  gα =  gN for all α ≥ N . By that is D gN =  (‡) this means gα + j = gN + j for all α ≥ N , and since for α ≥ δ we have gα = {0} it follows that gN + j = gω + j = j. Thus gω ⊆ j. Since j ∈  (g) was arbitrary and  (g) = {0} we have gω = {0}. From Lemma 7.69 we know that the filter basis of the D n g = gn , n ∈ N converges to 0. Now take y ∈ g arbitrary. Then D n y ∈ D n g and so limn (ad x)n y = limn D n y = 0. Since x, y ∈ g are arbitrary this proves the asserted condition (iv). Exercise E7.9. Prove the equivalence of the conditions (i), (ii), and (iii) in Proposition 7.57 and the implications (i) ⇒ (iv) ⇒ (b). [Hint. Emulate the proof of Proposition 7.45.] Exercise E7.10. Prove the following result. Proposition. The full categories of prosolvable and pronilpotent pro-Lie algebras in the category of pro-Lie algebras proLieAlg (Definition 3.6) are complete. [Hint. Applying the results of Chapter 1 such as Theorem 1.11 we have to show that the categories of prosolvable and pronilpotent Lie algebras are closed in proLieAlg under the formation of arbitrary products and passage to closed subalgebras. A (closed) subalgebra of a countably solvable, respectively, countably nilpotent Lie algebra is countably solvable, respectively, countably nilpotent, so the Equivalence Theorems 7.53 and 7.57 settle the passage to subalgebras. There remain products. Let {gj : j ∈ J } be a family of prosolvable, respectively pronilpotent pro-Lie algebras. Every cofinite-di def  mensional ideal of g = j ∈J gj contains one of the form i = j ∈J ij such that there is a finite subset F ⊆ J such that j ∈ J \ F implies ij = gj . ij ∈  (gj ) and  Then g/i ∼ = j ∈F gj /ij . Since each gj /ij is solvable, respectively, nilpotent as gj is prosolvable, respectively, pronilpotent and since F is finite, this product is solvable, respectively, nilpotent. It follows that g is prosolvable, respectively, pronilpotent.]

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7 Profinite-Dimensional Modules and Lie Algebras

The phenomena of solvability and nilpotency are largely parallel in the class of proLie algebras, but the proofs for the two equivalence theorems have to follow different strategies. The hardest part in either case is to get from the transfinite version to the rest. The main difficulty appears to be that transfinite solvability and transfinite nilpotency are not inherited by quotients. We observed (in a remark following Definition 7.37) that every free Lie algebra is countably nilpotent. Every Lie algebra whatsoever is a quotient of a free one and thus is a quotient of a countably nilpotent Lie algebra; thus the preservation of transfinite solvability and nilpotency by passing to quotients fails dramatically. This illustrates that our equivalence theorems are nontrivial statements on the class of pro-Lie algebras which fail to hold in the same generality in general. We will have similar issues when we discuss solvability and nilpotency concepts for pro-Lie groups in Chapter 10. However, in some ways nilpotency is more easily handled under module theoretical aspects than is solvability. Let us have a look at some of these aspects in the following section.

The Nilpotent Radicals Let g be a Lie algebra and E a g-module. We recall from 7.19 the notions of E0 = {v ∈ E | (∀x ∈ L) x · v = 0}, the maximal zero submodule, and Eeff = g · E = span{x · v | x ∈ L, v ∈ E}, the effective submodule. These are dual concepts in more than a superficial sense, as we have seen in Proposition 7.21 which we complement in the following: Lemma 7.58. Let E be a g-module and V its weakly complete dual module. Then the annihilator (V0 )⊥ of the zero-module of V is the effective module g · E and the annihilator (g · V )⊥ of the effective module g · V of V is the zero module E0 . The annihilator of the zero-module E0 is the closure of the effective submodule g · V . Proof. Let v ∈ E. Then we have v ∈ (g · V )⊥ iff (∀x ∈ g, w ∈ W ) v, x · w = −x · v, w = 0 iff (∀x ∈ g) x · v = 0 iff x ∈ E0 . Thus (g · V )⊥ = E0 . It follows that (E0 )⊥ = (g.V )⊥⊥ = g · V . Likewise, (g · E)⊥ = V0 , and consequently (V0 )⊥ = (g · E)⊥⊥ = g · E. Definition 7.59. Let E be a g-module with a submodule F , and V a topological gmodule with a submodule W . Define an ascending sequence of g-submodules E|F [α] with α ranging through the ordinals and a descending sequence of closed submodules gα · W by transfinite induction as follows. (i) E|F [0] = F , E|F [1] = {x ∈ E : g · E ⊆ F }, so that (E|F [1])/F is the zero module of E/F , and if E|Fα has been defined for α < β then " {x ∈ E : g · x ⊆ E|F [α]} if β = α + 1, E|F [β] =  if β is a limit ordinal. α<β E|F [α]

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301

(ii) g0 · W = W , g1 · W = g · W = Weff , and if gα · W has been defined for α < β then " g · (gα · W ) if β = α + 1, β g ·W =  α if β is a limit ordinal. α<β g For simplicity, if F = {0}, write E[α] in place of E|F [α]. For reasons of cardinality, there is a first ordinal γ such that E[γ +1] = E[γ ]; write E[∞] = E[γ ]. The g-module E is called a transfinite nilmodule, if E[∞] = E, an ω-nilmodule, if E[ω] = E, and a nilmodule, if there is a natural number n such that E[n] = E. def In a parallel way there is a smallest ordinal ρ such that gρ+1 · V = gρ · V = g∞ · V . ∞ Call V a transfinitely nilpotent module, if g · V = {0}, an ω-nilpotent module, if gω · V = V , and a nilpotent module, if there is a natural number n such that gn · V = V . Notice that (E/F )[α] = E|F [α]/F . If E is any g-module, it may be considered a topological g-module with respect to the finest locally convex topology on E for which every vector subspace is closed. In that sense g·E and inductively gα ·E is well defined in the sense of the preceding definition such that gα+1 · E = g · (gα · E). Proposition 7.60. Let g be a Lie algebra, E a g-module with a submodule F , and V its weakly complete dual module with W = F ⊥ . Then (i) for all ordinals α = 0, 1, . . . , E|F [α] is the annihilator of gα · W . (ii) V is a transfinitely nilpotent, respectively, countably nilpotent respectively, nilpotent module if and only if E is a transfinite nilmodule, respectively ω-nilmodule, respectively, nilmodule. V    W     .. . gα · W .. .    {0}

⊥

←−−−→

⊥

←−−−→

{0}     F    .. . E|F [α] .. .    E

Proof. (ii) is an immediate consequence of (i). We prove (i) by transfinite induction. There is no loss in generality assuming F = {0} and W = V . The step from α to α + 1 is clear from Lemma 7.58. Let β be a limit ordinal and assume that the assertion holds for α < β. Then E[β]⊥ =  ⊥   = α<β E[α]⊥ = α<β gα · V = gβ · V . α<β E[α]

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Lemma 7.61. Let E be a locally finite dimensional g-module, F a submodule such that E/F is a transfinite nilmodule and 0 = x ∈ g. Then there is a unique smallest nonnegative integer n(x) such that x ∈ E|F [n(x)]. Consequently, gn(x) · x ⊆ F and thus g ω · E ⊆ F . def

Proof. Take x ∈ E and note that M = g.x is a finite-dimensional submodule of E. Then (M + F )/F is finite-dimensional submodule of a transfinite nilmodule E/F and therefore is a nilmodule. Thus there is a smallest natural number n such that x + F ∈ ((M + F )/F )M[n] = (M[n] + F )/F . Thus n is the smallest natural number such that x ∈ M[n] + F = M ∩ E|F [n]. Hence n(x) = n satisfies the requirements. By finite induction we see that gm · x ⊆ E|F [n(x) − m], m = 0, 1, . . . , n(x). Proposition 7.62. (i) Every locally finite-dimensional transfinite nilmodule is an ω-nilmodule. (ii) Every profinite-dimensional weakly complete transfinitely nilpotent module is countably nilpotent. Proof. Lemma 7.61 implies that each x is contained in E|F [ω] and thus E|F [ω] = E. Taking F = {0} in Lemma 7.61 we get (i). Further, (ii) follows from (i) via duality. Example 7.63. Let g be a topological Lie algebra. Then g is transfinitely topologically nilpotent, respectively, countably topologically nilpotent, respectively nilpotent, iff the adjoint module is a transfinitely nilpotent g-module iff the coadjoint module is a transfinite nilmodule, respectively an ω-nilmodule, respectively, a nilmodule. Furthermore g is a pronilpotent pro-Lie algebra iff the adjoint module is a profinite-dimensional module such that every finite-dimensional quotient module is a nilpotent module iff its coadjoint module is a locally finite-dimensional module such that all finite-dimensional submodules are nilmodules. Exercise E7.11. Prove the claims made for Example 7.63. Exercise E7.12. Use 7.63 in order to give an alternative proof of Theorem 7.57. Let us say that a submodule F of a g-module is a conilsubmodule if E/F is a transfinite nilmodule, that is if there is an ordinal λ such that E|F [λ] = E. We shall use this terminology briefly; the accumulation of three prefixes like co, nil, and sub is less than elegant. But it will serve for the moment. In the meantime we note that for locally finite-dimensional modules λ ≤ ω. Lemma 7.64. Assume that E is an almost finite-dimensional g-module. (i) E has a unique smallest conilsubmodule EN , namely, the intersection of all conilsubmodules. (ii) E has a largest semisimple submodule Ess = E0 ⊕ (Ess ∩ Eeff ), its semisimple radical, namely, the union of all finite-dimensional semisimple submodules (see 7.19). (iii) Ess ∩ Eeff ⊆ EN . (iv) Every weakly complete g-module has a unique largest nilpotent submodule and a largest submodule modulo which it is reductive.

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303

Proof. (i) Let N be the set of all conilsubmodules and let EN be the intersection. We must show that for each x ∈ E there is a natural number n(x) such that x ∈ E|EN [n(x)]. Since E is a locally finite-dimensional module, the module M = g · x + R · x generated by x has a finite number n of dimensions. For F ∈ N there is a smallest nonnegative integer nF (x) such that E|F [nF (x)] = E. Therefore, gnF (x) · x ⊆ F . That is, M/(M ∩ F ) is nilpotent. Now {M ∩ F1 ∩ · · · ∩ Fm : F1 , . . . , Fm ∈ N } is a filter basis of vector subspaces of the finite-dimensional vector space M and thus has a minimal element. That is, there are conilsubmodules F1 , . . . , Fn such that g ∩ EN = g ∩ F1 ∩ · · · ∩ Fn . Then (M + En )/EN ∼ = M/(M ∩ EN ) has an isomorphic copy in the product M/(M ∩ F1 ) × · · · × M/(M ∩ Fn ) which is a nilpotent module. Thus (M + EN )/EN is a nilpotent module and so M + EN = (M + EN )|EN [k] ⊆ E|EN [k] for some natural number k, and hence x ∈ E|EN [ω]. Therefore E = E|EN [ω] and thus EN is a conilsubmodule. (ii) is just a recollection from 7.17 and 7.20. (iii) is immediate from the definitions. (iv) This follows at once from (i) and (ii) by duality. Definition 7.65. If a topological Lie algebra g has a unique largest transfinitely topologically nilpotent ideal then that ideal is called its nilradical and is denoted by n(g). If a pro-Lie algebra g has a smallest ideal ncored (g) such that g/ncored (g) is reductive then ncored (g) is called the coreductive radical. The terminology chosen here is at variance with that of [16] for finite-dimensional Lie algebras: Bourbaki calls the nilradical n(g) of g the largest nilpotent ideal (see [16, §4, no 4]) and the coreductive radical ncored (g) the nilpotent radical (see [16, §5, no 3 and §6, no 4]). We are now ready for several fundamental results on the structure of pro-Lie algebras in general. Firstly, applying 7.65 to the adjoint module of a pro-Lie algebra, in view of Proposition 7.57 and Theorem 7.27 we get at once: Theorem 7.66 (The Existence of the Nilradical and the Coreductive Radical). Every pro-Lie algebra g has a nilradical n(g), and the nilradical is countably topologically nilpotent. Every pro-Lie algebra g has a coreductive radical ncored (g), and g/ncored (g) is the product of a central ideal isomorphic to RI and a product of a family of finitedimensional simple real Lie algebras. Obviously, the nilradical n(g) is contained in the radical r(g). But now we have better information on the entire set-up due to the classical results on finite-dimensional Lie algebras so that we shall see presently that ncored (g) ⊆ n(g) ⊆ r(g). Specifically, the second in the row of important theorems reads as follows; recall the definition of a Levi summand from Definition 7.49.

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Theorem 7.67 (The Relation between the Nilradical and the Coreductive Radical). Let g be a pro-Lie algebra. Then the coreductive radical ncored (g) is contained in the nilradical n(g) and ncored (g) = [g, g] ∩ r(g) = [g, r(g)]. (†) Moreover, if s is any Levi summand of g, then [g, g] = ncored (g) ⊕ s.

(‡)

Proof. (a) We recall that for any pro-Lie algebra h, the coreductive radical ncored (g) is defined via duality as the annihilator of the ((gcoad )ss )⊥ (see Theorem 7.27). By duality, the finite-dimensional submodules of gcoad are the annihilators j⊥ , as j ⊆ gad ranges through  (g). By the definition of the semisimple radical of the locally  finite-dimensional module (gcoad )ss in 7.17 and 7.19 we now have (gcoad )ss = j∈ (g) (j⊥ )ss .  By duality, we note that ((j⊥ )ss )⊥ /j = ncored (g/j). Therefore j∈ (g) ((j⊥ )ss )⊥ = ncored (g) and thus ncored (g) = lim ncored (g/j). j∈ (g)

gcoad M MMM vv v MMM vv v M vv (gcoad )ss j⊥ GG GG rr r r GG r G rrr

{0} OO OOO uu u OOO uu OO u u u u ncored (g) j III II pp p II p pp II ppp ((j⊥ )ss )⊥

(j⊥ )ss

g

{0} As a consequence ncored (g/j) =

ncored (g) + j ∼ ncored (g) . = j ncored (g) ∩ j

(b) For a finite-dimensional Lie algebra h we know that ncored (h) = [h, h] ∩ r(h) = [h, r(h)]. (See e.g. [16, §5, no 3, Theorème 1 and §6, no 4, Proposition 6].) Returning to g we recall from Proposition 7.54 that for each j ∈  (g) we have r(g/j) = (r(g) + j)/j. Thus ncored (g/j) = [g/j, g/j] ∩ r(g/j) =

[g,g]+j j

∩

r(g)+j j

and ncored

g j

=

'g

 g & j,r j

=

'g

j,

r(g)+j & j

=

[g,r(g)]+j j

∼ =

[g,r(g)] [g,r(g)]∩j .

  From (a) we know that ncored gj = ncoredj(g)+j , whence ncored (g) + j = ([g, g] + j) ∩ (r(g) + j) = [g, r(g)] + j for all j ∈  (g). Since r(g) and ncored (g) are closed,

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305

  (ncored (g) + j) = ncored (g). Further j∈ (g) (r(g) + j) = r(g) and   j∈ (g)  j∈ (g) ([g, g]+j)∩(r(g)+j) = j∈ (g) ([g, g]+j)∩ j∈ (g) (r(g)+j) = [g, g]∩r(g)  and j∈ (g) ([g, r(g)] + j) = [g, r(g)], and so ncored (g) = [g, r(g)]. Since (ncored (g) + j)/j = ncored (g/j) ⊆ n(g/j) for all j ∈  (g), by [16, §5, no 3, Remarque 2], we know that ncoredj(g)+j is nilpotent. So there is a natural number nj such that ncored (g)[nj ] ⊆ j; since j is closed we conclude that in fact ncored (g)[[nj ]] ⊆ j, which implies ncored (g)[[ω]] ⊆ j for all j. Hence ncored (g)[[ω]] ⊆   (g) = {0}. Therefore ncored (g) is a countably topologically nilpotent ideal and is therefore contained in the unique largest countably topologically nilpotent ideal n(g). With this statement the first part of the theorem up through (†) is proved. In order to prove (‡), we first note that s is perfect by Lemma 7.26 and thus s ⊆ [g, g]. By (†), ncored (g) ⊆ [g, g] and thus ncored (g) ⊕ s ⊆ [g, g]. Conversely, [g, g] = (r(g) + s) = r(g) + [r(g), s] + s ⊆ ncored (g) + s by (†). But ncored (g) + s is closed by Lemma 3.17 (b). Hence [g, g] ⊆ ncored (g) + s and thus (‡) is proved. This completes the proof of the theorem. In any pro-Lie algebra g, just as in any finite-dimensional algebra, we have the closed characteristic ideals g   r(g)    n(g)    ncored  (g)   {0}. If g1 = R, then ncored (g1 ) = {0} = n(g1 ). If g2 is the 2-dimensional nonabelian Lie algebra spanned by X1 , X2 with [X1 , X2 ] = X2 then n(g2 ) = R.X2 = g2 = r(g2 ). def

And if g3 = sl(2, R), then r(g3 ) = 0 = g3 . Thus for g = g1 ⊕ g2 ⊕ g3 we have {0} = ncored (g) = n(g) = r(g) = g.

Special Endomorphisms of Pro-Lie Algebras Let E be a real vector space and V = E ∗ its weakly complete dual. Recall that E may be identified with the topological dual V  of V . x ∈ V and ω ∈ E. We write ω(x) in the form of ω, x. Every vector space endomorphism ϕ of E has an adjoint ϕ ∗ : V → V , an endomorphism of topological vector spaces defined by ϕ(ω), x = ω, ϕ ∗ (x).

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Lemma 7.68. (i) The vector subspaces ker ϕ ⊆ ker ϕ 2 ⊆ ker ϕ 3 ⊆ · · · E form an ascending sequence of vector spaces and the closed vector subspaces V ⊇ ϕ ∗ (V ) ⊇ (ϕ ∗ )2 (V ) ⊇ (ϕ ∗ )3 (V ) ⊇ · · · form a descending sequence. (ii) (ϕ ∗ )n (V ) = (ker ϕ n )⊥ for n = 0, 1, 2 . . . . (iii) If F is a finite-dimensional vector subspace invariant under ϕ, equivalently, if W = F ⊥ ⊆ V is a cofinite-dimensional closed vector subspace of V invariant under ϕ ∗ , then ϕ|F is nilpotent on W , and ϕ ∗ induces on V /W the adjoint endomorphism of ϕ|F , that is (ϕ|F )∗ (x + W ) = ϕ ∗ (x) + W . V    ϕ ∗ (V  )    .. . (ϕ ∗ )n .. .    {0}

⊥

←−−−→

⊥

←−−−→

{0}     ker ϕ    .. . ker ϕ n .. .    E

Proof. The assertions of (i) are immediate. (ii) Since (ϕ n )∗ = (ϕ ∗ )n , it suffices to prove the assertion for n = 1. Now ω ∈ ⊥ ϕ ∗ (V ) iff ω ∈ ϕ ∗ (V )⊥ iff (∀x ∈ V ) ω, ϕ ∗ (x) = 0; but ω, ϕ ∗ (x) = ϕ(ω), x, and so this last assertion is equivalent to ϕ(ω) ∈ V ⊥ = {0}, that is, to ϕ(ω) = 0. This means ω ∈ ker ϕ.  n (iii) If F is finite-dimensional and E = ∞ n=1 ker ϕ , then there is a natural number N N N such that F ⊆ ker ϕ , that is ϕ (F ) = {0}. Thus if F is invariant, then ϕ|F : F → F is a nilpotent endomorphism. Moreover, ϕ ∗ induces on V /W the endomorphism x + W  → ϕ ∗ (x) + W , while we defined (ϕ|F )∗ , upon identifying V /W with the dual of F in the standard way, by the following rule: (∀x ∈ V , ω ∈ F )

ω, (ϕ|F )∗ (x + W ) = ϕ(ω), x.

Since ϕ(ω), x = ω, ϕ ∗ (x) = ω, ϕ ∗ (x) + W  we get the identity we claimed. Lemma 7.69. Let V be a weakly complete topological vector space and F a filter basis consisting of closed vector subspaces. Then the following conditions are equivalent:

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307

(i) lim  F = 0. (ii) F = {0}. Proof. Let U denote the filter of zero neighborhoods of V . Then by definition, (i) is equivalent to (i) the filter F  generated by F contains U. This in turn is equivalent to (i) (∀U ∈ U)(∃F ∈ F ) F ⊆ U .   Now (i) ⇒ (ii) is clear since F ⊆ U = {0}. We prove (ii) ⇒ (i) . Let U be a zero neighborhood. Since V is weakly complete, the filter basis V of cofinite-dimensional vector spaces W converges to 0 (Appendix 2). Hence we may assume that there is a cofinite-dimensional vector space W contained in U , and we may even assume that W is the only vector spacecontaining W and contained in U . By Lemma 7.43 we have F ∈F (W + F ) = W + F = W + {0} = W . Now {(W + F )/W : F ∈ F } is a filter basis of vector subspaces of the finite-dimensional vector space V /W intersecting in {0}, and thus there is an F ∈ F such that (W +F )/W is zero, that is, F ⊆ W ⊆ U . This proves (i) . Lemma 7.70. Let vector space and let ϕ be a vector space endomorphism  E be a real n such that E = ∞ n=1 ker ϕ . Then (i) for each x ∈ E there is a natural number N (x) such that ϕ n (x) = 0 for n ≥ N(x); (ii) for each sequence (an )n=0,1,... of real numbers there is an endomorphism  : E → E defined by (x) =

∞ 6 n=0

def

an ϕ n (x) =



N6 (x)−1

an ϕ n (x).

n=0

Proof. (i) Let x ∈ E. Since x ∈ m∈N ker ϕ m , we find an N (x) ∈ N such that x ∈ ker ϕ N(x) . So ϕ N (x) (x) = 0 and then n ≥ N (x) implies ϕ n (x) = 0. 5N (x)−1 n (ii) Assume that (an )n∈N is given. 5∞ Definem (x) = m=0 am ϕ (x). We may write this expression formally as m=0 am ϕ (x). The function  : E → E defined in this way is an endomorphism of vector spaces as is readily checked. Having a sequence5 (an )n=0,1,... of real numbers is the same as having a formal real n power series f (ξ ) = ∞ n=0 an ξ . The endomorphism  constructed in Lemma 7.70 will be denoted by f (ϕ). Exercise E7.13. Prove that under5the hypotheses of 7.70, the function which assigns n all formal power to a formal power series f (ξ ) = ∞ n=0 an ξ in the algebra R[[ξ ]] of5 n series the endomorphism f (ϕ) ∈ Hom(E, E) given by f (ϕ)(x) = ∞ n=0 an ϕ (x) is a morphism of real algebras.

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Let us look at the dual side and consider a weakly complete topological vector space V . Definition 7.71. A continuous space endomorphism ψ : V → V is called  vector n (V ) = {0}. approximately nilpotent if ∞ ψ n=1 5∞ Let f (ξ ) = n=0 an ξ n be a formal power series and assume that a zero neighborhood U is given. If ψ is approximately nilpotent, then by 7.71 and by Lemma 7.69, there is an N such that ψ N (V 5) ⊆ U . Thus for any x ∈ V , for each m > N and any p ∈ N, the element pn=m an ψ n (x) is contained in U . Thus the sequence   5k n n=0 an ψ (x) k=0,1,... is a Cauchy sequence, and since V is complete, it has a limit def

f (ψ)(x) =

∞ 6

an ψ n (x).

n=0

The function f (ψ) : V → V is an endomorphism of topological vector spaces. Lemma 7.72. Let V be a weakly complete topological vector space and E = V  its topological dual. If ψ : V → V is an approximately nilpotent endomorphism of def topological vector spaces then its adjoint morphism ϕ = ψ  : E → E satisfies the 5∞ hypotheses of Lemma 7.70, and if f (ξ ) = n=0 an ξ n is a formal power series, then f (ϕ) is the adjoint of f (ψ).  ∞  ∞ ∞ n n ⊥ = n ⊥ Proof. Firstly, we have n=1 ker ϕ n=1 (ker ψ ) = n=1 ϕ (V ) = {0} by 7.68 and our assumption on ψ. In view of our definitions we compute f (ϕ)(ω), x =

∞ 96

9

∞ ∞ : 6 6 an ϕ n (ω), x = an ϕ n (ω), x = an ω, ψ n (x)

m=0 ∞ 6

= ω,

m=0

m=0

: an ψ n (x) = ω, f (ψ)(x)

m=0

for all ω ∈ E and x ∈ V and this shows that f (ϕ) and f (ψ) are adjoints of each other.

Now we apply this formalism to pro-Lie algebras. We recall that a derivation D of an algebra with a bilinear multiplication (x, y)  → xy is called a derivation, if 5 1 n D(xy) = (Dx)y +x(Dy) for all x and y in the algebra. It is understood that ∞ ξ n=0 n! is the formal power series exp ξ . Proposition 7.73. Let g be a pro-Lie algebra and D a continuous, approximately def nilpotent derivation. Then eD = exp(D) is an automorphism of the topological Lie algebra g.

Levi–Mal’cev: Uniqueness

309

Proof. Since D is an approximately nilpotent continuous vector spaces endomorphism for every formal power series f (ξ ), the continuous vector space endomorphism f (D) is well defined. We apply this to f (ξ ) = eξ , so that indeed eD is a well-defined continuous vector space endomorphism. Let j ∈  (g). Then by 7.68 (iii), (i) there is a number N(j) such that D n (g) ∈ j for all n ≥ N (j). We claim the following two assertions: (ii) e−D (eD x) ∈ x + j and eD (e−D x) ∈ x + j for all x ∈ g, (iiii) eD [x, y] ∈ [eD x, eD y] + j for all x, y ∈ g. Indeed by (i), on g/j the linear map D induces a nilpotent derivation Dj by Dj (x + j) = Dx + j. Then eDj (x + j) = eD x + j and by the elementary properties of the exponential function on Der(g) we know that eDj is an automorphism of the finite-dimensional Lie algebra g/j. This entails (ii) and (iii). (See also [16, §6, no 8].)  Since  (g) = {0}, (ii) shows that eD has the inverse e−D , and (iii) shows that D e is an automorphism of the Lie algebra g. Corollary 7.74. Let x ∈ n(g) be an element in the nilradical of a pro-Lie algebra. Then ad x : g → g is an approximately nilpotent continuous derivation. In particular, ead x is an automorphism of the topological Lie algebra g. def

Proof. Since n(g) is an ideal, ψ = ad x maps g into n(g); then ψ 2 maps g into [n(g), n(g)] = n(g)[1] , and by induction it follows that ψ m maps g into n(g)[m−1] for m = 2, 3, . . . . Therefore ψ n+1 (g) ⊆ g[n] ⊆ n(g)[[n]] . This shows that ad x = ψ is an approximately nilpotent continuous derivation. The preceding Proposition 7.73 shows that ead x is an automorphism of g. Since ncored (g) ⊆ n(g) by Theorem 7.67, if x ∈ ncored (g), then the continuous vector space endomorphism ad x is approximately nilpotent and thus ead x is a welldefined automorphism of topological Lie algebras. Definition 7.75. An automorphism of a pro-Lie algebra is called special if it is of the form ead x for some element x ∈ ncored (g).

Levi–Mal’cev: Uniqueness We shall need a simple lemma for the calculations which follow. Lemma 7.76. Let L be a topological Lie algebra and I a closed ideal. If x, y ∈ L − (ad y)n )z ∈ I . If are such5 that x − y ∈ I , then for all z ∈ L one has ((ad x)n5 ∞ n n f (ξ ) = n=15 an ξ is a formal power series and f (ad x)(z) = ∞ n=1 an (ad x) z and ∞ n f (ad y)(z) = n=1 an (ad y) z converge, then f (ad x)z − f (ad y)z ∈ I .

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7 Profinite-Dimensional Modules and Lie Algebras

5 Proof. In any associative algebra we compute ξ n − ηn = nm=1 ξ n−m (ξ − η)ηm−1 5 n n−m (ad x − ad y)(ad y)m−1 z. Now and thus ((ad x)n − (ad y)n )z = m=1 (ad x) m−1 z = [x − y, u] ∈ I (for u = (ad y)m−1 z) since x − y ∈ I . It (ad x − ad y)(ad y) n follows that ((ad x) − (ad y)n )z ∈ I . However, the element f (ad x)z − g(ad y)z = 5 n n n=1 an ((ad x) − (ad y) )z is the limit of all finite partial sums, each of which is in I by that which precedes; thus it is contained in I because I is closed. Levi–Mal’cev Theorem for Pro-Lie Algebras: Conjugacy Theorem 7.77. (i) Two Levi summands of a pro-Lie algebra are conjugate under a special automorphism. (ii) A pro-Lie algebra g has only one Levi summand s, if and only if g is the direct sum, algebraically and topologically, r(g) ⊕ s. (iii) If m is a semisimple closed subalgebra of a pro-Lie algebra g and s is a Levi summand of g, then a conjugate of m under an inner automorphism of g is contained in s, and m is contained in some Levi summand. (iv) A semisimple closed ideal is contained in every Levi summand. Proof. (i) Let g be a pro-Lie algebra and ncored (g) its coreductive radical. Assume that g = r(g) ⊕ s1 = r(g) ⊕ s2 with two Levi summands s1 and s2 . We are going to show the existence of an element x ∈ ncored (g) such that ead x s1 = s2 . def

(a) The pro-Lie algebra  g = g/ncored (g) is reductive by 7.67 and thus is the direct sum of its center and radical r(g)/n(g) and a unique closed Levi summand  s/n(g), where  s is a unique closed ideal of g such that g/ s∼ s/ncored (g)) ∼ = (g/ncored (g))/( = r(g)/ncored (g) is abelian. Hence  s contains all Levi summands. It therefore suffices to prove the conjugacy result in  g. Therefore we shall assume from here on that r(g) = ncored (g). Thus by this assumption the radical is countably topologically nilpotent. (b) Assume that the claim is true when r(g) = ncored (g) is abelian and assume that ncored (g) is nilpotent, that is, ncored (g)[[n]] = {0} and ncored (g)[[n−1]] = {0}. We then prove the claim by induction on n. Indeed if n = 1 we have assumed the claim to be true. Assume that it has been established for n ≥ 1 and we assume now that def ncored (g)[[n+1]] = {0} while z = ncored (g)[[n]] = {0}. By the induction hypothesis we find an xn ∈ ncored (g) such that ead x s1 ⊆ s2 + z. Now z is in the center of def ncored (g) and thus is abelian. For simplicity set s3 = ead x s1 . We note that (s2 + z)/z is a Levi summand in g/z and is therefore closed. Thus s2 + z is closed in g and is therefore a pro-Lie algebra with abelian radical z. By our assumption there is an element z ∈ z such that ead z s3 = s2 . Since z is central in n(g) we have [x, z] = 0 and thus [ad x, ad z] = ad[x, z] = 0, and ad x and ad z are commuting nilpotent continuous

Levi–Mal’cev: Uniqueness

311

def

derivations on g. Therefore ead z ead x = ead z+ad x = ead(x+z) and setting y = x + z we get y ∈ ncored (g) and ead y s1 = s2 . This completes the induction. (c) Assume that the claim is true when r(g) = ncored (g) is abelian. We show here that the claim is true. Indeed, the preceding section (b) and its proof yields inductively a sequence of elements x1 , x2 , . . . ∈ ncored (g) such that (for x0 = 0) (∀n ∈ N) (∀n ∈ N)

ead xn s1 ⊆ s2 + ncored (g)[[n]] , xn − xn−1 ∈ ncored (g)

[[n]]

.

(∗) (∗∗)

Since ncored (g) is countably topologically nilpotent by Theorem 7.66, Lemma 7.69 implies limn→∞ ncored (g)[[n]] = 0. We claim that (xn )n∈N is a Cauchy sequence. Indeed let U be a zero neighborhood of g. Then there is a natural number N such that ncored (g)[[N ]] ⊆ U . Then from (∗∗) for all n ≥ N and k ∈ N we have xn+k − xn = (xn+k − xn+k−1 ) + · · · + (xn+1 − xn ) ∈ ncored (g)[[n+k]] + · · · + ncored (g)[[n+1]] ⊆ ncored (g)[[N ]] ⊆ U ; this proves the claim. Since g is complete and ncored (g) is closed, x = limn→∞ xn ∈ ncored (g) exists and [[N ]] x−xn = limk→∞ x5 for all n ≥5N . Thus for each y ∈ g we have n+k −xn ∈ ncored (g) N −1 1 1 ad x ad x m m m n y = m=0 m! ((ad x) − (ad xn )m )y + ∞ e y −e m=N m! ((ad x) − (ad xn ) )y [[N ]] by the preceding Lemma 7.76, where the second summand is contained in ncored (g) and where the first summand tends to zero since g is a topological Lie algebra. Since limN→∞ ncored (g)[[N ]] = 0 this shows that limn→∞ ead xn y = ead x y. On the other hand, (∗) yields for n ≥ N the relation ead xn s1 ⊆ s2 + ncored (g)[[n]] ⊆ s2 + ncored (g)[[N ]] ⊆ s2 + U.

(∗∗∗)

Since s2 + ncored (g)[[N ]] is closed as the full inverse image of a Levi summand in g/ncored (g)[[N]] , for each s ∈ s1 by (∗∗∗) we have ead x s = limn→∞ ead xn s ∈ s2 + ncored (g)[[N]] ⊆ s2 + U . Since U was an arbitrary zero-neighborhood and s2 is closed we finally have ead x s1 ⊆ s2 . Since ead x is an automorphism of g, the closed subalgebra ead x s1 is a Levi summand and thus projects bijectively onto g/ncored (g) under the quotient map. The same is true for s2 . We conclude from this that ead x s1 = s2 . (d) It remains to prove the claim in the case that r(g) = ncored (g) is abelian. In this case, x ∈ ncored (g) implies (ad x)2 = 0 so that ead x = 1 + ad x. From the existence part of the Levi–Mal’cev Theorem 7.52 we may write g = a  s with a semisimple pro-Lie algebra s and an s-module a; addition is componentwise, and multiplication is given by [(v, s), (w, t)] = (s ·w −t ·v, [s, t]). An element (v, 0) ∈ ncored (g) = a×{0} then acts in the form ad(v, 0)(w, t) = (−t.v, 0). def The subalgebra s1 = {0} × s is a Levi summand; any other Levi summand is uniquely determined by a continuous linear function f : s → a such that s2 = {(f (s), s) : s ∈ s} and that (s · f (t) − t · f (s), [s, t]) = [(f (s), s), (f (t), t)] = (f ([s, t]), [s, t]). Thus f satisfies the functional equation f ([s, t]) = s · f (t) − t · f (s). Then our claim in the present situation is as follows:

(†)

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7 Profinite-Dimensional Modules and Lie Algebras

(‡) For every continuous linear map f : s → a satisfying (†) there is an element a ∈ a such that f (s) = s · a. This is true if s and a are finite-dimensional. (See [16, §6, no 2, Remarque 2].) (d1 ) We assume first that dim a < ∞. In this case a is a finite-dimensional s-module, that is, there is a morphism of topological Lie algebras π : s → gl(a) where gl(a) is the finite-dimensional Lie algebra of all vector space endomorphisms of a. Thus def

t = ker π is a cofinite-dimensional ideal of s, and since s is semisimple, we know that s is the ideal direct sum s∗ ⊕ t and thus g = a  (s∗ ⊕ t) = a  s∗ ⊕ {0}  t where the last direct sum is an ideal direct sum. If t1 and t2 are elements in t then f ([t1 , t2 ]) = t1 · f (t2 ) + t2 · f (t1 ) = 0 + 0 = 0, and since t is semisimple and thus is a product of simple finite-dimensional Lie algebras, the span of all [t1 , t2 ] is dense in t. Hence f vanishes on t. Thus for s ∈ s∗ and t ∈ t we have f (s ⊕ t) = f (s). Since claim (‡) is true in the finite-dimensional situation, there is an a ∈ a such that f (s) = s · a for s ∈ s∗ . If t ∈ t then f (s + t) = f (s) = s · a and (s + t) · a = s · a + 0 = s · a. Thus claim (‡) holds in the present case and the s-module a, being in effect an s∗ -module is semisimple (see [16, §6, no 2, Théorème 2.]). (d2 ) The case that a is infinite-dimensional. Since g is a pro-Lie algebra, for each j ∈  (g) we get a cofinite-dimensional submodule aj of a such that aj × {0} = (a × {0}) ∩ j and a/aj is a semisimple s-module by (d1 ). It then follows from Theorem 7.18 that a  s-modules ak where k ranges is isomorphic to a product k∈K ak of finite-dimensional through a suitable index set K. We shall identify a and k∈K ak . Therefore we may  write the function f in the form f : s → k∈K ak , f (s) = (fk (s))k∈K . Each fk satisfies the functional equation (†). By (d1 ) above, for each k ∈ K we find an element ak ∈ ak such that fk (s) = s · ak . If we now set a = (ak )k∈K ∈ k∈K ak = a, then f (s) = s · a and (d) is established in all cases. But this completes the proof of part (i) of the theorem. (ii) Assume that g is the direct sum of the radical r(g) and a Levi summand s. Then the projection g → r(g) with kernel s maps any simple finite-dimensional subalgebra f of g onto a simple finite-dimensional subalgebra of the prosolvable pro-Lie algebra r(g). By Theorem 7.53, this image is zero. If s∗ is any Levi summand, then by Corollary 7.29 it is the product of finite-dimensional simple subalgebras and thus projects onto {0}. Hence s∗ ⊆ s. By (i) above, s∗ is conjugate to s under an automorphism of g and thus is a direct summand. Hence the roles of s and s∗ may be exchanged and thus we also get s ⊆ s∗ . Hence s∗ = s. Conversely, assume that s is a unique Levi summand. Then ead x s = s for all x ∈ ncored (g). Let j ∈  (g) and set sj = s/j. Then we have (∀ξ ∈ g/j)

ead ξ sj = sj .

In the finite-dimensional Lie algebra g/j we can differentiate and for η ∈ g/j get [ξ, η] = lim

0 =t→0

1 t·ad ξ η − η). (e t

Direct and Semidirect Sums Revisited

313

Thus we get [ξ, sj ] = sj for all ξ ∈ g/j and so (∀j ∈  (g))

[x, s] ⊆ s + j.



Since j∈ (g) (s + j), this implies [x, s] ⊆ s and therefore s is an ideal. Thus the algebraic and topological semidirect sum g = r(g) ⊕ s according to Definition 7.49 and Theorem 4.52 is in fact an algebraic and topological direct sum. def

(iii) Let m be a closed semisimple subalgebra of g. Then gm = r(g) + m is a closed vector subspace of g by Lemma 3.17 (b). Since m is semisimple and r(g) is pro-solvable, by Corollary 7.29 and Theorem 7.53 we have r(g) ∩ m = {0}. Thus gm = r(g) ⊕ m (algebraically and topologically) by Lemma 3.17 again and m is a Levi summand of gm . Now let s be a Levi summand of g. Then g = r(g) ⊕ s. Since r(g) ⊆ gm we have gm = r(g) ⊕ (s ∩ gm ) and s ∩ gm is a Levi summand of gm . By Part (i) of this theorem there is an element x ∈ ncored (g) such that ead x m = s ∩ gm ⊆ s. Also, m ⊆ e− ad x s and e− ad x s is a Levi summand of g. Assertion (iv) is an immediate consequence of claim (iii), since any ideal is invariant under all inner automorphisms. This completes the proof of the theorem in its entirety.

Direct and Semidirect Sums Revisited We have seen that in fundamental structural results, direct and semidirect sums play a crucial role. We review these concepts with particular emphasis on how they are reflected by the duality theory of modules. Definition 7.78. Let g be a topological Lie algebra and let i and a be subalgebras and let σ : i × a → g be the morphism of topological vector spaces given by σ (x, a) = x + a; clearly this is a restriction of vector space addition. (i) The sum g = i + a is called direct, if, firstly, σ is an isomorphism of topological vector spaces and, secondly, both i and a are ideals, that is, submodules of the adjoint module. (ii) The sum g = i + a is called semidirect, if, firstly, σ is an isomorphism of topological vector spaces and, secondly, i is an ideal and a is a subalgebra. Remark 7.79. (i) Let a be a topological Lie algebra and i a Lie algebra and a-module with a continuous module action (a, x)  → a · x : a × i → i. Then the product i × a is a topological Lie algebra with respect to componentwise addition and the Lie bracket [(x, a), (y, b)] = ([x, y] + a · y − b · x, [a, b]).

(8)

(ii) Let i be an ideal of a topological Lie algebra g and a a subalgebra and equip i with the structure of an a-module via the adjoint action a · x = [a, x] and give i × a

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7 Profinite-Dimensional Modules and Lie Algebras

the Lie algebra structure introduced in (i). Then μ : i × a → g, μ(x, a) = x + a, is a morphism of topological Lie algebras. (iii) In the circumstances of (ii), the Lie algebra g is a semidirect sum of i and a if and only if μ is an isomorphism of topological Lie algebras. Further g is a direct sum of i and a if and only if it is a semidirect sum and [i, a] = {0}. Exercise E7.14. Prove Remark 7.79. [Hint. (i) Verify the anticommmutativity of the Lie bracket and the Jacobi identity. (ii) Verify that σ preserves Lie brackets. (iii) is then straightforward.] Let us observe how these concepts dualize in the case that g is a pro-Lie algebra. Proposition 7.80. Let g be a pro-Lie algebra and i and a closed subalgebras. Let gcoad denote the dual of the underlying topological vector space with the coadjoint g module structure; the vector subspaces i⊥ and a⊥ the annihilators in g. Then i⊥ is a g submodule of gcoad and a⊥ is an a-submodule of gcoad , and the following are equivalent: (i) g is the semidirect sum of i and a. (ii) gcoad = a⊥ ⊕ i⊥ . Also the following statements are equivalent: (I) g is a semidirect sum with i as the ideal summand. (II) The quotient morphism π : g → g/i is retraction, that is, there is a morphism of Lie algebras σ : g/i → g such that π  σ = idg/i . (III) The weakly complete topological vector space g has a g/i-module structure such that π is a retraction of g/i-modules with a coretraction σ : g/i → g, a = σ (g/i), satisfying (∀ξ ∈ g/i, Y ∈ a) ξ · Y = [σ (ξ ), Y ] = ad(σ (ξ ))(Y )

(9)

(IV) The vector space  g has a g/i-module structure such that the inclusion mapping incl : i⊥ → g is a coretraction of g/i modules. Proof. The annihilator of a closed vector subspace of a profinite-dimensional L-module V is an L-submodule of V iff its annihilator in the dual module V  is an L-submodule. (See 7.11 (v).) This applies to V = gad and i as the vector subspace, respectively, to V = g as the a-module obtained by restricting the adjoint action of g to a, and a as the vector subspace. Having observed this we readily see that the equivalence of (i) and (ii) is a consequence of 7.7 (v). (I) ⇒ (II): If g = i ⊕ a semidirectly, then the projection pr a : g → a is surjective and has kernel i and thus induces an isomorphism of Lie algebras σ0 : g/i → a. Let j : a → g denote the inclusion and set σ = j  σ0 . Let v = x ⊕ a with x ∈ i and a ∈ a. Then π(σ (v + i))π(a) = a + i = a + x + i = v + i.

Cartan Subalgebras of Pro-Lie Algebras

315

def

(II) ⇒ (I): Set a = σ (g/i). If v ∈ g, determine a = σ (π(v)). Then x = g − a satisfies π(x) = π(v − a) = π(v) − π σ π(v) = 0, whence x ∈ ker π = i. Thus v = x + a. If v ∈ i ∩ a, then v = σ (ξ ) for some ξ ∈ g/i and 0 = π(v) = π σ (ξ ) = ξ and thus v = σ (ξ ) = 0. Hence g = i ⊕ a. The proof of the equivalence of (II), (III), and (IV) is Exercise E7.15. Exercise E7.15. Prove the equivalence of (II), (III), and (IV) in Proposition 7.80. [Hint. (II) ⇒ (III): We define the bilinear map g/i × g → g by ξ · v = [σ (ξ ), v] for ξ ∈ g/i and v ∈ g. This bilinear map is readily seen to a be a module operation. Moreover, π(ξ · v) = π([σ (ξ ), v]) = [π(σ (ξ ), π(v)] = [ξ, π(v)], and σ ([ξ, η]) = [σ (ξ ), σ (η)] = ξ · σ (η). Hence if we let (g/i)ad be the adjoint module, then π and σ are g/i-module morphisms. Condition (9) follows readily from our definition of the g/i-module structure of g by taking v = σ (η). (III) ⇒ (I): Let σ : g/i → g be the g/i-module map satisfying π  σ = idg/i and set a = σ (g/i). Then g = i ⊕ a and we are finished if we can show that a is a subalgebra. We have [σ (ξ ), σ (η)] = ξ · σ (η) by (9) and ξ · σ (η) = σ ([ξ, η]), since σ is a g/i-module map. Thus σ is a morphism of Lie algebras and the assertion follows. (III) ⇔ (IV): By 7.11 (v), (g/i) ∼ π : (g/i) →  g module = i⊥ , and the dual  morphism of the quotient module morphism π : g → g/i may be identified with the module inclusion ι : i⊥ →  g = g . Then the equivalence of (III) and (IV) follows by duality.]

Cartan Subalgebras of Pro-Lie Algebras We recall that for a finite-dimensional vector space V the Fitting null-component of an endomorphism T of V is V 0 (T ) = {v ∈ V : (∃n ∈ N) T n v = 0} = ker T dim V . If h is a Lie algebra and V is a finite-dimensional h-module, then there is a representation π : h → gl(V ) given by π(X)(v) = X · v, and the finite-dimensional homomorphic image π(h) of h is nilpotent by 7.56 and 7.57 if h is a pronilpotent pro-Lie algebra. We set V 0 (h) = {v ∈ V : (∀h ∈ h)(∃n ∈ N) π(h)n (v) = 0} = {v ∈ V : (∀h ∈ h) π(h)dim V (v) = 0}  = ker π(h)dim V h∈h

and V0 (h) = {v ∈ V : (∀h ∈ h)π(h)(v) = 0}  = ker π(h). h∈h

Clearly, V0 (h) ⊆ V 0 (h) and if V is a semisimple h-module then equality holds.

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7 Profinite-Dimensional Modules and Lie Algebras

The following exercise is in essence one on finite-dimensional h-modules; yet we give complete instructions for its proof. Exercise E7.16. Prove the following Lemma. Let h be a pronilpotent pro-Lie algebra and f : V → W a surjective morphism of finite-dimensional h-modules. Then f (V 0 (h)) = W 0 (h). [Hint. The containment ⊆ is readily checked. In order to prove surjectivity, upon replacing V by f −1 (W 0 (h)) if necessary, we may assume that W = W 0 (h), that is that πW (h)dim W = 0. Let us write V 0 = V 0 (h) and set V ∗ = f −1 f (V 0 ), then f (V ∗ ) = f (V 0 ) by the surjectivity of f . Hence we must show V ∗ = V . If v ∗ ∈ V ∗ then there is a v 0 ∈ V 0 such that f (v ∗ ) = f (v 0 ) and thus v ∗ − v 0 ∈ ker f and so V ∗ = V 0 + ker f . On the other hand, since dim V < ∞, we see that there is an element k ∈ h such that πV (k) is regular (that is, in the characteristic polynomial det(πV (h) − λ · idV ) = (−1)n λn + a1 (πV (h))λn−1 + · · · + ar (πV (h))λn−r the value of the first coefficient not identically vanishing is nonzero: ar (πV (k)) = 0). Then V 0 (h) = ker πV (k)dim V , that is V 0 is the Fitting zero-component V 0 (πV (k)) of πV (k). Accordingly, V = V 0 ⊕ V 1 with the Fitting one-component on which πV (k) is invertible. Since πW (k) is nilpotent and f (πV (k)v) = πW (k)f (v), we conclude f (v) = 0 for v ∈ V1 . Thus V1 ⊆ ker f and so V = V 0 ⊕ V 1 ⊆ V 0 + ker f = V ∗ ⊆ V , and this is what we had to show.] A similar statement does not hold for V0 in place of V 0 . Definition 7.81. Let h be a Lie algebra and V a profinite-dimensional h-module (see Definition 7.8), and let M(V ) denote the filter basis of cofinite-dimensional submodules of V . We define V 0 (h) to be the set {v ∈ V : (∀h ∈ h, W ∈ M(V ))(∃n ∈ N) hnV · v ∈ W }. If qW : V → V /W denotes the quotient morphism of h-modules, then V 0 (h) =



−1 qW ((V /W )0 (h)).

W ∈M(V )

Since V is profinite-dimensional, we may identify V with limW ∈M(V ) V /W ; if we do this, then V 0 (h) becomes identified with limW ∈M(V ) (V /W )0 (h). Proposition 7.82. Let h be a pronilpotent pro-Lie algebra and V and W profinitedimensional h-modules. If f : V1 → V2 is a morphism of profinite-dimensional h-modules, then f (V10 (h)) ⊆ V20 (h), (V1) and if f is surjective, then f (V10 (h)) = V20 (h).

(V2)

Cartan Subalgebras of Pro-Lie Algebras

317

Proof. First we show (V1). So assume that v ∈ V10 (h), and let W2 ∈ M(V2 ). Then def

W1 = f −1 (W2 ) ∈ M(V1 ) and by the choice of v for each h ∈ h there is an n ∈ N such that hnV1 · v ∈ W1 . Since f is a morphism of h-modules we get hnV2 · f (v) = f (hnV1 · v) ∈ f (W1 ) = W2 . This shows that f (v) ∈ V20 (h). Thus f (V10 (h)) ⊆ V20 (h). Next we assume that f is surjective and show (V2). We simplify notation by replacing V2 by V20 (h) and V1 by f −1 V20 (h). We write V10 for V10 (h) and have to show def

f (V10 ) = V2 . Suppose this claim is false, that is, S = f (V10 ) is a proper submodule of V2 . We proceed in two steps. Case A. dim V2 < ∞. In this case, since lim M(V1 ) = 0 and V2 has no small vector subspaces, there is a W1 ∈ M(V1 ) such that W1 ⊆ ker f . Hence there is a morphism of h-modules fW1 : V1 /W1 → V2 such that f = fW1  qW1 . By the lemma in Exercise E7.16. we have fW1 ((V1 /W1 )0 ) = V2 . It follows that qW1 (V10 ) = V10 +W1 W1 ,

being contained in fW−11 S, is a proper submodule of (V1 /W1 )0 . Now V10 = limW ∈M(V1 ) (V1 /W )0 for the inverse system qW W  : (V1 /W  )0 → (V1 /W )0 for W ⊇ W  . If W1 ⊇ W then qW1 = qW1 W  qW , and if qW1 (V10 ) = (V1 /W1 )0 then there is a W ∈ M(V1 ), W1 ⊇ W , such that qW1 W ((V1 /W )0 ) = (V1 /W )0 . However, this contradicts the lemma in Exercise E7.16. This contradiction settles Case A. def Case B. V2 arbitrary. By 3.17 (a), the submodule S = f (V10 ) of V2 is closed and thus agrees with the intersection of all S + U as U ranges through all zero neighborhoods of V2 . Thus there is a U such that V2 = S + U . Hence there is a W2 ∈ M(V2 ) def such that V2 = S + W2 . Define W1 = f −1 W2 ; then W1 ∈ M(V1 ) and V2 = S + W2 = f (V10 ) + f (W1 ) = f (V10 + W1 ). The morphism f induces a morphism fW1 : V1 /W1 → V2 /W2 by fW1 (v + W1 ) = f (v) + W2 . By what we just saw we have fW1 (V10 + W1 /W1 ) = V2 /W2 .

(∗)

We illustrate the situation in the following commutative diagram, in which qWj |Vj0 : Vj0 → (Vj /Wj )0 (for V20 = V2 ) are both the restriction and corestriction of the quotient qWj : Vj → Vj /Wj for j = 1, 2, implementing at the same time the limit maps Vj0 = limW ∈M(Vj ) (Vj /W )0 → (Vj /Wj )0 : V10 ⏐ ⏐ qW1 |V10  (V1 /W1 )0

f |V10

−−−−→ fW1 |(V1 /W1 )0

−−−−−−−→

V2 ⏐ ⏐q |V  W2 2 (V2 /W2 ).

According to Lemma 7.81, fW1 (V1 /W1 )0 (h)) = (V2 /W2 )0 . By Case A, both vertical maps are surjective, notably, qW1 |V10 . Thus fW1 |(V1 /W1 )0  qW1 |V10 is surjective, yielding fW1 (V10 + W1 /W1 ) = V2 /W2 .

(∗∗)

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7 Profinite-Dimensional Modules and Lie Algebras

But (∗) and (∗∗) contradict each other. This shows that our supposition is false, and so the proposition is proved. We need a further analysis of V 0 (h) in terms of limits. Let us fix a Lie algebra h. Recall that for h ∈ h we let hV : V → V denote the vector space endomorphism given by hV (v) = h · v. For an h-module V and any nonzero integer n we write def

V [n] = {v ∈ V : (∀h ∈ h) hnV (v) = 0}. Then clearly every V [n] is a closed h vector subspace and  V [0] = {0} ⊆ V [1] = V0 (h) ⊆ V [2] ⊆ · · · , and if V is finite-dimensional then n∈N0 V [n] = V [dim V ] = V 0 (h). Now let V be a profinite-dimensional h module and notice that for U ⊃ W in M(V ) we have quotient morphism qU W : V /W → V /U which induces morphisms qU W [n] : (V /W )[n] → (V /U )[n] and thus produce for each n a projective system. We notice (V /W )[n] = {v +W ∈ V /W : (∀h ∈ h) hnV /W (v +W ) = W } = VW [n]/W def

where VW [n] = {v ∈ V : (∀h ∈ h) hnV (v) ∈ W }. We have lim (V /W )[n] ∼ =



W ∈M(V )





VW [n]

W ∈M(V )

= {v ∈ V : (∀h ∈ h, W ∈ M(V )) hnV (v) ∈ W } = V [n]. For each W ∈ M(V ) we let n(W ) = min{m ∈ N0 : (V /W )[m] = (V /W )[m + 1] = (V /W )[dim V /W } ≤ dim V /W. Now let U ⊇ W in M(V ) again. Then v + W ∈ (V /W )[n(W )]

iff

(∀h ∈ h)

n(W )

hV

(v) ∈ W.

Since W ⊆ U we have n(U ) ≤ n(W ) and v +U ∈ V /W [n(U )]. Thus qU W : V /W → V /U induces a morphism pU W : (V /W )[n(W )] → (V /U )[n(U )],

W ⊆ U in M(V ).

This is a projective system and we can form its limit:    lim (V /W )[n(W )] ∼ VW [n(W )] = W ∈M(V )

W ∈M(V ) n(W )

= {v ∈ V : (∀h ∈ h, W ∈ M(V )) hV

(v) ∈ W } = V 0 (h).

We observe that V [n] ⊆ V 0 (h) and visualize the entire situation in a diagram, representing the case U ⊇ W :

319

Cartan Subalgebras of Pro-Lie Algebras incl

V [0] U

−−−−→ · · ·

V [0] W

−−−−→ · · ·

V [0]

−−−−→ · · ·

⏐ ⏐qU W [0] ⏐

id

V U [n(U  )]

−−−−→ · · ·

V W [n(U  )]

−−−−→ · · ·

V [n(U )]

−−−−→ · · ·

⏐ ⏐

qU W [n(U )]⏐ incl

⏐ ⏐qW [0] ⏐

⏐ ⏐

−−−−→ · · ·

V W [n(W  )]

−−−−→ · · ·

V [n(W )]

−−−−→ · · ·

⏐ ⏐

qU W [n(W )]⏐ incl

qW [n(U )]⏐ incl

id

V U [n(W  )]

⏐ ⏐

⏐ ⏐

qU W [n(U )]⏐ id

qW [n(W )]⏐ incl

V U [n(U  )]

V W [n(W  )]

⏐ ⏐

0 ⏐ qW

incl

V 0 (h).

Lemma 7.83. Let h be a Lie algebra and V a profinite-dimensional h-module. If V0 (h) = V [1] = {0}, then V 0 (h) = {0}. Proof. We recall that V 0 (h) = limW ∈M(V ) (V /W )[n(W )] is tantamount to V 0 (h) =  W ∈M(V ) VW [n(W )]. Then letting U(0) denote the filter of zero-neighborhoods of V , we note (∀N ∈ M(V ))(∃U ∈ M(V ))(∀W ∈ M(V )) W ⊆ U ⇒ V (h) ⊆ VW [n(W )] ⊆ V 0 (h) + N

(∗)

Now assume that V [1] = {0}, that is, for all h ∈ h, the relation hV (v) = 0 implies v = 0. So if hnV (v) = 0 for n ≥ 2, then hn−1 V (v) = 0, and so on down, inductively until we find v = 0. Thus we have V [n] = {0} for all n ∈ N0 . For the next step it helps V to look at the diagram preceding the lemma and to recall that U [n] = VU [n]/U . We consider the set F = {VU [n] : U ∈ M(V ), n ∈ N0 }. If U, W ∈ M(V ) and m, n ∈ N0 min{m,n} n then hV (v) ⊆ U ∩ W implies hm V (v) ∈ U and hV (v) ∈ W , so VU ∩W [min{m, n}] ⊆ VU [m] ∩ VV [n]. Thus F is a filter basis on the weakly complete vector space V which intersects in {0}. So it converges to zero by Theorem A2.13 (b) of Appendix 2. Let N be a zero neighborhood of V . By the Fundamental Theorem on Projective Limits 1.29 (i) we may assume that there is a U ∈ M(V ) such that N + U = N and that N/U does not contain any nondegenerate vector subspaces. Since lim F = 0 there is a W ∈ M(V ), which we may assume to be contained in U , and an n ∈ N0 such that VW [n] ⊆ N. Since N/U has no nondegenerate vector subspaces this means VW [n] ⊆ VW [n(W )] ⊆ U.

(∗∗)

From (∗) and (∗∗) we obtain V 0 (h) ⊆ VW [n(W )] ⊆ U ⊆ N. Since N ∈ U(0) was arbitrary, we conclude V 0 (h) = {0}. Definition 7.84. Let g be a pro-Lie algebra and h a subalgebra. Then g is a profinitedimensional h-module with respect to adjoint action. Then Definition 7.81 applies and

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7 Profinite-Dimensional Modules and Lie Algebras

we set g0 (h) = {x ∈ g : (∀h ∈ h, j ∈  (g))(∃n = n(x, h, j) ∈ N) (ad h)n (x) ∈ j}   = ((ad h)n )−1 (j) h∈h,j∈ (g) n∈N

=



((ad h)dim g/j )−1 (j).

h∈h,j∈ (g)

We can also transport the definition of V0 (h) to the present case of V = g and obtain g0 (h) = {x ∈ g : (∀h ∈ h, j ∈  (g)) (ad h)(x) ∈ j} = {x ∈ g : (∀h ∈ h) [h, x] = 0} = z(h, g), the centralizer of h in g. It follows from the last two lines of the definition of the set g0 (h) that it is a closed vector subspace of g. Lemma 7.85. Let h be a closed subalgebra of the pro-Lie algebra g. Then g0 (h) is a closed subalgebra. If h is pronilpotent then the subalgebra g0 (h) contains h and is its own normalizer. Proof. By the preceding remarks g0 (h) is a closed vector subspace; so it remains to show that g0 (h) is closed under the bracket multiplication. Thus let x, y ∈ g0 (h) and h ∈ h. Then there is an n ∈ N such that (ad h)n x ∈ j and (ad h)n y ∈ j. So (ad h)[x, y] = [(ad h)x, y] + [x, (ad identity; that is, ad h is a  Jacobi 5 h)y] by the p x, (ad h)q y] by the Leibniz derivation. Then (ad h)m [x, y] = p+q=m m [(ad h) p rule. (This is easily verified by induction.) Therefore, (ad h)2n [x, y] = 0. Thus g0 (h) is a closed Lie subalgebra. Now assume that h is pronilpotent. Then for j ∈  (g) the quotient (h + j)/j is a nilpotent Lie algebra. Then we find an n = n(j) such that for all pairs of elements h, x ∈ h we have (adg/j (h + j))n (x + j) = j in g/j. This is equivalent to (ad h)n x ∈ j, which entails h ⊆ g0 (h). Finally, let x be in the normalizer of g0 (h) and h ∈ h. Then (ad h)x = [x, −h] ∈ [x, h] ⊆ h. Thus for j ∈  (x)g there is an n = n((ad h)x, h, j) such that (ad h)n+1 x = (ad h)n ((ad h)x) ∈ j. Thus x ∈ g0 (h) by the definition of g0 (h) in 7.84. Now we apply the definitions of V0 (h) and V 0 (h) for an h-module V to the quotient module V = g/h. If we first apply the definition of V0 to the h-module g/h, with the module action X · (Y + h) = [X, Y ] + h, we get (g/h)0 (h) = {x + h ∈ g/h : (∀h ∈ h, j ∈  (g)) hg/h · (x + h) ∈ (h + j)/h} = {x + h ∈ g/h : (∀h ∈ h) hg/h · (x + h) = h} = {x + h ∈ g/h : (∀h ∈ h) [h, x] ∈ h}.

Cartan Subalgebras of Pro-Lie Algebras

321

We define n(h, g) = {x ∈ g : [x, h] ⊆ h} to be the normalizer of h in g, i.e., the def largest subalgebra h is an ideal. We notice n(h, g) = {x ∈ g : (∀h ∈ h)  in which [x, h] ∈ h} = h∈h (ad h)−1 h and conclude that n(h, g) is a closed subalgebra if h is a closed subalgebra. We have observed that (g/h)0 (h) = n(h, g)/h.

(†)

Next let us apply the definition of V 0 (h) to g/h and find (g/h)0 (h) = {x + h ∈ g/h : (∀h ∈ h, j ∈  (g))(∃n ∈ N) hng/h · (x + h) ∈ (h + j)/h} = {x + h ∈ g/h : (∀h ∈ h, j ∈  (g))(∃n ∈ N) hng · x ∈ h + j} = {x + h ∈ g/h : (∀h ∈ h, j ∈  (g))(∃n ∈ N) (ad h)n x ∈ h + j}. Now we assume that h is pronilpotent. Then (h + j)/j is nilpotent as a finite-dimensional homomorphic image of h (see 7.56 and 7.57). Therefore, if (ad h)n x ∈ h + j then there is an m ∈ N such that (ad h)m+n x = (ad h)m ((ad h)n x) ∈ j. Consequently, x + h ∈ (g/h)0 (h) iff x ∈ g0 (h). Therefore, (g/h)0 (h) = g0 (h)/h.

(‡)

Lemma 7.86. Assume that f : g1 → g2 is a morphism of pro-Lie algebras. Let h1 be a closed subalgebra of g1 and set h2 = f (h1 ). Then the following conclusions hold. (i) f (n(h, g1 )) ⊆ n(f (h), g2 ) and if f is surjective and h contains the kernel of f , then equality holds. (ii) f (g01 (h1 )) ⊆ g02 (h2 ) and if f is surjective, then equality holds. Proof. (i) Let x ∈ n(h, g), then [x, h] ⊆ h and so [f (x), f (h)] = f [x, h] ⊆ f (h), that is f (x) ⊆ n(f (h), g2 ). Now let y ∈ n(f (h), g2 ). Then there is an x ∈ g1 such that y = f (x). Then f [x, h] = [f (x), f (h)] = [y, f (h)] ⊆ f (h). Hence [x, h] ⊆ h + ker f , and if h = f −1 f (h), then h + ker f = h. (ii) g1 is an h1 -module V1 . We make g2 into a h1 -module V2 by setting h1 · x2 = [f (h1 ), x2 ]. Indeed, ' & ' & & ' [h1 , h1 ] · x2 = f [h1 , h1 ], x2 = [f (h1 ), x2 ], f (h1 ) + f (h1 ), [f (h1 ), x2 ] = h1 · (h1 · x) − h1 · (h1 · x). The function f : g1 → g2 is a morphism of h1 -modules; we have f (h1 · x1 ) = f [h1 , x1 ] = [f (h1 ), f (x1 )] = h1 · f (x1 ). From Proposition 7.82 we obtain f (g0 (h1 )) = f (V10 (h1 )) ⊆ V20 (h1 ) = g0 (f (h1 )) = g0 (h2 ) with equality holding if f is surjective.

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7 Profinite-Dimensional Modules and Lie Algebras

Theorem 7.87. For a closed pronilpotent subalgebra h of a pro-Lie algebra g, the following conditions are equivalent: (i) g0 (h) = h. (ii) h is its own normalizer. (iii) (g/h)0 (h) = {0}. Proof. From (†) we know that (ii) and (iii) are equivalent. (i) ⇒ (ii): Since h is pronilpotent and g0 (h) = h, the subalgebra h is its own normalizer by Lemma 7.85. (iii) ⇒ (i): Since h is pronilpotent we know that h ⊆ g0 (h) by 7.85, and (g/h)0 (h) = 0 g (h)/h by (‡). This means g0 (h) = h as asserted. Definition of a Cartan Subalgebra of a Pro-Lie Algebra Definition 7.88. A subalgebra h of a pro-Lie algebra g is called a Cartan subalgebra if it is a closed pronilpotent subalgebra satisfying the equivalent conditions of Theorem 7.87. That is, a Cartan subalgebra of a pro-Lie algebra is a closed pronilpotent subalgebra that agrees with its own normalizer. We continue this definition for technical purposes as follows: We denote by P the category whose objects are pairs (g, h) consisting of a pro-Lie algebra g and a closed subalgebra h of g and whose morphisms f : (g1 , h1 ) → (g2 , h2 ) are morphisms f : g1 → g2 of pro-Lie algebras such that f (h1 ) ⊆ h2 . A Cartan pair is a P -object (g, h) where h is a Cartan subalgebra of g. The category C is the full subcategory of P of all Cartan pairs (g, h). Exercise E7.17. Prove: PropositionA. The center of a pro-Lie algebra is contained in every Cartan subalgebra. Proposition B. The only Cartan subalgebra of a pronilpotent pro-Lie algebra g is g itself. [Hint. A. If h is any subalgebra of g and z is the center of g, then z ⊆ n(h, g). If h is a Cartan subalgebra, then n(h, g) = h by the preceding paragraph, so z ⊆ h follows. B. By Definition 7.88, g is a Cartan subalgebra. Now let h be any Cartan subalgebra and let j ∈  (g). Let f : g → g/j denote the quotient morphism. Then h = g0 (h) and f (h) = f (g0 (h)) = g0 (f (h)) by 7.86 (ii), and since f(h) is nilpotent, f (h) = (h + j)/j is a Cartan subalgebra of g. A Cartan subalgebra of a finite-dimensional nilpotent Lie algebra is the whole algebra (see e.g. [19, Ch. VII, §2, no 1 Example 1)]). Thus h + j = g for all j ∈  (g). This implies h = g.] In a finite-dimensional Lie algebra the existence of Cartan subalgebras is based on the existence of regular elements. Indeed if x is regular in a finite-dimensional Lie algebra, then g0 (x) is a Cartan subalgebra and every Cartan subalgebra is so obtained. (See [19, Chap. VII, §2].) In the case of pro-Lie algebras we have to resort to our

Cartan Subalgebras of Pro-Lie Algebras

323

knowledge of the structure of pro-Lie algebras and the existence of Cartan subalgebras in the finite-dimensional case. In the following we shall refer to strict projective limits, whose definition was given in 1.24. The essence here is that a projective limit is strict if all limit morphisms are surjective. We noticed in a remark following Proposition 1.22 that for projective systems of weakly complete topological vector spaces (such as pro-Lie algebras) this is tantamount to saying that all bonding maps are surjective. Proposition 7.89. The category C of Cartan pairs is closed in P under the formation of arbitrary products, strict projective limits, and pullbacks of surjective morphisms. Proof. The category of pronilpotent pro-Lie algebras is closed in proLieAlg under passing to limits by Exercise E7.10. Moreover, products are strict projective limits of finite products by Theorem 1.5 (b). Therefore, we shall prove that C is closed under the formation of finite products and projective limits. (A) Finite products. Let {(gj , hj ) : j ∈ J } be a finite family of Cartan pairs. def  def  Set g = j ∈J gj and h = j ∈J hj ; we have to show that (g, h) is a Cartan pair, that is, that h is a Cartan subalgebra. By Exercise E7.10, h  is pronilpotent. Every cofinite-dimensional ideal of h contains one of the form i = j ∈J ij such that ij ∈   (hj ). Then h/i ∼ = j ∈J hj /ij . An element x = (xj )j ∈J ∈ g is in g0 (h) iff for each h = (hj )j ∈J ∈ h and each i as above there is a natural number n such that (ad h)n x = ((ad hj )n xj )j ∈J ∈ i, that is (ad hj )n xj ∈ ij for j ∈ F . This is the case iff xj ∈ g0j (hj ). Since hj is a Cartan subalgebra of gj we have g0j (hj ) = hj . Hence x ∈ g0 (h) iff xj ∈ hj for all j ∈ J iff x ∈ h. Thus h is a Cartan subalgebra. (B) Strict projective limits. Let {fj k : (gk , hk ) → (gj , hj ) | (j, k) ∈ J ×J, j ≤ k}, be a projective system of Cartan pairs and set g = limj ∈J gj and h = limj ∈J hj . We have to show that (g, h) is a Cartan pair, that is that h is a Cartan subalgebra of g. By Exercise E7.10, h is pronilpotent. Let fj : g → gj denote the limit morphisms. Then fj (g0 (h)) ⊆ g0j (fj (h)) = g0j (hj ) since fj is surjective as the projective system is assumed to be strict. Since all hj are Cartan subalgebras of gj , we have g0j (hj ) = hj .  Hence fj (g0 (h)) ⊆ hj for all j ∈ J . Thus h ⊆ g0 (h) ⊆ j ∈J fj−1 (hj ) = h (see for instance the Closed Subgroup Theorem for Projective Limits 1.34). Thus h = g0 (h) and so h is a Cartan subalgebra of g. (C) Pullbacks of surjective morphisms. Let π1

(g∗⏐ , h∗ ) −−−→ (g1⏐ , h1 ) ⏐ ⏐ π2  q1 (g2 , h2 ) −−−→ (g3 , h3 ) q2

be a pullback in P and assume that q1 and q2 are surjective and that (gj , hj ) are in C for j = 1, 2, 3. Then it is an easy exercise to verify that π1 and π2 are surjective, and h∗ is pronilpotent, since the full subcategory of proLieAlg of pronilpotent pro-Lie algebras is closed under passing to limits by Exercise E7.10. We claim that it is in fact a Cartan

324

7 Profinite-Dimensional Modules and Lie Algebras

subalgebra of g∗ . We consider (g∗ )0 (h∗ ) which contains h since h is pronilpotent and satisfies πj ((g∗ )0 (h∗ ) = g0j (πj (h∗ )) = g0j (hj ) by 7.86 and the surjectivity of πj for j = 1, 2. But g0j (hj ) = hj since hj is a Cartan subalgebra of gj for j = 1, 2. Hence πj ((g∗ )0 (h∗ )) = hj for j = 1, 2, and that implies h∗ ⊆ (g∗ )0 (h∗ ) ⊆ h∗ . Thus h∗ is a Cartan subalgebra of g∗ as asserted. Theorem 1.29 describes typical situations in which strict projective limits arise naturally. Exercise E7.18. Use the methods of proof applied for 7.89 to prove the following Proposition. The category C of Cartan pairs is closed in P under the formation of all limits whose limit morphisms are surjective. Proposition 7.90. Assume that f : g1 → g2 is a surjective morphism of pro-Lie aldef gebras. If h1 is a Cartan subalgebra of g1 , then h2 = f (h1 ) is a Cartan subalgebra of g2 . Proof. (i) As a homomorphic image of a pronilpotent Lie algebra h1 the pro-Lie algebra h2 is pronilpotent by 7.56 and 7.57. Since h1 = g0 (h1 ) by Definition 7.88, from Proposition 7.86 (ii) we obtain h2 = f (h1 ) = f (g0 (h1 )) = g0 (f (h1 )) = g0 (h2 ). Thus h2 is a Cartan subalgebra of g2 by Definition 7.88. Lemma 7.91. If h1 and h2 are Cartan subalgebras of a pro-Lie algebra g, and if h1 ⊆ h2 , then h1 = h2 . Proof. For each j ∈  (g) and the quotient morphism qj : g → g/j, the images qj (h1 ) and qj (h2 ) are Cartan subalgebras of g/j by Proposition 7.90. From qj (h1 ) ⊆ qj (h2 ) we conclude qj (h1 ) = qj (h2 ) since all Cartan subalgebras of the finite-dimensional Lie algebra g/j have the same dimension. Hence h1 + j = h2 + j and so h2 ⊆ h1 + j for all j ∈  (g). Since lim  (g) = 0 and h1 is closed, we have j∈ (g) (h1 + j) = h1 . Hence h2 ⊆ h1 and this completes the proof of the lemma.  Proposition 7.92. Let g = z × j ∈J sj be a reductive pro-Lie algebra with an abelian pro-Lie algebra z and a family of finite-dimensional simple Lie algebras sj according to Theorem  7.27. Then a subalgebra h of g is a Cartan subalgebra iff it is of the form h = z × j ∈J hj where hj is a Cartan subalgebra of sj for each j ∈ J . Exercise E7.19. Prove Proposition 7.92. [Hint. If h is of the form asserted, then it is a Cartan subalgebra by Proposition 7.89. Conversely, if h is a Cartan subalgebra of g, then it contains z × {1} by Proposition A of Exercise  E7.17 above. The projection into sj is a Cartan subalgebra hj of sj . So h ⊆ z × j ∈J hj . The latter is a Cartan subalgebra and thus equality must hold by Lemma 7.90.]

Cartan Subalgebras of Pro-Lie Algebras

325

Existence of Cartan Subalgebras Theorem 7.93. Let g be a pro-Lie algebra and i ∈  (g). Then for each subalgebra hi of g containing i such that hi /i is a Cartan subalgebra of g/i there is a Cartan subalgebra h of g such that h + i = hi . Proof. We consider families F = {hj : j ∈ J }, J ⊆  (g) of subalgebras of g such that (i) (ii) (iii) (iv)

j ⊆ hj and hj /j is a Cartan subalgebra of g/j, j2 ⊆ j1 in J implies hj2 ⊆ hj1 , i ∈ J and hi is the given subalgebra of g, and J is a filter basis.

It is clear from (i) and (ii) that j2 ⊆ j1 implies hj1 = hj2 + j1 . The singleton family {hi } satisfies all of these conditions. We let F be the set of all such families; we recall that each family is really a function j  → hj with domain J and the set of closed subalgebras of g as range. Then there is a partial order ≤ on F such that F1 ≤ F2 if J1 ⊆ J2 and the second function extends the first. Clearly, (F , ≤) is an inductive poset, and so Zorn’s Lemma allows us to choose a maximal element, say M = {hj : j ∈ M}. Then {qjk : (g/k, hk /k) → (g/j, hj /j) | (j, k) ∈ M × M, k ⊆ j}, is a strict projective system of Cartan pairs. Set gM = limj∈M g/j and hM = limj∈M hj /j. Then hM is a Cartan subalgebra of gM by 7.89. Moreover, if qj : gM → g/j,

j ∈ M,

denote the limit maps, then qi (hM ) = hi /i. The inclusion M →  (g) induces a unique morphism ηM : g → gM = lim g/j, j∈M

ηM (x) = (x + j)j∈M .

The quotient morphism ηM is an isomorphism iff M is cofinal in  (g). This is what we have to establish. Suppose this is not the case. Then there exists a k ∈  (g) such that no j ∈  (g) such that j ⊆ k is contained in M. Replacing k by i ∩ k, if necessary, we may assume that k ⊆ i. Since g/k is finite-dimensional, the filtered poset {j ∈ M : k ⊆ j ⊆ i} satisfies the finite descending chain condition and thus has a unique minimal element m. Since g/k and g/m are finite-dimensional, there is a subalgebra hk of g containing k such that hk /k is a Cartan subalgebra of g/k mapping onto the given subalgebra hm /m of g/m: see [19, Ch. vii, §2, no 3, Corollaire 2 de Théorème 1]. We form the pullback πM g⏐∗ −−−−→ g⏐M ⏐ ⏐q πk  m g/k −−−−→ g/m. qmk

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Since qm is surjective, πk is surjective, and since qmk is surjective, πM is surjective. In the category P of pairs of pro-Lie algebras we get a pullback πM

, h∗ ) −−−−→ (gM ⏐ , hM ) (g∗⏐ ⏐ ⏐q πk  m (g/k, hk /k) −−−−→ (g/m, hm /m). qmk

h∗

is a Cartan subalgebra of g∗ . For j ∈  (g) By Proposition 7.89, we know that we let pj : g → g/j denote the quotient morphism, and consider the two morphisms ηM : g → gM and pk : g → g/k. We notice that qm  ηM = pm = qmk  pk . Then the universal property of the pullback yields a unique morphism q ∗ : g → g∗ such def

that ηM = πM  q ∗ and pk = πk  q ∗ . Now let M ∗ = (q ∗ )−1 ( (g∗ )) ⊆  (g); then M ∪ {k} ⊆ M ∗ . For j∗ ∈  (g∗ ) let pj∗∗ : g∗ → g∗ /j∗ denote the quotient morphism. If j∗ ∈ M ∗ , then set j = q ∗−1 (j∗ ) and obtain a commuting diagram q∗

g⏐∗ g −−−−→ ⏐ ⏐ ⏐p ∗ pj   j∗ ∗ g/j −−−−→ g /j∗ , fj

where fj (x + j) = defines an isomorphism. The subalgebra pj∗∗ (h∗ ) of g∗ /j∗ is a Cartan subalgebra by 7.90, say h∗j∗ /j∗ for a subalgebra h∗j∗ of g∗ . Set hj = q ∗−1 (h∗j∗ ), then hj /j ∼ = h∗j∗ /j∗ is a subalgebra of g/j ∼ = g∗ /j∗ . The function j∗  → (q ∗ )−1 (j∗ ) : ∗ ∗  (g) → M is an isomorphism of posets. It follows that M ∗ is a filter basis. The family M∗ = {hj : j ∈ M ∗ } belongs to F and is properly larger than M. This contradicts the maximality of M. This contradiction shows that k does not exist and that M is cofinal in  (g) and that therefore ηM : g → gM is an isomorphism. Then q ∗ (x) + j∗

−1 (hM ) is a Cartan subalgebra of g such that pi (h) = qi (hM ) = hi /i. h = ηM def

Proposition 7.94. Assume that f : g1 → g2 is a surjective morphism of pro-Lie algebras. If h2 is a Cartan subalgebra of g2 , then there exists a Cartan subalgebra h1 of g1 such that f (h1 ) = h2 . Proof. Let k = f −1 (h2 ). Now k has a Cartan subalgebra h1 by Theorem 7.93. So, since the restriction f |k : k → h2 is surjective, we have h2 = f (h1 ) by Proposition 7.90. Since h2 is a Cartan subalgebra of k, it is pronilpotent. Now let us consider g01 (h1 ); since h is pronilpotent, we have h ⊆ g01 (h1 ). Now Proposition 7.87 shows f (g01 (h1 )) = g02 (f (h1 )) = g02 (h2 ) = h2 in view of the fact that h2 is a Cartan subalgebra. It follows that g01 (h1 ) ⊆ f −1 (h2 ) = k. But since h1 is a Cartan subalgebra of k, we have h1 ⊆ g01 (h1 ) ⊆ k0 (h1 ) = h1 . Thus h1 is a Cartan subalgebra of g1 . In Theorem 7.93 we proved the existence of Cartan subalgebras of pro-Lie algebras. In the following proposition we show that in fact they are as abundant as in the finitedimensional case. First we note that the Fundamental Theorem on Projective Limits gives us an easy characterisation of dense subsets in projective limits.

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327

Exercise E7.20. Prove the following assertion: Proposition. A subset D of a projective limit G = limj ∈J Gj of topological groups is dense if the image fj (D) is dense in fj (G) for each limit morphism fj : G → Gj . [Hint. Let x ∈ G and W a neighborhood of x. By the Fundamental Theorem 1.27 (i) there is an index k and an identity neighborhood U of Gk such that fk−1 (U ) is an identity neighborhood of G contained in x −1 W . So xfk−1 (U ) ⊆ W . Since fk (D) is dense in fk (G) there is a d ∈ D such that fk (d) ∈ fk (x)U . Then d ∈ xfk−1 (U ) ⊆ U .] Proposition 7.95. The union of all Cartan subalgebras in a pro-Lie algebra is dense. Proof. For a pro-Lie algebra g and each cofinite-dimensional ideal j ∈  (g), the quotient map qj : g → g/j is a limit map. Let D be the union of all Cartan subalgebras h of g. By Proposition 7.90, qj (h) is a Cartan subalgebra of g/j and therefore qj (D) contains the set Uj of all regular elements of the finite-dimensional Lie algebra g/j by [19, Chap. VII, §2, no 3 Théorème 1]. The set Uj is a dense open subset of g/j since it is the complement of the zero set of a nonzero polynomial function. Thus qj (D) is dense in g/j for each j ∈  (g) and indeed contains a dense open set. Then by the proposition in Exercise E7.20 above, D is dense in g. Lemma 7.96. Let g be a pro-Lie algebra and h a Cartan subalgebra. (i) Let f : g → n be a surjective morphism onto a pronilpotent pro-Lie algebra. Then f (h) = n and g = ker f + h. (ii) g + h = g. Proof. (i) By Proposition 7.90, f (h) is a Cartan subalgebra of n. Then by Proposition B of Exercise E7.17, f (h) = n follows. (ii) We apply (i) to the quotient homomorphism f : g → g/g onto an abelian pro-Lie algebra and conclude f (h) = g/g , that is g + h = g. Lemma 7.97. Let h1 and h2 be two Cartan subalgebras of a prosolvable pro-Lie algebra g and j ∈  (g). Then there is an element xj ∈ ncored (g) such that ead xj h1 ⊆ h2 + j. Proof. The continuous homomorphic image of ncored (g) in g/j is ncored (g)/j. If qj : g → g/j is the quotient map, then qj (h1 ) and qj (h2 ) are Cartan subalgebras of g/j by Proposition 7.90. Thus by Bourbaki, VII, §3, no 4, Théorème 3, there is a xj ∈ ncored (g) such that eadg/j xj qj (h1 ) = qj (h2 ), and this implies the assertion ead xj h1 ⊆ h2 + j. If g is a prosolvable pro-Lie algebra, we know from Theorem 7.67 that ncored (g) = [g, g] = g . We shall abbreviate this ideal by writing g˙ . Lemma 7.98. Let h1 and h2 be two Cartan subalgebras of a pro-solvable pro-Lie algebra g and assume that there is a finite-dimensional ideal a such that a + h1 = a + h2 = g. Then there is an x ∈ ncored (g) such that ead x h1 = h2 .

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Proof. Since dim a < ∞ and g = a + hm , the vector space g/hm is a homomorphic image of a and thus is finite-dimensional. Hence g/(h1 ∩h2 ) is finite-dimensional. Thus all sufficiently small j ∈  (g) are contained in h1 ∩ h2 . Then for these j, according to Lemma 7.97, we find an x ∈ g = ncored (g) such that ead x h1 ⊆ h2 +j = h2 . Since ead x for x ∈ ncored (g) is an automorphism by Corollary 7.74, ead x h1 is a Cartan subalgebra of g. Thus Lemma 7.92 shows ead x h1 = h2 . Recall that a Lie algebra is called metabelian, if its commutator algebra is abelian. If g is a metabelian pro-Lie algebra, then g˙ is abelian, and g = g˙ + h if h is a Cartan subalgebra. Assume now that a is a closed abelian ideal of g and that g = a + h for a Cartan subalgebra g. Now every vector subspace of a is an ideal in a. Since a ∩ h is an ideal in h for any subalgebra h, a ∩ h is an ideal of g = a + h. We observe that this ideal does not depend on the choice of h: Lemma 7.99. Let h1 and h2 be two Cartan subalgebras of a pro-Lie algebra g which is of the form g = a + h1 = a + h2 . Then a ∩ h1 = a ∩ h2 . Proof. For j ∈  (g), we set ˜j = a ∩ j. Then g/˜j is a solvable pro-Lie algebra in which dim(a/˜j) is finite, and (hm + ˜j)/˜j are Cartan subalgebras by 7.90. So by Lemma 7.97, ad (x+j) there is an x ∈ g˙ such that e g/j˜ (h1 + ˜j))/˜j = (h2 + ˜j)/˜j, that is, ead x (h1 + ˜j)) = ad x ˜ h2 + j. Now ϕ = e is an automorphism of g which leaves ideals invariant, as does ϕ −1 = e− ad x . Thus ϕ(h1 + ˜j) = ϕ(h1 ) + ϕ(˜j) = ϕ(h1 ) + ˜j. Since ˜j ⊆ a, by the modular law, h2 ∩ a = (h2 + ˜j) ∩ a = (ϕ(h1 ) + ˜j) ∩ a = ϕ(h1 ) ∩ a = ϕ(h1 ) ∩ ϕ(a) = ϕ(h1 ∩ a) = h1 ∩ a. Lemma 7.100. Let h1 and h2 be two Cartan subalgebras of a pro-Lie algebra g which is of the form g = a + h1 = a + h2 for some closed abelian ideal a. Then there is an x ∈ a such that ead x h1 = h2 . Proof. Assume that the assertion is true if h1 and h2 meet a trivially. By 7.98 we have a ∩ h1 = a ∩ h2 ; denote this ideal by i. Then the assumption applies to g/i and shows that there is an x ∈ a such that eadg/i x+i h1 /i = h2 /i; but that implies ead x h1 = h2 . Assume now that a ∩ h1 = a ∩ h2 = {0}. So we let g be a pro-Lie algebra and h = h1 a Cartan subalgebra and we assume that g is the topological vector space direct sum a ⊕ h for some abelian closed ideal a. By Lemma 7.98, any other Cartan subalgebra h2 is of the form h2 = {f (h) + h : h ∈ h} ⊆ a ⊕ h with a continuous linear function f : h → a satisfying the functional equation f ([h, k]) = [h, f (k)] + [f (h), k]. (10) Then ϕ : g → g, ϕ(a + h) = (a + f (h)) + h) is an automorphism of g such that ϕ(h1 ) = h2 . Therefore we must show that we find an x ∈ a such that f (h) = [x, h]. As (ad x)2 = 0, this implies ϕ = 1 + ad x = ead x with 1 = idg . The remainder of the proof concerns the search for this x. Let F denote the filter basis of all closed cofinite-dimensional vector subspaces j of a which are ideals of g. Then lim F = 0

Cartan Subalgebras of Pro-Lie Algebras

329

and for each j ∈ F , the quotient g/j is the topological vector space direct sum a/j ⊕ ((h ⊕ j)/j). By Proposition 7.90, the homomorphic image (h ⊕ j)/j is a Cartan subalgebra of g/j. We claim that for x ∈ a we have (∀j ∈ F )

[x, h] ⊆ j ⇒ x ∈ j.

(11)

Proof of (11). The relation [x, h] ⊆ j together with the commutativity of a implies that x + j is in the center of g/j = a/j + ((h + j)/j). By Proposition A of Exercise E7.17, this implies x + j ∈ (a/j) ∩ ((h + j)/j) = j/j, and that means x ∈ j as asserted. Now we claim that for each j ∈ F there is an element xj ∈ a such that (∀h ∈ h)

f (h) ∈ [xj , h] + j.

(12)

Proof of (12). We note that Lemma 7.98 applies to g/j and shows the existence of an xj + j ∈ a/j such that ({h + [xj , h] : h ∈ h1 } + j)/j = ((1 + ad xj )(h1 ) + j)/j = (((1 + adg/j (xj + j))(h1 + j))/j = eadg/j (xj +j) (h1 + j)/j ) = (h2 + j)/j = ({h + f (h) : h ∈ h1 } + j)/j. This shows that [xj , h] − f (h) ∈ j for all h ∈ h1 = h and so establishes (12). Now assume that j, k ∈ F such that j ⊇ k. Then for all h ∈ h we have [xj − xk , h] = [xj , h] − [xk , h] ∈ j + k = j. Then (11) implies xj − xk ∈ j. Since lim F = 0, this shows that (xj )j∈F is a Cauchy net in the weakly complete topological vector space a and therefore has a limit x = limj∈F xj . From (12), for k ⊆ j in F we have f (h) ∈ [xk , h] + k ⊆ [xk , h] + j. Fixing j and passing to the limit x = limk∈F xk yields f (h) ∈ [x, h] + j. Now lim F = 0 yields f (h) = [x, h] and this is what we had to show. Conjugacy of Cartan Subalgebras of Prosolvable Pro-Lie Algebras Theorem 7.101. Let h1 and h2 be two Cartan subalgebras of a prosolvable pro-Lie algebra g. Then there is an x ∈ ncored (g) such that ead x h1 = h2 . Proof. We know that ncored (g) = [g, g] is a pronilpotent ideal. If ncored (g) is abelian then Lemma 7.100 proves the assertion. We follow the strategy of the proof of Part (i) of Theorem 7.77. First assume that ncored (g) is nilpotent, that is, ncored (g)[[n]] = {0}

and

ncored (g)[[n−1]] = {0}.

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We then prove the claim by induction on n. Indeed if n = 1 we have assumed the claim to be true. Assume that it has been established for n ≥ 1 and we assume now def that ncored (g)[[n+1]] = {0} while z = ncored (g)[[n]] = {0}. By the induction hypothesis we find an xn ∈ ncored (g) such that ead x h1 ⊆ h2 + z. Now z is in the center of ncored (g) def and thus is abelian. For simplicity set h3 = ead x h1 . We note that (h2 + z)/z is a Cartan subalgebra of g/z. Thus h2 + z is closed in g and is therefore a pro-Lie algebra which is the sum of the abelian ideal z and the Cartan subalgebra h2 . By Lemma 7.100 there is an element z ∈ z such that ead z h3 = h2 . Since z is central in ncored (g) we have [x, z] = 0 and thus [ad x, ad z] = ad[x, z] = 0, and ad x and ad z are commuting nilpotent continuous derivations on g. Therefore ead z ead x = ead z+ad x = ead(x+z) , def

and setting y = x + z we get y ∈ ncored (g) and ead y h1 = h2 . This completes the induction. Now we proceed to the general case that ncored (g) = g˙ is pronilpotent. The preceddef

ing paragraph inductively yields a sequence of elements x1 , x2 , . . . ∈ n = ncored (g) such that (for x0 = 0) (∀n ∈ N) (∀n ∈ N)

ead xn h1 ⊆ h2 + ncored (g)[[n]] , xn − xn−1 ∈ ncored (g)

[[n]]

.

(∗) (∗∗)

Since ncored (g) is countably topologically nilpotent by Theorem 7.66, Lemma 7.69 implies limn→∞ ncored (g)[[n]] = 0. We claim that (xn )n∈N is a Cauchy sequence. Indeed let U be a zero neighborhood of g. Then there is a natural number N such that ncored (g)[[N ]] ⊆ U . Then from (∗∗) for all n ≥ N and k ∈ N we have xn+k − xn = (xn+k − xn+k−1 ) + · · · + (xn+1 − xn ) ∈ ncored (g)[[n+k]] + · · · + ncored (g)[[n+1]] ⊆ ncored (g)[[N ]] ⊆ U ; this proves the claim. Since g is complete and ncored (g) is closed, x = limn→∞ xn ∈ ncored (g) exists and [[N ]] x − xn = limk→∞ xn+k − x5 for all n ≥ N . 5 Thus for each y ∈ g n ∈ ncored (g) N −1 1 1 ad x ad x m n we have e y − e y = m=0 m! ((ad x)m − (ad xn )m )y + ∞ m=N m! ((ad x) − (ad xn )m )y where the second summand is contained in ncored (g)[[N ]] by Lemma 7.76, and where the first summand tends to zero since g is a topological Lie algebra. Since limN→∞ ncored (g)[[N ]] = 0 this shows that limn→∞ ead xn y = ead x y. On the other hand, for n ≥ N, statement (∗) yields the relation ead xn h1 ⊆ h2 + ncored (g)[[n]] ⊆ h2 + ncored (g)[[N ]] ⊆ h2 + U.

(∗∗∗)

Since h2 + ncored (g)[[N ]] is closed by Lemma 3.17 (ii), for each h ∈ h1 by (∗∗∗) we have ead x h = limn→∞ ead xn h ∈ h2 + ncored (g)[[N ]] ⊆ h2 + U . Since U was an arbitrary zero-neighborhood and h2 is closed, we finally have ead x h1 ⊆ h2 . Since ead x is an automorphism of g, the closed subalgebra ead x h1 is a Cartan subalgebra. Then by Lemma 7.92 we conclude that ead x h1 = h2 .  0  In the 3-dimensional simple Lie algebra sl(2, R) the subalgebras R · 01 −1 and  0 1 R · −1 0 are two Cartan subalgebras which are not conjugate as each element of the first has real and each of the second a purely imaginary spectrum.

Theorem of Ado

331

Theorem of Ado Let us recall Ado’s Theorem for finite-dimensional Lie algebras. Theorem 7.102. Let g be a finite-dimensional Lie algebra over a field K of characteristic 0 and let n denote its nilradical. Then there is a finite-dimensional K-vector space V and an injective representation π : g → gl(V ) such that π(x) is a nilpotent endomorphism of V for all x ∈ n. Proof. See for instance [16, §7 Théorème 3]. It is perhaps surprising that we have, as a fairly direct consequence, an easy version of an Ado-Theorem for pro-Lie algebras (over the reals). Let us introduce a suitable terminology. Definition 7.103. (i) Let g be a Lie algebra and V a g-module. Let π : g → gl(V ) be the morphism defined by x · v = π(x)(v). The ideal ker π of g is called the kernel of the module V . The g-module V is called faithful if the associated representation π is injective, that is, if the kernel of π is zero. (ii) Let V be a profinite-dimensional g-module V (see Definition 7.8 (i)) and let M be the filter basis of closed submodules M ⊆ V such that dim V /M < ∞. For each M ∈ M define πM : g → gl(V /M), πM (g)(v + M) = g · v + M. Then the g-module V will be called niladapted, if for each M ∈ M, and each x in n(g), the nilradical of g (see Definition 7.65), the endomorphism πM (x) of the finite-dimensional vector space V /M is nilpotent. Thus the Theorem of Ado 7.102, in this terminology and for real Lie algebras, may be reformulated as follows. Corollary 7.104 (Theorem of Ado for Finite-Dimensional Real Lie Algebras). Every finite-dimensional real Lie algebra g has a faithful niladapted g-module.  While up to this point g was an arbitrary (real) Lie algebra, we now consider g to be a pro-Lie algebra and  (g) the filter basis of its cofinite-dimensional ideals. For each j ∈  (g), let qj : g → g/j be the quotient morphisms. Every g/j-module W may be considered a g-module via x · w = qj (x) · w. The g/j-module W is faithful if and only if the g-module W has the kernel j. By Corollary 7.104, for a given proLie algebra there is a finite-dimensional faithful niladapted g/j-module real vector space Vj . Then Vj is also a g module with kernel j. Alternatively, there is a continuous representation πj : g → gl(Vj ) such that πj (g) consists of nilpotent endomorphisms of Vj and j = ker πj . def  Accordingly, V = j∈ (g) Vj is a profinite-dimensional g module (see Definition 7.8) for the module operation x · (vj )j∈ (g) = (x · vj )j∈ (g) . Thus we have a morphism ρ : g → gl(V ), ρ(x)((vj )j∈ (g) ) = (ρj (x)vj )j∈ (g) , (13) whose kernel is trivial. Thus V is a faithful profinite g-module.

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7 Profinite-Dimensional Modules and Lie Algebras

Ado’s Theorem for Pro-Lie Algebras Theorem 7.105. For each pro-Lie algebra g there exists a faithful, niladapted profinitedimensional g-module. Proof. We define the filter base B of cofinite partial products MF , defined for a finite subset F of  (g) by "  Vj if j ∈ / F, Wj , Wj = MF = {0} if j ∈ F . j∈ (g) Let M denote the filter basis of all cofinite-dimensional submodules and take M ∈ M. We claim that B is cofinal in M. We have lim B = 0 by the definition of the product topology, and the quotient morphism V  → V /M is continuous. The image of B by this quotient morphism converges to zero and thus limMF ∈B (MF + M)/M = M. Since V /M is finite-dimensional, this implies the existence of a finite set F such that MF ⊆ M. This shows that B is cofinal in M as claimed. Next we show that V is a niladapted g-module. Now let M ∈ M. Then we find a finite subset F of  (g) such that MF ⊆ M. Let x ∈ n(g) and let πMF : g → gl(V /MF ) and πM : g → gl(V /M) the representations associated with the g-modules V /MF , respectively, V /M. Since  V /MF ∼ = j∈F Vj , we have a natural number such that πMF (x)n = 0. There is a canonical g-module quotient morphism p : V /MF → V /M, p(v + MF ) = v + M. We have p πMF (x) = πM (x)p. Therefore πM (x)n p = p πMF (x)n = 0. Since p is surjective, πM (x)n = 0 follows, and this shows that V is niladapted.

Postscript We recall that a pro-Lie algebra is a topological Lie algebra which is, in general, infinite-dimensional without any cardinality restriction on the dimension. The core result of this chapter is the Levi–Mal’cev decomposition theorem which, in the context of pro-Lie algebras, has several components and aspects: • Every pro-Lie algebra g has a radical r which is countably topologically solvable; • every pro-Lie algebra has a Levi summand such that it is algebraically and topologically the semidirect sum of the radical and any Levi summand; • every Levi summand is semisimple in the sense that it is the product of a family of finite-dimensional simple Lie algebras indexed by a set whose cardinality is not subject to any restriction; • every pro-Lie algebra has a nilradical contained in r, which is countably topologically nilpotent; • every pro-Lie algebra has a coreductive radical [g, g] ∩ r(g) = [r, g] which is nilpotent and modulo which g is reductive, that is, g/([g, g]∩r(g)) the product of a central algebra isomorphic to RI for an index set I and a semisimple algebra;

Postscript

333

• any two Levi summands are conjugate under an automorphism of the well-defined form ead x for x in the coreductive radical. These propositions are complemented by basic results on Cartan subalgebras: • Every pro-Lie algebra has Cartan subalgebras whose union is dense. • The Cartan subalgebras of a prosolvable pro-Lie algebra are conjugate under inner automorphisms. While all of this has a familiar ring from the finite-dimensional classical Lie theory to which, of course, the result must correctly specialize, the proof takes an arduous route through all of these components, many of which first have to be based on appropriate definitions of solvability, nilpotency, reductivity, semisimplicity, and all of these definitions have their particularly tricky aspects in the context of an infinite-dimensional and topological Lie algebra theory unfettered by any countability assumptions, like that of pro-Lie algebras. For instance, solvability and nilpotency have algebraic aspects that force us into the transfinite domain. But it also has topological variants, since we are dealing with topological Lie algebras and we might just as well take closed commutator algebras in place of the mere algebraic commutator algebras. It is perhaps surprising that, firstly, the algebraic and the topological commutator series must terminate after countably many steps, even though the topological weight of the algebra may take any cardinal value, and secondly that all the potentially different concepts of solvability on the one hand and of nilpotency on the other are equivalent, respectively. Another tricky aspect of the Lie theory of pro-Lie algebras is that the concepts of reductivity and semisimplicity are based on cartesian (algebraic and topological) products of finite-dimensional simple algebras (where, in the case of reductivity we consider R temporarily as a simple algebra) rather than on direct algebraic sums. From hindsight, we understand that duality may be at the root of this, and indeed this is the reason that this chapter begins with a broad study of modules and their duality. Specifically we deal with the duality of locally finite-dimensional modules on the one hand and profinite-dimensional modules on the other. The eventual application aims at the fact that the adjoint module of a pro-Lie algebra is a profinite-dimensional module while the coadjoint module is its locally finite-dimensional dual. Many arguments are completed by moving over to the completely algebraic side of the locally finite-dimensional modules. If one were to look for analogies, one might observe that this duality of modules permits an algebraisation via duality not unlike the algebraisation of compact abelian groups via Pontryagin Duality. Every step of the way the presence of infinitely many dimensions forces us to devise special arguments that cannot be obtained by any naïve generalisation of methods from the classical theory of finite-dimensional Lie algebras with the final result that we find the central structure theory á la Levi–Mal’cev to persist in the pro-Lie algebra situation. Even the sheer definition of Cartan subalgebras is problematic; while in the end it turns out that the classical definition works, making it possible to define a Cartan subalgebra as a pronilpotent closed subalgebra agreeing with its own normalizer, for the most part we have to work with another characteristic property which assures us that the image of a Cartan subalgebra under a surjective morphism is again a Cartan subalgebra.

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Existing literature shows nothing of these things to the best of our knowledge. There are basic results in the theory of finite-dimensional Lie algebras which generalize with much less effort to the situation of pro-Lie algebras. One of these is Ado’s Theorem saying that a finite-dimensional Lie algebra over a field of characteristic 0 has a faithful linear representation. Once the concepts most suitable to the situation of pro-Lie algebras and their module theory are introduced, it is not very hard to deduce from the general definitions and the classical Theorem of Ado that every pro-Lie algebra has a faithful profinite-dimensional module which respects the nilradical in a sense to be expected.

Chapter 8

The Structure of Simply Connected Pro-Lie Groups

In Chapter 6 we recognized that the category of pro-Lie algebras has an equivalent copy in the category of topological groups in the form of the full subcategory of prosimply connected groups, and in Chapter 7 we determined a rather detailed structure theory of pro-Lie algebras. We use the results of these two preceding chapters in order to give a fairly explicit structure theory of prosimply connected pro-Lie groups and to show, in particular, that a pro-Lie group is prosimply connected if and only if it is simply connected. We use these insights to prove that a simply connected pro-Lie group is algebraically generated by its one-parameter subgroups, in other words, that G = expG L(G). Prerequisites. Apart from the material in the previous chapters we use at one point the Iwasawa decomposition G = KAN of a connected semisimple real Lie group G. The reader should have a basic knowledge of the ideas around simple connectivity such as they are, for instance, presented in [102, Appendix 2]. There will be a comment which we shall not use further; in it we make reference to the fact that π2 (G) = 0 for all connected Lie groups G.

The Adjoint Action The adjoint representation Ad : G → Aut(L(G)) of a topological group G with Lie algebra as discussed in Chapter 2, Definition 2.27 through Proposition 2.30, applies, in particular, to pro-Lie algebras. Notably, we have the formula (∀g ∈ G, X ∈ L(G))

g(expG X)g −1 = expG Ad(g)(X).

(Ad)

def

Proposition 8.1. Let G be a pro-Lie group and X ∈ g = L(G). Then the following diagram is commutative: adg ad⏐g g −−−−→ ⏐ ⏐ expG ⏐  δ→eδ G −−−−→ Inn(g), g →Ig

that is, (∀X ∈ g)

Ad(expG X) = eadg X .

(1)

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8 The Structure of Simply Connected Pro-Lie Groups

Proof. Let N ∈ N (G) and set j = L(N ) ∈  (g). (See Chapter 4, Corollary 4.21.) Then adg/j quot g −−−−→ g/j −−−−→ adg/j ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ expG/N  expG  (2) δ→eδ G −−−−→ G/N −−−−→ Inn(g/j) quot

and

AdG/N

G −−−−→ G/N ⏐ ⏐ quot ⏐ ⏐Ad AdG   G/N Aut j (g) −−−−→ Aut(g/j)

(3)

ϕj

are commutative, where Autj (g) is the set of all α leaving j invariant and ϕj (α)(Y +j) = α(Y ) + j. Now (1) means that for all X, Y ∈ g we have (∀X, Y ∈ g)

AdG/N (expG/N (X + j))(Y + j) = eadg/j (X+j) (Y + j)

and (3) means (∀g ∈ G, Y ∈ g)

ϕj (AdG (g))(Y + j) = Ad(g)(Y ) + j.

Also, eadg/j (X+j) (Y + j) = eadg X (Y ) + j. Therefore, (∀X, Y ∈ g) Since



AdG (expG X)(Y ) − eadg X (Y ) ∈ j.

 (g) = {0} and j ∈  (g) was arbitrary, assertion (1) follows.

Simply Connected Pronilpotent Pro-Lie Groups We recall from Proposition 7.57 and 7.66 that a pro-Lie algebra g is transfinitely topologically nilpotent iff it is pronilpotent iff it is countably topologically nilpotent iff it satisfies g = n(g). In this case, by Lemma 7.69 lim g[[n]] = 0.

n→∞

Lemma 8.2. Let V be a weakly complete topological  vector space and F a filter basis consisting of closed vector subspaces satisfying F = {0}. Let J be an infinite index set and let {aj : j ∈ J } be a family of elements of V such that for each finite 5 subset E ⊆ J we find an F ∈ F such that aj ∈ F whenever j ∈ / E. Then j ∈J aj is summable in V . Proof. See Lemma A2.20 of Appendix 2. We recall the definition of the Campbell–Baker–Hausdorff–Dynkin series called the Hausdorff series by Bourbaki [17, Chapitre II, §6, no 4, Definition 1].

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Lemma 8.3. If x and y are two elements then in the Q-algebra of formal 5 power series, the formal power series x∗y = log(exp x exp y) is of the form x∗y = r,s≥0 Hr,s (x, y) where Hr,s is a Lie polynomial of degree r in x and s in y which is computed as follows, H0,0 (x, y) = 0, H1,0 (x, y) = x, H0,1 (x, y) = y, H1,1 (x, y) = [x, y], where the higher terms are computed as follows. For each pair of nonnegative integers r and s with r + s ≥ 1, let J  (r, s) be the set of tuples (r1 , . . . , rm , s1 , . . . , sm−1 ) such that r1 + · · · + rm = r, s1 + · · · + sm−1 = s − 1, r1 + s1 , r2 + s2 , . . . , rm−1 + sm−1 ≥ 1 for m ∈ N, and let J  (r, s) be the set of tuples (r1 , . . . , rm−1 , s1 , . . . , sm−1 ) such that r1 + · · · + rm−1 = r − 1, s1 + · · · + sm−1 = s, r1 + s1 , r2 + s2 , . . . , rm−1 + sm−1 ≥ 1  (x, y) + H  (x, y) where for m ∈ N. Then Hr,s (x, y) = Hr,s r,s  (x, y) (r + s) · Hr,s 6 (−1)m−1 r s r s r = m m−1 (ad x) 1 (ad y) 1 . . . (ad x) m−1 (ad y) m−1 (ad x) m y, m r ! s ! j =1 j k=1 k J

where

5

J

is extended over all m ∈ N and (r1 , . . . , rm , s1 , . . . , sm−1 ) ∈ J  (r, s), and

 (r + s) · Hr,s (x, y) 6 (−1)m−1 r s r s = m−1 m−1 (ad x) 1 (ad y) 1 . . . (ad x) m−1 (ad y) m−1 x, m r ! s ! j =1 j k=1 k J 

where

5

J 

is extended over all m ∈ N and (r1 , . . . , rm−1 , s1 , . . . , sm−1 ) ∈ J  (r, s).

Proof. See [17, Chapitre II, §6, no 4, Theorème 2]. Given any Lie algebra L and elements X, Y ∈ L, the elements Hr,s (X, Y ) ∈ L are well defined, and thus (Hr,s (X, Y ))r,s∈N is a family of elements in L. If L is a topological Lie algebra, then it may or may not be summable. Lemma 8.4. Let g be a countably topologically nilpotent pro-Lie algebra. Then the family (Hr,s (X, Y ))r,s∈N is summable for all X, Y ∈ g. Therefore the element X ∗ Y is well defined. Proof. We apply Lemma 8.2 to the Hausdorff series and the filter basis F = {g[[n]] : n = 0, 1, . . . }. If n is given, then {J  (r, s) ∪ J  (r, s) : r + s < n} is finite and if r + s ≥ n then Hr,s (X, Y ) ∈ g[n] ⊆ g[[n]] . Thus Lemma 8.2 applies indeed and proves the assertion. Theorem 8.5 (Theorem on Pro-Lie Groups with Pronilpotent Lie Algebra). (i) Let g be a pronilpotent pro-Lie algebra. Then (g) ∼ = (g, ∗) and the following diagram commutes: ∼ =

(g,∗) −−−−→ (g)  ⏐ ⏐ id⏐ ⏐ exp(g) g −−−−→ g. id

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In particular, (g) is homeomorphic to RJ for some set J and thus is arcwise connected and simply connected. (ii) X ∗ Y ∗ (−X) = ead X Y for all X, Y ∈ g. (iii) Z((g, ∗)) = (z(g), ∗) = (z, +). Proof. (i) Let j ∈  (g). Then g/j is a finite-dimensional 5 nilpotent Lie algebra and (Y + j) = for all X, Y ∈ g the element (X + j) ∗ g/j r+s≥1 Hr,s (X + j, Y + j) = 5 H (X, Y ) + j = (X ∗ Y ) + j is a polynomial in X + j and Y + j. Thus the r,s r+s≥1 function (X, Y )  → (X ∗ (−Y )) + j : g × g → g/j is continuous for all j ∈  (g) and since g ∼ = limj∈ (g) g/j in the category of topological spaces and continuous maps, this means that the function (X, Y )  → X∗(−Y ) : g×g → g is continuous. Also we know that ∗g/j is a group operation. Hence (X ∗ Y ) ∗ Z − X ∗ (Y ∗ Z) − j = ((X + j) ∗g/j (Y + j)) ∗g/j (Z + j) − (X + j) ∗g/j ((Y + j) ∗g/j (Z + j)) = j in g/j. Hence (X ∗ Y ) ∗ Z − X ∗ (Y ∗ Z) ∈ j for all j ∈  (g) whence associativity of ∗ follows from  (g) = {0}. Thus we know that (g, ∗) is a topological group, and in the process we have learned that (g, ∗) ∼ = limj∈ (g) (g/j, ∗g/j ), naturally. The last relations shows, that (g, ∗) is in fact a prosimply connected pro-Lie group. Since the underlying space of g is a real topological vector space, it is contractible and therefore arcwise connected and simply connected. Since the underlying topological vector space of g is a weakly complete topological vector space, it is isomorphic to a product of reals, as the dual of a direct sum of reals. Since [X, Y ] = 0 implies X ∗ Y = X + Y from the definition of the Hausdorff series, every X ∈ g uniquely determines a member t  → t · X : R → (g, ∗) of Hom(R, (g, ∗)) = L((g, ∗)), and every one-parameter subgroup of (g, ∗) arises in this fashion. (This is an exercise.) Hence we may identify L((g, ∗)) with g and exp(g,∗) : g → g with the identity map. It remains to prove that (g) ∼ = (g, ∗). From 6.5 we get a unique morphism of pro-Lie groups f : (g) → (g, ∗) which induces on the Lie algebra level the identity id : L(γ (g)) = g → g = L((g, ∗)). From the construction of (g) in 6.4 we know that (g) = limj∈ (g) Gj , where Gj is a simply connected Lie group with Lie algebra g/j; the group Gj is unique up to natural isomorphism. We may therefore assume that Gj = (g/j, g/j ). The cone of quotient morphisms qj : (g, ∗) → (g/j, ∗g/j ), j ∈  (g), by the universal property of the limit gives us a unique morphism g : (g, ∗) → (g) such that qj  g = qj , where qj : (g) = limj∈ (g) (g/j, ∗g/j ) are the limit morphisms. Since f induces the identity on the Lie algebra level, for each j ∈  (g) we have a commutative diagram f

(g) −−−→ (g,⏐∗) ⏐ ⏐ ⏐q qj  j (g/j, ∗g/j ) −−−→ (g/j, ∗g/j ), id

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as both qj  f and qj induce the same Lie algebra morphism g → g/j, namely the quotient map qj . It follows from the uniqueness in the universal property of the limits, that f  g = id(g,∗) and g  f = id(g) . Hence f is an isomorphism with inverse g. (ii) From (i) we know that g = L((g, ∗)) and that idg : g → (g, ∗) is the exponential function. Hence Ad∗ (X)(Y ) = exp(g,∗) Ad(X)(Y ) = X ∗ (exp(g,∗) Y ) ∗ (−X) = X ∗ Y ∗ (−X) by (Ad) preceding 8.1. The assertion now follows from 8.1 (1). (iii) If X ∈ z(g), then [X, Y ] = 0 for all Y and thus X ∗ Y = X + Y and therefore X ∈ Z((g, ∗)). Conversely, let X ∈ Z((g, ∗)). Then for all Y ∈ g we have X = Y ∗ X ∗ (−Y ) = ead Y X. Then for all t ∈ R we get X = et·ad Y X = X + t · [Y, X] + t 1 2! [Y, [Y, X]] + · · · and this means [Y, X] = −t · ( 2! [Y, [Y, X]] + · · · ) for all t ∈ R. Taking t = 0 we get [Y, X] = 0 for all Y , that is, X ∈ z(g). Example 8.6. Let n = RN and define ν : n → n by ν(r1 , r2 , . . . ) = (0, r1 , r2 , . . . ). Define g to be the semidirect sum n  R with [(X, r), (Y, s)] = (r · α(Y ) − s · α(X), 0). We write g = RN0 , where N0 = {0, 1, 2, . . . }; and for any subset J ⊆ N0 we identify RJ with the obvious subgroup of g. The bracket operation of g is now written [(x, r1 , r2 , . . . ), (y, s1 , s2 , . . . )] = (x · (0, 0, s1 , s2 , . . . ) − y · (0, 0, r1 , r2 , . . . )) = (0, 0, xs1 − yr1 , xs2 − yr2 , . . . ). Then (i) g[n] = R{n+1,n+2,... } , n = 1, 2, . . . . In particular, g is countably nilpotent and countably topologically nilpotent, that is, pronilpotent, but not nilpotent. (ii) z(g) = {0}, that is, g is center-free. (iii) g = g[1] is abelian, that is, g is metabelian. (iv) g is the smallest closed subalgebra containing (1, 0, 0, 0, . . . ) and (0, 1, 0, 0, . . . ). Exercise E8.1. (a) Prove that for a pronilpotent pro-Lie algebra g, in (g, ∗) the oneparameter subgroups are exactly the maps t  → t · X : R → g for X ∈ g. (b) Verify the details of Example 8.6. [Hint. (a) Let f : R → (g, ∗) be a morphism. Set X = f (1). For each natural number the function x → x · · ∗ x+ = n · x is a bijection of g with inverse x  → n1 · x. We ( ∗ ·)* n times

conclude that f ( pq ) = pq · X for all q ∈ N, p ∈ Z. Then continuity of f and scalar multiplication proves f (r) = r · X. (b) The assertions of 8.6 are straightforward.] Example 8.6 shows, in particular, that center-free pronilpotent pro-Lie algebras exist.

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8 The Structure of Simply Connected Pro-Lie Groups

The Topological Splitting Technique Definition 8.7. A pro-Lie group G is said to be exponential if expG : L(G) → G is surjective; it is said to be strictly exponential if expG : L(G) → G is a homeomorphism. From the structure theory of abelian pro-Lie groups we argue that a prosimply connected abelian pro-Lie group is strictly exponential: By Theorem 5.20, a connected abelian pro-Lie group G is of the form V ⊕ comp(G) for a weakly complete vector group V and a compact connected subgroup comp(G). Since G is prosimply connected, there is a cofinal subset of closed subgroups N in N (G) such that G/N is a simply connected Lie group, that is, is a vector group isomorphic to some Rn in the present case. This entails that comp(G) has to be {0} and thus G = V . But then expV : L(G) → G is an isomorphism of abelian topological groups. However this is in fact a consequence of a more general result: Proposition 8.8. Every prosimply connected pro-Lie group G with a pronilpotent Lie algebra g is strictly exponential. Proof. This is immediate from Theorem 8.5. This fails even for three-dimensional solvable connected Lie groups: Example 8.9. Let G = C  R with multiplication (c, r)(d, s) = (c + e2π ir d, r + s). This group may be isomorphically represented as the group of all matrices ⎞ ⎛ 2π ir c 0 e (c, r)  → ⎝ 0 1 0 ⎠ , c ∈ C, r ∈ R. 0 0 er Then G is a three-dimensional metabelian Lie group in which (c, 2π ) is in the image of the exponential function iff c = 0. Thus G is not exponential. (The assertions are verified easily by straightforward calculations. A picture of the one parameter subgroups of this Lie group is to be found for instance in [113, p. 16, Figure 3], or in [84, p. 410, Figure 13]. We should mention that this group may be parametrized in a slightly different fashion so that the one parameter groups lie in planes: see [113, p. 14].) We say that a continuous function f : X → Y between topological spaces has a continuous cross section if there is a continuous function σ : Y → X such that f  σ = idY . This means that Y is homeomorphic to a retract σ (Y ) of X, the retraction being σ  f : X → X. Proposition 8.10. Let G be a pro-Lie group and N a closed normal subgroup such that G/N is a strictly exponential pro-Lie group. Then the quotient morphism q : G → G/N has a topological cross section.

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341

Proof. By Corollary 4.21 of the Strict Exactness Theorem 4.20 we know that L(q) : L(G) → L(G/N ) induces an isomorphism L(G)/L(N ) → L(G/N ) and from Theorem 7.7 (iv) we know that there is a morphism of weakly complete topological vector spaces s : L(G/N ) → L(G) such that L(q)  s = idL(G/N ) . By hypothesis, expG/N : L(G/N ) → G/N is a homeomorphism. We define σ : G/N → G by −1 σ = expG s  exp−1 G/N . Then σ is continuous and q  σ = q  expG s  expG/N = −1 expG/N L(q)  s  exp−1 G/N = expG/N  idL(G/N ) expG/N = idG/N . We note σ (N ) = expG s(0) = expG 0 = 1.

Recall that a function f : X → Y between sets on which a group N acts on the left is called equivariant if n · f (x) = f (n · x) for all n ∈ N and x ∈ X. Proposition 8.11. Let G be a topological group and N a closed normal subgroup. Let q : G → G/N = {Ng = gN : g ∈ G} be the quotient map and let N act on G on the left by multiplication. Let N act on N × (G/N ) by n · (n , Ng) = (nn , Ng). Assume that q has a continuous cross-section σ : G/N → G such that σ (N ) = 1. Then there is an N-equivariant homeomorphism ϕ : G → N × (G/N ) such that ϕ(n) = (n, N ) for n ∈ N and pr 2  ϕ = q. Proof. Observe that q(gσ q(g)−1 ) = q(g)qσ q(g)−1 = 1 whence q(g)σ q(g)−1 ∈ N . Thus we may define ϕ : G → N × (G/N ) by ϕ(g) = (gσ (q(g))−1 , q(g)). If n ∈ N then q(ng) = g(g −1 ngN ) = gN = q(N ) by the normality of N and thus n · ϕ(g) = (ngσ (q(g)), q(g)) = ϕ(ng). Moreover, σ q(n) = nσ N = n and thus ϕ(n) = (n, N ) and pr 2 ϕ = q. It is clear that ϕ is continuous and we claim that it has the inverse ψ : N × (G/N) → G given by ψ(n, ξ ) = nσ (ξ ). Indeed ψϕ(g) = ψ(gσ (q(g))−1 , q(g))gσ (q(g))−1 σ (q(g)) = g and ϕψ(n, ξ ) = ϕ(nσ (ξ )) = (nσ (ξ )σ q(nσ (ξ ))−1 , q(nσ (ξ )) = (n, q(ξ ))). Thus ϕ is a homeomorphism and G is homeomorphic to N × (G/N ). We can express the conclusion of the previous proposition by saying that the principal fibration q : G → G/N with fiber N is trivial. Corollary 8.12. Let G be a prosimply connected pro-Lie group and N a closed connected normal subgroup such that g/n is pronilpotent. Then: (i) The quotient morphism G → G/N has a continuous cross section. (ii) There is an N-equivariant homeomorphism G → N × RI for some set I . (iii) If n is also pronilpotent then G is homeomorphic to RJ for some set J and thus is, in particular, simply connected. Proof. (i) By Corollary 6.8 (iii), since N is connected, N is a closed prosimply connected normal subgroup and G/N is a prosimply connected pro-Lie group. By Proposition 8.8, G/N is strictly exponential. Then by Proposition 8.10, the quotient morphism q : G → G/N has a continuous cross section. (ii) By 8.11, the space G is N -equivariantly homeomorphic to N × (G/N ). But expG/N : g/n → G/N is a homeomorphism, and the topological vector space under-

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8 The Structure of Simply Connected Pro-Lie Groups

lying g/n, being a weakly complete topological vector space, is isomorphic to RI for some set I (Corollary A2.9). (iii) If n is pronilpotent, then N is strictly exponential by Lemma 8.8. Hence it is homeomorphic to a weakly complete topological vector space which in turn is topologically and algebraically isomorphic to a power of R. Hence by (ii) the group G itself is homeomorphic to a power of R. Theorem on the Topological Structure of Prosimply Connected Pro-Lie Groups with Prosolvable Lie Algebras Theorem 8.13. Let G be a prosimply connected pro-Lie group whose Lie algebra g = L(G) is prosolvable, that is, which is its own radical. Let n denote its Lie radical or its reductive radical, as the case may be. Then the following statements hold: (i) (n) ∼ = (n, ∗) may be considered as a closed normal subgroup N of G such that G/N is an abelian strictly exponential pro-Lie group and L(G/N ) is naturally isomorphic to g/n. (ii) expG/N : g/n → G/N is an isomorphism of weakly complete vector groups. (iii) The quotient morphism q : G → G/N admits a continuous cross section σ : G/N → G such that σ (N) = 1. (iv) There is an N-equivariant homeomorphism ϕ : G → N × (G/N ) such that ϕ(n) = (n, N ) for all n ∈ N , and pr 2 ϕ = q. (v) G is homeomorphic to RJ for some set J . (vi) G is simply connected in any sense for which the additive group of a weakly complete topological vector space is simply connected. Proof. By the Structure Theorem of Reductive Pro-Lie Algebras 7.27 (b) the factor algebra g/ ncored (g) modulo the coreductive radical is reductive on the one hand and prosolvable on the other. Thus it is abelian by Corollary 7.28. (See also Theorem 7.66.) By Theorem 7.67, the reductive radical ncored (g) is contained in the nilradical n(g). Therefore n is pronilpotent. Now we apply Proposition 8.12, which proves the theorem.

From the classical theory of finite-dimensional Lie groups we know that no improvement of Theorem 8.13 is to be expected: The 3-dimensional Heisenberg algebra and group shows that the reductive radical does not split in general as a semidirect factor. The 3-dimensional motion algebra R2 ⊕ R with bracket [(v, r), (w, s)] = (r ·w−s·v, 0) and the simply connected covering group of the group of motions of the euclidean plane show that when the reductive radical splits, the group may not be exponential, let alone strictly exponential. (See Example 8.9.) For references concerned with the surjectivity of the exponential function see for instance [47], [113], [203], and [204].)

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Simple Connectivity We finally prove the definitive result on simple connectivity of pro-Lie groups. Structure Theorem for Prosimply Connected Pro-Lie Groups Theorem 8.14. Let G be a prosimply connected pro-Lie group with Lie algebra g. Then: (i) G is the semidirect product R I S of a closed normal subgroup R whose Lie algebra L(R) is the radical r(g) and a closed subgroup S whose Lie algebra s is a Levi summand of g. (ii) Thereis a family of simply connected simple Lie groups Sj , j ∈ J such that S∼ = j ∈J Sj . (iii) There is a closed normal subgroup N of G contained in R such that the pro-Lie algebra L(N ) = ncored (g) is the coreductive radical of g and that there is an N -equivariant isomorphism ϕ : R → N × (R/N ), where N ∼ = (ncored (g), ∗) and where R/N ∼ = r(g)/ ncored (g) is a vector group. (iv) R is homeomorphic to RJ for some set J . (v) G is homeomorphic to a product of copies of R and of a family of simply connected real finite-dimensional simple Lie groups. (vi) If C denotes the identity component of the center of G, then G is C-equivariantly homeomorphic to C × G/C. Proof. (i), (ii) By the Levi–Mal’cev Theorem 7.52,  g is the semidirect sum r(g) ⊕ad s of its radical by a Levi summand s and s = j ∈J sj for a family of finite-dimensional simple real Lie algebras sj . By hypothesis, G is prosimply connected. Thus G may be identified with (g). By Theorem 6.11 on the preservation of semidirect products, G = (g) is the semidirect product of the group N ∼ = (n) and a closed subgroup S ∼ = (s). The full subcategory proSimpConLieGr of prosimply connected pro-Lie groups in the category TopGr of topological groups and continuous group homomorphisms is equivalent to the full subcategory proLieAlg of pro-Lie algebras in the category of topological real Lie algebras and continuous Lie algebra homomorphisms by Theorem 6.6; the equivalence is implemented by the two functors  : proLieAlg →  proSimpConLieGr and L : proSimpConLieGr → proLieAlg. Thus ∼ S = (s) = j ∈J Sj where Sj = (sj ) is a simply connected simple Lie group, uniquely determined up to natural isomorphism by the fact that L(Sj ) ∼ = sj . This proves (i) and (ii). (iii), (iv) By Theorems 7.27, 7.66, and 7.67 the coreductive radical ncored (g) is a pronilpotent characteristic ideal contained in the radical r(g) such that r(g)/ ncored (g) is abelian. Then the Theorem on the Topological Structure of Prosimply Connected Prosolvable Pro-Lie Groups with Prosolvable Lie Algebras 8.13 applies and shows that N∼ = (ncored (g), ∗) which proves, among other things, that N is homeomorphic to the additive group of the weakly complete topological vector space n. Moreover, 8.13 shows that there is an N-equivariant homeomorphism ϕ : R → N × (R/N ) such that

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ϕ(n) = (n, N) for all n ∈ N , and pr 2 ϕ = q, where R/N ∼ = (r(g)/ ncored (g), +). In particular, R is homeomorphic to (ncored (g), +) × (r(g)/ ncored (g), +) which is a weakly complete vector group and therefore is homeomorphic to some power RJ , for a suitable set J . (v) This is an immediate consequence of the preceding. (vi) By (i) and (iii), G is N-equivariantly homeomorphic to N ×G/N . By 8.12, N is C-equivariantly homeomorphic to C × N/C. So G is C-equivariantly homeomorphic to C × (N/C) × G/N and thus to C × G/C. In the Postscript of Chapter 6 we discussed those concepts of simple connectivity of topological spaces which are most current among topologists. The first one in terms of covering maps and universal properties is the one we adopted in [102] because in topological group theory the universal properties are most often used. The second one in terms of contractibility of loops is advantageous in classical algebraic topology and because it is easily seen that a product of simply connected spaces is simply connected. A space is called locally arcwise simply connected if it has a cover by open sets each of which is arcwise connected and in which every loop is contractible to a point. For both concepts the following is true as is verified for instance in [102, A2.11 (iii)]: Product Lemma for Simply Connected Spaces. Let {Sj : j ∈ J } be a family of simply connected, arcwise connected, locally arcwise connected, and locally arcwise simply connected spaces. Then the product j ∈J Sj is simply connected. In particular, every product of simply connected finite-dimensional topological manifolds (that is, locally euclidean spaces) is simply connected. Therefore, whichever of the two common concepts of simple connectivity we consider, on pro-Lie groups they agree as we shall now get as a consequence of Theorem 8.14. Recall from Theorems 2.26 and 6.6, that for a connected pro-Lie group G we have  = (L(G)) and that there is the universal morphism πG : G  → G. In the proof G of the following characterisation theorem we shall invoke the Quotient Theorem for Pro-Lie Groups 4.1 and the methods used in its proof. Theorem on the Simple Connectivity of Pro-Lie Groups Theorem 8.15. For a pro-Lie group G the following statements are equivalent: (i) (ii) (iii) (iv)

G is prosimply connected. G is simply connected.  → G is an isomorphism. πG : G  → G is bijective. πG : G

If f : G → H is a bijective morphism of pro-Lie groups and G is almost connected, →H  is an isomorphism of simply connected pro-Lie groups, and if G is then f: G simply connected, then f is an isomorphism.

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Proof. (i) ⇒ (ii): This is immediate from Theorem 8.14 (v) and the Product Lemma for Simply Connected Spaces. (ii) ⇒ (i): Assume that G is a simply connected pro-Lie group and assume that N ∈ N (G). Let q : L → G/N be the universal covering morphism and qN : G → G/N the quotient morphism. Since G is simply connected, there is a unique lifting  qn : G → L such that q  qN = qN . Since q is a covering map and thus induces a local isomorphism,  qN is open and so is a quotient map, as G is connected and thus  qN is def qN ∈ N (G). Also M ⊆ N . Hence G is prosimply connected surjective. So M = ker  by Definition 6.1. The equivalence of (i) and (iii) was shown in the Theorem on the Reflection into the Category of Prosimply Connected Pro-Lie Groups 6.6 (iii). The implication (iii) ⇒ (iv) is trivial. The remainder of the proof is devoted to the implication (iv) ⇒ (iii). Let us first prove a lemma which is a very mild version of an Open Mapping Theorem. Lemma A (The Lie-Pro-Lie Open Mapping Theorem). Let L be a connected Lie group, let P be a pro-Lie group with a finite-dimensional Lie algebra L(P ), and let f : L → P be a surjective morphism of topological groups. Then f is open and P is a Lie group. Proof. The canonical decomposition of morphisms gives f as a composition of the quotient morphism L → L/ ker f and the bijective morphism F : L/ ker f → P , F (g ker f ) = f (g). Since f is open iff F is open, it is no loss of generality if henceforth we assume that f is bijective. We must show that f is an isomorphism of pro-Lie groups. By Corollary 4.21, the filter basis {L(Q) : Q ∈ N (P )} converges to 0 in L(P ); since L(P ) is finite-dimensional this means that L(Q) = {0} cofinally. By Proposition 4.23 this means that the filter basis N (P ) has a basis of prodiscrete, equivalently, totally disconnected normal subgroups Q. For these Q the inverse image f −1 (Q) is a closed totally disconnected normal subgroup of the connected Lie group L and therefore is discrete central finitely generated. Since the f −1 (Q) intersect in the identity they are, cofinally, finitely generated free. Let Q ∈ N (P ) be such that Q is a finitely generated free closed profinite central subgroup of the pro-Lie group P . By Theorem 5.32 (iv), Q is discrete. Then P and P /Q are locally isomorphic and P /Q is a Lie group. Hence P is a Lie group. Then f : L → P is a bijective morphism between connected locally compact groups and is therefore an isomorphism by the Open Mapping Theorem for Locally Compact Groups (see for example [102, Appendix 1, EA1.21 following Definition A1.59]). This completes the proof of Lemma A.  → G is bijective. Since G  is simply connected and Assume now that πG : G so, in particular, prosimply connected according to Definition 6.1, it follows from  has a basis N S(G)  of Proposition 6.3 (c) and Definition 6.1 that the filter basis N (G) connected closed normal subgroups N such that G/N is simply connected.

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8 The Structure of Simply Connected Pro-Lie Groups

 we set N ∗ def For N ∈ N S(G) = πG (N ). Then N ∗ is a closed connected normal subgroup of G and then, by the Quotient Theorem for Pro-Lie Groups Revisited 4.28 (i), G/N ∗ is a pro-Lie group.  converges to 1G (see By the continuity of πG , the filter basis {N ∗ : N ∈ N S(G)} Theorem 4.1 and its proof). Thus for each M ∈ N (G) we find an open M-saturated identity neighborhood U of H such that U/M is an identity neighborhood of the Lie  group G/M not containing any nonsingleton subgroups. Then there is an N ∈ N (G) such that N ∗ ⊆ U , and so N ∗ M ⊆ U M = U gives a subgroup N ∗ M/M of U/M. Accordingly, it is singleton, that is, N ∗ ⊆ M. On the other hand, πG induces a def   → G/N ∗ , ϕN (gN ) = πG (g)N ∗ , whose domain L = G/N is a morphism ϕN : G/N def

Lie group, and whose range P = G/N ∗ is a pro-Lie group.  with g = L(G) and L(πG ) with the identity We may and shall identify L(G)   and q2 : G → G/N ∗ the respective morphism of g. Denote by q1 : G → G/N quotient maps. Consider the commutative diagrams πG  −−− G −→ G ⏐ ⏐ ⏐ ⏐q q1  2 L −−−− → P ∗

idg

and

g −−−−→ ⏐ ⏐ L(q1 ) L(L) −−−−→ ∗ L(p )

p

g ⏐ ⏐L(q )  .2 L(P ).

Since L(q2 ) is surjective by the Strict Exactness Theorem for L 4.20, the map L(p ∗ ) is surjective. It follows that L(P ) is finite-dimensional. Hence Lemma A above applies  and shows that ϕN : G/N = L → P = G/N ∗ is a quotient morphism of Lie groups. ∗  is cofinal in the filter basis N (G) and thus the Therefore, {N : N ∈ N S(G)} ∗ natural map G → limN ∈N S(G)  ∗ G/N is an isomorphism, where the limit is taken over the projective system def  N ⊆ M}. D = {pMN : G/N ∗ → G/M ∗ | (M, N ) ∈ N S(G),

 in Chapter 6 that from D we derive a projective We recall from the construction of G system of simply connected Lie groups  def  N ⊆ M}, D = { pMN : (G/N ∗ ) → (G/M ∗ ) | (M, N ) ∈ N S(G),  → D, that is a family of covering morphisms and a natural transformation c : D  ∗ ∗ ∗  = lim D  = lim cN : (G/N ) → G/N so that G  (G/N ) and that N ∈N S(G)  = lim D  → lim D = G is the limit of the c . πG : G N  by an equivalent one. For this purpose we It is convenient to replace the diagram D  the Lie group G/N  note that for each N ∈ N S(G), is simply connected, and thus the ∗   morphism ϕN : G/N → G/N lifts to a morphism ψN : G/N → (G/N ∗ ). Let N∗ quot ψN −−−−→G/N  −−−−→(G/N ∗ ), and N # the full denote the kernel of the composition G ∗ inverse image of N , def  N # = π −1 (N ∗ ) ∈ N (G). G

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Then we have a commutative diagram  G/N ⏐ ⏐   ∗ G/N ⏐ ⏐ covering  # G/N

ψN

∗ −−−−→ (G/N ⏐ ) ⏐ id ∗ −−−−→ (G/N ⏐ ) iN ⏐c  N −−−−→ G/N ∗ , jN

in which all morphisms are quotient morphisms of Lie groups and iN and jN are  ∗ is a simply connected Lie group, N∗ ∈ isomorphisms. In particular, since G/N  N S(G). This diagram in turn translates into Hasse diagrams of normal subgroups of Lie groups:  G G       # N ∗  N   discrete   πG (N N∗  ∗)     N πG (N ). Here groups on the same level correspond to each other under the bijective morphism πG . The normal subgroup N∗ is open in N # , and the factor group is finitely generated abelian. (Even though we do not need this for the proof, we observe that, by the definition of N ∗ , the subgroup πG (N ) is dense in N ∗ , giving us (N∗ )∗ = N ∗ .)  : G/N  ∗ → G/N ∗ is a universal covering morphism. Since The natural map cN  ϕN : G/N → G/N ∗ is a quotient morphism between Lie groups, the morphism #  G/N → G/N ∗ is an isomorphism of Lie groups. We may therefore replace the  by the equivalent diagram diagram D  def  ∗ → G/M  ∗ | (M, N ) ∈ N S(G),  N ⊆ M} E = { qMN : G/N and so we may write

limN ∈N S(G) 

 G ⏐ ∼⏐ =   ∗ G/N

πG

−−−−→ G ⏐ ⏐∼ =  #, −−−−→ limN ∈N S(G)  G/N

and record that the quotient morphism  ∗ → G/N  # cN : G/N is a covering morphism.

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8 The Structure of Simply Connected Pro-Lie Groups

 is a filter  # exists, we know that {N # : N ∈ N S(G)} Since limN∈N S(G)  G/N  repeating an argument in the  converging to 1  . So, given M ∈ N S(G), basis in G G  earlier part of the proof, we find an open M-saturated identity neighborhood U of G  such that U/M is an identity neighborhood of the Lie group G/M not containing any  such that N # ⊆ U , and then nonsingleton subgroups. Then there is an N ∈ N S(G) N # M ⊆ U M = U gives a subgroup N # M/M of U/M. Accordingly, it is singleton, that is, N # ⊆ M ⊆ M∗ .  G    M #   M ∗   N #   N∗ .. .  # → G/M  ∗ . Let πN : G → Accordingly, we get a quotient morphism fMN : G/N #   G/N∗ and ρN : G → G/N denote the limit maps. We then have a commutative diagram πG /G  G ρN

πN

  ∗ G/N

 cN / G/N  # yy fMN yy y qMN pMN y yy  |yy  / G/M # .  ∗ G/M cM

def

1 = f Consequently, fM = πM  πG mn  ρN is a morphism of pro-Lie groups. By  such that the universal property of the limit, there is a unique morphism f : G → G −1  This shows that π −1 = f and thus that π −1 πM  f = πM  πG for all M ∈ N S(G). G G is continuous. This is what we had to show.

Assume finally that f : G → H is a bijective morphism of pro-Lie groups and that G is almost connected. Then L(f ) is bijective and hence an isomorphism by →H  is an isomorphism. Now assume Corollary 4.22 (ii). Then f = (L(f )) : G  → G is an isomorphism by Theorem 6.6 (iii). that G is simply connected. Then πG : G  → H is an Then πH = f  πG  f−1 is bijective. Then, by (iii) ⇔ (iv) above, πH : H −1  isomorphism. Therefore f = πH  f  πG is an isomorphism as well.

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The implication (iv) ⇒ (iii) is a kind of Open Mapping Theorem. Its proof is remarkably technical. A better Open Mapping Theorem will become available in Theorem 9.60. After Theorem 8.15 we may safely replace the adjective “prosimply connected” as applied to a pro-Lie group by “simply connected”. However we also notice that the equivalence is based on the Structure Theorem for Prosimply Connected Pro-Lie Groups 8.14 which in turn rests on most of the content of Chapters 6 and 7. We also emphasize that the concept of simple connectivity as applied to pro-Lie groups is unproblematic in as far as the choice is concerned between the two definitions of simply connectivity which we discussed in the Postscript to Chapter 6. Connected Closed Normal Subgroups of Simply Connected Pro-Lie Groups Theorem 8.16. Let G be a simply connected pro-Lie group and N a closed connected normal subgroup. Then N and G/N are simply connected pro-Lie groups and there is an N -equivariant homeomorphism ϕ : G → N × (G/N ). If g/n is semisimple, then G is a semidirect product of N and a closed subgroup isomorphic to G/N. Proof. We set H = G/N, let F : G → H the quotient morphism, and j : N → H the inclusion morphism. Following Corollaries 6.8 (ii) and 6.9 (ii) we consider the strict exact sequence of pro-Lie groups j˜

 F

 −−−−→ G  −−−−→ H  → 1, 1→N  is a quotient. Since G is simply connected, we may where j˜ is an embedding and F  = G and if we consider j˜ as an inclusion map (which we may), then N ⊆N write G   is with pN : N → N being an inclusion map. Then by 6.6 (iv), the subgroup N  is closed, we have N  = N . Therefore dense in N0 , and since N is connected and N  and thus H is a simply connected pro-Lie group. H = G/N ∼ =H Next we assume that g/n is semisimple. Let r = r(g) denote the radical of g. Since g/n does not contain any solvable ideals, r ⊆ n. Let s be a Levi summand according to Theorem 7.52. Then g = r ⊕ s algebraically and topologically and s ∩ n is an ideal of s. Hence by Corollary 7.29, s = (s ∩ n) ⊕ t algebraically and topologically for a semisimple ideal t of s, given by a partial product of a product representation of s according to Corollary 7.29. Thus g = r ⊕ s = r ⊕ (s ∩ n) ⊕ t = n ⊕ t algebraically and topologically. The quotient morphism q : g → g/n maps t isomorphically onto g/n; the inverse of q|t : t → g/n followed by the inclusion t → g is a morphism σ : g/n → g satisfying q  σ = idg/n . But if g is the semidirect sum of n and t, then G = (g) is the semidirect product of N = (n) and T = im (σ ) ∼ = (t) by Theorem 6.11.

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8 The Structure of Simply Connected Pro-Lie Groups

Now we assume that g/n is solvable. If s is a Levi summand of g, then the quotient morphism g → g/n must annihilate s, that is, s ⊆ n. Since g = r ⊕ s we have n = (r ∩ n) ⊕ s and r ∩ n is the radical of n, while g/n =

r⊕s ∼ = r/(r ∩ n). (r ∩ n) ⊕ s

Thus g/n is abelian by Theorem 7.67. Now Corollary 8.12 proves the assertion. Finally, let g/n be arbitrary. Now let r be that ideal of g containing n for which r/n is the radical of g/n. Since G is simply connected and thus may be considered to be (g), we may consider (r) to be a closed normal subgroup R of G by 6.9; then L(R) = r. From the previous case we know that R is N -equivariantly homeomorphic to N × (R/N). The quotient algebra g/r is semisimple, then by what we proved above, G is the semidirect product R I S for a suitable closed subgroup S isomorphic to G/R. Thus G permits a homeomorphism onto R × (G/R) = N × (R/N ) × (G/R) and this homeomorphism is N -equivariant. Notice that the theorem says, in particular, that the quotient G/N is complete and is not only a proto-Lie group according to the Quotient Theorem 4.1, but is actually a pro-Lie group. It is, however, not even true for three-dimensional classical Lie groups that a simply connected G is a semidirect product of N and G/N as is illustrated by the Heisenberg group G = R2 × R with multiplication (v, r) ∗ (w, s) = (v + w, r + s + 21 · v ∧ w) where v ∧ w = det(v, w), and with the normal subgroup N = {0} × R. The following proposition is immediate from Theorem 8.14 (v): Let G be a simply connected pro-Lie group. Then G is arcwise connected. One could express Theorem 8.15 and this result by saying that A simply connected pro-Lie group satisfies π0 (G) = π1 (G) = 0. And indeed, since all finite-dimensional Lie groups have vanishing second homotopy, we have also π2 (G) = 0. The compact Lie group SU(2) of unitary 2 × 2 matrices of determinant 1 (see [102, 1.6 (iib) ff.]) is homeomorphic to SS 3 (see [102, E1.2]) and thus satisfies π3 (G) = Z. However, a better result than π0 (G) = 0 is available. Recall from Corollary 4.22 that we denote the subgroup of a topological group generated algebraically by the set of points lying on one parameter subgroups by E(G). Simply Connected Pro-Lie Groups Are Exponentially Generated Corollary 8.17. If G is a simply connected pro-Lie group, then G = E(G) = expG g. Proof. Recall the Theorem on the Preservation of Embeddings in Lie’s Third Theorem, Corollary 6.9; according to that result we have a unique simply connected closed normal subgroup C ∼ = (ncored (g)) of G ∼ = (g) whose Lie algebra L(C) is the coreductive ∼ radical ncored (g). Then C = (ncored (g), ∗) and thus is strictly exponential by 8.8. So

Simple Connectivity

351

C ⊆ expG g. By Corollary 6.8, G/C ∼ = (g/c). In particular, G/C is a simply connected pro-Lie group. We claim, that G/C = E(G/C). Indeed g/ ncored (g) is reductive, and thus its radical and its center agree  and thus L(G/C) = Z ⊕ S where Z is a weakly complete vector group and S = j ∈J Sj with simply connected simple Lie groups Sj . By the Iwasawa decomposition KAN of a semisimple Lie group in which each of the factors K, A, and N , is exponential, Sj = (expSj L(Sj ))3 . Thus G/C = (expG/C L(G/C))3 ⊆ E(G/C) ⊆ G/C, which establishes the claim. By the One-Parameter Subgroup Lifting Lemma 4.19, (expG g)/C = expG/C (g/ ncored (g)). Now E(G) is a characteristic subgroup of G containing C and E(G)/C = (expG g)/C = expG/C (g/ ncored (g)) = G/C.

It follows that E(G) = G which is what we had to show. By a theorem of Moskowitz and Sacksteder [152], and of Wüstner [205], a slightly more accurate reasoning shows that (expG g)2 C/C = (expG/C L(G/C))2 = G/C, whence G ⊆ (expG g)2 C ⊆ (expG g)3 , that is G = (expG g)3 . From Theorem 5.20 we know that an abelian connected pro-Lie group G is isomorphic to V × C for a weakly complete vector group V and a compact connected group C. From Chapter 8 of [102] we know that C is arcwise connected if and only  Z) = 0 for the torsion-free discrete character group C  (and by Pontryagin if Ext(C,   are called Duality every torsion-free discrete abelian group is a C). Such groups C Whitehead groups; in some models of ZFC (Zermelo–Fraenkel plus Axiom of Choice) these are free (in which case C has to be a torus, that is a product of circles); in other models they fail to be free (see for instance [102, Chapter 8]). Thus the precise nature of arc connectivity is a delicate matter. On the other hand, the character group C of the discrete group Q is not arcwise connected and indeed π0 (G) = G/ im expG is of the cardinality of the continuum. These remarks show, however, that for abelian pro-Lie groups at least, arc connectivity is a problem that is located within the realm of compact abelian groups; there we have much explicit information [102]. It will emerge that this remains true for arbitrary pro-Lie groups. Corollary 8.17 allows us to draw some useful consequences on the inner automorphism group of a simply connected pro-Lie group and of a pro-Lie algebra. Let g be a pro-Lie algebra, and G = g the attached simply connected pro-Lie group according to 6.6 and 8.15. Then g ∼ = L(G) and from 6.6 (vi) we know that there is a ∼ natural isomorphism Aut(g) = Aut(G) implemented by ϕ  → (ϕ) with the inverse

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8 The Structure of Simply Connected Pro-Lie Groups

α  → L(α). From Proposition 2.30 and Corollary 8.17 we get that the adjoint representation Ad : G → Aut(g) has the center Z(G) as its kernel. Recall that Ig ∈ Aut(G) denotes the inner automorphism given by Ig (x) = gxg −1 , and that by Definition 2.27 we have L(Ig ) = Ad(g); in other words g(expG X)g −1 = expG Ad(g)(X). If Inn(G) denotes the group of all inner automorphisms Ig , then there is an isomorphism of groups G/Z(G) → Inn(G), and the subgroup Inn(G) of Aut(G) is mapped isomorphically onto the image Ad(G) ⊆ Aut(g). By Corollary 8.17, we have G = expG g and so Ad(G) = Ad(exp g). In 8.1 (1) we showed (∀X ∈ g) Ad(expG X) = eadg X . Therefore we now have for a simply connected pro-Lie group G that Ad(G) = ead X : X ∈ g ⊆ Aut(g) and that this group is algebraically isomorphic with G/Z(G). We shall call the group ead X : X ∈ g ⊆ Aut(g) the group of inner automorphisms of the pro-Lie algebra g and denote it by Inn(g). (See also Definition 9.3) The next result is clear from the preceding discussion. Corollary 8.18. The group Inn(g) of inner automorphisms of a pro-Lie algebra g is algebraically isomorphic to the factor group (g)/Z((g)). If we give Inn(g) the topology which makes the natural algebraic isomorphism (g)/Z((g)) → Inn(g) an isomorphism, then Inn(g) becomes a proto-Lie group with this topology.

Universal Morphism versus Universal Covering Morphism  → G the In Chapter 6, after Theorem 6.6 we named the natural morphism πG : G  universal morphism. From Theorem 8.15 we know that G is simply connected. We have noticed for a compact abelian group G that πG equivalent to the exponential function expG : g → G and therefore often fails to be surjective (see for instance Example 14.4). Moreover, the Poincaré group P (G) = ker πG is frequently nondiscrete. Thus πG generally fails to be a covering morphism. But now let us assume that we are given a connected pro-Lie group G and the additional information that G, as a topological group, has a universal covering group G∗ and a universal covering morphism pG : G∗ → G. (See for instance [102, Definition A2.13].) Notably, G∗ is a simply connected topological group and ker pG is discrete. A priori it is not obvious, how the universal covering morphism pG and the universal morphism πG compare. However, from the very definition of simple connectivity as  is expressed for instance in [102, Definition A2.6] and the fundamental fact that G simply connected by Theorem 8.15 we easily get the following insight. Proposition 8.19. Let p : G+ → G be a morphism of connected topological groups with a discrete kernel and assume that G is a pro-Lie group. Then there is a unique

Universal Morphism versus Universal Covering Morphism

353

 → G+ of topological groups such that the following diagram commorphism ϕ : G mutes  −−−ϕ−→ G+ G ⏐ ⏐ ⏐ ⏐p πG   G −−−−→ G. idG

 is simply connected by Theorem 8.15 this is an immediate consequence Proof. Since G of the lifting definition of simple connectivity as expressed in [102, Definition A2.6]. This applies, in particular, to the universal covering morphism pG : G∗ → G, if it exists. Our Theorems 6.6.and 8.15 entail a first step towards elucidating the circumstances under which ϕ might be invertible. Lemma 8.20. Let f : H → G be a morphism of connected pro-Lie groups and assume  such that the that H is simply connected. Then there is a unique lifting f: H → G following diagram commutes: H ⏐ ⏐ f G

f

 G ⏐ ⏐π  G −−−−→ G. −−−−→

idG

Proof. In view of the fact, that prosimple connectivity is the same as simple connectivity for pro-Lie groups after Theorem 8.15, the lemma is just a reformulation of the adjunction statement of Theorem 6.6(ii). Theorem 8.21. If a connected pro-Lie group G has a simply connected covering group  = (g) are naturally isomorphic. G∗ , then G∗ and G Proof. Let p : G∗ → G be the universal covering morphism which we assume exists.  → G∗ of topological Then by Proposition 8.19 there is a unique morphism ϕ : G groups such that the following diagram commutes  −−−ϕ−→ G∗ G ⏐ ⏐ ⏐ ⏐p πG   G −−−−→ G. idG

By Lemma 3.32(ii), phism pG : commutes:

G∗

G∗

is a pro-Lie group. Then according to Lemma 8.20 the mor-

 such that the following diagram → G has a unique lifting ψ : G∗ → G G⏐∗ ⏐ pG  G

ψ

 G ⏐ ⏐π  G −−−−→ G.

−−−−→

idG

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8 The Structure of Simply Connected Pro-Lie Groups

The identity morphism of G has only the identity map of G∗ as lifting; since ϕ ◦ ψ is such a lifting we have ϕ ◦ ψ = idG∗ . Similar uniqueness properties yield ψ ◦ ϕ = idG . Therefore ψ = ϕ −1 and ϕ is the desired isomorphism. In other words, if a connected pro-Lie group G has a universal covering group,  is this universal covering group. then the universal group G

Postscript Before this chapter, the most pressing issue left open in the structure theory of connected pro-Lie groups is the question pertaining to prosimply connected pro-Lie groups: We do not know how the concept of prosimple connectedness, which appears to be quite appropriate for dealing with pro-Lie groups, is related to the several traditional concepts of simple connectivity of topological spaces and topological groups; we refer again, in this context to [102, Appendix 2, pp. 668–6869]. This issue is all the more pressing since we know after Chapter 6 of this book, that the category of pro-Lie groups contains a faithful copy of the category of pro-Lie algebras in the form of the full subcategory of prosimply connected pro-Lie groups. It is therefore a great relief when in the present chapter we show (Theorem 8.15.) that for a pro-Lie group G the following statements are equivalent: (i) G is prosimply connected; (ii) G is connected and every loop based at the identity can be homotopically contracted, that is, π1 (G) = 0; (iii) each covering map f : E → G of topological spaces maps every connected component of E homeomorphically onto G;  → G is bijective. (iv) The “universal covering morphism” πG : G Thus wherever in this book we have made a statement involving prosimply connected pro-Lie groups, the words prosimply connected may be replaced by simply connected without any ambiguity. In order to reinforce this confidence we recall that in Theorem 8.14 and its corollary we have infact shown that a pro-Lie group is simply connected iff it is homeomorphic to a space j ∈J Xj where Xj is homeomorphic to R or to the underlying space of a simple simply connected real Lie group. The proof of the fact that the bijectivity of the universal covering morphism implies simple connectivity is all but trivial. We can embellish this result if we allow ourselves to use the classical Iwasawa– Mal’cev Theorem that any connected Lie group H is homeomorphic to Rn × C, where n is a nonnegative integer and C is a maximal compact subgroup of G. (See for instance [86, p. 380, Theorem 3.1.].) If we use this we can say

Postscript

355

Theorem.  A pro-Lie group G is simply connected iff it is homeomorphic to a product j ∈J Xj where Xj is homeomorphic to R or to the underlying compact manifold of a simply connected simple compact Lie group. In particular, a simply connected pro-Lie group is homotopy equivalent to a semisimple compact simply connected group. Accordingly, the entire algebraic topology of a simply connected pro-Lie group (be it homotopy, cohomology, homology, in short any functor satisfying the homotopy axiom) is captured by that of a simply connected compact group. (For the structure of such groups see [102, Proposition 9.4ff. and Theorem 9.29]. In particular we recall that a simply connected compact abelian group is singleton. These results should be compared with the results of Chapter 5 (for instance 5.12) according to which a connected abelian pro-Lie group is isomorphic to a direct product of a vector group RJ and a compact connected abelian group. In both cases we have seen that pro-Lie groups tend to be homeomorphic to a product of lines and some compact connected groups. As the theory unfolds, we shall see more results of the same kind.

Chapter 9

Analytic Subgroups and the Lie Theory of Pro-Lie Groups

The information that we accumulated in the previous chapter on pro-Lie groups and algebras is now exploited for developing a basic Lie theory of pro-Lie groups. A lot will depend on how strongly the correspondence between pro-Lie algebras and proLie groups will promote the structure theory. We know from the classical theory of finite-dimensional Lie groups and Lie algebras that for a given Lie group G and its Lie algebra g = L(G), in general there are more Lie subalgebras of g than there are closed connected subgroups of G – a fact which one recognizes at once by considering √ the 2torus G = (R/Z)2 with its Lie algebra g = R2 in which the subalgebra R ·(1, 2) does not correspond to a closed connected subgroup of G. In response to this dilemma, in the classical theory one considers analytic subgroups so that indeed there is, in the end, a bijection between the set of all subalgebras of g and the set of all analytic subgroups of G. We will have to look for a generalisation of this set-up for pro-Lie groups; an apparatus which is already delicate in the classical situation is even more intricate on the level of pro-Lie groups where we have to cope with a generally infinite-dimensional setting. Prerequisites. We need the general theory of pro-Lie groups and pro-Lie algebras up to Chapter 8.

The Exponential Function on the Inner Derivation Algebra On a Lie algebra L, the inner derivations are those of the form ad x for an element x ∈ L, where (ad x)(y) = [x, y]. Assume that Der(L) denotes the algebra of all derivations, D is a derivation of L, and that x, y ∈ L. Then [D, ad x](y) = D(ad x)(y) − (ad x)(Dy) = D[x, y] − [x, Dy] = [Dx, y] + [x, Dy] − [x, Dy] = ad(Dx)(y), and def so ad L is an ideal of Der(L). Let Out(L) = Der(L)/ ad L be the outer derivation algebra of L. If C is the center of L, then C = ker ad and we have ad L ∼ = L/C as well as an exact sequence ad

0 → C → L −−−→ Der(L) → Out(L) → 0. Let G be a group. For each g ∈ G we define Ig : G → G by Ig (h) = ghg −1 for h ∈ G. If α ∈ Aut(G), the group of automorphisms of G, and g, h ∈ G, then αIg α −1 (h) = α(g(α −1 (h))g −1 ) = α(g)hα(g)−1 = Iα(g) (h). Let Inn(G) = {Ig : g ∈ G} denote the normal subgroup of Aut(G) consisting of all inner automorphisms of G and call

The Exponential Function on the Inner Derivation Algebra

357

def

it the inner automorphism group of G. The factor group Out(G) = Aut(G)/ Inn(G) is called the outer automorphism group of G. In the following discussion we let E be a vector space and V its dual with its weakly complete topological vector space dual (see Appendix 2). Let ϕ : E → E be an endomorphism and ψ : V → V its adjoint. Note that ψ is an endomorphism of topological vector spaces. Assume that E is a cofinal family of finite-dimensional vector subspaces F which are invariant under ϕ, that is, ϕ(F ) ⊆ F . Accordingly, ψ leaves invariant the annihilators W = F ⊥ , since ω ∈ E, x ∈ W implies ω, ψ(x) = ϕ(ω), x = 0 as ϕ(ω) ∈ F . Now the set F = {F ⊥ : F ∈ E } is a filter basis in V and is in fact a basis of the filter basis of all cofinite-dimensional closed vector subspaces of V . This is equivalent to saying that F converges to 0. 5 n If f is an entire function on the complex plane given by power series ∞ n=0 an z with 5∞ real coefficients an , then on each F ∈ E the endomorphism f (ϕ|F ) = n=0 an (ϕ|F )n is well defined by assigning to any x ∈ F an infinite series ∞ 6

an ϕ n (ω),

n=0

because for of the equivalent norms  ·  on the finite-dimensional vector space F , 5any ∞ the series n=0 an ϕ n (ω) converges absolutely with a majorant for the series of norms 5 n of the summands given by ∞ n=0 |an | · ϕ · ω where ϕ is the operator norm of ϕ with respect to the norm  ·  under consideration. If F1 , F2 ∈ E and ω ∈ F1 ∩ F2 , then ∞ ∞ 6 6 an (ϕ|F1 )n (ω) = an (ϕ|F2 )n (ω). n=0

n=0

As we obtain an endomorphism f (ϕ) : E → E given by f (ϕ(ω)) = 5∞a consequence n (ω). We note that f leaves all F ∈ E invariant. We denote by f (ψ) : V → V a ϕ n=0 n the adjoint of f (ϕ). Then f (ψ) is an endomorphism of topological vector spaces satisfying (∀ω ∈ E, x ∈ V )

ω, f (ψ)(x) = f (ϕ)(ω), x =

∞ 6 n=0

an ϕ (ω), x = n

∞ 6

an ω, ψ n (x)

n=0

where the infinite series appearing in the right half of this chain of equalities are absolutely convergent series of real numbers. If W = F ⊥ ∈ F , then the pairing of F and V /M identifying the two vector spaces as duals of each other is given by ω, x + W W = ω, x for ω ∈ F , and x ∈ V . The adjoint ψW : V /W → V /W of ϕ|F : F → F is then defined as usual by ω, ψW (x + W )W = ϕ(ω), x + W W = ϕ(ω), x for ω ∈ F , x ∈ V . If we select a norm on F and define on V /W the dual norm, then the operator norm of the adjoint ψW equals the operator norm of ϕ|F , and for each pair ω ∈ F and x ∈ V we can write

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

5∞ 5∞ 5∞ n n n n=0 an ω, ψ (x) = n=0 an ω, ψW (x + W )W = ω, n=0 an ψW (x + W )W = 5∞ ω, n=0 an ψ n (x) + W )W . Thus for each W ∈ F and all x ∈ V we get f (ψ)W (x + W ) =

∞ 6

n an ψW (x + W )

n=0

9

= ω,

∞ 6

n an ψW (x

:

+ W)

n=0

9

W

= ω,

∞ 6

an ψ (x) + W n

n=0

(1)

: W

.

In the sense of equation (1) we may also write f (ψ) =

∞ 6

an ψ n ,

(2)

n=0

so that when we write V = lim V /W ⊆ W ∈F



V /W,

W ∈F

 5∞  n the element f (ψ)((x + W )W ∈F ) is n=0 an (ψ|W ) (x + W ) W ∈F . A sequence, or more generally a net, converges in a projective limit if and only if the image net under each of the limit projections 5convergesn (see 1.27 (i)). Therefore the net of finite partial sums of the infinite series ∞ n=0 an ψ converges in the topology of V . Let us summarize one aspect of our discussion as follows: Lemma 9.1. Let ψ : V → V be an endomorphism of topological vector spaces on a weakly complete topological vector space and let f be an5 entire function given by the n everywhere absolutely convergent power series f (z) = ∞ n=0 an z with an ∈ R for all n. Assume that there is a filter basis of ψ-invariant neighborhoods converging to 5 n , where the series converges zero. Then there is an endomorphism f (ψ) = ∞ a ψ n=0 n in the topology of V in such a way 5 that fn(ψ) induces on every quotient V /W , W ∈ F the endomorphism x + W  → ∞ n=0 ψW (x + W ). On the locally finite-dimensional dual E of V , the adjoint ϕ of ψ leaves subspaces W ⊥ , W ∈ F 5∞ all finite-dimensional n  invariant and is given by f (ϕ) = n=0 an ϕ . This is now applied to pro-Lie algebras. Proposition g be a pro-Lie algebra. Then for each entire function f : C → C, 5 9.2. Let n with a ∈ R for all n, and each x ∈ g there is an endomorphism f (z) = ∞ a z n n n=0 5 n of topological vector spaces f (ad x) = ∞ n=0 an · (ad x) , where the sum converges with respect to the topology of g and where for each ideal j ∈  (g) an endomorphism f (ad x)j : g/j → g/j is induced via f (ad x)j (y + j) =

∞ 6 n=0

an (adg/j (x + j))n (y + j).

(3)

The Exponential Function on the Inner Derivation Algebra

359

In particular, for each x ∈ g there is an automorphism of topological Lie algebras ead x =

∞ 6 1 · (ad x)n . n!

(4)

n=0

The function t → et·ad x : R → Aut(g)

(5)

is a homomorphism. Proof. Since each j ∈  (g) is an ideal, we have (ad x)(j) = [x, j] ⊆ j, and thus all members of the filter basis  (g) are ad x-invariant for all x ∈5g. Thus Lemma 9.1 ∞ n applies with  (g) in place of F and shows that f (ad x) = n=0 an · (ad x) is a well-defined endomorphism of topological vector spaces. We apply this to the entire function f = exp and it remains to show that, firstly, ead x respects brackets and that, secondly, e(s+t)·ad x = es·ad x et·ad x ; this last fact will show, in particular, that e− ad x is the inverse of ead x and that ead x , therefore, is an automorphism. Firstly, on each finite-dimensional real Lie algebra g/j for j ∈  (g), def

the topological vector space endomorphism eadg x induces ϕj = eadg/j (x+j) by (3). Now ϕj is an automorphism of finite-dimensional Lie algebras for each j and thus ϕj ([u + j, v + j]) = [ϕj (u + j), ϕj (v + j)]. Therefore (∀u, v ∈ g, j ∈  (g))

ead x [u, v] − [ead x u, ead x v] ∈ j

 def and since  (g) = {0} we conclude that ead x respects brackets. Similarly, j (t) = et·adg/j (x+j) defines a homomorphism  : R → Aut(g/j) for each j. Thus (∀r, s ∈ R, u ∈ g, j ∈  (g))

e(s+t)·ad x u, −es·ad x et·ad x u ∈ j.

We conclude that t  → et·ad x : R → Aut(g) is a homomorphism Definition 9.3. The automorphisms which are finite products of automorphisms of the form ead x ∈ Aut(g) are called inner automorphisms of g. The subgroup of Aut(g) generated by all inner automorphisms is called the inner automorphism group of g and is written Inn(g). We note that in the form of special automorphisms we encountered inner automorphisms of a pro-Lie algebra in 7.73 through 7.76. Furthermore we remind the reader that when g is the Lie algebra of a pro-Lie group G we discussed inner automorphisms of the form ead x by approximation of G by Lie groups G/N and by approximation of L(G) by the finite-dimensional quotient algebras L(G/N ) ∼ = L(G)/L(N ). We proved the formula Ad(expG x) = ead x in Aut(g) for all x ∈ g. (See Proposition 8.1 (1).) The concept of inner automorphisms of a pro-Lie algebra was also discussed in Corollary 8.18.

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Analytic Subgroups The close relationship between the structure of a connected pro-Lie group and its proLie algebra is one of the keys to the fine structure theory of connected pro-Lie groups. Recall that for any topological group G one defines L(G) = Hom(R, G) with the compact open topology and that for a subgroup H of a topological group G, accordingly, one has L(H ) = {X ∈ L(G) : X(R) = expG R · X ⊆ H }. (See Chapter 2.) Definition 9.4. A morphism of topological groups f : G → H between pro-Lie groups is called an analytic morphism of pro-Lie groups if G is connected and L(f )(L(G)) is a closed Lie subalgebra of L(H ). Definition 9.5. (i) Let G be a pro-Lie group and H a subgroup. Then H is said to be an analytic subgroup of G if H is the image under a strict morphism of a connected pro-Lie group into G (see definition of strict morphism in the paragraph preceding 4.20). That is, there is a morphism f : C → G of topological groups from a connected pro-Lie group C into G such that H = f (C) and L(f )(L(C)) is closed. The morphism f is said to be a defining morphism of the analytic subgroup H . (ii) A subgroup H of a pro-Lie group G is said to be exponentially generated if def

h = L(H ) is a closed Lie subalgebra of L(G) and H = exp h. For Lie groups, in particular for finite-dimensional ones, a subgroup is analytic in the sense of Definition 9.5 if and only if it is analytic in any of the equivalent classical meanings of the word. By Corollary 8.17 of the Structure Theorem of Prosimply Connected Pro-Lie Groups 8.14 we know that every simply connected pro-Lie group (see Theorem 8.15 for the equivalence of simple connectivity and prosimple connectivity for pro-Lie groups) is exponentially generated. The vector space h in Definition 9.5 (ii) is arcwise connected and so is its continuous image exp h; a topological group generated by an arcwise connected subset containing the identity is arcwise connected. Hence every exponentially generated subgroup of a pro-Lie group is arcwise connected. From Definition 9.5 (i) and (ii), for any analytic or exponentially generated subgroup H we get a subset L(H ) of L(G), from any analytic morphism f : C → G (Definition 9.4) defining H we get a closed subalgebra im L(f ) ⊆ h, because for X ∈ L(C) def

the one-parameter subgroup Y = L(f )(X) = f  X : R → G is a one-parameter subgroup of H . Note that we have not said that we have to have im L(f ) = L(H ). However, this is the case for finite-dimensional Lie groups and indeed for separable Banach Lie groups (see e.g. [102, Chapter 5, Theorem 5.52]; the theorem holds for Lie groups instead of linear Lie groups). But there is a commutative Banach Lie group whose exponential function maps a proper closed hyperplane in its Lie algebra surjectively onto the group (see [102, Chapter 5, paragraph following the proof of Theorem 5.52]). The following result shows that this complication does not occur in pro-Lie groups. Moreover, it shows that proper analytic subgroups of pro-Lie groups

Analytic Subgroups

361

in some respects behave similarly to analytic subgroups of (finite-dimensional) Lie groups. Proposition 9.6. Let H be an analytic subgroup of a pro-Lie group G and assume that f : C → G is a defining morphism of H . Then the following conclusions hold: (i) For each closed normal subgroup N such that G/N is a Lie group, H N/N is an analytic subgroup of G/N. (ii) im L(f ) = L(H ). That is, the Lie algebra of H is a pro-Lie algebra, and H determines im L(f ) uniquely. (iii) There is a group topology τproto-Lie on H making H into a proto-Lie group Hproto-Lie such that the identity map Hproto-Lie → H is continuous and induces an isomorphism L(Hproto-Lie ) → L(H ). def

Proof. (i) We let N ∈ N (G) and set n = L(N ). By 4.21 (i) we may identify L(G/N ) with g/n. The quotient morphism p : G → G/N gives us a quotient morphism def

L(p) : g → g/n. Define K = ker(p  f ) = f −1 (N ). By the Quotient Theorem for Pro-Lie Groups 4.1 (i), the quotient group C/K is a proto-Lie group and p  f induces an injective morphism of the proto-Lie group C/K into the Lie group G/N . Hence C/K has an identity neighborhood in which {1} is the only subgroup. Since N (C/K) converges to 1 we conclude that {1} ∈ N (C/K) and that therefore C/K is a Lie group. We let p : C → C/K denote the quotient map and f  : C/K → G/N the injective morphism of Lie groups induced by f . Then H N/N = p(H ) = f (p(C)) = (p  f )(C) = (f   q)(C) = f  (C/K). Thus H N/N is an analytic subgroup of the Lie group G/N, and its Lie algebra L(H N/N) agrees with L(f  )(L(C/K)). (ii) By 4.21 (i) again we identify L(C/K) with c/k where c = L(C), and k = L(K). Using p  f = f   q we calculate L(f )(c)+n n

= L(p)(L(f )(c)) = L(p  f )(c) = L(f   q)(c) = L(f  )(c/k) = L(H N/N ).

Then L(p)(L(H )) ⊆ L(H N/N ) = (L(f )(c) + n)/n, and so (∀N ∈ N (G)) L(f )(c) ⊆ L(H ) ⊆ L(f )(c) + L(N ). By 4.21 (ii), {L(N ) : N  ∈ N (G)} is cofinal in  (g). Thus, since L(f )(c) is closed in g, we have L(H ) = N ∈N (G) (L(f )(c) + L(N )) = L(f )(c) = im L(f ). This completes the proof of (ii). (iii) By the Quotient Theorem for Pro-Lie Groups 4.1 (i), the quotient C/ ker f is a proto-Lie group. The morphism f induces a bijective morphism of topological groups F : C/ ker f → H . By the Strict Exactness Theorem 4.20 for L, the quotient morphism Q : C → C/ ker f induces a surjective morphism L(Q) : L(C) → L(C/ ker f ) of pro-Lie algebras. By (ii) we have L(H ) = im f . Thus F induces an isomorphism L(F ) : L(C/ ker f ) → L(H ). Now we define τproto-Lie to be that topology on H def

which makes F a homeomorphism. Then Hproto-Lie = (H, τproto-Lie ) satisfies all the conditions specified in (iii).

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

The following theorem deals with closed analytic subgroups: The first part shows that every closed connected subgroup is analytic, and the second part shows how these arise as subgroups of projective limit of Lie groups; this portion generalizes and completes Lemma 3.22 and Corollary 4.22 (vi). Theorem 9.7. Let G be a pro-Lie group and H a closed subgroup. (i) If the subgroup H is connected, then it is an analytic subgroup. (ii) Assume that G is the projective limit of a projective system of finite-dimensional Lie groups {ϕj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} with limit morphisms ϕj : G → Gj . Assume further that (a) for a cofinal subset K ⊆ J , the induced morphism (∀k ∈ K) L(ϕk ) : L(G) → L(Gk ) is surjective,

(∗)

(b) for each j ∈ J there is a closed subgroup Hj of Gj such that ϕj k (Hk ) ⊆ Hj for all j ≤ k in J , such that (∀k ∈ K) Hk is connected.

(∗∗)

Then the well-defined limit H = limj Hj is a closed connected, hence analytic, subgroup of G. (iii) Assume that for a cofinal set of N ∈ N (G) the subgroup H N /N of G/N is connected. Then H is connected, hence analytic. Proof. (i) If H is a closed connected subgroup of a pro-Lie group G, then H is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35, and L(H ) = Hom(R, H ) is a closed subspace of L(G) = Hom(R, G). Hence we may take C = H , and the inclusion map f : C → G is analytic and thus H = f (C) is an analytic subgroup. (ii) Since {ϕj k |Hk : Hk → Hj | (j, k) ∈ J × J, j ≤ k} is a projective system, its limit H is indeed a well-defined closed subgroup of G and def

we have to show that it is connected. We write gj = L(Gj ), g = L(G), hj = L(Hj ), and h = L(H ). In order to show connectedness it suffices to show that expG h is dense in H , and this is what we shall prove. For each k ∈ K set ak = ϕk−1 (hj ) ⊆ gj . Then (∗) implies def

(∀k ∈ K) 

L(ϕk )(ak ) = hk .

(†)

We notice that k∈K ak = limk∈K hk = h since L preserves limits by Theorem 2.25 (ii). Let h ∈ H and let U be an identity neighborhood of G. We claim that

Analytic Subgroups

363

h ∈ expG hU , and if this claim is proved, the lemma is proved. By the Fundamental Theorem on Projective Limits 1.27 (i) and by the cofinality of K in J there is no loss of generality to assume that there is an i ∈ K and an identity neighborhood W of Gi such that (‡) ϕi−1 (W ) = U. Now we invoke Lemma 5.18 (b) for L(G)/L(H ) in order to conclude from  ak /L(H ) = {0} in L(G)/L(H ) k∈K

that there is a k ∈ K, k ≥ i such that ak ⊆ L(H ) + U . By (∗∗) and Corollary 4.22, there are elements X1 , . . . , Xn ∈ hk such that −1 (W ). exp X1 . . . exp Xn ∈ ϕk (h)ϕik

By (†) there are elements Y1 , . . . , Yn ∈ ak such that L(ϕ)(Yp ) = Xp for p = 1, . . . , n. Now expGk Xp = expGk L(ϕk )(Yp ) = ϕk (expG Yp ). Hence −1 ϕk (h−1 expG Y1 . . . expG Yn ) = ϕk (h)−1 exp X1 . . . exp Xn ∈ ϕik (W )

and so

−1 (W ) = ϕi−1 (W ) = U h−1 expG Y1 . . . expG Yn ∈ ϕk−1 ϕik

by (‡), which proves the claim and thereby finishes the proof. (iii) We apply (ii) with J = N (G), Gj = G/j and Hj = Hj /j . By the Closed Subgroup Theorem for Projective Limits 1.34 (ii) we have H = limN ∈N (G) H N /N , and so (ii) proves the claim. Before we continue, let us look at some examples in a familiar environment. Of course, for any finite-dimensional Lie group G, our definition of an analytic subgroup agrees with any of the equivalent definitions current for finite-dimensional real Lie groups. Every analytic subgroup H therefore is uniquely determined by its Lie algebra L(H ), and the assignment H  → L(H ) is a bijection from the set of analytic subgroups of G to the set of subalgebras of the Lie algebra L(G). In the context of pro-Lie groups the abelian specimen of Chapter 5 yield a good illustration of the concepts we have introduced. We shall fall back on the results of Chapter 5 and use the information provided in [102], Chapters 7 and 8. Let G be an abelian pro-Lie group. For the purposes of discussing analytic subgroups which are always connected by their very definition we only have to consider G0 ; it is therefore no loss of generality to assume that G is connected. By the Vector Subgroup Splitting Theorem 5.20, G is the direct sum V ⊕ comp(G) algebraically and topologically of a weakly complete vector subgroup complement V (isomorphic to RJ for some set J def

by Corollary A2.9) and the characteristic largest compact subgroup C = comp(G), a compact connected abelian group. Moreover, we know from section (viii) of that theorem that g = L(G) = L(V ) ⊕ L(comp(G)) and that L(V ) and V may actually be

364

9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

identified; we shall do that, write g = V ⊕ c where c = comp(L)(G) = L(comp(G)) (see Definition 5.4) and where for v ∈ V and X ∈ c we write expG (v + X) =  R) is the Lie algebra of C. (See v + expC X. Recall that c = Hom(R, C) ∼ = Hom(C, for instance [102, Chapters 7 and 8].) Proposition 9.8. Let G be an abelian pro-Lie group. (i) Each closed connected subgroup H is an analytic subgroup. The arc component of the identity in H is Ha = expG L(H ). (ii) If A = expG h is exponentially generated by a closed subalgebra h of g and if A is the identity arc component of some closed subgroup H of G, then h = L(A) = L(H ). In these circumstances A is an analytic subgroup and expG |h : h → A is a defining morphism of the analytic subgroup A. (iii) If G is a compact connected but not arcwise connected group, then A = G is an analytic subgroup which is not exponentially generated. Every compact group G  is countable torsion-free but not free is of this type. for which G (iv) The compact connected metric group whose character group is Q(N) has a continuum cardinality of different analytic subgroups all of whose Lie algebras agree with g. Proof. (i) If H is a closed connected subgroup, then it is an analytic subgroup by Theorem 9.7 (i). The arc component Ga of the identity is im expG by Theorem 5.20 (vii). Accordingly, Ha = im expH = expG L(H ). (ii) In order to simplify notation, it is no loss of generality to assume G = H and to show h = g. Now Ga = exp g = E(G) by Theorem 5.20 (vi), and by hypothesis expG h = A = E(G). Then from Corollary 4.21 (i) it follows that h = g. Also h ⊆ L(A) ⊆ L(G). The additive topological group of h is a pro-Lie group and expG |h : h → G is a morphism of pro-Lie groups with image A. Hence A is an analytic group with defining morphism expG |h. (iii) Let G be a compact connected group which is not arcwise connected; then G is an analytic subgroup of itself by (i) and L(G) = g. But Ga = G is analytic by (ii) and L(Ga ) = g as well. def  is not arcwise connected, and S is (iv) Typically, a group like the solenoid S = Q the only closed, hence maximal analytic subgroup with Lie algebra s ∼ = R. The arc component Sa is a bijective image of s and is equal to expS s; it is the smallest analytic def

subgroup expS s. On the metrizable compact connected abelian group G = S N , for every subset X ⊆ N we can form HX = SaX × S N\X considered in the obvious way as a subgroup of G = S N . Then HX is an analytic subgroup with Lie algebra g = sN ∼ = RN . Thus G has 2ℵ0 different analytic subgroups with the same Lie algebra as the whole group. We have seen in Chapter 4 in 4.9 and 4.10 an example of a compact connected abelian group G (namely, the character group of the discrete group ZN ) for which the corestriction of the exponential function exp : g → Ga is a quotient morphism. Thus,

Analytic Subgroups

365

in this case, the proto-Lie group topology τproto-Lie on the analytic group Ga = expG g with Lie algebra g is the subgroup topology, and it is not a pro-Lie topology because it is not a complete topology. If G is the additive subgroup of a weakly complete topological vector space, then a subgroup A of G is analytic if and only if it is the additive group of a closed vector subspace. For instance, if G = RN , then the subgroup H = R(N) of all real sequences of finite support is an arcwise connected, but not analytic subgroup. In this context we record the Theorem of Yamabe and Gotô saying that a subgroup of a finite-dimensional Lie group is analytic if and only if it is arcwise connected (see e.g. [17, Chap. 3, §7, Exercise 4], or [70]). When dealing with infinite-dimensional pro-Lie groups we have to get accustomed to circumstances that are rather different from those one is used to in handling finitedimensional Lie groups. For the moment we denote by C(g) the set of closed Lie subalgebras of g = L(G) for a pro-Lie group G and by A(G) the set of analytic subgroups of G. We have seen that H  → L(H ) : A(G) → C(g) is a well-defined function which is not injective in general. Is it surjective? We shall provide an affirmative answer in the following. For this purpose, it may be a good idea to recall the set-up of Lie’s Third Fundamental Theorem (Theorem 5.4, through Corollary 5.9) and the universal properties surrounding it. We have a natural isomorphism of pro-Lie algebras ηg : g → L((g)) (see 5.5) and  → G, G  = (L(G)) → G whose image a natural morphism of pro-Lie groups πG : G is dense in G0 (see 5.6 (iv)), and the universal properties which are best visualized in the following diagrams: proLieAlg ηg

proLieGr

g ⏐ ⏐ ∀f 

−−−−→

L((g)) ⏐ ⏐  L(f )

L(H )

−−−−→

L(H )

idL(H )

(g) ⏐ ⏐  ∃!f

()

H,

and proLieAlg

proLieGr πG

L(G)  ⏐ ∃!f  ⏐

(L(G))  ⏐  ⏐(f )

−−−−→

G  ⏐∀f ⏐

h

(h)

−−−−→

(h).

id(h)

(⊥)

In particular from () we instantly get the following result: def

Let G be a pro-Lie group and h a closed subalgebra of the Lie algebra g = L(G). Let i : h → g be the inclusion map. Since ηh : h → L((h)) is an isomorphism, we may identify h with L((h)).

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

We claim that in these circumstances there is a unique analytic morphism ih : (h) → G inducing the morphism i : h = L((h)) → G. Indeed we specialize () to the present situation: proLieAlg

proLieGr

h = L((h)) ⏐ ⏐ ∀i 

(h) ⏐ ⏐∃!i  h

g = L(G)

G

and obtain ih as asserted: the domain (h) is connected and the range h of L(ih ) = i is closed. Definition 9.9. Let G be a pro-Lie group with Lie algebra g and h a closed Lie subalgebra. (i) The analytic morphism ih : (h) → G just introduced is referred to as the morphism defined by h ⊆ g. (ii) The analytic subgroup im ih is denoted by A(h). If reference must be made to the containing group it shall be written A(h, G). It follows from the definition of the analytic subgroup A(h) that it has the Lie algebra h.  and ig : (g) → G is exactly the universal If h happens to be g, then (g) = G,  → G of Theorem 6.6, and A(g, G) = im πG . morphism πG : G Proposition 9.10. Assume that G is a pro-Lie group with Lie algebra g and the universal  = (g) → G of Theorem 5.6; the Lie algebras of G  and G may morphism πG : G and often will be identified. Assume that h is a closed Lie subalgebra of g and let i : h → g denote the inclusion. Then there is at least one analytic subgroup H such that L(H ) = h, namely, H = A(h) = A(h, G), for which the following conclusions hold: (i) There is a commutative diagram (i)

(h) −−−−→ (g) ⏐ ⏐ ⏐ ⏐ c id incl A(h,⏐(g)) −−−−→ (g) ⏐ ⏐ ⏐π d  G A(h, G) −−−−→ G, incl

where the surjective morphism c is a corestriction of (i) : (h) → (g) and the surjective morphism d is a restriction and corestriction of πG , and where d  c is a corestriction of ih . If h is an ideal, then c is an isomorphism and (h) may and will be identified with a closed normal subgroup A(h, (g)) of (g).

Analytic Subgroups

367

(ii) The identity map A(h)proto-Lie → A(h) defines an inclusion j : A(h)proto-Lie → G such that there is a commutative diagram (L(j ))

(L(A(h)⏐proto-Lie )) −−−−→ (h) ⏐ ⏐ ⏐ ⏐ih iL(Hproto-Lie ) ⏐   −−−−→ A(h)

A(h)proto-Lie

j

in which the top horizonal map is an isomorphism. (iii) The analytic subgroup A(h) is exponentially generated and satisfies L(A(h)) = h and so expG h = A(h). In particular, the analytic subgroup A(h) is arcwise connected. Proof. (i) The statements on the diagram are straightforward from the definitions. If h is normal we apply Corollary 6.9 for the remaining assertions. (ii) We let ih∗ : (h) → A(h) denote the surjective corestriction of ih and consider the canonical decomposition (h) ⏐ ⏐ quot⏐  (h)/ ker ih

ih∗

−−−−→

A(h)  ⏐ ⏐id ⏐

−−−−→ im ih∗ β

for some continuous bijection β. We recall that τproto-Lie is that topology on A(h) which makes β : (h)/ ker ih → (H, τproto-Lie ) a homeomorphism. Note that ker ih∗ = ker ih . The functor L preserves kernels, hence L(ker ih ) ∼ = ker L(ih ) = ker i, where i : h → g is the inclusion map. But i as an inclusion map has zero kernel. Hence L(ker ih ) = {0} and therefore ker ih is totally disconnected by 4.23. Hence L((h)/ ker ih∗ ) ∼ = L((h)) = h and thus the identity Hproto-Lie → H induces an isomorphism of Lie algebras L(Hproto-Lie ) → h and thus an isomorphism of pro-Lie groups (L(Hproto-Lie )) → (h). This proves (ii). (iii) By Definition 9.9 (ii), the analytic subgroup A(h) is the image of the analytic morphism ih : (h) → G which induces on the Lie algebra level the inclusion mapping i : h → g. By Corollary 8.17, (h) is exponentially generated, that is, (h) = exp(h) h. The commutative diagram h ⏐ ⏐ exp(h) ⏐ 

i

−−−−→

g ⏐ ⏐ ⏐expG 

(h) −−−−→ G ih

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

shows that expG h = ih (exp(h) ) = ih ((h)) = A(h). As remarked after Definition 9.5, every exponentially generated subgroup of a pro-Lie group is arcwise connected, and so A(h) is arcwise connected. From (i) above we know L(A(h)) = h. It is now clear that the function H → L(H ) : A(G) → C(g) is surjective for each pro-Lie group G. If there is a closed connected subgroup C of G with L(C) = h then C is obviously the largest analytic subgroup having h as its Lie algebra. At any rate, the analytic subgroup A(h) is the smallest analytic subgroup having h as its Lie algebra as we shall see now. Proposition 9.11. Let H be any analytic subgroup of a pro-Lie group G and set def h = L(H ). Then A(h) ⊆ H and A(h) is dense in Hproto-Lie and thus in H . Proof. Let f : C → G be an analytic morphism of pro-Lie groups such that H = f (C). Let q : c → h be the surjective morphism which we have in this set-up. By Corollary 6.8, the morphism (q) : (c) → (h) is a quotient morphism. There is a commuting diagram (q)

−−−−→ (h) (c) ⏐ ⏐ ⏐ ⏐i πC  h C −−−−→ G, f

where H = im f and A(h) = im ih . We conclude that H = f (C) ⊇ (f πC )((c)) = (ih  (q))((c)) = ih ((h)) = A(h). It remains to show that A(h) is dense in Hproto-Lie . Let c = L(C) and E(C) = expC c. We have seen that h = L(f )(c) in 9.6 (ii) and thus A(h) = expG h = exp L(f )(c) = f (E(C)). Further C = E(C) by 4.22 (i). Let Cl = Clproto-Lie denote the closure with respect to τproto-Lie . We recall that the natural bijection C/ ker f → Hproto-Lie is an isomorphism of topological groups. Thus we compute H = f (C) = f (E(C)) ⊆ f (E(C)) = Cl(A(h)), as was asserted. Definition 9.12. The analytic subgroup A(h) = ih ((h)) is called the minimal analytic subgroup with Lie algebra h. Scholium. Let G be a pro-Lie group and g its Lie algebra. Denote the set of all minimal analytic subgroups of G by A0 (G). Then the assignment H  → L(H ) : A0 (G) → C(g) is a bijection with inverse function h → A(h). In particular, every pro-Lie group G contains a minimal analytic subgroup A(g) with Lie algebra g and a maximal analytic subgroup G0 (the identity component of G) with Lie algebra g. In the case of the example of an abelian pro-Lie group G discussed in Proposition 9.8 and g = L(G) = Hom(R, G), we know that (g) is the additive topological group underlying g and that πG = ig : (g) → G and expG : g → G are one and the same function. Thus Hg = im ig = expG g = Ga . Thus the arc component of the

Automorphisms and Invariant Analytic Subgroups

369

identity in G is the minimal analytic subgroup A(g) having g as Lie algebra and is the exponentially generated subgroup with Lie algebra g, while G0 is the maximal analytic subgroup with Lie algebra g. In Proposition 9.8 (iv) we saw that there may be many analytic subgroups between Ga and G0 all having the same Lie algebra g.

Automorphisms and Invariant Analytic Subgroups Recall from Definition 9.10 that a closed subalgebra h of the pro-Lie algebra L(G) of a pro-Lie group G gives us a morphism ih : (h) → G which defines the minimal analytic subgroup A(h) with Lie algebra h by A(h) = ih ((h)). Lemma 9.13. Let G be a pro-Lie group and α an automorphism of the topological group G. Assume that h is a closed subalgebra of L(G) and that L(α)(h) = h. Then α(A(h)) = A(h). Proof. Since L(α)(h) = h, there is an automorphism of topological Lie algebras β : h → h defined by β(Y ) = L(α)(Y ), and using the inclusion map i : h → g we have a commutative diagram in proLieAlg: β

h −−−−→ ⏐ ⏐ i g −−−−→ L(α)

h ⏐ ⏐ i g.

We claim that the following diagram in proLieGr is commutative: (β)

(h) −−−−→ (h) ⏐ ⏐ ⏐ ⏐  i i G −−−−→ G. α

Indeed, α i  induces L(αi  ) = L(α)L(i  ) = L(α)i : h → g, while i  (β) induces L(i   (β)) = L(i  )  L((β)) = i  β : h → g. By the commutativity of the first diagram, these two maps are equal. By the uniqueness in Lemma 9.9, α  i  = (β)  i  follows, and that is the asserted commutativity of the second diagram. From the commuting of this diagram, however, we conclude that α(A(h)) = (α  i  )((h)) = (i   (β))((h)) = i  ((h)) = A(h) (as (β) is an automorphism of topological groups). Proposition 9.14. Let G be a pro-Lie group and H an analytic subgroup. Assume that def α is an automorphism of G and that the automorphism L(α) of g = L(G) leaves the def closed subalgebra h = L(H ) invariant. Then α(H ) = H , provided at least one of the following conditions is satisfied:

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(a) H is the minimal analytic subgroup A(h) with Lie algebra h. (b) H is a closed analytic subgroup with Lie algebra h. (c) Hproto-Lie is complete, that is, is a pro-Lie group Hpro-Lie , and α|A(h) is an automorphism with respect to the topology τ (Hpro-Lie ) of Hpro-Lie on A(h). Proof. In the case (a) we have α(H ) = α(expG L(H )) = exp L(α)(L(H )) = exp L(H ) = H . See also Lemma 9.13. In the case (b), H is a connected pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Hence H = expG h by 4.22 (i). Then α(H ) = α(expG L(H )) = exp L(α)(L(H )) = exp L(H ) = H. In the case (c) we recall from Lemma 9.13 that α(A(h)) = A(h). We denote by τ (Hproto-Lie )|A(h) the topology induced by τ (Hproto-Lie ) on A(h). Then def

β = α|A(h) : (A(h), τ (Hproto-Lie )|A(h)) → (H, τ (Hproto-Lie )) is a morphism of topological groups into a complete group. Since A(h) is dense in H , it has a unique extension to a morphism β : Hproto-Lie → Hproto-Lie . Now α|H : Hproto-Lie → G is a morphism since τproto-Lie is finer than or equal to the induced topology on H , and it agrees with β on the group A(h) which is dense in Hproto-Lie . Hence they agree on Hproto-Lie , and this implies α(H ) = H . 2 , We cannot do much better as is illustrated by the following example. Let G = Q )a × Q  ⊆ G. Then H is an analytic subgroup, and Hproto-Lie is isomorphic and H = (Q , and thus is a pro-Lie group. Now let α be the automorphism of G which to R × Q ))2 ∼ switches the two factors, that is, α(x1 , x2 ) = (x2 , x1 ). Since L(H ) = (L(Q = R2 , clearly L(α) leaves L(H ) invariant. But it does not leave H invariant.

Centralizers When we now address the concept of centralizers and normalizers, dealing with the former is fairly elementary, but dealing with the latter requires the ideas concerning analytic subgroups. Lemma 9.15. Let g be a pro-Lie algebra and let X and Y be two vectors in g. Assume that C is a closed vector subspace of g. Consider the following two conditions: (i) (∀s ∈ R) es·ad X Y − Y ∈ C. (ii) [X, Y ] ∈ C. Then (i) implies (ii), and if [X, C] ⊆ C, then both conditions are equivalent. In particular, [X, Y ] = 0 iff (∀s ∈ R) es·ad X Y = Y .

Centralizers

371

Proof. (i) ⇒ (ii): Assume (i) and define ϕ : R → C by ϕ(s) = es·ad X Y − Y . Let j ∈  (g) and set ϕj (s) = ϕ(s) + j. Write X = X + j and Y = Y + j. Then ϕj (s) = es·ad X Y − Y in the finite-dimensional Lie algebra g/j. Accordingly, ϕj : R → (C +j)/j is a function with values in a finite-dimensional real vector space and [X, Y ] =lim0 =s→0 1s · ϕj (s) ∈ C. Hence [X, Y ] ∈ C + j for all j ∈  (g). So [X, Y ] ∈ j∈ (g) (C + j) = C since C is closed. & 2 ' (ii) ⇒ (i): Let [X, Y ] ∈ C. Then es·ad X Y −Y = s·[X, Y ]+ s2! · X, [X, Y ] +· · · ∈ C ' & since C is a closed vector subspace and X, [. . . , [X, Y ] . . . ] ∈ C by induction from [X, C] ⊆ C. Thus the equivalence of (i) and (ii) is established, and the last assertion of the lemma follows by letting C = {0}. Proposition 9.16. Let G be a pro-Lie group and H any subset. Then the centralizer or commutant Z(H, G) = {g ∈ G : (∀h ∈ H ) gh = hg} is closed in G and is therefore a pro-Lie group. Proof. The claim follows directly from the continuity of the function g  → ghg −1 : def

−1 G → G; indeed for each h ∈ H the set Fh = {g ∈ G : ghg = h} is closed, and thus the set Z(H, G) = h∈H Fh is closed.

Proposition 9.17. Let H be a subgroup of a connected pro-Lie group G and assume that H ⊆ expG h, where h = L(H ). (This assumption is automatically satisfied if H is exponentially generated or analytic.) Then the following conclusions hold: (i) An automorphism α of G satisfies α(h) = h for all h ∈ H iff L(α)X = X for all X ∈ h. (i ) An element g ∈ G is in Z(H, G) iff Ad(g)X = X for all X ∈ h. (ii) L(Z(H, G)) = z(h, g). (iii) Z(H, G)0 = expG z(h, g). Proof. (i) Assume that α(h) = h for all h ∈ H . Let X ∈ h. Then expG r · L(α)(X) = α(expG r · X) = expG r · X for all r ∈ R. Consequently L(α)(X) = X. Conversely, assume that L(α)(X) = X for all X ∈ h. Then α(expG X) = expG L(α)(X) = expG X shows that α fixes expG h elementwise. Since α is continuous, its fixed point set F is a closed subgroup of G containing expG h. Then expG h ⊆ H ⊆ expG h ⊆ F . Thus α(h) = h for all h ∈ H . (i ) follows from (i) by setting α = Ad(g). (ii) By (i ) we have X ∈ L(Z(H, G)) iff Ad(g)X = X for all g ∈ G. Since g  → Ad(g)X : G → g is continuous by 2.27, and G = expG g by 4.22 (i), this is equivalent to saying that Ad(expG Y )X = X for all Y ∈ g. Thus X ∈ L(Z(H, G)) iff (∀Y ∈ g) · ead Y X = X in view of 8.1(1). This is the case iff (∀t, ∈ R, Y ∈ g)

et·ad Y X = X.

(∗)

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

By Lemma 9.15, Condition (∗) is equivalent to (∀Y ∈ g)

[Y, X] = 0.

(∗∗)

But (∗∗) is equivalent to X ∈ z(h, g). This proves (ii). (iii) By 4.22 (i) we have Z(H, G)0 = expZ(H,G) L(Z(H, G)), and thus (iii) is a consequence of (ii) and this fact. Proposition 9.18 (The Bicommutant Lemma). (i) If A is any subgroup of a topological def group, then the bicommutant B(A, G) = Z(Z(A, G), G) of A is a closed subgroup of G containing A. (ii) The centralizer Z(B(A, G), G) of the bicommutant is Z(A, G). (iii) If A is dense in G, then the centralizer Z(A, G) of A is the center Z(G) of G. (iv) If A is normal, then so are the centralizer Z(A, G) and the bicommutant. def

(v) If a is a subset of a topological Lie algebra g, then the bicommutant b(a, g) = z(z(a, g), g) is a closed subalgebra containing a, and if a is an ideal, so is the bicommutant. The centralizer of b(a, g) is z(a, g). (vi) If A is an analytic subgroup of a pro-Lie group G and a = L(A) is its Lie algebra, then L(B(A, G)) = b(a, g) and A ⊆ B(A, G). If A is dense in G then Z(A, G) is the center of G and z(a, g) = z(g). (vii) If A is an analytic subgroup of a pro-Lie group G and AZ(G) is dense in G, then G ⊆ A . So G/A is abelian. Proof. (i) By Lemma 9.16, the bicommutant is a closed subgroup of G. If a ∈ A, and z ∈ Z(A, G), then az = za and thus a ∈ B(A, G). (ii) For this proof only, abbreviate Z(A, G) = Ac . From (i) we know A ⊆ Acc . If A1 ⊆ A2 , then Ac2 ⊆ Ac1 . Hence Accc = (Acc )c ⊆ Ac . By applying (i) now to Ac we get Ac ⊆ (Ac )cc = Accc . Therefore Ac = Accc . (iii) Since B(A, G) is closed, from (i) we get A ⊆ B(A, G). If A = G, then G = B(A, G), and thus Z(G) = Z(G, G) = Z(B(A, Z), G) = Z(A, G) by (ii). (iv) and (v) are easy exercises and (vi) follows at once from the preceding. (vii) G is generated by the commutators comm(x1 , x2 ) = x1 x2 x1−1 x2−1 , where x1 and x2 range through G. Since G = AZ, Z = Z(G), for every neighborhood U of comm(x1 , x2 ) there are elements aj ∈ A, zj ∈ Z, j = 1, 2, such that U contains comm(a1 z2 , a2 z2 ) = comm(a1 , a2 ) ∈ A . Thus U ∩ A = Ø. Hence comm(x1 , x2 ) ∈ A . Therefore G ⊆ A Exercise E9.1. Prove 9.18 (iv), (v), and (vi). A variant of 9.18 (vii), which in most respects is more general, will be proved in Theorem 9.32 below.

Normalizers

373

Normalizers Definition 9.19. Let H be a subgroup of a group G. The normalizer of H in G is the set N(H, G) = {g ∈ G : gH g −1 = H }. Let h be a subalgebra of a Lie algebra g. The normalizer of h in g is the set n(h, g) = {X ∈ g : [X, h] ⊆ h}. Sometimes n(h, g) is said to be the idealizer of the subalgebra h in g. It is easily seen that N(H, G) is the largest subgroup of G containing H in which H is normal. Likewise n(h, g) is the largest subalgebra containing h in which h is an ideal. Proposition 9.20. Let H be a subgroup of a pro-Lie group G and assume that H satisfies at least one of the following conditions: (a) H is a minimal analytic subgroup of G. (b) H is a closed connected subgroup. Then the following conclusions hold: (i) (i ) (ii) (iii)

An automorphism α of G satisfies α(H ) = H iff L(α)(h) = h. Let g be an element of G. Then gH g −1 = H iff Ad(g)h = h. The normalizer N(H, G) is closed in G. L(N(H, G)) = n(h, g).

Proof. (i) Assume that α(H ) = H . Let X ∈ h. Then (∀r ∈ R)

expG r · L(α)(X) = α(expG r · X) ∈ H.

Consequently L(α)(X) ∈ h. Therefore L(α)h ⊆ h, and since the same holds for L(α −1 ) = L(α)−1 , equality follows. Conversely, assume that L(α)(h) = h. Then α(expG h) expG L(α)(h) = expG h. So expG h = α(expG h) = αexpG h, and since α is a homeomorphism, H = expG h = α(expG h) = α(H ) We claim that α(H ) = H follows in each of the cases (a), (b): In Case (a), this is clear in view of the preceding paragraph because H = expG h. See also Proposition 9.14. In Case (b) we have H = H and thus the claim was just established. (i ) follows from (i) by setting α = Ad(g). (ii) By (i ), N(H, G) = {g ∈ G : Ad(g)h = h}. Now for each X ∈ h let def

FX = {g ∈ G : Ad(g)X ∈ h}. Then N(H, G) =



FX ∩ FX−1 .

X∈h

It therefore suffices to notice that FX is closed. But this follows from the fact that g  → Ad(g)X : G → L(G) is continuous by Proposition 2.27. (iii) A vector X ∈ g is contained in L(N (H, G)) iff (∀s ∈ R)

H = (expG s · X)H (expG s · X)−1 .

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

If X satisfies this condition so does −X, and thus it is tantamount to (∀s ∈ R, h ∈ H )

(expG s · X)h(expG −s · X) ∈ H.

We note expG es·ad X Y = expG Ad(expG s · X)Y = (expG s · X) expG Y (expG −s · X). Since H is a closed analytic subgroup, it is a connected pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35, and thus H = expG h by 4.22 (i). Hence X ∈ L(N(H, G)) iff (∀s ∈ R, Y ∈ h)

expG es·ad X Y ∈ H.

(∗)

By Lemma 9.15, this is equivalent to (∀Y ∈ h)

[X, Y ] ∈ h.

(∗∗)

But (∗∗) means X ∈ n(h, g). It is noteworthy that normalizers of certain not necessarily closed subgroups are closed as is illustrated by Case (a) in the previous result. This phenomenon, of course, is known from the theory of finite-dimensional Lie groups.

Subalgebras and Subgroups The following theorem complements the information we gave in Corollary 8.16 on closed connected normal subgroups of simply connected pro-Lie groups. Theorem 9.21 (Normal Analytic Subgroups of Simply Connected Pro-Lie Groups). Let G be a simply connected pro-Lie group and N a connected normal analytic subgroup. (i) N is closed and simply connected. (ii) G/N is a simply connected pro-Lie group. Proof. (i) By the results of Chapter 7 we may assume that G is of the form G = (g). Let n = L(N) and let i : n → g be the inclusion and q : g → g/n the quotient map. Then by Corollaries 7.8 and 7.9, (i) : (n) → (g) = G is an embedding onto a closed subgroup and (q) : (g) → (g/n) is a quotient morphism with ker(q) = im(i), and we may write (n) = A(n) ⊆ G for the closed minimal analytic subgroup A(n) with Lie algebra n. (See also 9.10 (i).) A portion of these pieces of information is encapsulated in the exact sequence (i)

(q)

1 → (n) −−−−→ (g) −−−−→ (g/n) → 1.

(∗)

By 9.12 we have A(n) ⊆ N ⊆ A(n) = A(n). Thus N = A(n) is closed. Because N∼ = (n), the group N is prosimply connected. Thus we can rewrite (∗) as (i)

(q)

1 → N −−−−→ G −−−−→ (g/n) → 1. The remainder now follows from Theorem 8.18.

(∗∗)

Subalgebras and Subgroups

375

We have shown in Corollary 8.18 that N is topologically a direct factor. This generalizes the fact in the theory of finite-dimensional Lie groups that a normal analytic subgroup of a simply connected Lie group is closed and simply connected. (See e.g. [17, Chap. 3, §6, no 6, Proposition 14 and Corollaire 2].) The following theorem summarizes most of what we have proved so far on the relationship between subgroups and subalgebras. The Correspondence Theorem of Subalgebras and Subgroups Theorem 9.22. Let G be a pro-Lie group and g its Lie algebra; then g is a pro-Lie algebra. (i) For every analytic subgroup H of G, the set L(H ) is a uniquely determined closed Lie subalgebra of g and every closed Lie subalgebra of g occurs in this fashion. The assignment H → L(H ) : A(G) → C(g) is a surjective order preserving map from the partially ordered set of analytic subgroups of G onto the partially ordered set of closed subalgebras of g. (ii) If h is a closed Lie subalgebra of g = L(G), then there is a unique minimal analytic subgroup A(h) of G such that L(A(h)) = h, and it is exponentially generated by h. The function h → A(h) : C(g) → A0 (G) is a lattice isomorphism from the lattice of closed subalgebras of g to the lattice of minimal analytic subgroups with inverse morphism H  → L(H ) : A0 (G) → C(g). (iii) Assume that h is a closed ideal of g and H is a minimal analytic, or a closed analytic subgroup satisfying L(H ) = h. Then H is normal. If, moreover, G is simply connected, then H is closed and simply connected. (iv) A(g) = E(G) = expG g is a normal subgroup which is dense in G0 . def

(v) If H is a closed subgroup of G, then H is a pro-Lie group and h = L(H ) = {X ∈ L(G) : exp R · X ⊆ H } is a closed Lie subalgebra such that A(h) = H0 . If H and G are connected then H is normal in G iff h is an ideal in g. Proof. (i) This follows from 9.6 and 9.10. (ii) The existence of A(h) and its properties were established in 9.6, 9.10, 9.11. (iii) Next assume that h is an ideal of g. Then n(h, g) = g, and thus by 9.20 (iii), L(N(H, G)) = g and so E(G) = expG g ⊆ N (H, G). Since N (H, G) is closed from G = E(G) (by 4.22 (i)) we get G = N(H, G), that is H is normal in G. If G is simply connected and H is analytic, then H is closed by 9.21. This proves (i). (iv) This follows from (i) by specializing h to g. (v) By the Theorem on Closed Subgroups of Pro-Lie Groups 3.35, H is a proLie group and thus L(H ) = Hom(R, H ) is a pro-Lie algebra. In the obvious way, def

Hom(R, H ) is a subspace of Hom(R, G), that is, L(H ) ⊆ L(G). Since h = L(H ), as the Lie algebra of a pro-Lie group, is a complete topological vector space, it is closed in L(G). From 4.22 (i) we know that E(H ) = expH h = expG h is dense in H0 . The

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

minimal analytic group A(h) with Lie algebra h contains expG h and is contained in H0 , the maximal analytic subgroup with Lie algebra h. Thus H0 = A(h) follows. Now let H be connected. Then 9.20 (b) applies and 9.20 (iii) shows that L(N(H, G)) = n(h, g). Now H is normal iff N (H, G) = G and h is an ideal iff n(h, g) = g. Thus h is an ideal iff L(N (H, G)) = g, and N (H, G) is closed by 9.20 (ii). If H is normal, then N(H, G) = G and then L(N (H, G)) = g; thus h is an ideal. If h is an ideal then by the preceding N (H, G)0 = expG L(N (H, G)) = expG g = G0 = G and this implies N(H, G) = G, that is, that H is normal in G.

The Center The center of a group G, we recall, is the set of elements commuting with all elements of G. From 7.25 we remember that the center of a Lie algebra L is the set of all x ∈ L such that [x, y] = 0 for all y ∈ L. Proposition 9.23. Let G be a connected pro-Lie group and g its Lie algebra. If Z = Z(G) denotes the center of G, and z = z(g) the center of g, then (i) z = L(Z), and (ii) exp z = Z0 . Proof. We observe that z = z(g, g) and Z = Z(G, G). Then (i) is a consequence of 9.17 (ii) and (ii) follows from 9.17 (iii). By Lemma 9.23, L(Z(G)) = z(g). By Proposition 4.23, L(Z(G)) is zero iff the center Z(G) of G is totally disconnected. Corollary 9.24. A connected pro-Lie group G is abelian if and only if its Lie algebra L(G) is abelian. Proof. G is abelian iff G = Z(G), and g is abelian iff g = z(G). If G = Z(G) then trivially g = L(G) = Z(G) = z(g). Conversely, if g = z(g) implies Z(G)0 = expG z(g) = expG g = G, and thus G ⊆ Z(G) ⊆ G. In 9.51 we shall show that every connected abelian pro-Lie group G such that comp(G) is compact can occur as the center of a connected pro-Lie group.

The Commutator Subgroup For a pro-Lie algebra g we recall that g((1)) denotes the closed commutator subalgebra [g, g]. At this time we shall not consider any higher commutator algebras. Therefore let us abbreviate the closed commutator subalgebra by g˙ . This subalgebra is universal

The Commutator Subgroup

377

in the sense that every morphism of topological Lie algebras g → a into an abelian Lie ˙ def algebra factors through the quotient morphism g → g/˙g. Likewise we write G = G  for the closure of the algebraic commutator subgroup G of a topological group G. Recall that for a pro-Lie group G, we denote by N (G) the filter basis of closed normal subgroups N such that G/N is a Lie group. Lemma 9.25. (i) Let G be a pro-Lie group. Then ˙ g˙ ⊆ L(G)

(*)

and ˙ 0. A(˙g) ⊆ A(˙g) ⊆ (G)

(†)

(ii) Let G be a connected pro-Lie group. Then (∀N ∈ N (G)) G N = expG g N ⊆ A(˙g)N. In particular,

G ⊆



A(˙g)N ⊆ A(˙g).

N ∈N (G)

˙ = G ⊆ A(˙g). (iii) G ˙ Then A is a proto-Lie group by the Quotient TheoProof. (i) We set A = G/G. ˙ contains the commutator subgroup. Then also the rem 4.1 and it is abelian because G completion B of A is abelian. By the Strict Exactness Theorem 4.20 for L we have ˙ Since L(A) = L(B) and L(B) is abelian by 9.24. Also by 4.20, L(A) ∼ = L(G)/L(G). ˙ g/L(G) is abelian, containment (∗) follows. ˙ = expG L(G) ˙ ⊆ (G) ˙ 0 by 4.21 (i). This containment implies A(˙g) ⊆ A(L(G)) ˙ Since (G)0 is closed, assertion (†) is proved. (ii) Assume now that G is connected. Then for each N ∈ N (G) the Lie group G/N is connected and the image G N/N of the algebraic commutator subgroup G of G in G/N is the algebraic commutator subgroup (G/N ) of G/N . The commutator subgroup of a finite-dimensional connected Lie group is an analytic subgroup whose Lie algebra is the commutator subalgebra (for linear Lie groups see for instance [102, Theorem 5.60]; the proof works for Lie groups in place of linear Lie groups as well; or see [17, Chap. 3, §9, no 2, Corollaire de Proposition 4]). Thus G N/N = (G/N ) = expG/N L(G/N )  = (expG L(G) )N/N = expG g N/N ⊆ expG g˙ N/N =   A(˙g)N/N . Thus G ⊆ N ∈N (G) A(˙g)N ⊆ N ∈N (G) A(˙g)N = A(˙g). (iii) is now directly deduced from (ii). def

As an immediate consequence we obtain ˙ Theorem 9.26. For a connected pro-Lie group G, the closed commutator subgroup G is a closed analytic subgroup which agrees with the closure A(˙g) of the unique smallest analytic subgroup whose Lie algebra is the closed commutator subalgebra g˙ of the Lie algebra g = L(G) of G.

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Proof. This is a consequence of Lemma 9.25. For simply connected pro-Lie groups the situation is even clearer. Recall that for a simply connected pro-Lie group G we let N S(G) denote the filter basis of closed normal subgroups N such that G/N is a simply connected Lie group. Proposition 9.27. Let G be a simply connected pro-Lie group and g = L(G) its Lie algebra. Then ˙ is the minimal analytic subgroup A(˙g) with Lie algebra g˙ . If G is written (g) (i) G and (˙g) is naturally identified with a closed normal subgroup of (g), then ˙ = g˙ . ˙ (g) = (˙g) and L(G) (ii) For each N ∈ N S(G), the algebraic commutator subgroup G satisfies G N = ˙ ˙ is a closed normal subgroup of G. GN, and GN Proof. (i) We may assume that G = (g). The exact sequence 0 → g˙ → g → a → 0,

a = g/˙g

yields an exact sequence {1} → (˙g) → (g) → (a) → {1} where we may consider (˙g) ⊆ (g) and (a) ∼ = (g)/ (˙g): see 6.7, 6.8 and 6.9. Since L((a)) = a it follows from 9.24 that (a) is abelian, and thus ˙ (g) ⊆ (˙g).

(∗∗)

˙ Now 9.25(∗) and (∗∗) taken together prove that (g) = (g). ˙ Since L((˙g)) = g˙ , assertion (i) is proved. (ii) For each N ∈ N S(G) the image G N/N in G/N is the algebraic commutator subgroup (G/N ) of G/N. Since G/N is simply connected, G N/N is closed in G/N (see 9.21 above or [17, loc. cit.]). Thus G N is closed in G, and thus G N = G N = ˙ GN. It is time to look at some examples that illustrate what happens, typically. Example 9.28. The ground field will be the reals, Sl(2) = Sl(2, R) the group of all real 2 × 2 matrices of determinant 1, and sl(2) = sl(2, R) the Lie algebra of all real 2 × 2 matrices of trace 0. The Lie algebra of Sl(2) is sl(2).   def  = Sl(2), the universal coveringgroup Let H  of Sl(2) and let p : H → Sl(2) be 1 0 the universal covering morphism. Set E = 0 1 . The center Z(Sl(2)) is {E, −E} and def

C = p −1 Z(Sl(2)) is an infinite cyclic group with identity e and a generator c such that p(e) = E and p(c) = −E. The simply connected 4-dimensional reductive Lie  has center R × C. =R×H group G

The Commutator Subgroup

379

√ √ Let D be the subgroup generated by (1,√e) and ( 2, c), that is, D = {(m+n 2, cn ) : m, n ∈ Z}. The projection P = {(m + n 2, cn ) : m, n ∈ Z} into the first component def  The Lie algebra is g = R × sl(2). Set is dense in R. Now we define G = G/D.  πG : (g) → G h = {0} × sl(2). In terms of our current notation we have (g) = G,  . We note (g) = agrees with p, the commutator algebra g is h, and (h) = {0} × H   (g ). The unique analytic subgroup with Lie algebra h = g is A(h) = (h)D/D =

 P ×H = G . D

Then ih : (h) → G, ih (0, x) = (0, x)D induces a bijective morphism (h) → A(h), and the image is dense in G. Also G = A(h). The center Z(G) is (R × C)/D ∼ = R/Z = T, and /C ∼ G/Z(G) ∼ = Sl(2)/{±E} = PSl(2, R). =H Further, Z(G) ∩ A(h) = Z(G) ∩ G = P × C/D = ((P × {e}) + D)/D = (P × {e})/(P × {e} ∩ D) ∼ = P /Z. Let K and B be the subgroups of Sl(2) given by / ., r / ., s e cos t − sin t and B = : r, s ∈ . R K= :t ∈R 0 e−t sin t cos t  = p −1 B. Then C ⊆ K.  The maps (x, y)  → xy :  = p−1 (K) ∼ Set K = R and B ×B →H  and p|B : B  → B are diffeomorphisms. If we identify H  and R × B K then we have G homeomorphic with R×R×B √ {(m + n 2, n, E) : m, n ∈ Z}

which is diffeomorphic to T2 × R2 . The group T × PSl(2) is likewise homeomorphic to R×R×B Z × Z × {E} which is diffeomorphic to T2 × R2 . The example of this group G is still of low dimension, namely, 4, but shows that the commutator group G which is the analytic subgroup whose Lie algebra is the unique Levi complement of the radical may be nonclosed and dense, intersecting the center in a dense subgroup. The group T × PSl(2, R) has the same center and the same factor group modulo the center; the maximal compact subgroups are isomorphic to T2 in both cases, and their topological structures are the same, but the two groups are radically different in their global algebraic structure.

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Example 9.29. Let α : R2 → Gl(2, C) be the representation given by , 2π ir e 0 α(r, s) = . 0 e2π is  be the real Lie group C2 α R2 of dimension 6 with the multiplication Let G  is g = C2 × R2 with (v, w)(v  , w ) = (v + α(w)v  , w + w  ). The Lie algebra of G     where δ(r [(v, w), (v , w )] = (δ(w)v √ −δ(w )v, 0) √ 1 , r2 )(c1 , c2 ) = 2π i ·(r1 c1 , r2 , c2 ). 2 2   = (h) is a 5-dimensional Now let h = C ×R(1, 2) and H = C ×R·(1, 2). So H subgroup.  is D def = {(0, 0)} × Z2 , and accordingly, g is center-free. Set The center of G 2 2 ∼ D/D. Then H is the analytic subgroup A(h) of G  G = G/D = C  T and H = H with Lie algebra h. The center of G is trivial, and the commutator groups of G and H agree and equal (C2 × {(0, 0)})D/D ∼ = C2 . The general principle of this last example deserves to be generalized in the framework of pro-Lie groups.  = Example 9.30. Let C be any (additively written) compact abelian group and C  Hom(C, R/Z) its character group. Each character χ ∈ C gives a simple C-module Vχ ∼ = R2 with the module operation , , - , -, x x cos 2π χ (c) − sin 2π χ (c) 2π χ (c)·I x , c· = =e y y sin 2π χ (c) cos 2π χ (c) y   where I = 01 −1 0 and where we have written sin 2π(r + Z) as sin 2π r and so on. We consider the commutative diagram L(χ )

L(C) −−−−→ ⏐ ⏐ expC  C −−−−→ χ

R = L(T) ⏐ ⏐quot  T = R/Z

which may be expressed as χ (exp X) = (χ  X)(1) = L(χ )(X) + Z in formula terms. The derivative of t  → e2π χ(exp tX) v = e2π tL(X)v : R → Vχ at t = 0 is 2π L(X)v. Let us form the weakly complete module W = χ ∈C Vχ with componentwise operations c · (vχ )χ ∈G  = (χ (c)vχ )χ ∈C  and X∗(vχ )χ ∈C  = (2π L(χ )vχ )χ ∈C . We define G = W  C to be the semidirect product with multiplication given by (v, c)(w, d) = (v + c · w, c + d). Then g = W × L(C) with the Lie bracket [(v, X), (w, Y )] = (X∗w − Y ∗v, 0) is its Lie algebra. We have g = span{Xv : X ∈ L(C), v ∈ W } and ˙ = W × {0}. The g˙ = W × {0}; likewise G = span{c · w : c ∈ C, w ∈ W } and G group (g) is the semidirect product W L(C) with the multiplication (v, X)(w, Y ) = (v + X∗w, X + Y ) and πG : (G) → G is given by πG (v, X) = (v, expC X). Now we note that Ga = W × Ca is the minimal analytic subgroup of G with Lie algebra g. It is dense in G but not equal to G in general. If G is a group such that G ⊆ Z(G), we say that G is nilpotent of class 2.

The Commutator Subgroup

381

; Example 9.31. (i) Let V = Rn and C = 2 V the second exterior power of V and  let (v, w) → v ∧ w : V × V → C the canonical symplectic map. Then dim C = n2 and C = span im ∧. Let N = V ×C with the multiplication (v, c)(w, d) = (v +w, c+d + 1 2 v ∧w). Then N is a nilpotent group of class 2. The commutator comm((v, c), (w, d)) of two elements equals (0, v∧w). The commutator subgroup N  of N is {0}×C and this equals the center Z(N). Now let B be the Bohr compactification of C and η : C → B the Bohr compactification morphism. In the product N × B let D be the closed central def

subgroup {((0, −c), η(c)) : c ∈ C}. Set G = (N × C)/D. Then G is a locally compact nilpotent pro-Lie group of class 2 and f : N → G, f (v, c) = ((v, c), 0)D is an injective morphism of pro-Lie groups with dense image (N × {0})D/D = def

(N × f (C))/D. In particular, H = im f is a dense analytic subgroup. The commuta# tor subgroups H  and G agree, and equal (({0}×C)×{0})D/D = (({0}, c), f (c))D : $ ˙ ∼ c ∈ C . The center is Z(G) = ({0} × B)D/D = G = (G) = B. ; (ii) We let n ≥ 2 be a natural number and denote the group Rn × 2 Rn in (i) ; above by Nn . The set Sn = {x ∧ y : x, y ∈ Rn } is a variety in 2 Rn ; it is in fact n the image under the function (x, y)  → x ∧ y : R2n → R(2) (where we identified ; n Rn × Rn with R2n and 2 Rn with R(2) ). The derivative of this function at (u, v) is (x, y)  → x ∧ v + u ∧ y. The variety Sn + · · · + Sn is the image of the function ( )* + k-times n

(x1 , y1 , . . . , xk , yk )  → x1 ∧ y1 + · · · + xk ∧ yk : R2kn → R(2) ,

(∗)

whose derivative in the point (u1 , v1 , . . . , uk , vk ) is n (x1 , y1 , . . . , xk , yk )  → x1 ∧v1 +u1 ∧y1 +· · ·+xk ∧vk +uk ∧yk : R2kn → R(2) . (∗∗) n Thus if k < n−1 2 we have 2kn < 2 and the derivative (∗∗) fails to be surjective, whence ; the polynomial function (∗) fails to be surjective; therefore, if every element of 2 Rn is a sum of m elements x ∧ y then n−1 4 ≤ m. Therefore, the commutator subgroup Nn of the class 2 nilpotent group Nn contains an element an which is a product ; of no fewer than n−1 commutators. Notice that Nn = Z(Nn ) = {0} × 2 Rn . 4 ∞  Now let G = ∞ n=2 Nn . Then Z(G) = n=2 Z(Nn ). Set

comm(G, G) = {xyx −1 y −1 : x, y ∈ G}.  Then comm(G, G) = ∞ n=2 comm(Nn , Nn ) G = We identify

n

∞ 

comm(G, G)n =

n=1

m=1 Nm

∞ ∞  

comm(Nm , Nm )n .

n=1 m=2

with its canonical image in G so that ∞ 6 n=1

Nn =

n ∞   n=1 m=1

Nm ⊆ G.

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Now G =



 n=2 Nn

= Z(G) and

5∞

 n=2 Nn

⊆ G . Since an ∈ / comm(Nn , Nn )k for def

k < n−1 p, the element a = 4 , it is readily seen that for any given natural number ∞ p  ˙ (an )n=2,3,... ∈ G = G is not contained in comm(G, G) = n=2 comm(Nn , Nn )p ; ˙ \ G . therefore a ∈ G ; The vector space nn = Rn × 2 Rn together with the bracket [(x, v), (y, w)] = (0, x ∧ y) may be identified with the Lie algebra of Nn such that expNn : nn → Nn is none other thanthe identity function. Accordingly, G is a pro-Lie group with Lie algebra g = ∞ n=2 nn and the identity function as exponential function. The commutator algebra g is not closed and agrees as a topological space with G . The ˙ = G . Lie algebra g˙ = g agrees as a space with G Thus the algebraic commutator subgroup G of the simply connected pronilpotent pro-Lie group G is not an analytic subgroup. The Example 9.31 (ii) is a pro-Lie group analog of the example in [102, Exercise E6.6, following Corollary 6.12], of a compact, totally disconnected class 2 nilpotent group in which the algebraic commutator subgroup is not closed. Theorem on Commutator Subgroups of Dense Analytic Subgroups Theorem 9.32. Let G be a connected pro-Lie group and h a closed subalgebra of g = L(G). Then (i) L(A(h)) ˙ ⊆ h˙ ⊆ h. (ii) In particular, L(A(h))/h is abelian. (iii) The abstract group A(L(A(h)))/A(h) is abelian. Proof. (i) By the Closed Subgroup Theorem 3.35, A(h) is a pro-Lie group with a Lie algebra containing L(A(h))) = h. Since A(h) is connected, so is A(h). Hence there is no loss in generality if we assume A(h) = G, that is, A(h) is dense in G. From 9.19 (ii) we know that A(h) is normal in G, and from 9.19 (iii) that h is an ideal. Since h˙ is a characteristic ideal of h, that is, it is invariant under all continuous derivations, it is an ideal of g. Then by the Theorem on the Preservation of Embeddings by  5.9 we may ˙ and (h) are closed minimal analytic normal subgroups of (g) = G,  consider (h) and for the inclusion j : h → g there is a commutative diagram (j )

(h) −−−−→ ⏐ ⏐ ε A(h) −−−−→ incl

 (g) =G ⏐ ⏐π  G G.

with an algebraic and topological embedding (j ), and a surjective morphism ε such that incl  ε = ih . Let ˙ K = {g ∈ G : (∀X ∈ h) Ad(g)X − X ∈ h}.

383

The Commutator Subgroup

Since g  → Ad(g)X − X : G → g is continuous by 2.27, K is closed. It is clearly a subgroup. Clearly Ad(K)h ⊆ h; thus Ad(k)h ⊆ h and Ad(k −1 )h ⊆ h, that is, h ⊆ A(k)h) for all k ∈ K. Hence Ad(k)h = h for all k ∈ K. If 'Y ∈ h, then & by 8.1(1) we have Ad(expG Y )X − X = ead Y X − X = [Y, X] + 2!1 · Y, [Y, X] ∈ h˙ since (ad Y )n X ∈ [h, h] by induction. Therefore expG Y ∈ K for all Y ∈ h, and thus A(h) ⊆ K. By 9.12, G = A(h) ⊆ K, and thus K = G. Hence ˙ Ad(g)X − X ∈ h.

(∀g ∈ G, X ∈ h) In the process we saw

˙ ead Y X − X ∈ h,

(∀Y ∈ g, X ∈ h) for t = 0 we see 1 · (et·ad Y X − X) − [Y, X] = t t

,

1 t · [Y, [Y, X]] + · (ad Y )3 X + · · · 2! 3!

.

Modulo any cofinite-dimensional ideal of g the right side converges to 0, and so the left side converges to zero. Since h˙ is closed, we conclude [Y, X] ∈ h˙ for all Y ∈ g, X ∈ h. We now define ˙ c = {Y ∈ g : (∀X ∈ g) [Y, X] ∈ h}. def

This is a closed subalgebra which, after the preceding, contains h. Therefore the analytic subgroup A(c) contains A(h) and thus is dense. In particular, c is an ideal of g. Let Y ∈ c. Then (∀X ∈ g)

ead Y X − X = [Y, X] +

& 1 ' ˙ · Y, [Y, X] + · · · ∈ h. 2!

Since ead Y = Ad(expG Y ) and expG c = G we deduce ˙ Ad(g)X − X ∈ h.

(∀g ∈ G, X ∈ g)

Now we retrace the steps and let Y ∈ g, g = expG Y and obtain (∀X, Y ∈ g)

˙ ead Y X − X ∈ h.

We have seen before that this implies (∀X, Y ∈ g)

˙ [Y, X] ∈ h.

˙ and so g˙ = g ⊆ h˙ ⊆ h. And this in turn immediately implies g ⊆ h, (ii) Accordingly, g/h is a homomorphic image of the abelian pro-Lie algebra g/˙g and is therefore abelian. ˙ ⊆ (h). From the exact (iii) From (i) we have g˙ ⊆ h and so (g) ˙ = (˙g) ⊆ (h) sequence {0} → h → g → g/h → {0},

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

by the Strict Exactness Theorem of  in 6.7, we obtain a strict exact sequence {1} → (h) → (g) → (g/h) → {1} → 0, that is

 (h) = (g)/ (h) ∼ G/ = (g/h). ∼  (h) is By (ii) above L((g/h)) = g/h is abelian, and by 9.24 this implies that G/ abelian. We have a commutative diagram (h) −−−→ ⏐ ⏐  A(h) −−−→

 (g) = G ⏐ ⏐π  G A(g) ⊆ G.

 → G has a dense image A(g) and by Proposition 9.10 (i) The morphism πG : G  (h) maps (h) onto A(h). The image of πG is A(g), and the commutativity of G/ implies the commutativity of A(g)/A(h). We do not know whether the abstract group A(h)/A(h) is abelian. Corollary 9.33. Let H be any analytic subgroup of a pro-Lie group. Then L(H ) ˙ = L(H ) = L(H ) = L(H ) ˙ . Proof. Set h = L(H ). Then A(h) ⊆ H by 9.11. Set C = H . Then C = A(h) by ˙ Trivially 9.11. Define c = L(C). From the preceding Theorem 9.32 we get c ⊆ h. h˙ ⊆ c˙. Thus h˙ = c˙ follows. Since h = L(H ) and L(H ) = L(C) = c, the corollary is proved. The statement 9.32 (iii) exhibited consequences on the group level saying that for a dense analytic subgroup H of a pro-Lie group G the group A(g)/A(h) is abelian, that is, A(g) ⊆ A(h) ⊆ H . The group A(g) is dense in G, but unless we know that A(g) = G (which is the case if G is a Lie group), we do not know at this stage, whether G ⊆ H , that is whether G/H is abelian. What we can say with ease is the following. Corollary 9.34. Let G be a pro-Lie group and H a dense analytic subgroup with Lie algebra h, then for any N ∈ N (G), the algebraic commutator subgroup G of G is ˙ contained in A(h)N ⊆ H N. As a consequence  ˙ ˙ G ⊆ A(h)N ⊆ A(h), N ∈N (G)

and

˙ ⊆ A(˙g) = G. ˙ A(˙g) = A(h)

˙ = A(˙g). Now we simply apply 9.26 Proof. From 9.33 we obtain h˙ = g˙ and so A(h) and 9.25. ˙  = g¨ . Corollary 9.35. Let G be a pro-Lie group with Lie algebra g. Then L(G)

Finite-Dimensional Connected Pro-Lie Groups

385

˙ and the analytic subgroup Proof. We apply Corollary 9.33 to the pro-Lie group G def ˙ by Theorem 9.26. On H = A(˙g). With these definitions we have H = A(˙g) = G the other hand, h = L(H ) = L(A(˙g)) = g˙ by Definition 9.12. Now 9.33 yields ˙  = L(H ) = L(H ) = (˙g) ˙ = g¨ . L(G)

Finite-Dimensional Connected Pro-Lie Groups A clear conceptual understanding of the theory of finite-dimensional pro-Lie groups, to be defined shortly, requires some preparations. As a first step, a very simple group theoretical lemma of the diagram type is helpful. We are dealing with a morphism of topological groups π : T → G which factors through a group G1 via morphisms ϕ : T → G1 and ρ : G1 → G so that π = ρ  ϕ. def

We define A = ϕ(T ) = im ϕ and  = ρ −1 (1) = ker ρ and denote the corestriction of ϕ by ϕ1 : T → A. Then we have the following diagram pr 

× ⏐ A −−−−→ pr A ⏐  incl A −−−−→  ⏐ ϕ1 ⏐ T −−−−→

 ⏐ ⏐ incl G⏐1 ⏐ρ  G.

π

Since  is normal, the set A is a subgroup and T acts automorphically on  by g · c = ϕ(g)cϕ(g)−1 ; this may be expressed by saying that we have a morphism α : T → Aut() from T into the automorphism group of T given by α(g)(c) = g · c. The function μ : α T → G1 , μ(c, g) = cϕ(g) is a morphism of topological groups. def

Let P = ker π be the kernel of π . Lemma 9.36. (i) Assume that π is surjective. Then A = G1 and μ is surjective. The function δ : P →  α T , δ(g) = (ϕ(g)−1 , g) is an isomorphism onto ker μ and there is a strict exact sequence δ

μ

1 → P −−−→  α T −−−→ G1 → 1. (ii) The function d : P → , d(g) = ϕ(g) is a well-defined morphism mapping P surjectively onto  ∩ A. If ϕ(P ) is dense in , then A is dense in G1 . (iii) Assume in addition that ρ has a local cross section. Then μ :  α T → G1 is open and the induced morphism ( α T )/ ker μ → G1 is an isomorphism. Under these conditions, if A is dense in G1 , then ϕ(P ) =  ∩ A is dense in .

386

9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Proof. (i) Let g1 ∈ G1 . Since π is surjective, there is a g ∈ T such that ρ(g1 ) = def

π(g) = ρ(ϕ(g)). Thus c = g1 ϕ(g)−1 is in the kernel of ρ, that is, in , and g1 = cϕ(g) ∈ A. Thus A = G1 and μ is surjective. An element (c, g) is in the kernel of μ iff cϕ(g) = 1, that is iff c = ϕ(g)−1 ∈ ∩A. This relation implies π(g) = ρ(ϕ(g)) = ρ(g)−1 = 1, that is, g ∈ ker π = P . Conversely, if g ∈ P , then ϕ(g) ∈ ker ρ =  and μ(ϕ(g)−1 , g) = 1, and so δ(g) ∈ ker μ. The inverse of δ is given by δ −1 (c, g) = g, that is δ −1 = pr T | ker μ. (ii) If 1 = π(g) = ρ(ϕ(g)), then ϕ(g) ∈ ker ρ = , and so d maps P into  ∩ A. Conversely, if g1 ∈  ∩ A, then there is a g ∈ T such that ϕ(g) = g1 and ρ(g1 ) = 1. This implies g ∈ ker π = P and g1 = d(g). Now suppose that im d = ϕ(P ) =  ∩ A is dense in . Abbreviate ker μ = {(ϕ(g)−1 , g) : g ∈ P } by D. Then by (i) there is )D ϕ(P )×T = ({1}×T is mapped onto A. a bijective morphism ×T D → G1 by which D D Since ϕ(P ) × T is dense in  × T , it follows that A is dense in G1 . (iii) Now assume that there is an open identity neighborhood W of G and a continuous function σ : W → G1 such that ρ(σ (w)) = w for all w ∈ W . Then f :  × W → ρ −1 (W ), f (c, w) = cσ (w) is a homeomorphism with inverse g1  → (g1 σ (ρ(g1 ))−1 , ρ(g1 )), that is, ρ defines a locally trivial fibration of G1 over G with fiber . We define a continuous function κ : W →  by κ(w) = σ (π(w))ϕ(w)−1 ; indeed ρ(κ(w)) = ρσ π(w) · ρϕ(w)−1 = π(w)π(w)−1 = 1; so κ is well defined and (∀w ∈ W )

κ(w)ϕ(w) = σ π(w).

Now let U be an open identity neighborhood of  and V an open identity neighborhood of T such that π(V ) ⊆ W . Now we let U1 be an identity neighborhood of  and V1 an identity neighborhood of  contained V such that U1 κ(V1 ) ⊆ U . Then u1 ∈ U1 and v1 ∈ V1 implies (∀(u1 , v1 ) ∈ U1 × V1 )

u1 σ π(v1 ) = u1 κ(v1 )ϕ(v1 ) ∈ U ϕ(V ) = μ(U × V ).

But U1 σ (V1 ) is a product neighborhood of the identity in the trivially fibered group G1 , and this shows that μ is an open morphism. As a consequence μ induces an isomorphism of topological groups (c, g)D  → cϕ(g) :

×T → G1 D

(×{1})D )D isomorphically to  and ϕ(TD)×T = ({1}×T isomorphimapping ×P D = D D ϕ(T )×T ×T cally to A. So if A is dense in G1 then D is dense in D , and so ϕ(T ) × T is dense in  × T . This finally implies that ϕ(T ) is dense in .

The essence of the lemma is that we can, under the given circumstances, approximate the group G1 by a product  × T modulo a group that is isomorphic to P = ker π . The subgroup  is always closed and normal in G1 and will, in the applications, have many more special properties, while the group = ϕ(T )A is not closed in general. This is illustrated in the first application we are exposing now.

Finite-Dimensional Connected Pro-Lie Groups

387

 → G its universal Proposition 9.37. Let G be a connected pro-Lie group and πG : G morphism with the prodiscrete Poincaré group P (G) as kernel. Assume that (i) there is a morphism ρ : G1 → G of connected pro-Lie groups inducing an isomorphism L(ρ) : L(G1 ) → L(G), and that  → G is surjective. (ii) πG : G Let  denote the kernel of ρ. Then  is a central prodiscrete subgroup of G1 and  → G1 such that ρ  ϕ = πG . The image (a) there is a unique morphism ϕ : G of ϕ is the minimal analytic subgroup A(L(G1 ), G1 ) of G1 which is dense in G1 .  → G1 , μ(c, g) = cϕ(g) is a surjective morphism The function μ :  × G whose kernel is isomorphic to the Poincaré group P (G) under the morphism  δ(g) = (ϕ(g)−1 , g) so that there is a strict exact sequence δ : P (G) →  × G, of pro-Lie groups δ

μ

1 → P (G) −−−→  × G −−−→ G1 → 1. def Accordingly, if we write D = im δ ∼ = P (G), then there is a bijective morphism of proto-Lie groups

β:

×G → G1 , D

β(c, g)D = cϕ(g).

(b) The function d : P (G) → , d(g) = ϕ(g) maps P (G) surjectively onto  ∩ im ϕ. Assume that, in addition, the following hypothesis is also satisfied: (iii) The morphism ρ : G1 → G has local cross sections. Then (c) μ is an open morphism and β is an isomorphism of topological groups and  = ϕ(P (G)). Hypothesis (iii) and hence conclusion (c) is implied by (iv) ρ is a quotient morphism and dim L(G) < ∞.  = (g). By hypothesis on ρ : G1 → G, Proof. Let g = L(G). Then by definition, G we know that L(ρ) : L(G1 ) → L(G) is an isomorphism of pro-Lie algebras. We now invoke the universal property of  in Theorem 6.5 in order to get a unique

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

 → G1 such that L(ϕ)  ηg = L(ρ)−1 : morphism ϕ : g = G proLieAlg ηg

g ⏐ ⏐ −1 L(ρ)  L(G1 )

proLieGr

−−−−→

 L(⏐G) ⏐L(ϕ) 

−−−−→

L(G1 )

idL(G1 )

 (g)⏐= G ⏐ϕ  G1

 → G such that Moreover, ηG = L(πG )−1 . We have two morphisms πG , ρ  ϕ : G −1 L(πG ) = ηg = L(ρ)  L(ϕ) = L(ρ  ϕ). Therefore, πG  expG˜ = expG L(πG ) =  = E(G)  = exp ˜ L(G)  by Corollary 4.22 (i), expG L(ρϕ) = ρϕ  expG˜ . Since G G we conclude πG = ρ  ϕ. By the universal property of  there are unique morphisms  ϕ and ρ  fitting correctly into the following diagram: ϕ  −−− <1 G −→ G ⏐ ⏐ ⏐ ⏐ π idG G1    −−−−→ G1 G ϕ

ρ 

−−−−→ −−−−→ ρ

 G ⏐ ⏐π  G G.

From ρ  ϕ = πG we conclude ρ   ϕ = idG  is a retraction of topologi . Thus ρ <1 is a semidirect product of ker ρ and a group isomorphic cal groups showing that G  However, all groups in the diagram induce isomorphisms on the Lie algebra to G. level, whence ker ρ, by the Strict Exactness Theorem and Proposition 4.22, is totally <1 is connected, and so ker ρ = {1}. Thus ρ  is a monic retracdisconnected. Yet G tion and therefore an isomorphism. It follows that  ϕ and ρ  are inverses of each other. From ϕ = πG1   ϕ we conclude that im ϕ = im πG1 = A(L(G1 ), G1 ) according to the paragraph preceding 9.10. The implication (iv) ⇒ (iii) is a consequence of Corollary 4.22 (iv). All other conclusions of the proposition are immediate from Lemma 9.36 Let us now see what happens if we know more about P (G) and, consequently, about . Such additional information may arise, for instance, from the additional information that G is a Lie group in Proposition 9.37. Corollary 9.38. Let G be a connected Lie group, ρ : G1 → G a morphism of connected pro-Lie groups and assume that hypotheses (i), (ii) of Proposition 9.37 and the following condition are satisfied: (iv ) ρ is a quotient morphism. Then all conclusions of Proposition 9.37 hold and in addition the following conclusions hold as well:

Finite-Dimensional Connected Pro-Lie Groups

389

(d) comp() is a compact metric totally disconnected characteristic subgroup of , and there is a subgroup f isomorphic to Zn for a nonnegative integer n such that (c, f )  → cf : comp() × f →  is an isomorphism of topological groups. The Poincaré group P (G) contains a finitely generated subgroup F1 and a free group F2 of rank n such that (f1 , f2 )  → f1 f2 : F1 × F2 → P (G) is an isomorphism of finitely generated abelian groups and such that ϕ(F1 ) is dense in comp() and ϕ|F2 : F2 → f is an isomorphism: pr F1

pr F2

pr comp()

pr f

F⏐1 ←−−−− P (G) −−−−→ ⏐ ⏐ ⏐ ϕ|F1  ϕ|P (G) comp() ←−−−−  −−−−→

F⏐2 ⏐ϕ|F  2 f .

def   and L  may be identified in such a fashion that the (e) Set L = G/F 1 . Then G  → L is πL . There is a covering morphism κ : L → G quotient morphism G such that πG = κ  πL , and ker κ ∼ = Zn . The Poincaré group P (L) of L is isomorphic to F1 . (f) The morphism ρ : G1 → G lifts to a morphism ρ1 : G1 → L with kernel ker ρ1 = comp(). Define D1 = {(ϕ(g)−1 , g) : g ∈ F1 }; then there is a commutative diagram of morphisms

 G ⏐ ⏐ ϕ quot  comp() ⏐ × G −−−−→ ⏐ incl  ×G −−−−→ quot

 comp()×G D ⏐1

⏐ δ

 ×G D

idG˜

−−−−→

β1

ρ1

 G ⏐ ⏐π  L

−−−−→ G⏐1 −−−−→ ⏐id  G1

L ⏐ ⏐κ 

−−−−→ G1

G,

β

−−−−→ ρ

(∗)

where β1 (c, g)D1 ) = cϕ(g), and δ((c, g)D1 ) = (c, g)D; the morphisms β1 , δ, and β are isomorphisms. All groups in the diagram are locally compact metric and all maps induce isomorphisms on the Lie algebra level. (g) The following statements are equivalent: – comp() is discrete, that is, a finite abelian group, – F1 /(F1 ∩ ker ϕ) is finite, and – G1 is a Lie group, and if these conditions are satisfied, ρ and ρ1 are covering morphisms. These conditions hold whenever the Poincaré group P (L) of L is finite.

390

9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Proof. (d) Since G is a Lie group, P (G) is finitely generated. By Proposition 9.37 (c), we know that  is a prodiscrete group in which the finitely generated abelian group ϕ(P (G)) is dense. Then by the Compact Generation Theorem 5.32 (iv), comp() is compact metric and  ∼ = comp() × Zp . By [102, Proposition 12.28], comp(G) is metric. Since ϕ(P (G)) is dense in  by 9.37 and comp() is open by what we just saw, we have  = comp()ϕ(P (G)). Since ϕ(P (G))/(ϕ(P (G)) ∩ comp()) is free, there is a free subgroup F2 of P (G) mapping isomorphically onto a discrete free subgroup f of , and F2 is isomorphic to Zn such that  is the direct product of comp() and f . Set F1 = ϕ −1 (comp()). We note ϕ(ϕ −1 (f ) = f ∼ = Zp . Now πG (F1 F2 ) = ρϕ(F1 F2 ) = ρ() = {1} since  = ker ρ by definition. Thus F1 F2 ⊆ ker πG = P (G). If g ∈ P (G) then ϕ(g) ∈  = comp() · f , whence g ∈ F1 F2 . If g ∈ F1 ∩F2 then ϕ(g) ∈ comp()∩1 = {1} and so g ∈ ker ϕ ∩F2 = {1}, since ϕ|F2 is injective. Because ϕ(F1 )f = ϕ(F1 )ϕ(F2 ) = ϕ(F ) is dense in  = comp()f and the last product is direct, it follows that ϕ(F1 ) is dense in comp(). def  (e) After the definition of the Lie group L = G/F 1 , the assertions are all straightforward. (f) Consider the commutative diagram of maps dense

F⏐1 −−−−→ ⏐ incl  −−−ϕ−→ G ⏐ ⏐ q  1 −−−−→ F = G/F ψ

comp() ⏐ ⏐open incl  G⏐1 ⏐q 1 G1 / comp(),

where ψ(gF1 ) = ϕ(g) comp(). The map ϕ, and the quotient maps q and q1 induce isomorphisms of (finite-dimensional) Lie algebras since F1 is discrete and comp() is totally disconnected. Thus L(ψ) is an isomorphism as well. We notice that G ∼ = G1 / since ρ is a quotient morphism and  = ker ρ. Further ∼ G1 / = (G1 / comp())/(/ comp()) and recall that / comp() is isomorphic to the discrete group Zn . Hence G1 / comp() is a connected Lie group. Since L(ψ) is an isomorphism it now follows that ψ is open surjective. From F1 = ϕ −1 (comp() in (d) above we conclude that ψ is injective. Therefore ψ : L → G1 / comp() is an isomorphism. There is a quotient morphism p : G1 / comp(G) → G such that ρ = p  q1 and we recall πG = κ  πL so that we have a commutative diagram  −−−ϕ−→ G⏐1 G ⏐ ⏐ ⏐q πL  1 ψ L ⏐ −−−−→ G1 / comp() ⏐ ⏐ ⏐p κ  G −−−−→ G. idG

Finite-Dimensional Connected Pro-Lie Groups

391

def

Therefore, ρ1 = ψ −1  q1 : G1 → L is the required lifting of ρ with kernel comp(). Now Proposition 9.37 applied to

G1

 G ⏐ ⏐π  L ρ1 −−−−→ L

G1

 G ⏐ ⏐π  G −−−−→ G

in place of

ρ

shows that β1 is an isomorphism, and it is already known from 9.37 that β is an isomorphism. The commutativity of diagram (∗) therefore shows that δ is an isomorphism as well. (This can easily be verified directly.)  L, and G are Lie groups, Since  was seen to be locally compact metric and G, all groups in sight are locally compact metric and it is straightforward to see that all morphisms in diagram (∗) induce isomorphisms on the Lie algebra level. (g) Since comp() = {1} = ϕ(F1 ), the groups comp() and ϕ(F1 ) ∼ = F1 /(F1 ∩  is a Lie group iff ker ϕ) are finite at the same time. This occurs iff comp() × G  comp()×G is a Lie group (since D1 is discrete) iff G1 is a Lie group, in which case ρ D1 and ρ1 are covering morphisms. These conditions are implied by the finiteness of the Poincaré group P (L) ∼ = π1 (L). = F1 ∼ The gist of this body of information contained in the diagram

G1

 G ⏐ ⏐π  G −−−−→ G ρ

if G is a Lie group is that we can, more or less, replace G by a covering group L and assume without great loss of generality, that the group , which represents the obstruction to G1 being a Lie group is compact; it is always totally disconnected  times this metric and second countable. The group G1 is locally isomorphic to G compact totally disconnected abelian group in which some finitely generated subgroup (indeed a homomorphic image of the Poincaré group of L is dense. Now we exploit this information for the general structure theory of connected proLie groups. Theorem 9.39. Let H be a connected pro-Lie group and N an arbitrary member of the filter basis N (H ). Then there is a characteristic subgroup N1 ∈ N (H ) that is def open in N such that the connected Lie group L = H /N1 and its universal covering  → L, together with the quotient map (ρ1 )N : H /N0 → L satisfy the following πL : L conditions:

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

 → H /N0 such that πL = ρ  ϕ, (∗) There is a morphism ϕ : L (∗∗) N1 /N0 = ker(ρ1 )N = ϕ(P (L)) = comp(H /N0 ) is a compact metric totally disconnected central subgroup of H /N0 . The group (P (L)/(P (L)∩ker ϕ) is finite iff N0 has finite index in N1 iff N0 ∈ N (H ). This is the case if P (L) is finite and this is the case if P (H /N ) = ker πH /N is finite. def (∗∗∗) Let D1 = {(ϕ(g)−1 , g) : g ∈ P (L)} ∼ = P (L); then

(nN0 , g)D → nN0 · ϕ(g) :

 N1 /N0 × L → H /N0 D1

 is a well-defined isomorphism of locally compact metric groups and N/N1 × L,  N/N0 × L and H /N0 are all locally isomorphic.  (†) The subgroup (ϕ(P (L))× H /N )/D1 is isomorphic to the minimal analytic subgroup A(L(H /N0 ), H /N0 ) of H /N0 with Lie algebra L(H /N0 ) ∼ = L(H /N ). Proof. Let H be a connected pro-Lie group and N ∈ N (H ). Then H /N0 is a pro-Lie group by Theorem 4.28 (i), and N is a pro-Lie group by Theorem 3.35. The strict exact sequence 1 → N/N0 → H /N0 → H /N → 1 gives rise to a strict exact sequence of pro-Lie algebras 0 → L(N/N0 ) → L(H /N0 ) → L(H /N ) → 0 by Theorem 4.20. Since N/N0 is a totally disconnected pro-Lie group by the Closed Subgroup Theorem 3.35, it is prodiscrete and satisfies L(N/N0 ) = {0} by Proposition 4.23. Thus, if ρN : H /N0 → H /N denotes the quotient morphism, then L(ρN ) : L(H /N0 ) → L(H /N ) is an isomorphism of pro-Lie algebras, and since H /N is a Lie group because N ∈ N (H ), we have dim L(H /N0 ) = dim L(H /N ) < ∞. Now the hypotheses (i), (ii) of 9.37 and (iv ) of 9.38 are satisfied and therefore all conclusions of Proposition 9.37 and Corollary 9.38 apply to the universal covering  morphism πH /N : H /N → H /N with its discrete finitely generated Poincaré group def

P (H /N), and to the quotient map ρN : G1 = H /N0 → H /N = G. In particular, the quotient morphism ρN : H /N0 → H /N factors through some Lie group quotient H /N1 , N1 ∈ N (H ), in the form (ρ1 )N

κN

H /N0 −−−−→ H /N1 −−−−→ H /N. The situation is summarized in the following diagram, taken from 9.38 and being

393

Finite-Dimensional Connected Pro-Lie Groups

appropriately renamed:  H /N ⏐ ⏐ ϕN ⏐  quot

 comp(N/N0 ) × H /N ⏐ ⏐ incl⏐ 

−−−−→

 H /N0 × H /N

−−−−→ quot

 comp(N/N0 )×H /N D ⏐1

β1

idH  /N

 H /N ⏐ ⏐ πH /N1 ⏐ 

−−−−→

(ρ1 )N

δ⏐ 

⏐

−−−→ H /N0 ⏐ ⏐ idH /N0 ⏐ 

−−−−→ H /N1 ⏐ ⏐ κN ⏐ 

 H /N0 ×H /N D

−−−→ H /N0

−−−−→ H /N.

β

ρN

In particular,  comp(N/N0 ) × H /N → H /N0 , D1 D1 = {(ϕ(g)−1 , g) : g ∈ P (H /N1 )} ∼ = P (H /N1 ),

(nN0 , g)  → nN0 · ϕN (g) :

is an isomorphism. Note that N1 is the open characteristic subgroup of N containing N0 such that N1 /N0 = comp(N/N0 ), and abbreviate this group by . Then N1 ∈ N (H ) and H /N1 is a covering Lie group of H /N while N1 /N0 is a compact metric totally disconnected central subgroup of H /N0 . The conditions |P (H /N1 )/(P (H /N1 ) ∩ ker ϕ)| < ∞ and |N1 /N0 | < ∞ are equivalent and these in turn are equivalent to N1 /N0 and thus H /N0 being a Lie group, which means N0 ∈ N (H ). This the case if P (G/N1 ) is finite and this is the case if P (H /N) = ker πH /N is finite. d is isomorphic to Q 2ℵ0 and thus has The universal solenoidal compact group R continuum weight and therefore is not metric. For the conclusion that H /N0 is metric it is therefore essential that we have that N/N0 is totally disconnected. This theorem provides a useful sufficient condition for the nonexistence of proper dense analytic subgroups having the full Lie algebra as their Lie algebra. Given a pro-Lie group G, recall from 9.22 (iv) that A(g, G) = E(G) = expG g is the unique minimal analytic subgroup having g as its Lie algebra. The following result will be applied in Chapter 11 in important considerations of the structure theory of pro-Lie groups. (See Theorem 11.27.) Proposition 9.40. Let G be a connected pro-Lie group with the following property: (•) The filter basis N (G) has a cofinal subset F such that N/N0 is finite for all N ∈F. Then G = A(g, G). Proof. For N ∈ N (G) let qN : G → G/N0 be the quotient (that is, limit map when G is considered as limN ∈N (G) G/N0 ). Since qN is surjective, by 4.22 (iii), we have

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

qN (A(g, G)) = A(L(G/N0 ), G/N0 ). In the terminology of 9.39 we have an isomorphism  N1 /N0 × G/N → G/N0 D1 for a suitable compact metric totally disconnected central subgroup N1 /N0 of G/N0  for some open subgroup N1 of N and for a discrete subgroup D1 of N1 /N0 × G/N that is isomorphic to the Poincaré group P (G/N1 ). We have  ϕ(G/N1 ) × G/N A(L(G/N0 ), G/N0 ) ∼ = D ∼ 1 → G/N0 . for the morphism ϕ : G/N = G/N Now, by way of contraposition, we suppose that A(g, G) = G. Then we find an M ∈ N (G) such that N ∈ N (G), N ⊆ M implies G/N0 = qN (A(g, G)) = A(L(G/N0 ), G/N0 ). We use the notation of the proof of 9.39 and conclude   N1 /N0 × G/N ϕ(G/N1 ) × G/N = , D1 D1  = N1 /N2 × G/N  and so ϕ(P (G/N1 )) = N1 /N0 . By that is ϕ(P (G/N1 )) × G/N Theorem 9.38, this signifies that N1 /N0 is infinite. However this implies that N/N0 has a nonsingleton compact totally disconnected central subgroup and so the negation of (•) holds. Condition (•) can, of course, be replaced by (•) The filter basis N (G) has a cofinal subset F such that N0 ∈ N (G) for N ∈ F . One might be led to believe that in connected pro-Lie groups, the nonexistence of compact central subgroups might imply the nonexistence of proper dense analytic subgroups having the full Lie algebra as their Lie algebra. This is not the case. In order to have an illustrative example, let us prove a general lemma which is of independent interest because it confirms the general impression that the category of pro-Lie groups is very rich. If a group K acts linearly on a vector space V , then the set V1 = {v ∈ V : G·v = {v}} of fixed vectors is a submodule as is any of its vector subspaces. The action and the G-module are said to be fixed point-free if V1 = {0}. If V is a nonzero irreducible G-module then it is certainly fixed point free. The Center-Free Embedding Lemma Theorem 9.41. Let K be any pro-Lie group possessing enough finite-dimensional fixed point-free representations to separate the points. This is the case for all compact groups K. Then there is a center-free pro-Lie group G with a normal subgroup V such that G/V ∼ = K.

Finite-Dimensional Connected Pro-Lie Groups

395

Proof. Let πj : K → Gl(Rn(j ) ), j ∈ J be a family of finite-dimensional representan(j ) for all j ) implies tions of K such that  πj (g)n(j=) 1 (the identity automorphism of R and define a continuous automorphic action of K on V g = 1. Let V = j ∈J R by g · (vj )j ∈J = (πj (g)(vj ))j ∈J making V a K-module and giving us a morphism def

π : K → Aut(V ) by π(g)(v) = g · v. We form the semidirect product G = V π K with the multiplication (v, g)(w, h) = (v + g · w, gh). This is a completetopological group.  Now let F be an arbitrary finite subset of J and define VF = j ∈J Vj , WF = j ∈J Wj , where " " Rn(j ) , if j ∈ F , {0}, if j ∈ F , Wj = Vj = n(j ) R if j ∈ J \ F . {0}, if j ∈ J \ F , Then V = VF ⊕ WF and VF and WF are K-submodules. Now Gl(Rn(j ) ) is a Lie group for all j ; since lim N (K) = 1 in K we find an MF such that πj (MF ) = {1} for all j ∈ F . For any N ⊆ M, consequently, G/M acts on VF via gN ·v = g ·v, unambiguously. Now let N ∈ N (K). We find an MF,M ∈ N (K) such that MF,N ⊆ MF ∩ N. Then WF × M is a closed normal subgroup of G and G/(WF ×MF,N ) ∼ = VF G/MF,N is a Lie group. It follows that G is a pro-Lie group. If (v, g) ∈ Z(G), then (v, g) = (w, h)(v, g)(w, h)−1 = (w+h·v, hg)(−h−1 ·w, h−1 ) = (w − hgh−1 · w + h · v, hgh−1 ), that is w − hgh−1 · w = v − h · v and hg = gh for all w ∈ V and h ∈ K. Taking w = 0 we see that v is a fixed vector of the K-module V . Since the representations πj are fixed point-free, v = 0 follows. Secondly, taking h = 1 we see that πj (g) = 1 for all j , that is g = 1 since the family of representations πj separates the points. Thus (v, g) = (0, 1) and we showed that G is center-free. For the following special case we refer to [102], Chapters 7 and 8, notably to Theorem 8.30 as concerns the identity arc component of a compact abelian group and Example 9.30 above of which the following consequence of 9.41 is just a reformulation, justified by the significance of the example. Example 9.42 (= 9.30). Let K be any compact connected abelian group. Then each  determines an irreducible representation πχ : K → C ∼ character χ ∈ K = R2 . The construction of 9.41 provides us with a center-free metabelian pro-Lie group G =   R), and the image exp k CK π K. The Lie algebra k of K is isomorphic to Hom(K, of the exponential function is the dense proper analytic subgroup A(k, K), which is  exactly the arc component Ka of K and Ga = CK π |Ka Ka is the unique dense subgroup A(g, G) of G with Lie algebra g.  Z). Thus The abstract quotient group π0 (K) = K/K0 is isomorphic to Ext(K, whenever this group is nonzero, the analytic subgroup A(k, K) is proper and dense. It then follows that A(g, G) is a proper subgroup as well. All nontrivial Lie group homomorphic images G/N of G are of the form R2p × Tq with q > 0 and thus have an infinite Poincaré group. d of the discrete additive group A good special case is the character group K = Q  of K may be of rational numbers. Then by Pontryagin Duality the character group K

396

9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

identified with Qd . Here for each N ∈ N (G) the factor group G/N is a circle group and so P (G/N) ∼ = Z while G/N0 is of the form R2p × K and thus has a nontrivial center. These examples show that much improvement of Proposition 9.39 cannot be expected. On several occasions before we noted that the category of compact connected abelian groups K provides us with the prototype of pro-Lie groups with a unique minimal analytic subgroup A(k, K) having the same Lie algebra k as the group K itself, but being a proper dense subgroup, namely the arc component Ka = exp k of  Z) = {0} (see [102, Chapter 8, Theorem 8.30]). Since the identity whenever Ext(K, d is a good special case. In Example 9.42 we saw Ext(Q, Z) = {0} the group K = Q that the Center-Free Embedding Lemma 9.41 permits us to create for each such group K a center-free metabelian group G = V π K with a dense proper analytic group A(g, G). On the other hand we also know that inside a commutative pro-Lie group G, by the Vector Group Splitting Lemma 5.12 for Connected Abelian Pro-Lie Groups, it is the maximal compact subgroup comp(G) alone that is responsible for possible dense proper analytic subgroups A(g, G). For g = L(V ) ⊕ k with k = L(comp(G)), and A(g, G) contains V = L(expG (L(V )) and thus is of the form A(g, G) = V ⊕ A(k, comp(G)). In a provisional definition we shall now lay down when we call a pro-Lie group finite-dimensional: Definition 9.43. A pro-Lie group G will be called finite-dimensional if the vector space dimension of L(G) is finite. We shall argue later that this definition is in agreement with topological concepts of dimension. (See also [102, Scholium 9.54] and [103].) For the time being, the present definition will be perfectly adequate. Let us merely observe that from the results of Chapter 8 such as Theorems 8.13 and 8.14, notably Theorem 8.14 (v), it follows that a simply connected pro-Lie group that is infinite-dimensional by Definition 9.43 is certainly infinite-dimensional with respect to all standard topological definitions of dimension (compare [102, Scholium 8.25ff]). Now we can apply Theorem 9.39 to prove quickly the following theorem on finitedimensional connected pro-Lie groups: Theorem 9.44. Let H be a finite-dimensional connected pro-Lie group with Lie algebra h. Then H is locally compact metric, and there is a compact metric totally def disconnected member  ∈ N (H ) such that the Lie group G = H / and the quotient morphism ρ : H → G satisfy the following conditions:  → H such that πG = ρ  ϕ. (∗) There is a morphism ϕ : G (∗∗)  = ϕ(P (G)). def (∗∗∗) Let D = {(ϕ(g)−1 , g) : g ∈ P (G)} ∼ = P (G); then (c, g)D  → cϕ(g) :

 ×G →H D

Finite-Dimensional Connected Pro-Lie Groups

397

is a well-defined isomorphism of locally compact metric groups and  × G,  and H are all locally isomorphic. ×G  G is isomorphic to the minimal analytic subgroup (†) The subgroup ϕ(P (G))× D A(L(H ), H ) of H with Lie algebra h ∼ = L(G). Proof. Since h = L(H ) is finite-dimensional, by Corollary 4.21 (ii) there is an N ∈ N (G) such that dim L(N ) = 0. Then by Proposition 4.23 (=3.30), N is prodiscrete and totally disconnected. That is, N0 = {1}. Now we apply 9.38 with  = N1 and obtain the theorem. Corollary 9.45. (i) Let {Nj : j ∈ J } be a filtered family of normal subgroups of an almost connected complete topological group G such that G/Nj is a Lie group and assume limj ∈J Nj = 1. Then g  → (g(Nj )0 )j ∈J : G → lim G/(Nj )0 j ∈J

is an isomorphism and each quotient group G/(Nj )0 is an almost connected finite-dimensional locally compact group. (ii) Every almost connected pro-Lie group G is isomorphic to limN ∈N (G) G/N0 where each quotient group G/N0 is locally compact finite-dimensional and almost connected. (iii) For a connected pro-Lie group L and each N ∈ N (L), the factor group N/N0 is central in the finite-dimensional locally compact group L/N and is a direct product of a compact totally disconnected abelian subgroup comp(N/N0 ) and a finitely generated free discrete group. Proof. (i) The assumptions guarantee that G is a pro-Lie group. We have the isomorphism γG : G → G{Nj :j ∈J } = limj ∈J G/Nj . By Theorem 4.28 (i), G/(Nj )0 is a pro-Lie group, and there is a strict exact sequence {1} → Nj /(Nj )0 → G/(Nj )0 → G/Nj → {1}. Then, by the Strict Exactness Theorem 4.21, we have a strict exact sequence {0} → L(N − j/(Nj )0 ) → L(G/(Nj )0 ) → L(G/Nj ) → {0}. As L(Nj /(Nj )0 ) = {0} we have L(G/(Nj )0 ) ∼ = L(G/Nj ), and thus G/(Nj )0 is finitedimensional. By Theorem 9.44, G/(Nj )0 is locally compact. Since limj ∈J Nj = 1, we have limj ∈J (Nj )0 = 1 as well, and therefore γG,{(Nj )0 :j ∈J } : G → G{(Nj )0 :j∈J } = limj ∈J G/(Nj )0 is an isomorphism by Theorem 1.33. (ii) follows from (i) upon taking J = N (G) and Nj = j for j ∈ J .  where we may assume (iii) We apply Theorem 9.44 to H = L/N0 ∼ = ( × G)/D, ∼ L/N to be a covering group of of G in the notation of Theorem 9.44. Then G = (L/N ) and N/N0 is isomorphic to a quotient of ( × P (G))/D. Since P (G) is a finitely generated abelian group,  × P (G) is a compactly generated (see Definition 5.12(ii) and 5.22–5.27) totally disconnected locally compact abelian group. This class is closed

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under passage to quotients, and every member of this class is the extension of a compact totally disconnected group by a free discrete group of finite rank. Since free discrete quotients of any topological abelian group split, the assertion follows. For further conclusions on finite-dimensional pro-Lie groups, we also need the following lemma: Lemma 9.46. Let C be a compact normal subgroup of a connected topological group T . Assume that the identity component C0 is abelian. Then: (i) C is central, that is, C ⊆ Z(T ). (ii) If K is a normal subgroup of G containing C, and if K/C is compact central in G/C, then K is compact central in G. Proof. (i) Since comm(c × T ) is connected for each c ∈ C and contains 1, we have comm(C ×T ) ⊆ C0 . By Iwasawa’s Theorem on the Automorphism Group of Compact Groups (see [102, Theorem 9.82]), the identity component of the automorphism group Aut(C)0 of C is the group Inn0 (C) of inner automorphisms of C implemented by elements c ∈ C0 . Let χ : C0 → T = R/Z be any character of C0 and define g · χ by 0 is a continuous function from a (g · χ)(c) = χ (g −1 cg). Then g  → g · χ : G → C connected set to a discrete set (see for instance [102, Proposition 7.5]). It is therefore constant. Thus χ (g −1 cgc−1 ) = 0 for all g ∈ G and c ∈ C0 . Since the characters of a compact abelian group separate the points, comm(G × C0 ) = {1}, and so C0 is central. Since C0 is central (even in G) we conclude that Inn0 (C) and therefore Aut(C)0 is singleton. That is, Aut(C) is totally disconnected. The representation g  → (c  → gcg −1 ) : G → Aut(C) is continuous. It has a connected domain and a totally disconnected range and is therefore constant. Thus gcg −1 = c for all g ∈ G and c ∈ C. Thus C is central. (ii) Let K be a closed normal subgroup of G containing C such that K/C is compact central in G/C. We claim that K is central in G. As an extension of a compact group by a compact group, K is compact normal. Since K/C is central in G/C we know that comm(K × G) ⊆ C, and since comm(K × G) is connected and contains 1, we have indeed comm(K × G) ⊆ C0 . Therefore gkg −1 ∈ kC0 for all k ∈ K and g ∈ G. In particular, K  ⊆ C0 and C0 is central in G, hence is central in K. Thus K is normal in G and compact nilpotent of class 2. Therefore K0 is central in G. Now part (i) applies at once to show that K is central in G as claimed. In particular, any compact normal abelian subgroup of a connected topological group is central. By the Levi–Mal’cev Structure Theorem for Connected Compact Groups (see [102, Theorem 9.24]) in Lemma 9.46, instead of demanding that C0 be abelian it suffices to require that C does not contain any compact connected simple Lie group. If C1 and C2 are two compact solvable normal subgroups of any topological group G, then C1 C2 is a compact solvable normal subgroup containing both C1 and C2 . Thus

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the set CN of compact solvable normal subgroups is directed. Consequently, the union of CN is a characteristic normal subgroup of G that is a union of solvable subgroups. Definition9.47. For any topological group G, let KZ(G) denote the characteristic subgroup CN. In Definition 5.4 we denoted the set of all x ∈ G such that the smallest closed subgroup x containing x is compact by comp(G). In a connected topological group, KZ(G) is the union of all compact normal abelian subgroups by Lemma 9.46 and thus agrees with comp(Z(G)). The additive group G of the field Qp of p-adic rationals (see Example 1.20 (A)) is a locally compact noncompact nondiscrete abelian group with KZ(G) = comp(G) = G. In particular, G has no largest compact subgroup. Lemma 9.48. Let H be a finite-dimensional connected pro-Lie group with Lie algebra h. Then the union KZ(H ) of compact subgroups of the center Z(H ) of H is a compact fully characteristic subgroup of H and is the unique largest compact normal abelian subgroup of H .  is a connected Lie group, Proof. Let G be as in the proof of 9.38 and 9.45. Since G  is of the form Rp × Tq × A with a discrete group A that is isomorphic the center Z(G)   0 and is therefore finitely generated and so to a discrete central subgroup of G/Z( G) r  =  × Z(G)  is of the is of the form Z × E with a finite group E. Thus Z( × G) p q r form  × R × T × Z × E.  Now we remember that H ∼ where D is the finitely generated abelian = ( × G)/D −1  subgroup {(ϕ(g) , g) : g ∈ P (G)}. Thus we must show that ( × G)/D has a largest  is central modulo D, then compact central subgroup. If (c, g) ∈  × G  (∀c ∈ , g  ∈ G)

comm((c, g), (c , g  )) ∈ D.

This means (1, comm(g, g  )) ∈ D which in turn is equivalent to saying that  Now comm({g} × G)  comm(g, g  ) ∈ P (G) and ϕ(comm(g, g  )) = 1 for all g  ∈ G. is connected and contains comm(g, 1) = 1, while P (G) is discrete. Therefore  = {1} and so g ∈ Z(G).  Thus Z(( × G)/D)   comm({g} × G) = ( × Z(G))/D. def Hence we are dealing with a quotient group of Z =  × (Rp × Tq × Zr × E) def

modulo a subgroup of the form A = {(f (x), x) : x ∈ F } for a finitely generated discrete subgroup F of Rp × Tq × Zr × E and a morphism f : F → . The kernel of the quotient morphism Z/A → Z/( × F ) is the compact group ( × F )/A = ( × {0})A/A ∼ = ( × {0})/(( × {0}) ∩ A) ∼ = . Thus Z/A has a maximal compact subgroup iff Z/( × F ) does; but this latter group is isomorphic def

to L/F , where L = Rp ⊕ Tq ⊕ Zr ⊕ E is an abelian Lie group such that L/L0 is finitely generated. It is now an exercise to verify that every quotient group of such an L is again of the same type (Exercise E9.2). Every such L, however, has a maximal compact subgroup, and that is what remained to be shown.

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Exercise E9.2. Prove the following: Lemma. Every quotient group of an abelian Lie group L with finitely generated component factor group L/L0 is again of the same type. [Hint. Let F be a closed subgroup of L and let F1 = L0 ∩ F . Then L/F ∼ = (L/F1 )/(F /F1 ), and L/F1 is of the same type as L by the Characterisation Theorem of Closed Subgroups and Quotient groups of Rn (see for instance [102, Theorem A1.12]). Thus we may assume without loss of generality that F ∩ L0 = {0}. Since F is isomorphic to a subgroup of the finitely generated group L/L0 , the group F is finitely generated. Thus L0 ⊕ F is an abelian Lie group L1 such that L1 /F is connected. Since L/L1 ∼ = (L/L0 )/(L1 /F ) is finitely generated, L/F is finitely generated.] We remark that the cardinality of  is ≤ 2ℵ0 as is that of L. We therefore have the following additional Remark 9.49. The cardinality of a nondegenerate finite-dimensional connected proLie group G is that of the continuum. Exercise E9.3. Follow through with the details of the following discussion of a covering construction utilizing the techniques provided in this subsection on finite-dimensional pro-Lie groups. By the Retraction Theorem for Full Closed Subcategories 1.41, the inclusion functor of the category ZCompGr of compact zero-dimensional (equivalently, compact totally disconnected) groups into the category TopGr of all topological groups has a left adjoint denoted Fz in Chapter 1 with the zero-dimensional Bohr compactification morphism ζG : G → Fz G. From the fact that Fz G is zero-dimensional it follows at once that the identity component G0 of G is contained in the kernel of ζG . Now let G be a pro-Lie group and Z(G) its center. The universal covering mor → G, G  = (L(G)) of Chapter 6 has a prodiscrete kernel and induces phism πG : G an isomorphism of Lie algebras. We construct canonically a group G# attached functorially to a pro-Lie group G as follows: Where no confusion is possible we abbreviate the prodiscrete Poincaré group group ker πG = P (G) of G by P , and we write  = (G) = Fz (P (G)) for the compactified Poincaré group.We shall also abbreviate  contains the graph the natural injective morphism ζP : P →  by ζ . The group  × G def

D = D(G) = {(ζ (z)−1 , z) : z ∈ P } of z  → ζ (z)−1 in  × P as a central subgroup, and z  → (ζ (z)−1 , z) : P → D(G) is an isomorphism of prodiscrete groups. We set G# =

 ×G . D

 is a pro-Lie group, but because of the notorious difficulties with quotients Clearly × G discussed in Chapter 4, we cannot be sure that G# itself is a pro-Lie group; but it is a proto-Lie group by Theorem 4.1, and it has a completion which is a pro-Lie group. Remark. If G is a connected pro-Lie group and if P (G) is locally compact, then G# is a pro-Lie group.

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Proof. This is a consequence of Theorem 4.28 (iii) and the isomorphism P (G) ∼ = D.  → G# given by dG (g) = (1, g)D is an injective morphism of The function dG : G #  G into G , and the map δG :  → G# , δG (c) = (c, 1)D implements an isomorphism of compact zero-dimensional groups from  onto its image δG (). Indeed what is constructed in this fashion is in fact a pushout incl

P (G)  −−−−→ G P (G) ⏐ ⏐ ⏐ ⏐d ζP (G)  G (G) −−−−→ G#

δG

of natural maps, thereby ensuring, that G  → G# is a functor from the category proLieGr to the category of proto-Lie groups protoLieGr. The fact that every morphism f : G → H of pro-Lie groups induces a morphism f # : G# → H # of proto-Lie groups can be verified explicitly by verifying that ((f ) × f)(D(G)) ⊆ D(H ).  → L(G# ), and im dG = The morphism dG induces an isomorphism L(dG ) : L(G)  # # #  (G) → G the natural bijective morphism A(L(G ), G ) = E(G ). Let πG : G/P  induced by πG : G → G. Moreover, since  is compact, (( × {1})D)/D is a closed normal subgroup of G# and thus we have a morphism βG : G# =

  ∼  πG  ∼ ×G ( × G)/D = ×G = G → → −−−−→G, → D (( × {1})D)/D) ×P P

giving us a commutative diagram dG  −−− −→ G ⏐ ⏐ πG  G −−−−→ idG

G⏐# ⏐β  G G.

(∗)

 is From the definition of βG it is clear that the morphism βG is surjective iff πG surjective iff πG is surjective. Note that this is automatically the case if G is a connected Lie group. The natural morphism πG is injective iff the Poincaré group P (G) is singleton. It is straightforward to verify that dG is an isomorphism iff ζP (G) : P (G) → ∼ (G) is an isomorphism iff P (G) is compact. In this case the two groups G = G# may be identified. If G is a connected Lie group, then P (G) is a finitely generated abelian ∼  is a finite cover of G iff the fundamental group group and then G = G# if and only if G π1 (G) is finite. d provides an example of a connected compact (hence pro-Lie) The group G = Q group for which βG : G# → G fails to be surjective. If G is a finite-dimensional connected locally compact group, then G# is a finitedimensional connected locally compact group.

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Compact Central Subgroups The results of the preceding discussion allows us to conclude several basic results. The first concerns the existence of a largest compact normal abelian subgroup. More definitive information will emerge in Theorem 12.59 below. Also recall from Lemma 9.46 that a compact abelian normal subgroup of a connected topological group is central. A topological group is said to be compactly simple if it has no compact normal subgroups except the trivial one. Theorem 9.50 (Existence of the Largest Compact Normal Abelian Subgroup). Let G be a connected pro-Lie group. (i) Then KZ(G) is compact and is the unique largest compact central subgroup. The factor group G/ KZ(G) does not have nondegenerate compact central subgroups. (ii) The center Z(G) is a direct product of a weakly complete vector group V and a subgroup A of Z(G) containing the characteristic compact subgroup KZ(G); moreover, the factor group Z(G)/V KZ(G) ∼ = A/ KZ(G) is prodiscrete and compactly simple. The characteristic closed subgroup Z(G)0 comp(Z(G)) is the direct product of V and KZ(G). Proof. (i) The projective system {qMN : G/M0 → G/N0 : (M, N ) ∈ N (G) × N (G), M ⊆ N } has G as its limit by 9.45. It induces a projective subsystem pMN : KZ(G/M0 )) → KZ(G/N0 ) : (M, N ) ∈ N (G) × N (G), M ⊆ N } whose limit C is acompact subgroup of G. Since it is central modulo N0 for all N ∈ N (G), and N (G) = {1}, it is central. So C ⊆ KZ(G). Conversely, the limit map qN maps KZ(G) into KZ(G/N0 ), and so KZ(G) ⊆ CN0 . Now we have limN∈N (G) N0 = 1 since lim N (G) = 1. Therefore, since C is closed we conclude KZ(G) ⊆ N∈N (G) CN0 = C. Thus equality holds and C = KZ(G) is compact. We claim that G/C has no nondegenerate central compact subgroups. Indeed let K be a closed normal subgroup of G containing C such that K/C is compact central in G/C. Then by 9.46 (ii), K is central compact in G, that is K ⊆ KZ(G) = C and thus K/C is singleton. Therefore G/ KZ(G) has no nondegenerate compact central subgroups, as claimed. (ii) Since comp(Z(G)) = KZ(G) by a remark following Definition 9.47, we know from Part (i) above that comp(Z(G)) is compact. The assertions of (ii) therefore are immediate consequences of the Vector Group Splitting Theorem 5.20. According to this theorem, a group like Qp cannot be contained in a connected finite-dimensional pro-Lie group as a normal subgroup. We say that a topological group G has no compact central subgroups if comp(Z(G)) = {1}. If G is connected this is the same as saying KZ(G) = {1}.

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Exercise E9.4. Prove the following Proposition. Let f : G → H be a surjective morphism of connected topological groups with a compact kernel. Assume that KZ(G) and f (KZ(G)) are closed subgroups; this is certainly the case whenever KZ(G) is compact. Then f maps KZ(G) surjectively onto KZ(H ). [Hint. Clearly f (KZ(G)) ⊆ KZ(G). We must prove equality. Since KZ(G) and f (KZ(G)) are closed, we may factor these groups and consider the induced surjective morphism G/ KZ(G) → H /f (KZ(G)). Then the domain has no compact central subgroups by Theorem 9.50. Thus we simplify notation and assume henceforth that KZ(G) = {1}, that is, that G has no central compact subgroups. Then we must show that KZ(H ) = {1}. def def Define C = f −1 (KZ(H )) and N = f −1 ({1}) = ker f . Then C/N, being mapped bijectively onto KZ(H ) is central in G/N, and thus comm(G × C) ⊆ N . Since G is connected, comm(G × N) is connected and contains 1, whence comm(G × N ) ⊆ N0 . Therefore [G, N] ⊆ N0 in particular, it follows that C˙ = C  ⊆ N0 . Let K1 be a compact central subgroup of H and set K = f −1 (K). Then K is compact normal in G, containing N and being contained in C.. The center Z(K) is characteristic in C thus is normal in G; hence it is contained in Z(G) by 9.46 and thus is singleton. Therefore K is center free; also, K0 is semisimple (see for instance [102, Theorem 9.24]). Since K˙ ⊆ C˙ ⊆ N0 we know that K/N0 is abelian. In particular, K0 /N0 is connected abelian, but also semisimple, hence is singleton. Thus K0 = N0 . The centralizer Z(K0 , G) satisfies G = K0 Z(K0 , G) by Iwasawa’s Theorem (coming directly out of the fact that every automorphism of K0 implemented by an inner automorphism of the connected group G is in fact implemented by an inner automorphism of K0 (see for instance [102, Theorem 9.82 (i)] or [120])); since K0 is center-free, we have K0 ∩ Z(K0 , G) = {1} and therefore K is the direct product of K0 and Z(K0 , K). Since Z(K0 , K) is a normal totally disconnected subgroup of the connected group G it is central compact and therefore singleton. Thus K = K0 = N0 and so K/N ∼ = K1 is singleton. Hence KZ(H ) = {1} as asserted.] def

def

Let G = R and f : G → H = R/Z be the quotient morphism. Then f is a surjective morphism of connected abelian Lie groups such that KZ(G) = {0} and KZ(H ) = H ; so f (KZ(G)) = KZ(H ). The kernel is not compact and not connected. For an example with connected kernel let us consider a simple example of the Center-Free Embedding Lemma 9.41 def

and take G = R2 ι SO(2) with ι(T )(v) = T (v); that is in G we multiply according to (v, S)(w, T ) = (v + T (w), ST ). Then G is the group of rigid motions of the euclidean plane. Let H = SO(2) and let f : G → H be the projection onto the second factor. Since G is center free, we have KZ(G) = {(0, idR2 )} and KZ(H ) = H . Again f (KZ(G)) = KZ(H ), and ker f = R2 × {idR2 } ∼ = R2 . Let G be a connected pro-Lie group. Then its center Z(G) is an abelian pro-Lie group by the Closed Subgroup Theorem 3.35. Hence the structure of Z(G) is known from the Vector Group Splitting Theorem for Abelian Pro-Lie Groups 5.20. However,

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not every abelian pro-Lie group can be the center of a connected pro-Lie group. By Theorem 9.50, we know that comp(Z(G)) is compact. Let us assume momentarily that G has no compact central subgroups. Then comp(Z(G)) = {1} and Z(G)0 = Z(G)1 (in the terminology of 5.20) is a weakly complete vector group, and Z(G)/Z(G)0 is a prodiscrete group without compact subgroups. By 5.20, Z(G) ∼ = Z(G)0 × Z(G)/Z(G)0 . Using the Center-Free Embedding Lemma we can construct the following Example 9.51. Let A be an abelian pro-Lie group such that comp(A) is compact. Then there is a connected metabelian pro-Lie group G such that A is the center of G. [Hint. By 5.10 we may consider A to be a closed subgroup of a group RJ × TK for suitable sets J and K. By Theorem 4.1, the quotient (RJ × TK )/A is a proto-Lie group whose completion Q is a pro-Lie group, which is connected, since it contains a dense connected subgroup. The natural morphism f : RJ × TK → Q has the precise kernel A. The abelian connected pro-Lie group Q is isomorphic to V ×C with a weakly complete vector group V and a compact connected group C. By Theorem 5.36, Q is reflexive. In particular, the characters of Q separate the points, and we can form  V = CQ and get a faithful and fixed point free action of Q on V , corresponding to a faithful representation π : Q → Aut(V ). Now we form the semidirect product def

G = V π f (RJ × TK ). Then G is a connected metabelian pro-Lie group, whose center is {0} × A ∼ = A.]

Divisibility of Groups and Connected Pro-Lie Groups For what is to follow soon we need a new line of algebraic properties of pro-Lie groups and groups in general. Recall that a divisible group is one in which the equation x n = g has a solution for every group element g and every natural number n. Clearly homomorphic images of divisible groups are divisible. Exercise E9.5. Prove the following Proposition. (i) Let g be an element of a divisible group G. Then there is a group homomorphism f : Q → G such that f (1) = g. Accordingly, g is contained in the divisible abelian subgroup f (Q) of G. (ii) Homomorphic images of Q are either singleton or infinite. (iii) A finite group has no divisible subgroups other than the singleton one. [Hint. (i) Using divisibility, recursively define elements g1 = g, g2 , . . . such that gnn = gn−1 , n = 2, 3, . . . . Every rational number q ∈ Q can be written in the form q = ±m/n!. The function f : Q → G sending q = ±m/n! to gn±m is well defined

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and satisfies the requirements. Alternatively, use the injectivity of G in the category of abelian groups to extend the morphism n  → n · g : Z → G to f : Q → G. (ii) Let S be a subgroup of Q. If S = {0} then Q/S ∼ = Q. Assume that S contains a member s = 0; then x  → s −1 x : Q → Q is an automorphism and Q/S ∼ = Q/s −1 S. Moreover, 1 = s −1 s ∈ s −1 S. We may and will assume that 1 ∈ S. Then Q/S ∼ = (Q/Z)/(S/Z) is a homomorphism image of Q/Z = p prime Z(p∞ ) where ∞ Z(p ) = {m/p n : m ∈ Z, n ∈ N}/Z is the Prüfer group for the prime p (cf. paragraph preceding 5.1). Since this is the decomposition into p-primary components or Sylow groups, S/Z = p prime Sp for Sp = (S/Z) ∩ Z(p∞ ), and Q/S ∼ =

8

Z(p ∞ )/Sp .

p prime

Show that a quotient group of Z(p∞ ) is either singleton or isomorphic to Z(p∞ ). Thus Q/S is either singleton or infinite as a direct sum of Prüfer subgroups. (iii) is a consequence of (i) and (ii). Alternatively, if an abelian group G, for which there is a natural number n such that n · x = 0 for all x ∈ G, contains a nonzero g ∈ G, then divisibility would entail the existence of an x ∈ G such that g = n · x = 0, a contradiction.] The structure of abelian divisible groups is completely known: See for instance [102, Appendix, 1, Theorem A1.42]. In any compact connected group, every element is contained in a maximal protorus (see for instance [102, Theorem 9.32]). Compact connected abelian groups are divisible (see e.g. [102, Corollary 8.5]). Therefore if a compact group G is connected, that is G = G0 , then it is divisible. The finite quotients of G/G0 separate the points; thus is G = G0 , by E9.5 (iii), G fails to be divisible. Therefore we have the Fact. A compact group is connected iff it is divisible. (For compact abelian groups see [102, Theorem 8.4.]) The additive group of the field Qp of p-adic numbers is a nondiscrete locally compact noncompact divisible group. By Lemma 5.12, and the Fact above, every connected abelian pro-Lie group is divisible. For an arbitrary (not necessarily topological) group G, let us denote by D(G) the subgroup that is algebraically generated by all divisible subgroups. Then clearly for any group homomorphism f : G → H the containment f (D(G)) ⊆ D(H ) holds. In particular, recalling that a subgroup of a group is said to be fully characteristic if every endomorphism maps it into itself (cf. paragraph following 2.21) we have the following observation: for any group G the subgroup D(G) is fully characteristic. This is understood in the sense of the category of groups; that is, no continuity is involved. However, let us now consider a connected pro-Lie group G with Lie algebra g. The minimal analytic subgroup with full Lie algebra A(G, g) = expG L(G) satisfies D(A(G, g)) = A(g, G).

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We shall formulate here a Lemma which we shall use, but whose proof we shall complete later when more information on compact subgroups of connected pro-Lie groups is known. Lemma 9.52. Let G be a connected pro-Lie group. Then G = D(G). That is, G is generated by its divisible subgroups. Proof. (i) In Theorem 12.65 we shall show that there is a compact connected abelian, hence divisible, subgroup C of G such that G = C · A(G, g). Since A(G, g) is generated by its one parameter subgroups, G is generated by divisible subgroups. We have to be very careful that in the proof of 12.65 we shall not use the Open Mapping Theorem for Almost Connected Pro-Lie Groups which we are about to prove given this information. We have some immediate consequences. Proposition 9.53. Let G be a connected pro-Lie group. Then G has no subgroups of finite index. Proof. By 9.52 we have G = D(G) and so for any homomorphism f : G → H the image f (G) is contained in D(H ). If N is any normal subgroup of G, closed or not, then G/N = D(G/N). If G/N is finite, then D(G/N ) = {N} and so N =G. If H is any −1 subgroup of G of finite index, say, G = g1 H ˙∪ · · · ˙∪ gn H , then N = nm=1 gm Hgm is a normal subgroup of finite index, and thus must equal G by what we saw. So H = G. The proof shows that G cannot contain any proper normal subgroup N such that G/N does not contain any nontrivial divisible subgroups. Proposition 9.54. Let C be a closed central totally disconnected subgroup of a connected pro-Lie group L. If G is any subgroup of L such that G = CG and C ∩G = {1}. Then G = L. Proof. Let N ∈ N (L). Then CN/N is a closed central subgroup of the Lie group L/N . By the Closed Subgroup Theorem for Projective Limits 1.34(i), C is canonically isomorphic to limN ∈N (L) CN/N where CN/N is discrete since C is prodiscrete. Also, if L is connected, then CN/N is a closed central subgroup of the connected Lie group L/N . Hence it is finitely generated abelian, and so CN/N is finitely generated abelian, that is, it is isomorphic to a direct product of finitely many cyclic groups. By Theorem 1.34 (iv) we know that C/(C ∩ N) ∼ = CN/N as topological groups, and so there is a closed subgroup BN = C ∩ N of C such that C/BN is finitely generated discrete, and limN ∈N (L) BN = limN ∈N (L) N = 1. Now any direct product of nondegenerate cyclic groups is profinite, that is, the subgroups of finite index separate the points. Finally we suppose that C = {1} and derive a contradiction. From the preceding paragraph we get a subgroup B of C such that C/B is finite.

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Since L is algebraically the direct product C · G and B is contained in C, the factor group L/B is algebraically the direct product (C/B) · (GB/B). Thus the connected pro-Lie group L has a normal subgroup GB of index |C/B|. This is impossible by Proposition 9.53 above. We introduce a technical, but convenient terminology: Definition 9.55. A proto-Lie group P has a stable Lie algebra if L(P ) = L(P ) for its completion P . The following is now an important technical step in the proof of the Open Mapping Theorem which we shall attack presently. Corollary 9.56. Assume that G is a proto-Lie group with a stable Lie algebra and assume that its completion G is connected. If there is a bijective morphism f : G → H onto a complete topological group H , then G is complete, that is, G is a pro-Lie group. Proof. Consider G as a subgroup of its completion G. The continuous morphism f : G → H into the complete group H has a unique extension to a morphism F : G → H . of complete topological groups by the universal property of the completion. Let C = ker F . Since L preserves kernels by Theorem 2.20 or 4.20 (ii), L(C) = ker L(F ) = {X ∈ L(G) : (∀t ∈ R)F (X(t)) = 1} = {X ∈ L(G) : (∀t ∈ R)f (X(t)) = 1} = {0} since L(G) = L(G), F |G = f , and f is injective. Thus C, being a pro-Lie group as a closed subgroup of G by the Closed Subgroup Theorem for Pro-Lie Groups 3.35, is totally disconnected by Proposition 3.30. Since G is connected, C is central (see 12.55; this is elementary). Now f is bijective, and thus if j : G → G is the inclusion morphism, then ϕ = j  f −1  F : G → G defines an algebraic endomorphism of G satisfying ϕ 2 = ϕ such that C = ker ϕ and G = im ϕ. Then G = CG and C ∩ G = {1}. Thus Proposition 9.54 shows G = G. Clearly, the completion G of a topological group G is connected if G is connected but Q in the induced topology of R is a totally disconnected topological group, whose completion R is connected. It was for this result and its consequences that we had to find an algebraic property of connected pro-Lie groups, such as being generated by divisible subgroups and thus having no finite algebraic homomorphic images. But now we improve it by considering almost connected proto-Lie and pro-Lie groups in place of connected ones; for a definition see Definition 4.24. Corollary 9.57. Assume that G is an almost connected proto-Lie group with a stable Lie algebra. If there is a bijective morphism f : G → H onto a pro-Lie group H , then G is a pro-Lie group and f (G0 ) = H0 . Moreover, f is an isomorphism of pro-Lie groups if f |G0 : G0 → H0 is open. Proof. Let P = f −1 (H0 ). Then G0 ⊆ P . Thus P is an almost connected proto-Lie group and f |P : P → H0 is surjective since f is surjective. We claim that P is connected and therefore equals G0 . As P is almost connected, the factor group P /G0 is profinite. So if P = G0 , then P has an open normal proper subgroup Q of finite

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index. Since f is surjective, f (Q) is a normal subgroup of finite index of H0 . Now H0 is a closed subgroup of H which is assumed to be a pro-Lie group. Thus H0 is a connected pro-Lie group. By Proposition 9.53, this implies f (Q) = H0 . Since f is also injective, Q = P follows, in contradiction to the assumption that Q is a proper subgroup of P . Thus G0 = f −1 (H0 ) as asserted. We note that G0 ⊆ G and L(G0 ) = L(G) = L(G), whence L(G0 ) = L(G0 ). Now f |G0 : G0 → H0 is a bijective morphism, H0 is complete, and G0 is a connected proto-Lie group with stable Lie algebra. Hence Corollary 9.56 applies to f |G0 and shows that G0 is a pro-Lie group. Observe quite generally that a topological group T with a complete normal subgroup N such that T /N is complete is itself complete (see for instance [176, p. 225, 12.3]). Therefore, since G/G0 is compact and thus complete, G is a pro-Lie group. Hence f : G → H is a bijective morphism between almost connected groups such that f |G0 : G0 → H0 is a bijective morphism between connected pro-Lie groups. Now we assume that f |G0 : G0 → H0 is open and show that f is open. For a proof let U be an open identity neighborhood of G; we must show that f (U ) is open in H . First we find a normal subgroup N ∈ N (G) contained in U , and, by making U smaller if needed, we do not restrict generality by assuming that U N = U . (See e.g. 1.27 (i).) Observe that G0 N/N is an analytic subgroup of the Lie group G/N whose Lie algebra agrees with L((G/N )0 ) and which therefore is (G/N )0 , whence ∼ G0 N/N is a Lie group. Thus M def = N ∩ G0 ∈ N (G0 ). Hence G0 /(G0 ∩ N) = G0 /M is a Lie group and G/G0 is compact since G is almost connected. So the factor group G/M is a locally compact almost connected and therefore σ -compact group. Since f |G0 : G0 → H0 is an isomorphism, H0 /f (M) is isomorphic to G0 /M and is therefore a Lie group, whence H /f (M) is locally compact. Thus the induced map fM : G/M → H /f (M) is open by the classical Open Mapping Theorem for Locally Compact Groups (see [79, p.42, Theorem 5.29]). Thus fM (U/M) = f (U )/f (M) is open in H /f (M) and so f (U ) is open in H as asserted. Thus f is open and therefore an isomorphism. From our discussion of finite-dimensional pro-Lie groups we can now turn to an Open Mapping Theorem for which we were on the lookout at so many points in this book. A lot of information that we accumulated in the meantime enters its proof. An Open Mapping Theorem in the theory of topological groups deals with a surjective morphism f : G → H between topological groups and states sufficient conditions for f to be open. Recall from the elementary theory of topological groups that any surjective morphism f : G → H of topological groups defines a quotient morphism q : G → G/ ker f and a unique bijective morphism f  : G/ ker f → H such that f = f   q; indeed f  (g · ker f ) = f (g) unambiguously. We note once and for all that f : G → H is open iff f  : G/ ker f → H is open iff f  is an isomorphism of topological groups. In Proposition 5.2 it was shown that the free abelian group Z(N) has a nondiscrete nonmetrizable prodiscrete (and thus pro-Lie) topology τ . If δ is the discrete topology

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409

on Z(N) , then the identity morphism f : (Z(N) , δ) → (Z(N) , τ ) is a nonopen bijective morphism between two countable pro-Lie groups. So, in general, an Open Mapping Theorem for Pro-Lie Groups is not available generally. We will see in the following lemma that our chances are much better if connectivity is present. Lemma 9.58. Assume that f : G → H is a surjective morphism of pro-Lie groups and that G is almost connected. Then the following conclusions hold: (i) The quotient group G/ ker f is a pro-Lie group, and the induced bijective morphism f  : G/ ker f → H maps (G/ ker f )0 bijectively onto H0 . (ii) f is open if the induced bijective morphism f  induces an open morphism (G/ ker f )0 → H0 between connected pro-Lie groups. Proof. Let P = G/ ker f . Then P is a proto-Lie group by Theorem 4.1, and by Theorem 4.20(i0 ), it has a stable Lie algebra. Now Corollary 9.57 applies to f  : G/ ker f → H and proves claims (i) and (ii). This lemma reduces the Open Mapping Theorem for surjective morphisms between almost connected pro-Lie groups to the Open Mapping Theorem for bijective morphisms between connected pro-Lie groups. The following observation rounds off our general orientation on almost connected groups. Proposition 9.59. Let f : G → H be a morphism of topological groups and assume, firstly, that G is almost connected and, secondly, that f (G) is dense in H . Then H is almost connected. Proof. Since f (G0 ) is connected and contains the identity, f (G0 ) ⊆ H0 and therefore the morphism ϕ : G/G0 → H /H0 , ϕ(gG0 ) = f (g)H0 is well-defined. By assumption G/G0 is compact. Thus the continuous image ϕ(G/G0 ) is compact and therefore, since H /H0 is Hausdorff, is closed in H /H0 . Since f (G) is dense in H , it follows that ϕ(G/G0 ) is dense in H /H0 . Therefore, H /H0 = ϕ(G/G0 ), and so H /H0 is compact.

The Open Mapping Theorem In this section we shall prove the following main result: The Open Mapping Theorem for Almost Connected Pro-Lie Groups Theorem 9.60. A surjective morphism between pro-Lie groups is open if its domain group is almost connected.

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Proof. Let f : G → H be a surjective morphism between pro-Lie groups and assume that G is almost connected. We must show that f is open. By Lemma 9.58 it is no loss of generality to assume that both G and H are connected and that f is bijective. We will do this from now on prove the theorem through a sequence of steps. (1) Claim: G and H have the same Lie algebra g. The morphism L(f ) : L(G) → L(H ) is injective as L preserves kernels by Theorem 2.20; it is surjective by Corollary 4.22 (ii) and so is an isomorphism by Theorem A2.12 (b) in Appendix 2. We may therefore set g = L(G) = L(H ) and keep the commutative diagram id g −−−−→ ⏐ g ⏐ ⏐exp expG ⏐ (∗)   H G −−−−→ H f

in mind. (2) Claims: (i) (∀N ∈ N (H ))f −1 (N ) ∈ N (G), (ii) f induces an isomorphism def

G/f −1 (N) → H /N, and (iii) L(f −1 (N ) = L−1 (f −1 (N )0 ) = n = L(N ). The structural invariants of G and H are the filters N (G) and N (H ), respectively. def

Consider an N ∈ N (H ) and n = L(N ) ∈  (g). Let V be an open identity neighborhood of H containing N such that V N = V and V /N contains no subgroups other that def

the singleton one. Since f is continuous, U = f −1 (V ) is an identity neighborhood of def

G. Set M = f −1 (N ). Then U M = U and U M/M contains no subgroup other than the singleton one. Since lim N (G) = 1 in G there is a P ∈ N (G) such that P ⊆ U . Then P M/M is a subgroup of G/M contained in U/M and thus agrees with M/M, that is, P M = M, and so P ⊆ M. Therefore we have a surjective morphism fN P : G/P → H /N,

fN P (gP ) = f (g)N

of topological groups. By the definition of N (G) we know that G/P is a Lie group; it is connected since G is connected. Thus G/P is a locally compact group which is the union of a countable collection of compact subsets. Also, H /N is a Lie group since N ∈ N (H ). Hence it is locally compact. Thus the Open Mapping Theorem for Locally Compact Groups (see [79, p. 42, Theorem 5.29]) applies and shows that fN P is open. Since M/P = ker fN P , we know that G/M ∼ = (G/P )/(M/P ) ∼ = H /N is a Lie group. Therefore M ∈ N (G). Thus f −1 (N (H )) ⊆ N (G) def

and fN : G/f −1 N → H /N , fN (gf −1 (N )) = f (g)N is an isomorphism. From Corollary 4.21(i) we know that g/L(f −1 (N )) ∼ = L(G/f −1 (N )) = L(H /N ) ∼ = g/n. −1 From (∗) it now follows that L(f (N )) = n. We shall abbreviate (f −1 (N ))0 by f −1 (N)0 and recall L(f −1 (N )0 ) = n. (3) Claim: {f −1 (N )0 : N ∈ N (H )} is cofinal in {M0 : M ∈ N (G)}.

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From (∗) and Corollary 4.21(ii) we know that in the filter basis  (g) of cofinite dimensional closed ideals of g both of the filterbases G = {L(M) : M ∈ N (G)} and H = {L(N) : N ∈ N (H )} are cofinal in id g; we shall use the cofinality of the latter. def

If M ∈ N (G), then m = L(M) ∈ G and M0 = expG m by Corollary 4.22 (i). Since H is cofinal in  (g) there is an NM ∈ N (H ) such that L(f −1 (NM )) ⊆ m. Consequently, using 4.22(i) again we get f −1 (NM )0 = expG n ⊆ expG m = M0 , more specifically, (∀M ∈ N (G))(∃NM ∈ N (H ))f −1 (N )0 ⊆ M0 . (4) Claim: G ∼ = limN ∈N (H ) G/f −1 (N )0 , that is, we have a limit representation of G indexed by N (H ). By 3, the filterbasis {f −1 (N )0 : N ∈ N (H )} is cofinal in the filterbasis {M0 : M ∈ N (G)}. From Corollary 9.45 we know that G ∼ = limM∈N (G) G/M0 . Thus by the Cofinality Lemma 1.21, γ:G→

lim

N ∈N (H )

G/f −1 (N )0 ,

γ (g) = (gf −1 (N )0 )N ∈N (H )

is an isomorphism. (5) Claim: (∀N ∈ N (H ))f −1 (N0 ) = f −1 (N )0 . This is a subtle but important point. Abbreviate f −1 (N ) by M. The bijective morphism f induces a bijective morphism f |M : M → N between pro-Lie groups; it def

clearly maps M0 into N0 , and we claim that it maps M0 onto N0 . Let P = f −1 (N0 ) ⊆ M, then M0 = f −1 (N )0 ⊆ P and we have to show equality. From Corollary 9.45(iii) it follows that P /M0 ⊆ M/M0 is isomorphic to a direct product C × Zn for a compact totally disconnected abelian group C and a discrete free group of rank n. These groups are residually finite, that is, the finite homomorphic images separate the points. Now suppose that P = M0 , then P contains a normal subgroup Q of finite positive index (containing M0 ). Then f (Q) is a normal subgroup of f (P ) = N0 of finite positive index since f is bijective. But this contradicts Proposition 9.53. Thus P = M0 and the claim is proved. (6) Claim: The bijection f induces a natural isomorphism of topological groups α : limN∈N (H ) G/f −1 (N )0 → limN ∈N (H ) H /N0 . Let N ∈ N (H ). Then f −1 (N )0 = f −1 (N0 ) by 5, and thus f induces a bijective morphism f −1 (N )0 → N0 and then also a bijective morphism αN : G/f −1 (N )0 → H /N0 ,

αN (gf −1 (N )0 ) = f (g)N0 .

Since f −1 (N) ∈ N (G) by 2 above the factor group G/f −1 (N )0 is locally compact by Corollary 9.45, as is the factor group H /N0 . Since the group G is connected, G/f −1 (N)0 is σ -compact. The Open Mapping Theorem for Locally Compact Groups applies and shows that αN is an isomorphism for each N ∈ N (H ). This gives us an

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

isomorphism lim G/f −1 (N )0 → lim H /N0 , N ∈N (H ) N ∈N (H ) −1 α((gN f (N )0 )N ∈N (H ) ) = (f (gN )N0 )N ∈N (H )

α:

which is represented in the following diagram: μN

G/f −1 ⏐ (N )0 ⏐ αN  H /N0

←−−−−− limQ∈N (H )⏐G/f −1 (Q)0 ⏐α  ←−−−−− limQ∈N (H ) H /Q0 νN

for the respective limit morphisms μ, ν. (7) Claim: f is an isomorphism of topological groups. The function γ : G → limN ∈N (H ) G/f −1 (N )0 , γ (g) = (gf −1 (N )0 )N ∈N (H ) is an isomorphism by 4, and γN0 (H ) : H → limN ∈N (H ) H /N0 , N0 (H ) = {N0 : N ∈ N (H )} is an isomorphism by Corollary 9.45 (ii). By 6, the map α is an isomorphism. We have a commutative diagram f

−−−−→

G ⏐ ⏐ γ limN ∈N (H )

G f −1 (N )0

H ⏐ ⏐N (H )  0

−−−−→ limN ∈N (H ) α

H N0 .

It follows that f is an isomorphism. And this last step completes the proof of the theorem. We record some immediate consequences; later applications will abound. Corollary 9.61 (Closed Graph Theorem for Pro-Lie Groups). Assume that G and H are pro-Lie groups and that f : G → H is a morphism of groups (algebraically) with graph {(x, f (x)) : x ∈ G} ⊆ G × H . Consider the following statements: (i) f is continuous. (ii) The graph of f is closed in G × H . (iii) The graph of f is closed in G × H and is almost connected. Then (iii) ⇒ (i) ⇒ (ii). Proof. (i) ⇒ (ii) is a consequence of the general fact that the graph of any continuous function into a Hausdorff space is closed. Trivially (iii) ⇒ (ii). We must show that (iii) ⇒ (i). We define γ : G → graph(f ) by γ (x) = (x, f (x)) and decompose f as follows: γ

pr H | graph(f )

G −−−−→ graph(f ) −−−−−−−→ H. We see that f is continuous if γ is continuous. The continuity of γ is equivalent to the openness of γ −1 = pr G | graph(f ). By (ii), graph(f ) is a closed subgroup of the

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413

pro-Lie group G × H . By the Closed Subgroup Theorem of Pro-Lie Groups 3.35, it is a pro-Lie group. Now the Open Mapping Theorem 9.60 applies to the continuous morphism pr G | graph(f ) : graph(f ) → G and shows that it is open. If G is a topological group which happens to be the product N H of a closed normal subgroup N and some closed subgroup H , then there is a natural bijective continuous morphism β : H /(H ∩ N) → G/N, β(h(H ∩ N )) = hN. Whenever β is an isomorphism of topological groups, then one refers to this statement as the Second Isomorphism Theorem of Group Theory. Unfortunately, in general, the Second Isomorphism Theorem is not guaranteed for topological groups without additional hypotheses. For pro-Lie groups we can now formulate the following result: Corollary 9.62 (Second Isomorphism Theorem for Pro-Lie Groups). Assume that a pro-Lie group G is a product of an almost connected closed normal subgroup N and an almost connected closed subgroup H . Then β : H /(H ∩ N ) → H N/N is an isomorphism. Proof. By the Closed Subgroup Theorem 3.35, the closed subgroup H of the proLie group G is a pro-Lie group. By Theorem 4.28 (i), G/N is a pro-Lie group since both G and N are almost connected. Hence the function f : H → G, f (h) = hN is a surjective morphism of pro-Lie groups whose domain is almost connected by hypothesis. Therefore, by the Open Mapping Theorem 9.60, the morphism f is open. We have ker f = H ∩ N , and if q : H → H / ker f is the quotient morphism we have f = β  q and β is open and therefore is an isomorphism.

Completing Proto-Lie Groups If G ⊇ G is the completion of a proto-Lie group G, one might surmise that we had to have L(G) = L(G) making all proto-Lie groups have stable Lie algebras.. However this equality fails rather grotesquely as we shall show presently. Another seemingly reasonable conjecture would be that every bijective morphism f : G → H from a connected proto-Lie group G to a pro-Lie group H had to be an isomorphism. Unfortunately, this conjecture is also false. These things we learn, among other things, from the following examples, which we find useful to keep in mind. Since their discussion involves real vector spaces and their vector space dimensions, we preface it with the following remarks which are less obvious than their proofs. Remarks 9.63. (a) Let W be a real vector space of infinite dimension ℵ, that is, ℵ dimR W = ℵ. Then dimR W ∗ = 2ℵ and dimR W ∗∗ = 22 . (b) The additive group of any vector subspace of a weakly complete vector space is a proto-Lie group.

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

Proof. (a) Pick a basis B of W ; then card B = ℵ. Now the algebraic dual W ∗ is given by W ∗ = Hom(W, R) ∼ = Hom(R(B) , R) ∼ = RB . Thus dimR W ∗ = 2ℵ . (Indeed if E = R ⊗ EQ for a Q-vector space EQ then dimR E = dimQ EQ , and if Q(X) is any infinite dimensional vector space then dimQ Q(X) = card(X) = card Q(X) . Now R ⊗ QB is isomorphic to a vector subspace of RB , and so 2card B = card QB = dimQ QB = dimR R ⊗ QB ≤ dimR RB ≤ card RB = (2ℵ0 )card B = 2ℵ0 card B = 2card B ℵ since B is infinite.) Therefore, dim W ∗∗ = 22 . Furthermore, if W is a real vector ℵ B space of infinite dimension 2 , then W ∼ = R for a set B of cardinality ℵ because dimR Rℵ = dimQ Qℵ = 2ℵ . (b) Let V be a weakly complete vector space and E any vector subspace. Let UE be a neighborhood of 0 in E and UV a neighborhood of 0 in V such that UE = E ∩ UV . Then there is a closed vector subspace W contained in UV such that dim V /W < ∞. Now E ∩ W ⊆ E ∩ UV = UE and E/(E ∩ W ) ∼ = (E + W )/W ⊆ V /W . Since (E + W )/W is a vector subspace of a finite dimensional vector space, E/(E ∩ W ) is a finite dimensional vector space. It follows that E is a proto-Lie group. The following examples are now straightforward from the preceding remarks. Example 9.64. There is a connected proto-Lie group G whose Lie algebra L(G) is 2 ℵ0 algebraically isomorphic to Rℵ0 , while L(G) ∼ = R2 . Proof. let V be the vector space RN and let θV : V → V ∗∗ be the natural morphism into its algebraic bidual, given by θV (v), ω = ω, v for v ∈ V , ω ∈ V ∗ . Let G be the additive group of V ∗∗ with the weak-∗ topology. Then G is a pro-Lie group and 2ℵ0

L(G) = V ∗∗ . Thus L(G) = R2 . Now let G = θV (RN ) with the subgroup topology of G. Then G is a proto-Lie group with L(G) algebraically isomorphic to Rℵ0 . Examples 9.65. Let ω : H → G1 be a group homomorphism between connected pro-Lie groups such that the graph def

def

G = {(g, ω(g)) : g ∈ G} ⊆ G = H × G1 def

is a proper dense subgroup. Then f = pr H |G : G → H is a bijective but nonopen def

morphism which extends to a surjective open morphism F = pr H G → H of pro-Lie groups. Specifically, let K denote either (a) R, or, (b) the discrete field Z(p) = Z/pZ for some prime p, and consider H = KN with its product topology, in Case (a) a proLie group agreeing with L(H ) (up to a natural isomorphism), in case (b) a compact totally disconnected vector space over the field of p elements. Let G1 = K and let ω : H → G1 be a discontinuous K-linear form. Such a linear map certainly exists, N is isomorphic to K(N) ∼ since the topological dual K = R(ℵ0 ) , while the algebraic dual 2ℵ0

(KN )∗ is isomorphic to K2 . Now the graph G of ω in H × G1 is a K- hyperplane, being the image of KN × {0} under the linear automorphism (x, y)  → (x, y + ω(x))

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415

of KN × K. Since ω is discontinuous, G is dense, since hyperplanes in a topological vector space are either closed or dense. Case (a): G = L(G), as the additive group of a vector subspace of a weakly complete vector space, is a connected proto-Lie group which is not a pro-Lie group. G does not have a stable Lie algebra. H = L(H ) is a connected pro-Lie group. The morphism f : G → H is bijective and not open. Case (b): G is a precompact protofinite (hence proto-Lie) group but is not almost connected, while H is an abelian compact totally disconnected, hence almost connected, group. By default, both L(G) and L(H ) are singleton and so G has a stable Lie algebra and L(f ) is an isomorphism. The morphism f : G → H is continuous bijective and not open. It is worth noting in passing, that the construction in Example 9.65 yields an interesting example in the world of finite dimensional Lie groups: We may take H = G1 = R and find a bijective Q-linear map ω : R → R that is not a multiplication by a real number. This can even be done in such a fashion that the graph G of ω is a connected subgroup of R2 . (For the existence of such an f see [124].) Here G is a connected abelian topological group which is arcwise totally disconnected, causing L(G) to be zero; its completion is R2 , but it is certainly not a proto-Lie group because it has no small subgroups and is not a Lie group. We have L(G) ∼ = R2 , L(H ) ∼ = R.

Unitary Representations Let U denote the group of unitary operators of a given Hilbert space H with the strong operator topology, that is, the topology of pointwise convergence. Let G be a pro-Lie group and N (G) the filter basis of normal subgroups N such that G/N is a Lie group. For each N ∈ N (G) there is a Hilbert space HN and a faithful representation λN : G/N → UN for the unitary group UN on HN . Indeed we may take for HN the space L2 (G/N, C) of square integrable functions on G with respect to a left invariant Haar measure and define λN (g)(f )(x) = f (g −1 x); this is the so-called left-regular representation of G/N. Define πN : G → UN by composing the quotient morphism G → G/N with λN , that is, πN (g) = λN (gN ). Then ker πN = N . Now let H = N ∈N (G) HN be the Hilbert space direct sum of the family of the Hilbert spaces HN , N ∈ N (G). There is a natural injection i : N ∈N (G) UN → U given by i((TN )N ∈N (G) )(xN )N ∈N (G) ) = (TN (xN ))N ∈N (G) def

for (xN )N∈N (G) ∈ N ∈N (G) HN . Clearly T = i((TN )N ∈N (G) ) is a unitary operator on H, and so i is well defined and is readily seen to be an isomorphism onto its image. Theorem 9.66 (Theorem on the Existence of Faithful Unitary Representations). Every pro-Lie group has a faithful unitary representation.

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9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups

 Proof. Let G be a pro-Lie group. First define  f : G → N ∈N (G) UN by f (g) = (λN (g))N∈N (G) . Recall the embedding i : N ∈N (G) UN → U, where U is the unitary group of the Hilbert space H defined in the preceding paragraph. Now we set π : G → U,

π = i  f.

Then π is a morphism of topological groups and is a unitary representation, and ker π = ker f = {1}.

Postscript One of the most significant aspects of classical Lie theory of a finite-dimensional real Lie group G and its Lie algebra g is the correspondence between the set of subalgebras h of g and the set of subgroups H of G that correspond to them via the map h  → expG h and the inverse function H  → L(H ) = Hom(R, H ). Historically, one learnt that while every closed connected subgroup H of G corresponded to a Lie subalgebra in this way, but that not every subalgebra of g is so obtained. However, every Lie subalgebra h corresponds to what is called an analytic subgroup A(h) = expG h of G, the group that is algebraically generated by expG h. One shows that there is another and unique topology on A(h) making it into a Lie group H such that the inclusion morphism j : H → G induces a morphism L(j ) of Lie algebras L(H ) → L(G) with im L(j ) = h. Frequently, j is not an isomorphism of topological groups onto the image. Specifically, j fails to be an isomorphism onto its image exactly when the Lie group topology on A(h) is properly finer than the induced topology, and this is the case iff A(h) is not closed in G. One has to admit that the situation for analytic subgroups is awkward in that they have two often distinct topologies, the topology they inherit as a subgroup of a Lie group, and their Lie group topology. However, this awkwardness is more than compensated for by having the correspondence between subgroups and subalgebras. For pro-Lie groups and pro-Lie algebras we seek a correspondence between closed subalgebras of the pro-Lie algebra and certain subgroups of the pro-Lie group. We are able to define analytic subgroups of a pro-Lie group G such that for every analytic subgroup H of G, the set L(H ) is a closed Lie subalgebra of the Lie algebra L(G) of G, and every closed Lie subalgebra of L(G) occurs in this fashion. To be precise, a subgroup H of a pro-Lie group G is called an analytic subgroup of G if it is the image under a strict morphism of pro-Lie groups in G. The correspondence is not immediately as nice as we might hope, as distinct analytic subgroups can and often do correspond to the same subalgebra. However, we define minimal analytic subgroups, and these have a one-to-one correspondence with closed subalgebras. The behavior of analytic subgroups with respect to automorphisms is described as follows: If H is an analytic subgroup of a pro-Lie group G and α is an automorphism of G whose induced automorphism of the Lie algebra of G leave the Lie subalgebra of H invariant, then α(H ) = H if H is minimal analytic or is closed in G. This allows us to deal with centralizers and normalizers of analytic subgroups

Postscript

417

in a way that might be expected from the theory of finite-dimensional Lie groups. In particular, under either of these two conditions the normalizer of H is closed whether or not H itself is closed. If G is a simply connected Lie group, then a connected normal analytic subgroup is closed and simply connected and G/N is a simply connected pro-Lie group. While the center is treated with expected results, the relation between the commutator algebra and the commutator group is more complicated than it already is in the finite-dimensional theory. Yet some significant aspects remain: If G is a connected pro-Lie group then the closed commutator subgroup is a closed analytic group which is the closure of the minimal analytic subgroup whose Lie subalgebra is the closed commutator subalgebra of the Lie algebra of G. We do not discuss topological dimension here. But saying that a pro-Lie group is finite-dimensional iff the vector space dimension of its Lie algebra is finite gives us a preliminary definition of finite-dimensionality that is going to be consistent with all sensible topological definitions of dimension. Accepting this definition we have shown here that a finite-dimensional connected pro-Lie group G is not too far from a canonically attached simply connected Lie group L; indeed G is locally compact metric and there is a central totally disconnected compact subgroup  of G such that G is a quotient group of  × L modulo a discrete central subgroup. The image of L in G is the unique minimal analytic subgroup A(g, G) of G which has the same Lie algebra as G and L. This is the significance of the context of the theory of finite-dimensional pro-Lie groups and proper dense analytic subgroups having the full Lie algebra. That is the difference between Lie groups and finite-dimensional locally compact groups: Lie groups cannot have proper analytic subgroups having the full Lie algebra as their Lie algebra; finite-dimensional locally compact groups have such analytic subgroups if and only if they fail to be Lie groups. We give an example of an infinite-dimensional pro-Lie group G such that A(g, G) is a proper dense subgroup and G is center-free, that is, no such result as the one we quoted as determining the structure of finite-dimensional pro-Lie groups holds for infinite-dimensional ones. Every connected pro-Lie group is the strict projective limit of the projective system of all finite-dimensional quotients G/N0 where N ranges through our standard filter basis N (G) of closed normal subgroups such that G/N is a Lie group. The group G/N0 is a Lie group iff N/N0 is finite. The fact G = limN ∈N (G) G/N0 is used to show that every connected pro-Lie group has a unique largest compact central subgroup KZ(G) and we show that G/ KZ(G) has no nontrivial compact central subgroup. The center of a connected pro-Lie group without compact central subgroup is the direct product of a unique central vector subgroup isomorphic to RI for some set I and a prodiscrete group without compact subgroups. Abelian pro-Lie groups without compact subgroups always embed as a closed subgroup into some weakly complete vector group (see 5.10). We observe that every abelian pro-Lie group C in which comp(C) is compact occurs as the center of some connected pro-Lie group G, and indeed G may be taken to be metabelian, that is, solvable with abelian commutator subgroup. We use some of the insights gained from discussing finite-dimensional pro-Lie groups and the topics surrounding this discussion to prove two important results:

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– Firstly, we show that a connected pro-Lie group always contains a unique, hence characteristic, largest central compact subgroup, irrespective of its connectivity. – Secondly, we show that a surjective morphism from an almost connected pro-Lie group to a pro-Lie group is open, which entails that – Thirdly, a quotient G/N of a pro-Lie group G modulo a normal subgroup is a pro-Lie group if G is almost connected and N is the kernel of a morphism with a pro-Lie image group. None of these three results is trivial. Open Mapping Theorems are usually proved by some form of a Baire category argument and requires in all concrete cases metric completeness or local compactness. While pro-Lie groups are complete by definition, neither local compactness nor metrizability are guaranteed. Progress towards an Open Mapping Theorem proceeded piecemeal through preceding chapters, notably Chapter 4 and Chapter 8. The proof of the Open Mapping Theorem 9.60 is based in an essential way on the foundations and the Lie theory of pro-Lie groups. While we do not discuss unitary representation theory of pro-Lie groups in this book at length, we do show, rather effortlessly from first principles, that every pro-Lie group has a faithful unitary representation. This opens the gate to a unitary representation theory. In Exercise E4.7 we argued that the set of representations Hom(G, U) into a fixed unitary group contains a subset colimN ∈N (G) Hom(G/N, U) which contains all representations that factor through a unitary representation of one of the Lie groups G/N . If G is not itself a Lie group, then Theorem 9.66 provides an example of a representation that is not contained in colimN ∈N (G) Hom(G/N, U).

Chapter 10

The Global Structure of Connected Pro-Lie Groups

We have had first substantial insights into the global structure of connected pro-Lie groups: for instance the structure of connected abelian pro-Lie groups in Chapter 5 and the structure of simply connected pro-Lie groups in Chapter 8. The example of abelian pro-Lie groups shows that in these circumstances the structure theory is reduced to that of weakly complete vector groups on the one hand and to compact connected abelian groups on the other. All the true complications are identified to be in the compact portion, and for compact abelian groups we have a highly developed structure theory that was elucidated in Chapters 7 and 8 of [102]. The example of simply connected pro-Lie groups was very explicitly discussed in Chapter 8 with the result that, due to a bijective functorial correspondence between pro-Lie algebras and simply connected pro-Lie groups, our knowledge of the structure of simply connected pro-Lie groups is just as good as our knowledge of pro-Lie algebras. Moreover, in Chapter 6 we saw in Theorem 6.6 that every connected pro-Lie group G has a universal  → G from a simply connected pro-Lie group G  = (L(G)) into morphism πG : G G which induces an isomorphism of Lie algebras and which, in the finite-dimensional case is the universal covering morphism. Yet the example of compact connected abelian groups shows that this universal morphism may not be as immediately revealing as a universal covering morphism. Thus a good deal of global structure theory remains to be uncovered, notably as far as the role of compact groups is concerned. This chapter is devoted to this global structure theory. As an example of a task to be done we point out that in Chapter 7 we have a good deal of algebraic theory of what, in the absence of finite-dimensionality, the solvability and nilpotency of (topological) Lie algebras had to mean, and how the appropriate concepts contracted to the concept of prosolvability and pronilpotency in the case of pro-Lie algebras. We further recall that we used this information in Chapter 8 for the structure theory of simply connected Lie groups without ever discussing solvability and nilpotency on the group level. Therefore, from a vantage point of group theory, one of the tasks to address is that of identifying the issue of solvability and nilpotency and its Lie theory for pro-Lie groups in a general vein. We shall, of course, be guided by what we already know in the case of pro-Lie algebras. From Chapter 9, in which we discussed the Lie theory of commutator groups (see 9.25 ff.) we know that this cannot be an entirely trivial matter; after all: the Lie theory of commutator subgroups is not simple even in the finite-dimensional case. Prerequisites. We have reached a point where almost everything that has been discussed in earlier chapters of the book is coming together to be used. On the other hand, no further outside information is being fed into this chapter.

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Solvability of Pro-Lie Groups The commutator subgroup G of a group G is the subgroup which is algebraically generated by all commutators comm(g, h) = ghg −1 h−1 , g, h ∈ G. The group G is abelian if and only if G = {1}. For arbitrary subsets A and B of G, we let [A, B] denote the subgroup generated by all commutators comm(a, b), a ∈ A and b ∈ B. Definition 10.1. Let G be a group. Set G(0) = G and define sequences of subgroups G(α) indexed by the ordinals α, card α ≤ card G via transfinite induction. Assume that G(α) is defined for α < β.  (i) If β is a limit ordinal, set G(β) = α<β G(α) . (ii) If β = α + 1, set G(β) = [G(α) , G(α) ]. For cardinality reasons, there is a smallest ordinal γ such that G(γ +1) = G(γ ) . Set = G(γ ) . Let ω denote the first infinite ordinal. Then G is said to be transfinitely solvable, if G(∞) = {0}. If G is transfinitely solvable and γ ≤ ω, then G is called countably solvable. If γ is finite and G(γ ) = {0}, then G is called solvable. If γ = 0, that is, [G, G] = G, then G is called perfect. G(∞)

Exercise E10.1. Prove the following observations: Proposition. A subgroup of a transfinitely solvable group is transfinitely solvable. A transfinitely solvable group does not contain a nontrivial perfect subgroup. [Hint. Assume that H ≤ G and that G(γ ) = {1}. Prove by transfinite induction, that H (α) ⊆ G(α) for all α. This is certainly true for α = 0; assume that it is true for α < β. ] ⊆ [G(α) , G(α) ] = G(β) . Indeed, if β = α + 1 then H (β) = H (α+1) = [H (α) , H (α)   Assume now that β is a limit ordinal. Then H (β) = α<β H (α) ⊆ α<β G(α) = G(β) . This completes the induction. If H ≤ G, further H is perfect, and G is transfinitely solvable, then H = H (α) ≤ G(α) for all α and thus H = G(γ ) = {1}.] Definition 10.2. Let G be a group. A sequence of normal subgroups (Nα )α≤ρ indexed by an initial section of ordinals is said to be an abelian sequence of normal subgroups if it is descending and the following conditions hold. Let β < ρ.  (i) If β is a limit ordinal, then Nβ = α<β Nβ . (ii) If β = α + 1, then Nβ /Nα is abelian. A sequence (Nα )α≤ρ of normal subgroups is said to be terminating if it is descending and Nρ = {1}. The sequence (G(α) )α≤γ is an abelian sequence of normal subgroups; it is that abelian sequence of normal subgroups which descends fastest. Indeed we have Lemma 10.3. Let (Nα )α≤ρ be an abelian sequence of normal subgroups of G. Then (∀α ≤ ρ) G(α) ⊆ Nα .

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Proof. We prove the assertion by transfinite induction. Let assume that  β ≤ ρ and  G(α) ⊆ Nα for α < β. If β is a limit ordinal, then G(β) = α<β G(α) ⊆ α<β Nα = Nβ . If, however, β = α + 1, then G(α) ⊆ Nα and there is homomorphism ϕ : G(α) → Nα /Nβ given by ϕ(x) = xNβ . Since the range of ϕ is abelian, G(β) = [G(α) , G(α) ] ⊆ ker ϕ = Nβ . This completes the induction. Clearly G is transfinitely solvable if (G(α) )α≤ρ is terminating. If so, then γ ≤ ρ. Proposition 10.4. For a group G, the following statements are equivalent: (i) G is transfinitely solvable. (ii) G has a terminating abelian sequence of normal subgroups. If (Nα )α≤ρ is a terminating sequence of closed normal subgroups, then G(ρ) = {1}. Proof. Since clearly (i) ⇒ (ii) we must prove that (ii) implies (i). Let (Nα )α≤ρ be a terminating abelian sequence of closed normal subgroups. By Lemma 10.3 we have G(ρ) ⊆ Nρ = {1}, and thus the sequence (G(α) )α≤ρ is terminating; therefore G is transfinitely solvable. As we unfold the definitions for solvability, it is efficient to prepare for the subsequent section dealing with nilpotency by putting the definitions of solvability and nilpotency side by side, much as we have done in dealing with pro-Lie algebras in Chapter 7. Definition 10.5. Let G be a group. Set G[0] = G and define sequences of normal subgroups G[α] indexed by the ordinals α, card α ≤ card G via transfinite induction. Assume that G[α] is defined for α < β.  (i) If β is a limit ordinal, set G[β] = α<β G[α] . (ii) If β = α + 1, set G[β] = [G, G[α] ]. For cardinality reasons, there is a smallest ordinal δ such that G[δ+1] = G[δ] . Set = G[δ] . Then G is said to be transfinitely nilpotent, if G[∞] = {0}. If G is transfinitely nilpotent and δ ≤ ω, then G is called countably nilpotent. If δ is finite and G[δ] = {0}, then G is called nilpotent.

G[∞]

Since G(α) ⊆ G[α] , any transfinitely nilpotent Lie group is transfinitely solvable. The sequence is said to be a nilsequence of normal subgroups if it is descending and condition (i) above and the following conditions are satisfied:  (i ) If β is a limit ordinal, then Nβ = α<β Nβ . (ii ) If β = α + 1, then Nα /Nβ is central in G/Nβ . The sequence G[α] as α ranges through the ordinals (whose cardinality is bounded by the cardinality of G), is a nilsequence of normal subgroups. In fact, it is the nilsequence descending fastest:

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Lemma 10.6. Let (Nα )α≤ρ be a nilsequence of normal subgroups of G. Then (∀α ≤ ρ) G[α] ⊆ Nα . Proof. We prove the assertion by transfinite induction. Let β ≤ ρ and assume that G[α] ⊆ Nα for α < β. If β is a limit ordinal, then we proceed as in the proof of Lemma 10.3. If β = α + 1, then Nα /Nβ is central in G/Nβ by Definition 10.2 (ii ). In other words, [G, Nα ] ⊆ Nβ . Then G[β] = [G, G[α] ] ⊆ [G, Nα ] ⊆ Nβ . The transfinite induction is complete. Proposition 10.7. For a group G, the following statements are equivalent: (i) G is transfinitely nilpotent. (ii) G has a terminating nilsequence of normal subgroups. Proof. Exercise. Exercise E10.2. Provide the details of the proof of Proposition 10.7. [Hint. Emulate the proof of Proposition 10.4 using Lemma 10.6 this time.] Next let us turn to the topological variants of solvability and nilpotency! Definition 10.8. Let G be a subgroup of a topological group H . (For instance, H = G.) Set G((0)) = G and define sequences of normal subgroups G((α)) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that G((α)) is defined for α < β.  (i) If β is a limit ordinal, set G((β)) = α<β G((α)) . (ii) If β = α + 1, set G((β)) = [G((α)) , G((α)) ]. For cardinality reasons, there is a smallest ordinal γ such that G((γ +1)) = G((γ )) . Set G((∞)) = G((γ )) . Let ω denote the first infinite ordinal. Then G is said to be transfinitely topologically solvable, if G((∞)) = {1}. If g is transfinitely topologically solvable and γ ≤ ω, then G is called countably topologically solvable. If γ is finite and G((γ )) = {0}, then G is called topologically solvable. If γ = 0, that is, [G, G] = G, then G is said to be topologically perfect. Definition 10.9. Let G be a subgroup of a topological group. Set G[[0]] = G and define sequences of closed normal subgroups G[[α]] indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that G[[α]] is defined for α < β.  (i) If β is a limit ordinal, set G[[β]] = α<β G[[α]] . (ii) If β = α + 1, set G[[β]] = [G, G[[α]] ].

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For cardinality reasons, there is a smallest ordinal δ such that G[[δ+1]] = G[[δ]] . Set G[[∞]] = G[[δ]] . Then G is said to be transfinitely topologically nilpotent, if G[∞] = {0}. If G is transfinitely topologically nilpotent and δ ≤ ω, then G is called countably topologically nilpotent. If δ is finite and G[[δ]] = {0}, then G is called topologically nilpotent. Lemma 10.10. Let G be a topological group and let A and B subsets of G. Then (i) [A, B] = [A, B], & ' & ' (ii) [G, G], [G, G] = [G, G], [G, G] , ' & ' & (iii) G, [G, G] = G, [G, G] , (iv) G(n) = G((n)) for all n ∈ N, (v) G[n] = G[[n]] for all n ∈ N. Assume that G is a subgroup of a topological group H ; then (vi) G((α)) = G((α)) for all α, (vii) G[[α]] = G[[α]] for all α. Proof. (ii) and (iii) are consequences of (i), and (iv) and (v) follow by finite induction from (i). By definition we have G((0)) = G = G = G((0)) , and similarly G[[0]] = G[[0]] . Now (i) implies G(1)) = G((1)) and then G((α)) = G((α)) from there on out. Similarly G[[α]] = G[[α]] . Thus (i) implies (vi) and (vii). So we have to establish (i). Obviously the left side of (i) is contained in the right side. The function f : G × G → G given by f (a, b) = [a, b] is continuous since G is a topological group. Then f (A × B) = f (A × B) ⊆ f (A × B), and this set is contained in the closed subgroup [A, B] generated by f (A × B). Thus the closed subgroup [A, B] is contained in [A, B]. Proposition 10.11. Let G be a topological group. (i) If G is transfinitely topologically solvable, then G is transfinitely solvable. If G is countably topologically solvable, then G is countably solvable. (ii) If G is transfinitely topologically nilpotent, then G is transfinitely nilpotent. If G is countably topologically nilpotent, then G is countably nilpotent. (iii) G is solvable if and only if it is topologically solvable. (iv) G is nilpotent if and only if it is topologically nilpotent. (v) If G is a subgroup of a topological group H , then G is transfinitely topologically solvable iff G is transfinitely topologically solvable, and G is transfinitely topologically nilpotent iff G is transfinitely topologically nilpotent. Proof. (i) Assume that G is transfinitely topologically solvable. The sequence (G((α)) )α≤γ is a terminal abelian sequence of (closed) normal subgroups. Hence G is transfinitely solvable by 10.4.

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(ii) Exercise. (iii) This is a consequence of 10.10 (iv). (iv) Exercise. (v) By 10.10 (vi) we have G((α)) = G((α)) for all ordinals, and so G((∞)) = {1} iff ((∞)) G = {1}. The nilpotent case is analogous. Exercise E10.3. Fill in the proofs of Proposition 10.11 (ii) and (iv). [Hint. Use 10.7, respectively, 10.10 (v).] The case of pro-Lie groups. Of course, in this book we are interested in pro-Lie groups. In Chapter 7 we were rather successful in dealing with the appropriate concepts of solvability and nilpotency for pro-Lie algebras. As a consequence we anticipate that we might be successful on the group level in dealing with solvability and nilpotency of connected pro-Lie groups. First the appropriate concepts for pro-Lie groups. Definition 10.12. A pro-Lie group G is called prosolvable if every (finite-dimensional) quotient Lie group G/N, N ∈ N (G) is solvable. It is called pronilpotent if every (finite-dimensional) quotient Lie group G/N , N ∈ N (G), is nilpotent. Lemma 10.13. (i) Let G be a pro-Lie group and N a closed normal subgroup. Further let F be a filter basis of closed subgroups. Then N



F =



NH.

(∗)

H ∈F

(ii) If, in addition, G is connected and G/N is a simply connected Lie group and the members of F are closed connected normal subgroups, then   N H. (∗∗) N F = H ∈F

Proof. (i) For  a filter basis of closed subsets such as F or {N H : H ∈ F }, the intersection F is at the same time the set of points of adherence, that is, the set of all x such that for all identity neighborhoods U of G we find some H ∈ F such that xU ∩ H = Ø, respectively xU ∩ NH . Therefore, the left side of (∗) consists of all pointsx ∈ G such that for each identity neighborhood U there are elements n ∈ N and d ∈ F such that nd ∈ xU , and d is characterized by saying that for each identity neighborhood V in G there is an H ∈ F such that dV ∩ H =  Ø, that is, that dv = h for some v ∈ V and h ∈ H . This amounts to saying that x ∈ N F iff for each identity neighborhood W there is an n ∈ N and an H ∈ F and an h ∈ H such that xw = nh for some w ∈ W ; indeed if W is given we find identity neighborhoods V and U such that V U ⊆ W , and thus dv = h and nd = xu gives nh = xw with w = uv ∈ U V ⊆ W .

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The right hand side of (∗) consists of all points x ∈ G such that for each identity neighborhood U there is an H ∈ F and an h ∈ H and an n ∈ N such that xu = nh for some u ∈ U . In this fashion it becomes evident that (∗) is satisfied. (ii) Now assume that G is connected and that G/N is a simply connected Lie group. All normal analytic subgroups of a simply connected Lie group are closed. Hence every normal analytic subgroup of G containing N is the full inverse image under the quotient morphism G → G/N of a normal analytic subgroup of G/N and thus is closed. Thus (∗∗) is equivalent to (∗) in the present situation. Theorem 10.14. Assume that G is a pro-Lie group. (i) Then for each closed normal subgroup N of G, (G/N )((α)) = G((α)) N /N.

(∗)

(ii) Every quotient of a transfinitely topologically solvable pro-Lie group is transfinitely topologically solvable. (iii) Assume that G is a connected transfinitely topologically solvable pro-Lie group. Then G is prosolvable. Proof. (i) We prove (∗) by transfinite induction. Assume the assertion to be proved for  all ordinals α < β. If β = α + 1, then (G/N )((β)) = (G/N )((α)) = (G((α)) N )/N ) = 

(G((α)) ) N)/N = (G((α)) N )/N = G((α+1)) N /N = G((β)) N /N. This completes the induction in this case. Now assume that β is a limit ordinal. Then     ((α)) N /N = (G/N)((β)) = α<β (G/N )((α)) = α<β G((α)) N /N = α<β G  N α<β G((α)) /N = G((β)) N/N by the definitions and by (∗) in the preceding Lemma 10.13 (ii). This completes the induction. Thus (∗) is proved. (ii) If G is transfinitely topologically solvable, we have G((γ )) = {1}, and thus (G/N )((γ )) = G((γ )) N /N = {1}, by (i) above. This shows that G/N is transfinitely topologically solvable. (iii) In the circumstances of (ii), if G/N is a connected Lie group, then G/N is transfinitely topologically solvable on the one hand and satisfies the descending chain condition for closed connected subgroups and thus is topologically solvable and so is solvable by 10.10 (iv). Proposition 10.15. Let G be a simply connected pro-Lie group and g = L(G) its Lie algebra. Then L(G((α)) ) = g((α)) and G((α)) ∼ = (g((α)) )

for all ordinals α.

In particular, G is transfinitely topologically solvable, respectively, countably topologically solvable, respectively, topologically solvable if and only if g is transfinitely topologically solvable, respectively, countably topologically solvable, respectively, topologically solvable.

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Proof. Assume for the moment that we have established L(G((α)) ) = g((α)) . A closed connected normal subgroup of a simply connected pro-Lie group is simply connected by Corollary 8.16. Hence G((α)) is a simply connected pro-Lie group and thus may be identified with (L(G((α)) )) = (g((α)) ) by Theorem 6.6 (iii) and Theorem 8.15. It remains therefore to prove the first assertion. This is shown by transfinite induction. The claim is true for α = 0, 1 by 9.27 (i). Assume that the claim holds for all β < α. First  assume that αis a limit ordinal. Since L preserves intersections, L(G((α)) ) = L( β<α G((β)) ) = β<α L(G((β)) ) =  ((β)) = g((α)) in view of the definitions of G((α)) and g((α)) . β<α g Next assume that α = β + 1. Then we apply 9.27 (i) to derive L(G((α)) ) = L((G((β)) ) ˙ ) = L(G((β) )) ˙ = g((β)) ˙ = g((α)) . It follows that G((γ )) = {1} iff g((γ )) = {0}. In a simply connected pro-Lie group G = (g), the analytic subgroup (g((α)) ) may also be written A(g((α)) ). With this terminology, the result in the preceding proposition may also be written (∀α)

G((α)) = A(g((α)) ) = A(g((α)) ).

We shall have more to say about this equation in the more general case of connected but not necessarily simply connected pro-Lie groups later in Theorem 10.20. Remark 10.16. Let G be a connected pro-Lie group and g = L(G) its Lie algebra. Then L(G((α)) ) ⊇ g((α)) for all ordinals α. In particular, if G is transfinitely topologically solvable, respectively, countably topologically solvable, respectively, topologically solvable then g is transfinitely topologically solvable, respectively, countably topologically solvable, respectively, topologically solvable. Proof. We recall Lemma 9.25 (∗) and note that the proof is analogous to the proof of Proposition 10.15. Exercise E10.4. Provide the details of the proof of Remark 10.16. Theorem 10.17. Let G be a connected pro-Lie group. (i) If N ∈ N (G) is a closed normal subgroup such that dim G/N is a ( finite-dimensional) Lie group, then for every limit ordinal β there is an αN < β such that for all α the relation αN ≤ α < β implies G((β)) N = G(α) N .

(∗∗)

(ii) γ ≤ ω, that is G((ω+1)) = [G((ω)) , G((ω)) ] = G((ω)) . (iii) For every pro-Lie group G, the characteristic subgroup G((ω)) is topologically perfect and for each N ∈ N (G) there is a natural number nN such that for all natural numbers n satisfying nN ≤ n we have G((ω)) N = G(n) N

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427

and [G((ω)) , G((ω)) ]N = G((ω)) N. Proof. (i) Let N ∈ N (G) be a closed normal subgroup of G such that G/N is a Lie group. Since G is connected, G/N is connected. Then the descending chain (G(α) N)α<β of pullbacks of the closed connected subgroups G(α) N /N has a smallest element, say G(αN ) N for a smallest ordinal αN < β. Then (#) G(αN ) N = G(α) N for all α with αN ≤ α < β. We claim that G((α)) ⊆ G(α) N for all ordinals α. For α = 0 this is trivially true. Let us assume that this claim holds for α, we shall show that it holds for β = α + 1. Indeed, G((β)) = [G((α)) , G((α)) ] ⊆ [G(α) N, G(α) N ] ⊆ [G(α) , G(α) ]N = G(β) N . So induction works for the step from α to α +1. Now assume that β is a limit ordinal and that the claim is true for all α < β. Then from the definition of G((α)) by transfinite     induction we have G((β)) = α  <β G((α )) ⊆ α  <β (G(α ) N ) = G(α) N for all α satisfying αN ≤ α < β by the induction hypothesis and by (#). Thus for all these α we       obtain G((β)) N ⊆ G(α) N = α  <β G(α ) N = N α  <β G(α ) ⊆ N α  <β G((α )) = G((β)) N by 10.13 (i)(∗) and the inductive definition of G((β)) . Thus equality holds throughout and this is the assertion (∗∗) for all α satisfying αN ≤ α < β. (ii) Let β be any limit ordinal and let N ∈ N (G) be any closed normal subgroup N ∈ N (G). For these data we determine αN as in (i) above. In view of 10.10 (i), we then compute [G((β)) , G((β)) ]N = [G((β)) N, G((β)) N]N = [G((β)) N , G((β)) N ]N = [G(αN ) N, G(αN ) N]N = [G(αN ) N, G(αN ) N ]N = [G(αN ) , G(αN ) ]N = G(αN +1) N = G(αN ) N = G((β)) N . For fixed β we form the intersection over all N ∈ N (G) on both sides and find G((β)) on the right side since this is a closed subset of G, and [G((β)) , G((β)) ] = G((β+1)) on the left. Thus G((β+1)) = G((β)) for all limit ordinals β. In particular, this applies to β = ω. (iii) This is an immediate consequence of (ii) and (i). In the following theorem we see that for connected pro-Lie groups, luckily, all the various reasonable concepts for infinite solvability agree. A completely analogous phenomenon we have seen for pro-Lie algebras in Chapter 7 (see the Equivalence Theorem for Solvability 7.53). The Equivalence Theorem for Solvability of Connected Pro-Lie Groups Theorem 10.18. Let G be a connected pro-Lie group and g its Lie algebra L(G). Then the following assertions are equivalent:

428 (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

10 The Global Structure of Connected Pro-Lie Groups

G is countably topologically solvable. G is transfinitely topologically solvable. G is prosolvable. g is prosolvable. G is countably solvable. G is transfinitely solvable. G does not contain a finite-dimensional analytic simple subgroup. g does not contain a finite-dimensional simple Lie algebra.

Proof. The implication (i) ⇒ (ii) is trivial. The implication (ii) ⇒ (iii) was shown in 10.14. (iii) ⇒ (i): Let N ∈ N (G). Then since G/N is solvable by is a natural  (iii), there ((n)) ⊆ N . Thus number msuch that G(n) ⊆ N for all n ≥ m. Then G((ω)) = ∞ G n=1 G((ω)) ⊆ N (G) = {0}. This proves (i). Thus (i), (ii), and (iii) are equivalent. (iii) ⇔ (iv): Condition (iii) means that for each N ∈ N (G) the connected Lie group G/N is solvable. This is the case if and only if L(G/N ) is solvable. By Corollary 4.21 to the Strict Exactness Theorem for L we have L(G/N ) ∼ = L(G)/L(N ) canonically and {L(N) : N ∈ N (G)} is cofinal in the set  (g) of all ideals j of g such that g/j is finite-dimensional. Thus L(G/N ) is solvable for all N ∈ N (G) iff g/j is solvable for all j ∈  (g) iff g is prosolvable, and this is Condition (iv). The implication (i) ⇒ (v) was proved in 10.11 (i). The implication (v) ⇒ (vi) is trivial. (vi) ⇒ (vii): If S is a simple analytic subgroup, then the Lie algebra L(S) is perfect, that is agrees with its commutator subgroup and thus S is perfect. Thus the implication follows from Exercise E10.1. (vii) ⇒ (viii): Suppose that s is a finite-dimensional simple subalgebra of g. Then s ∩ r(g) = {0} and thus g/r(g) = {0} for the radical r(g) (see Chapter 7). Hence the Levi summands of g are not trivial by the Levi Mal’cev Theorems 7.52 and 7.77. By the Structure Theorem of Simply Connected Pro-Lie Groups 8.14, the simply connected  = (g) contains at least one simple simply connected finite-dimenpro-Lie group G  → G the canonical morphism (see Chapter 6). Then sional subgroup  S. Let ϕG : G def S) is a simple analytic finite-dimensional subgroup of G. Thus (vii) fails. S = πG ( The equivalence (viii) ⇔ (iv) is a small portion of the Equivalence Theorem for Solvability 7.53. The Equivalence Theorem for Solvability of Connected Pro-Lie Groups 10.18 can at once be complemented by the equivalent statements characterizing the prosolvability of the Lie algebra g as specified in the Equivalence Theorem for g in 7.53. So far we have no explicit description of the Lie theory of the commutator sequences of pro-Lie groups. Even in the case of finite-dimensional connected Lie groups this is a somewhat delicate issue due to the fact that the algebraic commutator group of a connected Lie group is not closed in general; at least it is an analytic subgroup. In the case of simply connected pro-Lie groups, the situation is comparatively clear as

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was shown in Proposition 10.15. Remark 10.16 gave some indication of the general direction. The following observations give more insight for countable commutator series. We observe that we have no general insight into the Lie algebras of the characteristic proLie subgroups G((α)) . We recall from 10.10 (iv) that for natural numbers n we have G(n) = G((n)) . Proposition 10.19. Let G be a connected pro-Lie group with Lie algebra g = L(G). Then for all natural numbers m the following conclusion holds L(G((m)) ) = g((m+1)) .

m

Proof. We shall prove this by induction with respect to m. Thus assume that m holds for all m < n and show that n holds. So let n = m + 1. Set h = g((n)) and let K = G((m)) and k = L(K). We assume m; this means k˙ = L(K) = h. By ˙ Now K˙ = G((n)) ˙  = k¨ = h. Corollary 9.35, applied with K in place of G, we get L(K) and h˙ = g((n+1)) and so L(G((n)) ) = g((n+1)) , and this completes the induction. Before we formulate the next major theorem we recall from Lemma 7.44 (iv) and from Theorem 10.17 (ii) that for any connected pro-Lie group G and any pro-Lie algebra g the smallest ordinals γ G and γ g such that the equations G((γ G +1)) = G((γ G )) , respectively, g((γ g +1)) = g((γ g )) hold, satisfy γG ≤ ω

and γ g ≤ ω.

(1)

Theorem on the Commutator Series of Pro-Lie Groups Theorem 10.20. Let G be a connected pro-Lie group. Then G((α)) = A(g((α)) ) for all ordinals α.

(2)

Proof. For α = 0 there is nothing to prove, and for natural numbers α = m we prove the claim by induction. We assume that the claim holds for m and abbreviate g((m)) by h; note that so G((m)) = A(h). Recall that ih : (h) → G with image A(h) is  = (g) → G (see Definition 9.9ff., notably the restriction of the morphism πG : G Proposition 9.10 (i)). For n = m + 1 we compute that G((n)) = [G((m)) , G((m)) ] = [A(h), A(h)] = [A(h), A(h)] = [ih ((h)), ih ((h))] (by Definition 9.9) = ih [(h)), (h)] = ˙ (by Proposition 9.27 (i)) = i ˙ ((h)) ˙ (by Proposiih ([(h)), (h)]) = ih ((h)) h

˙ = A(g((m+1)) ) = A(g((n)) ). This concludes the proof for natural tion 9.10 (i)) = A(h) number α. In view of the result (1) which precedes the theorem, it now remains to prove the claim for the limit ordinal α = ω.

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10 The Global Structure of Connected Pro-Lie Groups def

From Theorem 10.17 (iii) we know that H = G((ω)) is topologically perfect for any connected pro-Lie group G. Abbreviate A(g((ω)) ) by N. Then L(N ) ⊇ L(A(g((ω)) ) = g((ω)) . From Corollary 4.21 (i) of the Strict Exactness Theorem for L we have L(G/N ) ∼ = L(G)/L(N ) as a homomorphic image of the factor algebra g/g((ω)) which is topologically countably solvable by Lemma 7.44 (i). Hence G/N is topologically countably solvable by Theorem 10.18. The subgroup G((ω)) /N is isomorphic to (G/N )((ω)) by Lemma 10.14 (i) and thus is topologically perfect. But then this quotient is singleton by Exercise E10.1. It follows that G((ω)) = N = A(g((ω)) ), and this is the claim. The algebraic commutator subgroup G of a connected Lie group G is the analytic subgroup A(g ) which is generated by the algebraic commutator subalgebra g of the Lie algebra g = L(G). In the more general case of a pro-Lie group, the best thing we can say is that G ⊆ A(˙g)N for any N ∈ N (G) and thus G ⊆ A(g) ˙ (see 9.25 (ii)). On the surface it would have been conceivable, that there is a big discrepancy between the topological commutator groups G((m)) and the next best thing we have to be close to the algebraic commutator groups, namely, the minimal analytic subgroups A(g((m)) ) corresponding to the topological commutator algebras g((m)) , and indeed that this discrepancy might increase as m grows. That this in fact is not the case, is the content of the preceding theorem, saying that the discrepancy is not more than the topological closure. In particular, the minimal analytic subgroup A(g((m))) with Lie algebra g((m)) is dense in G((m)) . Thus by 9.33, L(G((m)) ) ˙ = (g((m)) ) = g((m+1)) ⊆ L(G((m+1)) ). From Theorem 9.32 (i) and 10.20 (2) above we know that L(G((α+1)) ⊆ g((α+1)) ; thus, by induction, we have the equality L(G((m)) ) = g((m)) for all m ∈ N. From this theorem we can easily deduce once again the result that is contained in the Equivalence Theorem 10.18: Proposition 10.21. Let G be a connected pro-Lie group and g = L(G) its Lie algebra. Then G is transfinitely topologically solvable if and only if g is transfinitely topologically solvable. Proof. By Theorem 10.20 we have (∀α)

G((α)) = A(g((α)) ).

(∗)

If g is transfinitely topologically solvable, then there is an ordinal δ such that g((δ)) = {0}. By 9.10 (iii) we have L(A(h)) = h and A(h) = exp h for all closed subalgebras h of g. Then (∗) implies G((δ)) = {1}, that is, G is transfinitely topologically solvable. Conversely, if there is an ordinal δ such that G((δ)) = {1}, then (∗) implies A(g((δ)) ) = {1} and this entails g((δ)) = 0, that is, g is transfinitely topologically solvable.

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The Radical We know from Theorem 7.48 that every pro-Lie algebra g has a unique largest prosolvable ideal r(g) which is closed such that g/r(g) is a product of simple finite-dimensional Lie algebras. We now attempt to show that a similar phenomenon occurs on the level of pro-Lie groups. Firstly we observe the following fact: Lemma 10.22. If N is a normal subgroup of a group G and both N and G/N are transfinitely solvable, then G is transfinitely solvable. Proof. There are ordinals γ and ν such that N (ν) = {1} and (G/N )(γ ) = {N }. We claim that for all ordinals α we have G(α) N/N ⊆ (G/N )(α) . This is proved by transfinite induction. Let us assume that the assertion holds for all (α) α < β. For ordinal β, for all α < β G(β) N ⊆ thus  a limit we have  G N and(α) (β) (α) β (α) G N ⊆ α<β G N, whence G N/N ⊆ α<β G N/N ⊆ α<β (G/N ) = (G/N)(β) . If β = α + 1, then G(β) N/N = [G(α) , G(α) ]N/N = [G(α) N/N, G(α) N/N] ⊆ [(G/N )(α) , (G/N )(α) ] = (G/N )(β) . Transfinite induction then proves the claim. Using the claim we get G(γ ) N/N ⊆ (G/N )(γ ) = {N } and thus G(γ ) ⊆ N . Also, (ν) N = {1}. Thus, if γ + ν denotes the ordinal sum, we get G(γ +ν) = {1}, and so G is transfinitely solvable. It is a natural reflex to claim that every  group G has a unique largest normal transfinitely solvable subgroup R(G), namely, {N ' G : N is transfinitely solvable}. A proof would proceed by showing that if M, N ' G are transfinitely solvable normal subgroups, then MN is transfinitely solvable: Firstly, MN is a normal subgroup and MN/M ∼ = N/(M ∩ N). Now M ∩ N is transfinitely solvable as a subgroup of N and one would like to argue that M is transfinitely solvable by hypothesis and one has the reflex to argue that then N/(M ∩ N) and thus MN/M is transfinitely solvable which would allow us to apply the preceding Lemma 10.22. However, what gives us pause is the fact that every free group is countably (hence transfinitely) nilpotent and thus transfinitely solvable, while every group, for instance, the simple, hence perfect group A5 of 120 elements is a quotient of a free group. Therefore we cannot pass to quotients and believe that transfinite solvability is preserved, in fact not even countable solvability is preserved in general. It is true that [MN, MN] = [M, M] · [N, N] · [M, N], verified by elementary calculations with commutators, but the hypothesis that M and N are transfinitely solvable does not yield any information on [M, N]; so again there is no immediate argument that a product of two normal transfinitely solvable subgroups is solvable.

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10 The Global Structure of Connected Pro-Lie Groups

The attempt to show that the set S of all transfinitely solvable subgroups is inductive goes awry because we come up against infinite distributivity: If (Cj )j ∈J is a chain in S indexed by a totally ordered set J and C its union; we would have to (β)  show that C (σ ) = {1}. But for a limit ordinal β we have C (β) = = j ∈J Cj     (α) α = f ∈J [0,β[ α<β Cf (α) by the infinite law of distributivity. Why α<β j ∈J Cj  (β) this should be j ∈J Cj is not to be seen even if β = ω. These remarks are meant to show that the existence of a radical is not at all a trivial proposition in the general environment of groups or topological groups. However, the topological concepts of solvability behave better than the algebraic ones as we have already seen in Theorem 10.14. The Equivalence Theorem for Solvability of Connected Pro-Lie Groups is a powerful tool to secure the existence of radicals at least in the context of pro-Lie groups. Now let G be a pro-Lie group. Let g = L(G) be its Lie algebra and r(g) the solvable radical of g according to Definition 7.46 and Theorem 7.48. From an observation preceding 7.47 we recall that r(g) is always closed in g. Definition 10.23. The closed subgroup expG r(g) will be denoted by R(G). This group is called the radical of G or, if more clarity is required, the solvable radical of the group G. Lemma 10.24. (i) Let G be a simply connected pro-Lie group with Lie algebra g. Then R(G) = A(r(g)) = expG r(g) and L(R(G)) = r(g). The radical R(G) is prosolvable.  = (g) → G the canonical (ii) Let G be a connected pro-Lie group and πG : G  and L(πG ) maps morphism constructed in Chapter 8. Then R(G) = πG (R(G))  isomorphically onto L(R(G)) = r(g). L(R(G)) (iii) For every pro-Lie group G, the radical is a closed connected prosolvable subgroup. Proof. (i) Since G is a simply connected pro-Lie group, A(r(g)) = (r(g)) in view of Definition 9.9 and the fact that (r(g)) may be considered a closed normal subgroup of (g) = G by Corollary 7.9. By Proposition 9.10 (ii) we have expG r(g) = A(r(g)). Thus R(G) = expG r(g) = A(r(g)) = A(r(g)) is a closed normal subgroup of G such that L(R(G)) = L(A(r(g))) = r(g) by 9.10. Now r(g) is prosolvable by definition. Then by the Equivalence Theorem for Solvability 10.18, R(G) is a prosolvable pro-Lie group. (ii) By Definition 10.23, R(G) = expG r(g); now expG r(g) = A(r(g)) and by Definition 9.9, this equals im ir(g) = πG ((r(g))) where (r(g)) is again identified  namely, the unique one having r(g) as with a closed normal subgroup of (g) = G,  follows.  Lie algebra, and that is R(G) by (i) above. Thus R(G) = πG (R(G)) From Theorem 6.6 (i) we know that L(πG ) is an isomorphism of pro-Lie algebras, and thus maps the unique largest prosolvable ideal isomorphically onto the largest prosolvable ideal.

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(iii) Let G be a pro-Lie group. By definition R(G) is a connected closed subgroup of G and thus is a connected pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.36. By (ii) we know L(R(G)) = r(g). Since r(g) is a prosolvable pro-Lie algebra, R(G) is a prosolvable pro-Lie group by the Equivalence Theorem for Solvability 10.18. Theorem 10.25. If G is a pro-Lie group, then the radical R(G) is the largest connected transfinitely topologically solvable normal subgroup and is a closed connected characteristic subgroup of G such that L(R(G)) = r(g). If f : G → H is a quotient morphism of connected pro-Lie groups, then f (R(G)) = R(H ). Proof. Let S be a transfinitely topologically solvable connected normal subgroup. Then S is transfinitely topologically solvable by 10.11 (v) and S is a connected normal subdef

group. Then s = L(S) is a prosolvable pro-Lie algebra by the Equivalence Theorem 10.18. Since S is normal, s is an ideal by the Correspondence Theorem of Subgroups and Subalgebras 9.22 (v). By Definition 7.46 and Theorem 7.48, r(g) is the unique largest topologically transfinitely solvable subgroup. Therefore s ⊆ r(g). Then S ⊆ S = expG s ⊆ expG r(g) = R(G). This shows that R(G) contains all connected normal prosolvable subgroups. Since R(G) is prosolvable by Lemma 10.24 (iii). Hence R(G) is the unique largest connected normal prosolvable subgroup. Thus it is preserved under all automorphisms of topological groups. Hence R(G) is characteristic. Let f : G → H be a quotient morphism of connected pro-Lie groups. Then L(f ) : g → h, g = L(G) and h = L(H ), is surjective by Corollary 4.21. Now according to Lemma 7.47 and Corollary 7.29 we know L(f )(r(g)) = r(h). As expH L(f ) = f  expG we get f (expG r(g) = expH r(h) ⊆ R(H ). Since R(G) = expG r(g), we conclude f (R(G)) ⊆ R(H ). On the other hand we compute R(H ) = expH r(h) = expH L(f )(r(g)) ⊆ expH L(f )(r(g)) ⊆ f (expG r(g)) ⊆ f (expG r(g)) = f (expG r(g)) = f (expG r(g)) = f (expG r(g)) = f (R(G)). The remainder of the theorem follows from Lemma 10.24. Example 10.26. (a) Let SS 1 denote the complex unit circle group and SS 3 the group of quaternions of norm 1. We have SS 1 ⊆ SS 3 . For n ∈ Z we set " SS 3 if n ≥ 0, Gn = SS 1 if n < 0.

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 Define G = n∈N Gn . Define ϕ : G → G by ϕ((gn )n∈N ) = (gn−1 )n∈N . Then ϕ is an injective endomorphism of a compact connected group. The radical is "  {1} if n ≥ 0, R(G) = Hn , Hn = SS 1 if n < 0. n∈Z Then ϕ(R(G)) =



" Kn ,

Kn =

n∈Z

{1} SS 1

if n ≥ 1, if n < 1.

Thus R(G) is not left invariant by ϕ. Therefore the radical R(G) is not fully characteristic. (b) Let S denote the simply connected covering group of Sl(2, R) and let ϕ : Z → S be one of the two possible isomorphisms from Z onto Z(S) and let ψ : Z → Zp be the injection of Z into its p-adic completion. Set def

D = {(0, ψ(m), −ϕ(m)) : m ∈ Z} ⊆ R × Zp × S, and def

 = {(n, ψ(m − n), −ϕ(m)) : m, n ∈ Z} ⊆ R × Zp × S. Define G = (R × Zp × S)/D, H = (R × Zp × S)/ and let f : G → H be the quotient morphism given by f (xD) = x. Note that G is a four-dimensional locally compact connected reductive group whose center is isomorphic to R × Zp , while H is a quotient of G modulo a subgroup isomorphic to Z, and Z(H ) ∼ = SSp where SSp is the p-adic solenoid. Thus R(G) = Z(G)0 ∼ = R and R(H ) = Z(H ) ∼ = SSp . In this case f (R(G)) is dense in, but not equal to, R(H ).

Semisimple and Reductive Groups We recall from Theorem 10.25 that for a pro-Lie group G, its (solvable) radical R(G) is the largest connected normal prosolvable subgroup and that L(R(G)) = r(g). Consequently, if Z(G) denotes the center of G, then Z(G)0 ⊆ R(G). Definition 10.27. A connected pro-Lie group is called semisimple if R(G) = {1}. It is called reductive if R(G) ⊆ Z(G), that is, if the radical is the identity component of the center. Theorem 10.28. Let G be a connected pro-Lie group. Then G/R(G) is semisimple. Proof. The factor group G/R(G) is a pro-Lie group by Theorem 4.28 (i). Suppose that N is a closed connected normal subgroup of G containing R(G) such that N/R(G) = R(G/R(G)). Then N is a prosolvable connected normal subgroup of G by Lemma 10.22. Hence N ⊆ R(G) by Theorem 10.25. Thus R(G/R(G)) is singleton.

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Characterisation of Semisimple and Reductive Connected Pro-Lie Groups Theorem 10.29. Let G be a connected pro-Lie group. (i) G is semisimple iff g is semisimple,and G is reductive iff g is reductive.  is a product (ii) G is semisimple iff G j ∈J Sj of simply connected simple finite-di  is a product mensional Lie groups Sj , j ∈ J . Also G is reductive iff G j ∈J Sj of pro-Lie groups Sj , j ∈ J which are either simply connected simple finite-dimensional Lie groups or copies of R. (iii) Assume that P is a connected proto-Lie group, embedded into its completion G according to Theorem 4.1 and assume that g = L(P ) = L(G) such that g is a semisimple pro-Lie algebra. This assumption is satisfied if P = G is a semisimple pro-Lie group by (i) above. Then we have the following conclusions:  =P  = (g) ∼ (a) G = j ∈J Sj where all Sj are simply connected simple Lie groups, and  = Sj → P πP : G j ∈J

is  a morphism with dense image whose kernel D is a closed subgroup of  j ∈J Z(Sj ) and thus is a totally disconnected  central subgroup of G. (b) There is a quotient morphism f : P → j ∈J Sj /Z(Sj ) with a totally disconnected kernel ker f = Z(P ). The quotient P /Z(P ) is a center-free semisimple pro-Lie group. The completion G of P is Z(G)P , a semisimple connected pro-Lie group satisfying P /Z(P ) ∼ = G/Z(G). (c) (Sandwich Theorem) The group P is ‘sandwiched’ between two products via two morphisms   f πP Sj −−−→ P −−−→ Sj /Z(Sj ) j ∈J

j ∈J

whose composition is just the quotient morphism obtained by passing to the quotient Sj → Sj /Z(Sj ) in each factor. (d) Let G be a semisimple pro-Lie group and let A(g) be the minimal analytic subgroup with Lie algebra  g = L(G). Then G = Z(G)A(g), and ∼ ∼ A(g)/Z(A(g)) = G/Z(g) = j ∈J Gj /Z(Gj ). If A(g) is center-free then A(g) is complete and therefore equal to G. (e) Assume that G is semisimple. Then there is a closed connected abelian  commutative subgroup T ⊆ G containing Z(G) such that f (T ) = j ∈J Tj where Tj is a maximal torus of the semisimple centerfree group Sj /Z(Sj ). (f) If G is semisimple and K is a compact subgroup of Z(G) then K is contained in a compact connected abelian group. Proof. (i) By Definition 10.27, the semisimplicity of G is equivalent to R(G) = {1}. By Lemma 10.24, this is equivalent to r(g) = {0}. By Theorem 7.48 this means that g is semisimple.

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The reductivity of G is equivalent to R(G) = Z(G)0 by Definition 10.27. Since Z(G)0 = A(z(g)) = expG z(g) and L(Z(G)) = z(g) by Proposition 9.23, this is equivalent to r(g) = z(g).  (ii) By (i) above, G is semisimple, respectively, reductive, iff g ∼ = j ∈J sj with a family of finite-dimensional Lie algebras sj which are simple, respectively simple or isomorphic to R. The functor  : proLieAlg → proSimpConLieGr of Theorem   6.6(vi) ∼  = (g) ∼  implements an equivalence of categories, whence G = j ∈J sj =  j ∈J (sj ), and setting Sj = (sj ), this is the assertion.  (iii) (a) By assumption we may identify L(P ) with j ∈J sj for a family of simple real Lie algebras sj . By Theorem 6.6 (vi) and Theorem 8.15,  implements an equivalence proLieAlg → proSimpConLieGr from the category of pro-Lie algebras to the   category  of simply connected pro-Lie groups. Thus we may identify P = G = (g) with j ∈J Sj , Sj = (sj ) being a simply connected simple real Lie group and the  → P of Theorem 2.26 (i ) (and for pro-Lie groups in universal morphism πP : G  Theorem 6.6 (i)) with a morphism j ∈J Sj → P which induces  an isomorphism  λ : j ∈J sj → g on the Lie algebra level. Hence the kernel D of j ∈J Sj → P has the Lie algebra {0} since L preserves kernels. Thus D is totally disconnected  by Propo is connected, D is central, contained in sition 3.30 (or 4.23), and since G j ∈J Z(Sj ). (See [102, Proposition A4.27].)  (b) Since L(πP ) : j ∈J sj → g = L(P ) is an isomorphism, the subgroup im πP of P contains expP g and therefore also E(P ) = expP g = expG g = E(G). We consider P as a subgroup of its completion G according to Theorem 4.1. But then E(G) is dense in G by Corollary 4.22 (i) and thus E(P ) is dense in P . Fix a k ∈ J and find an N ∈ N (P ) such that the image of sk ∼ = L(Sk ) in L(P ) is not contained in L(N ); since lim N (P ) = 1, such a choice is certainly possible. Then P /N is a semisimple Lie group such that there is a finite subset F ⊆ J such that L(P /N) = j ∈F sj where k ∈ F , and P /N is a product of a finite family finitedimensional simple closed normal Lie subgroups (j )j ∈F such that L(j ) ∼ = sj for j ∈ F . Then the product of all j with j ∈ F \ {k} is a closed normal Lie subgroup of P /N whose full inverse image in P is a closed normal subgroup Nk of P such that P /N = k (Nk /N ) and that k ∩ (Nk /N ) is a discrete subgroup of the center Z(k ) of k . Hence there is a unique morphism of Lie groups P /N → Sk /Z(Sk ) such that for the induced morphism fk : P → Sk /Z(Sk ) sk →



λ

L(fk )

sj −−−→ L(G) −−−−→ L(Sk /Z(Sk )) = L(Sk ) = sk

j ∈J

is the identity map with an obvious identification of the Lie algebras of Sk /Z(Sk ) and Sk . The family (fk )k∈J  , by the universal property of the product, gives rise to a unique such that pr k f = fk for the projection pr k onto morphism f : P → k∈J Sk /Z(Sk ) the component with index k. Since k∈J S k /Z(Sk ) is complete, f extends uniquely to a morphism of topological groups G → k∈J Sk /Z(Sk ) which we shall denote by f ∗ , that is f ∗ |P = f .

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preserves products, we get a morphism L(f ) = L(f ∗ ) : g →  Since the functor L  ∗ −1 k∈J L(Sk /Z(Sk )) = k∈J sk such that pr k L(f ) = L(fk ) = pr k λ . We con∗ −1 ∗ clude L(f ) = λ and thus that L(f ) is an isomorphism. Hence L(ker f ∗ ) = {0} and thus ker f ∗ is totally disconnected and therefore central, since G is connected. (See again [102, Proposition A4.27].) There is a natural isomorphism of topological groups    j ∈J Sj (sj )j ∈J Sj /Z(Sj ). → Z(Sj )  → (sj Z(Sj ))j ∈J :  j ∈J Z(Sj ) j ∈J

j ∈J

We have a sequence of quotient maps     Sj → Sj /D → Sj /Z(Sj ) j ∈J

j ∈J

j ∈J

whose composition is the natural quotient morphism (sj )j ∈J  → (sj Z(Sj ))j ∈J . So we have a commutative diagram     q1 q2 Sj −−−−→ Sj /D −−−−→ j /Z(Sj ) j ∈J j ∈J j ∈J S ⏐ ⏐ ⏐ ⏐ ⏐ ∗ πP  id ⏐f  −−−→ P −−−−→ G, j ∈J Sj − πP

incl

πP (xD)

= πP (x). for two quotient morphisms q1 and q2 and with  −1 where π  is continuous and q is open. Hence f is open. Now f = q  π 2 2 P P  S /Z(S ) is connected, this implies that f is surjective. Thus P / ker f ∼ Since = j j j ∈J  ∗ and this group is center-free since each factor S /Z(S ) ∼ S /Z(S ) G/ ker f j = j j j ∈J j is a center-free simple Lie group. Since Z(P )/ ker f ⊆ Z(P / ker f ) = {1}, it follows that Z(P ) = ker f . So P /Z(P ) is a center-free pro-Lie ker f ∗ ⊆  group. We have ∗ ∼ Z(G). Moreover, since f is open, so is f , whence j ∈J Sj /Z(Sj ) = G/ ker f ∗ and thus G/ ker f ∗ ∼ = P /(ker f ∗ ∩ P ) is center-free, showing ker f ∗ = Z(G); also G = Z(G)P . Since L(G) = g, the Lie algebra of the connected pro-Lie group G is semisimple and thus G is semisimple by (i) above. (c) is a straightforward summary of what has been proved. (d) Here we apply (b): The group A(g) is a proto-Lie group whose completion is G, and the inclusion A(g) → G induces an isomorphism of Lie algebras. (e) If Kj Aj Nj is the Iwasawa decomposition of the centerfree semisimple group Sj /Z(Sj ) then Kj is a maximal compact subgroup and Kj has a maximal torus Tj . Its full inverse image in Sj is a maximal compactly embedded connected abelian def  subgroup Tj containing Z(Sj ). Thus T = Tj is a maximal j ∈J  compactly embedded connected abelian subgroup containing Z( j ∈J Sj ) = j ∈J Z(Sj ). Now   Z(G) = ker f ⊆ f −1 ( j ∈J Tj ); we define the closed subgroup f −1 ( j ∈J Tj ) of   −1  (T ) = (f  πG )−1 ( j ∈J Tj ) = j ∈J Tj = T. Since G to be T . Then πG  T is connected, im πG ⊆ T0 , and since f  πG is surjective, we have f (T0 ) = j ∈J Tj ,

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whence T = (ker f )T0 = Z(G)T0 and T0 /(T0 ∩ Z(G)) is abelian. Since Z(G) is totally disconnected, and [T0 , T0 ] is connected on one hand and contained in Z(G) on the other, [T0 , T0 ] = {0} follows. So T0 is abelian. Then T = Z(G)T0 is abelian as well. If N ∈ N (G), then q : G → G/N is a quotient morphism onto a finite dimensional connected semisimple Lie group and q  πG : j ∈J Sj → G/N is a quotient embedded morphism. Then the image q(πG (T)) in G/N is a maximal compactly  abelian connected subgroup of G/N containing q(T ) = q(f −1 ( j ∈J Sj /Z(Sj ))) as we see by a bit of diagram chasing. through  f G −−−−→ j) j ∈J Sj/Z(S ⏐ ⏐ ⏐ ⏐proj q   fN G/N −−−−→ j ∈F Sj/Z(Sj ) for a suitable finite subset F of J . Thus T ⊆ πG (T)N. Since N (G) converges to the identity, we conclude T ⊆ πG (T) ⊆ T0 . Thus T is connected. This completes the proof. (f) Let T ⊆ G be as in (e). Then T contains a unique maximal compact subgroup comp(T ) which is connected by the Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups 5.12. It follows that K ⊆ comp(G). If G is the group of all 3 × 3 matrices ⎛ ⎞ 1 x z ⎝0 1 y ⎠ , 0 0 1

x, y, z ∈ R,

then Z(G) is the set of matrices with x = y = 0 and thus is isomorphic to R. For this group G/Z(G) ∼ = R2 and thus is its own nontrivial center; so it is all but center-free. We therefore note that it is a special property of a group G for G/Z(G) to be center-free.  → G has to be surjective, perhaps even a quotient One might surmise that πG : G morphism. However, this fails to be true even for three-dimensional locally compact groups as the following example shows.  be the universal covering group of Sl(2, R). (See for instance Example 10.30. Let G  Let ϕ : Z → Z(G)  be one of the two possible [113] for details on the structure of G.)  ∼  isomorphisms onto the center G. Let β : Z → ζ (Z) = q prime Zq be the universal totally disconnected compactification of Z where Zq is the additive group of q-adic integers. (See for instance Example 14.2, and [102], Chapter 8, notably Theorem 8.67ff., and see also Chapter 1, The Retraction Theorem for Full Closed Subcategories of TopGr 1.41 and the examples following the proof of that theorem.) The subgroup  = {(ϕ(n), β(n)−1 ) : n ∈ Z} def

 × ζ (Z) is discrete central and n  → (ϕ(n), β(n)−1 ) : Z →  is an isomorphism of G of topological groups. We set def  × ζ (G))/. G = (G

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439

Then G and G × ζ (G) are locally isomorphic. On the Lie algebra level,  × L(ζ (Z)) ∼  × ζ (Z)) ∼ L(G = L(G) = sl(2, R), and (in view of Corollary 4.21 (i))  × z(Z))/L() ∼ L(G) ∼ = L(G = sl(2, R)  × ζ (Z) → G induces an isomorphism L(q). The and the quotient morphism q : G morphism →G g  → (g, 0) : G may be identified with πG . Since ζ (Z) = β(Z), the morphism πG is not surjective, and thus A(sl(2, R)) = G. Therefore, (i) G is semisimple and L(G) = sl(2, R).  = A(L(G)) = (G  × β(Z))/ is a bijective image of G,  is dense, but is (ii) πG (G) not equal to G. In particular, πG is not a quotient map. For a topological group T , let Td be the discrete group obtained from T by endowing the underlying group with the discrete topology. If A is a topological abelian group def

and α(A) is the Bohr compactification of A, then ζ (A) = α(A)/α(A)0 is the uni is the character group of A, then versal zero-dimensional compactification of A. If A ∼  α(A) = ((A)d ). Now we may return to the idea of the construction of Example 10.30. Obviously, it generalizes to the following proposition. Proposition 10.31. Assume that {Sj : j ∈ J } is a family of simple simply connected Lie groups Sj withLie algebras sj . Then there exists  a semisimple  pro-Lie group G ∼ such that L(G) ∼ s and such that Z(G) ζ Z(S ) = j ∈J j = j . j ∈J If at least one of the Z(Sj ) is infinite, then it holds that A(L(G)) = G, the morphism  → G is not surjective, and the corestriction to its image is not open. πG : G Proof. Exercise. Exercise E10.5. Prove Proposition 10.31. [Hint. Generalize the construction of Example 10.30.] Theorem on the Closure of Semisimple Analytic Subgroups Theorem 10.32. Let G be a pro-Lie group and s a closed semisimple subalgebra of g. def Set H = A(s). Then the following conclusions hold: (i) H is topologically perfect, that is, [H, H ] = H .

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10 The Global Structure of Connected Pro-Lie Groups

(ii) [H, H ] ⊆ A(s). (iii) H is reductive such that [L(H ), L(H )] = s. (iv) In particular,  L(H ) ∼ sj = RI × s, s ∼ = j ∈J

for some set I and J and a family  of simple finite-dimensional Lie algebras sj , ∼ j ∈ J . Therefore, H = RI × j ∈J Sj , Sj = (sj ). (v) H = Z(H )A(s) and H /Z(H ) is a center-free pro-Lie group.  then (vi) Let S be the image of j ∈J (sj ) in G; (a) A(s) = πG (S), and (b) Z(H ) = Z(A(s)). (c) If L(H ) = s, that is, if H is semisimple, then KZ(H )A(s, H ) = H . (vii) The minimal analytic subgroup A(s) is closed in G if and only if its center Z(A(s)) is closed in G. This is the case for instance if Z(A(s)) is compact. Proof. There is clearly no harm in assuming H = A(s) = G. Let us do that; then L(H ) = g. (i) By Corollary 9.33 and since s is semisimple, we have [g, g] = g = s = s. By Theorem 9.26 we have [G, G] = A([g, g]) = A(s) = G, so G is topologically perfect. (ii) This assertion follows from [g, g] = s and Lemma 9.25 (ii). (iii) In order to show that G is reductive, we have to show that R(G) is central. By Lemma 10.24 this is tantamount to saying that r(g) is central in g. Any Levi summand (see Definition 7.47 and Theorem 7.52) is semisimple and thus is contained in g . Since s = g , the subalgebra s is an ideal and is a unique Levi summand. Accordingly, g is the direct sum r(g) ⊕ s by Theorem 7.77 (ii). Then s = g = r(g) ⊕ s = r(g) ⊕ s, and this implies r(g) = {0}. That is, r(g) is abelian. Since [r(g), s] = {0}, we have shown that r(g) is central. (iv) The structure of a reductive pro-Lie algebra was determined in Theorem  7.27.  = (g) = RI × This gives us the asserted structure of g. It follows that G j ∈J Sj , Sj = (sj ).  → G. The kernel D = ker πG is a totally (v) We consider the morphism πG : G  and is therefore central. The disconnected normal subgroup of the connected group G −1 subgroup πG (Z(G)) is therefore a central extension by an abelian group and thus is a   = RI × class 2 nilpotent group. The largest normal solvable subgroup of G j ∈J Sj is  −1 I  =R × easily identified as Z(G) j ∈J Z(Sj ). Thus πG (Z(G)) is in fact contained  ⊆ Z(G). Thus π −1 (Z(G)) = Z(G).  Then  On the other hand, πG (Z(G)) in Z(G). G  ∼   the group G/Z(G) = j ∈J Sj /Z(Sj ) is a center-free semisimple pro-Lie group with  is a center-free Lie algebra s, and its image A(s, G/Z(G)) = πG (G)Z(G)/Z(G) proto-Lie group whose completion is the completion (G/Z(G))∗ of G/Z(G); we have L((G/Z(G))∗ ) = L(G/Z(G)) by 4.20. Thus by 10.29 (iii)(d), A(s, G/Z(G)) is complete and thus A(s, G/Z(G)) = G/Z(G) = (G/Z(G))∗ . In particular, by

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441

Theorem 10.29 (iii), G/Z(G) is a center-free pro-Lie group and is isomorphic to a product of connected simple center-free Lie groups Sj /Z(Sj ) = expSj /Z(Sj ) sj , j ∈ J . Therefore 0 1 G/Z(G) = expG/Z(G) L(G/Z(G)) . Since L(G/Z(G)) ∼ = g/z(g) by Corollary 4.21 (i), we conclude G/Z(G) = expG gZ(G)/Z(G). Since g = z(g)+s, further expG (z(g)+s) = expG z(g) expG s, and expG z(g) ⊆ Z(G), we have G/Z(G) = Z(G)expG s/Z(G), and consequently G = Z(G)expG s = Z(G)A(s). (vi) (a) By Definition 9.9 we have A(s, G) = πG (S), where we identify S with (s).   G). (b) We saw in (v) that G = Z(G)A(g) and G/Z(G) ∼ = G/Z( = A(s)/Z(A(s)) ∼ def We have Z(A(s)) = A(s)∩Z(G) since A(s) is dense in G. Now let C = A(s) ∩ Z(G). Then C ⊆ Z(G), and C ∩ A(s) = Z(G) ∩ A(s) and so A(s)/(A(s) ∩ C) is center-free. Then CA(s)/C is a dense analytic subgroup in the completion (G/C)∗ of the protoLie group G/C. The bijective continuous morphism A(s)/(A(s) ∩ C) → CA(s)/C is an algebraic isomorphism. So CA(s)/C, which by a slight abuse of notation we can write as A(s, (G/C)∗ ), is center-free. But we saw in 10.29 (iii)(d) that this implies the completeness of the group CA(s)/A(s) and that it equals G/C. Thus CA(s) = G. We have C ⊆ Z(G) and A(s)/(A(s) ∩ C) center-free, and thus C = Z(CA(s)) = Z(G). This completes the proof of (b). (c) Since L(G) = s we know that R(G) = {1} and so Z(G)0 = {1}. Thus Z(G) is totally disconnected. By Theorem 9.50, comp(Z(G)) is the unique maximal compact central subgroup KZ(G) of G, and G/ KZ(G) has no compact central subgroups. We factor KZ(G), assume that Z(G) is compact-free and must show that G = A(g), def

g = s. Let N ∈ N (G) and n = L(N ). Then g = f ⊕ n for a finite-dimensional semisimple ideal f by Corollary 7.29. Let F = A(f, G); then L(F ) = f by (iv), since R(F ) ⊆ R(G) = {1}. Thus F is a finite-dimensional closed connected normal subgroup of G. Let C = Z(F, G) denote the centralizer of H in G. Then L(C) = z(f, g) by Proposition 9.17 (ii). We know that z(f, g) = n = L(N0 ) = n. Thus N0 = exp n = exp L(C) by Corollary 4.22 (i). Thus N0 = C0 ; in particular, L(F C) ⊇ f + n = g and so A(g, G) ⊆ F C, whence F C is dense in G. Since C ∩ F = Z(H ) we have Z(H ) ⊆ Z(G), and so KZ(F ) = comp(Z(F )) ⊆ comp(Z(G)) = KZ(G) = {1}. From the structure theory of finite-dimensional connected pro-Lie groups in Theorem 9.44 (∗∗∗) we conclude that F is a Lie group. Thus F ∩ C, having the Lie algebra L(F ) ∩ L(C) = {0}, is discrete. The morphism λ : F → G/N0 , λ(h) = hN0 is a morphism of connected locally compact groups by Theorem 9.44 and has the

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kernel F ∩ N0 = F ∩ C0 ⊆ F ∩ C and therefore is discrete. We claim that λ is surjective, that is that G = F N0 . Assume that this is shown: Then the Open Mapping Theorem for Locally Compact Groups shows that the induced bijective morphism F /(F ∩N0 ) → G/N0 is an isomorphism and thus proves that G/N0 is a Lie group, that is N0 ∈ N (G). But then we can apply Proposition 9.40 with F = {N0 : N ∈ N (G)} and the assertion is proved. In order to prove surjectivity we may consider G/(F ∩ C) in view of Theorem 4.28 (iii) and therefore assume that F ∩ C = {1} and we have to show that F C = G. Since F is a Lie group, and limM∈N (G) F ∩ M = {1}, for all sufficiently small M ∈ N (G) we have F ∩ M = {1}. Both F and M are normal in G and then M ⊆ Z(F, G) = C. Then the Lie group contains the analytic subgroup F M/M, a bijective homomorphic image of F and the closed connected Lie subgroup C/M. Moreover we know that L(G/M) = L(F M/M) ⊕ L(C/M). Then we may conclude that G/M = (F M/M) · C/M, that is, G = F C, and this is what we had to show. Thus (c) is established. (vii) If A(s) is closed in G, then Z(A(s)), being closed in A(s) is closed in G. Conversely, assume that Z(A(s)) is closed in G. Then (vi) above implies that Z(G) ⊆ A(s) and thus by (v) above we have A(s) = G = Z(G)A(s) = A(s). Exercise E10.6. Prove the following Proposition. Let G be a connected reductive Lie group. Then G/Z(G) is center-free.  → G is a quotient morphism [Hint. The universal covering homomorphism πG : G ∼   with Rm × S1 × · · · × Sn and so G = G/D for D = ker πG . We may identify G for simply connected simple Lie groups S1 , . . . , Sn . The kernel D is a discrete central  contained in Z(G)  = Rm × Z(S1 ) × · · · × Z(Sn ). Now G/Z(G) ∼ subgroup of G, = ∼     (G/D)/(Z(G)/D) = G/Z(G) ∼ = S1 /Z(S1 ) × · · · × Sn /Z(Sn ), and this group is center-free.] Assume that G = A(s) for a semisimple closed subalgebra of L(G). Then R(G) = Z(G)0 by Theorem 10.32. One might surmise that G = R(G)A(s) as is the case for Lie groups, whence G = R(G)A(s)N for all N ∈ N (G). However, Example 10.30 shows that this is not even the case when G is semisimple (that is, R(G) = {0}) and g is simple. Thus Theorem 10.32 (v) illustrates that there is a significant difference between Z(G) and Z(G)0 = R(G) for a reductive pro-Lie group G. Example 10.33. Let A be the abelian Lie group R × Z × Z and define the subgroup D ⊆ A by √ D = {(n1 + n2 2, n1 , n2 ) : (n1 , n2 ) ∈ Z2 }. Then D is a discrete subgroup of A such that D ∩ (R × {0} × {0}) = {(0, 0, 0)} = D ∩ ({0} × Z × Z). The group A is the direct sum algebraically and topologically of A0√= R × 0 × 0 and D. Thus A/D ∼ = R. The subgroup ({0} × Z × Z) + D = (Z + 2Z) × Z × Z is dense in A  = R ×  × . Now let  be the universal covering group of Sl(2, R) and set G Let α : Z → Z() be one of the two isomorphisms from Z onto the center of  and

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√  : (n1 , n2 ) ∈ Z2 }. Then  is a discrete define  = {(n1 + n2 2, α(n1 ), α(n2 )) ∈ G   The subgroup of G and is contained in the center R × Z() × Z() ∼ = A of G. √ def  subgroup T = ({0} ×  × ) equals (Z + 2Z) ×  ×  and thus is dense in G.  Now we set G = G/. Then G is a reductive Lie group in which the anadef lytic subgroup S = T / is dense where L(S) ∼ = sl(2, R)2 . The radical R(G) is ∼ (R ×{1}×{1})/ = (R ×Z()×Z())/ = A/D ∼ = R. Note that Z(G) = R(G). Thus G is a 7-dimensional connected reductive Lie group with a dense analytic Levi subgroup of dimension 6 and a radical equal to the center, isomorphic to R. In particular, G has no nontrivial compact normal subgroup. However, the analytic subgroup A(g, G) with Lie algebra g is G itself. It would therefore be wrong to surmise that, for some reason or another, the radical of the closure of a semisimple analytic subgroup of a Lie group would have to be compact. For a topological group G we let α(G) be the Bohr compactification of G. Proposition 10.34. Let J and K be sets. Then there is a pro-Lie group G and a minimal analytic Levi subgroup A(s) = expG s which is dense in G0 such that the radical R(G) = R(G0 ) is isomorphic to RJ × α(ZK )0 . Proof. Exercise. Exercise E10.7. Prove Proposition 10.34 [Hint. A product of J copies of the group constructed in Example 10.33 realizes a connected reductive group with dense minimal analytic Levi subgroup whose radical is isomorphic to RJ . Next consider a product of K copies of the simply connected covering group  of Sl(2, R) in order to have a simply connected semisimple pro-Lie group with center ∼ = ZK . Then use the construction of Example 10.33 with α(ZK ) in place√of R and the injective morphism η : ZK → α(ZK ) in place of (n1 , n2 )  → n1 + n2 2 : Z2 → R.]

Nilpotency of Pro-Lie Groups We recall from Proposition 7.57 and 7.66 that a pro-Lie algebra g is transfinitely topologically nilpotent iff it is pronilpotent iff it is countably topologically nilpotent iff it satisfies g = n(g). In this case, by Lemma 6.69 lim g[[n]] = 0.

n→∞

(3)

Lemma 10.35. Assume that G is a pro-Lie group. (i) Then for each closed normal subgroup N of G, (G/N )[[α]] = G[[α]] N /N.

(∗)

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(ii) Every quotient of a transfinitely topologically nilpotent pro-Lie group is transfinitely topologically nilpotent. (iii) Assume that G is a connected transfinitely topologically nilpotent pro-Lie group. Then G is pronilpotent. Proof. (i) We prove (∗) by transfinite induction. Assume the assertion to be proved for all ordinals α < β. If β = α + 1, then (G/N )[[β]] = [G/N, (G/N )[[α]] ] = [G/N, (G[[α]] N )/N] = [G, G[[α]] ]N )/N = ([G, G[[α]] ]N )/N = G[[α+1]] N /N = G[[β]] N /N. This completes the induction in this case.  Now assume that β is a limit ordinal. Then (G/N )[[β]] = α<β (G/N )[[α]] =     [[α]] N/N = [[α]] N /N = N ((α)) /N = G((β)) N /N by the α<β G α<β G α<β G definitions and by (∗) in Lemma 10.13. This completes the induction. Thus (∗) is proved. (ii) If G is transfinitely topologically nilpotent, we have G[[γ ]] = {1}, and thus (G/N )[[γ ]] = G[[γ ]] N /N = {1}, by (i) above. This shows that G/N is transfinitely topologically nilpotent. (iii) In the circumstances of (ii), if G/N is a connected Lie group, then G/N is transfinitely topologically nilpotent on the one hand and satisfies the descending chain condition for closed connected subgroups and thus is topologically nilpotent and so is nilpotent by 10.11 (iv). The Equivalence Theorem for Nilpotency of Connected Pro-Lie Groups Theorem 10.36. Let G be a connected pro-Lie group and g its Lie algebra L(G). Then the following assertions are equivalent: (i) (ii) (iii) (iv) (v)

G is countably topologically nilpotent. G is transfinitely topologically nilpotent. G is pronilpotent. g is pronilpotent. G is countably nilpotent.

These conditions imply the following ones: (vi) G is transfinitely nilpotent. Proof. The implication (i) ⇒ (ii) is trivial. The implication (ii) ⇒ (iii) was shown in 10.35. (iii) ⇒ (i): Let N ∈ N (G). Then since G/N is nilpotent by is a natural  (iii), there [[n]] ⊆ N . Thus number msuch that G[[n]] ⊆ N for all n ≥ m. Then G[[ω]] = ∞ G n=1 G((ω)) ⊆ N (G) = {0}. This proves (i). Thus (i), (ii), and (iii) are equivalent. (iii) ⇔ (iv): Condition (iii) means that for each N ∈ N (G) the connected Lie group G/N is nilpotent. This is the case if and only if L(G/N ) is nilpotent. By Corollary 4.21,

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a corollary of the Strict Exactness Theorem for L we have L(G/N ) ∼ = L(G)/L(N ) canonically and {L(N ) : N ∈ N (G)} is cofinal in the set  (g) of all ideals j of g such that g/j is finite-dimensional. Thus L(G/N ) is nilpotent for all N ∈ N (G) iff g/j is nilpotent for all j ∈  (g) iff g is pronilpotent, and this is Condition (iv). The implication (i) ⇒ (v) was proved in 10.11 (ii). The implication (v) ⇒ (vi) is trivial. Problem P10.1. Is a transfinitely nilpotent connected pro-Lie group pronilpotent? Such a group has to be prosolvable since it is transfinitely solvable and then the Equivalence Theorem for Solvability of Connected Pro-Lie Groups 10.18 applies. The impediment for a proof is the failure of transfinite nilpotency to be preserved by passing to quotients. Free topological groups are free groups in the algebraic sense and thus are countably nilpotent; but every topological group is a quotient of a free topological group and thus of a transfinitely countably nilpotent topological group. We now establish a counterpart of Lemma 10.17. Lemma 10.37. Let G be a connected pro-Lie group. (i) If N ∈ N (G) is a closed normal subgroup such that dim G/N is a (finite-dimensional) Lie group, then for every limit ordinal β there is an αN < β such that for all α the relation αN ≤ α < β implies G[[β]] N = G[[α]] N . (ii) δ ≤ ω, that is

(∗∗)

G[[ω+1]] = [G, G[[ω]] ] = G[[ω]] .

As a consequence G[[α]] = G[[ω]] for all ordinals α ≥ ω. (iii) For every pro-Lie group G, the closed subgroup G[[ω]] is characteristic and for each N ∈ N (G) there is a natural number nN such that for all natural numbers n satisfying nN ≤ n we have G[[ω]] N = G[n] N and [G, G[[ω]] ]N = G[[ω]] N. Proof. (i) Let N ∈ N (G) be a closed normal subgroup of G such that G/N is a Lie group. Since G is connected, G/N is connected. Then the descending chain (G[α] N )α<β of pull-backs of the closed connected subgroups G[α] N /N has a smallest element, say G[αN ] N for a smallest ordinal αN < β. Then (#) G[αN ] N = G[α] N for all α with αN ≤ α < β. We claim that G[[α]] ⊆ G[α] N for all ordinals α. For α = 0 this is trivially true. Let us assume that this claim holds for α, we shall show that it holds for β = α + 1. Indeed, G[[β]] = [G, G[[α]] ] ⊆ [G, G[α] N] ⊆ [G, G[α] ]N = G[β] N . So induction works for the step from α to α + 1. Now assume that β is a limit ordinal and that the claim is true for all α < β. Then from the definition of G[[α]]

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    by transfinite induction we have G[[β]] = α  <β G[[α ]] ⊆ α  <β G[α ] N = G[α] N for all α satisfying αN ≤ α < β by the induction hypothesis and by (#). Thus     for all these α we obtain G[[β]] N ⊆ G[α] N = α  <β G[α ] N = N α  <β G[α ] ⊆   N α  <β G[[α ]] = G[[β]] N by 10.13 (i)(∗) and the inductive definition of G[[β]] . Thus equality holds throughout and this is the assertion (∗∗) for all α satisfying αN ≤ α < β. (ii) Let β be any limit ordinal and let N ∈ N (G) be any closed normal subgroup N ∈ N (G). For these data we determine αN as in (i) above. In view of 10.10 (i), we then compute [G, G[[β]] ]N = [G, G[[β]] N]N = [G, G[[β]] N ]N = [G, G[αN ] N]N = [G, G[αN ] N ]N = [G, G[αN ] ]N = G[αN +1] N = G[αN ] N = G[[β]] N . For fixed β we form the intersection over all N ∈ N (G) on both sides and find G[[β]] on the right side since this is a closed subset of G, and [G, G[[β]] ] = G[[β+1]] on the left. Thus G[[β+1]] = G[[β]] for all limit ordinals β. In particular, this applies to β = ω. (iii) This is an immediate consequence of (ii) and (i). Theorem on the Descending Central Series of Pro-Lie Groups Theorem 10.38. Let G be a connected pro-Lie group. Then G[[α]] = A(g[[α]] ) for all ordinals α. Proof. For α = 0 there is nothing to prove, and for natural numbers α = m we prove the claim by induction. We assume that the claim holds for m and abbreviate g[[m]] by h; note that so G[[m]] = A(h). Recall that ih : (h) → G with image A(h) is  = (g) → G (see Definition 9.9ff., notably the restriction of the morphism πG : G Proposition 9.10 (i)). For n = m+1 we compute G[[n]] = [G, G[[m]] ] = [A(g), A(h)] = [A(g), A(h)] = [ig ((g)), ih ((h))] (by Definition 9.9) = ig [(g)), (h)] = ig ([(g)), (h)]) = ig (([g, h])) (by Proposition 9.27 (i)) = i[g,h] (([g, h])) (by Proposition 9.10 (i)) = A([g, h]) = A(g[[m+1]] ) = A(g[[n]] ). This concludes the proof for natural numbers α. We now prove the claim for the limit ordinal α = ω. Abbreviate A(g[[ω]] ) by N. Then L(N ) ⊇ L(A(g[[ω]] )) = g[[ω]] . From Corollary 4.21 (i) of the Strict Exactness Theorem for L we have L(G/N ) ∼ = L(G)/L(N ) as a homomorphic image of the factor algebra g/g[[ω]] which is topologically countably nilpotent by Lemma 7.44 (i). Hence G/N is topologically countably nilpotent by Proposition 10.35 (i). The subgroup G[[ω]] /N is isomorphic to (G/N )[[ω]] = {1}. It follows that G[[ω]] = N = A(g[[ω]] ),

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which is the claim. Since for ordinals α ≥ ω we have that G[[α]] = G[[ω]] by Lemma 10.23 (ii), the proof is complete, and since g[[α]] = g[[ω]] , the theorem follows. Proposition 10.39. Let G be a pronilpotent pro-Lie group and H a closed subgroup. Define inductively C 0 H = H and " [G, C β H ] if α = β + 1, α C H =  β β<α C H if α is a limit ordinal. Then H [[α]] ⊆ C α H ⊆ G[[α]] . If G is topologically nilpotent then H is topologically nilpotent, and the relation [G, H ] = H implies H = {1}. Proof. By transfinite induction we prove that C α H ⊆ G[α] for all α. This is certainly true for α = 0; assume that it is true for α < β. We claim that it holds for β. Indeed, if β = α + 1 then H [[β]] = [H, H [[α]] ] ⊆ C β H = [G, H[α] ] ⊆ [G, G[α] ] = G[β] . Assume now that β is a limit ordinal. Then H [[β]] = α<β H [[α]] and C β H =   α [α] = G[β] . This completes the induction. α<β C H ⊆ α<β G If G[[δ]] = {1}, then H [[α]] ⊆ C δ H = {1} by the preceding. Thus H is topologically nilpotent. On the other hand, C 1 H = [G, H ] = H . Inductively, it follows that C α H = H for all ordinals α. It follows that H = {1}. Exercise E10.8. Prove the following proposition: If H is a closed subgroup of a pro-Lie group G, then H [α] ⊆ G[α] and H [[α]] ⊆ G[[α]] . [Hint. Emulate the proof of Proposition 10.25.]

The Nilradical and the Coreductive Radical We know from Theorem 7.27 that every pro-Lie algebra g has a unique smallest ideal ncored (g) which is closed and is such that g/ ncored (g) is a product of simple finite-dimensional Lie algebras and copies of R. According to Theorem 7.67 this ideal, called the coreductive radical, is pronilpotent and equals [g, g] ∩ r(g) and [g, r(g)]. We will now emulate this situation on the level of pro-Lie groups. So let G be a pro-Lie group. Let g = L(G) be its Lie algebra, n(g) its nilradical, and ncored (g) the coreductive radical of g according to Theorem 7.27, Definition 7.65 and Theorem 7.66. From an observation preceding 7.47 we recall that ncored (g) is always closed in g. Definition 10.40. The closed subgroups expG n(g) and expG ncored (g) will be denoted by N(G), respectively Ncored (G). These groups are called the nilradical, respectively, coreductive radical of G.

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Lemma 10.41. (i) Let G be a simply connected pro-Lie group with Lie algebra g. Then N(G) = A(n(g)) = expG n(g) and L(N (G)) = n(g) and Ncored (G) = A(ncored (g)) = expG (ncored (g)). The nilradical N (G) and the coreductive radical ncored (g) are pronilpotent and are therefore isomorphic to (n(g), ∗), respectively, (ncored (g), ∗).  = (g) → G the canonical (ii) Let G be a connected pro-Lie group and πG : G  and L(πG ) maps morphism constructed in Chapter 8. Then N (G) = πG (N (G))   L(N(G)) isomorphically onto L(N (G)) = n(g); also Ncored (G) = πG (ncored (G)) and L(Ncored (G)) = ncored (g). (iii) For every pro-Lie group G, the nilradical and the coreductive radical are closed connected pronilpotent normal subgroups. Proof. (i) Since G is a simply connected pro-Lie group, A(n(g)) = (n(g)) in view of Definition 9.9 and the fact that (n(g)) may be considered a closed normal subgroup of (g) = G by Corollary 7.9. By Proposition 9.10 (ii) we have expG n(g) = A(n(g)). Thus N(G) = expG n(g) = A(n(g)) = A(n(g)) is a closed normal subgroup of G such that L(N (G)) = L(A(n(g))) = n(g) by 9.10. The same arguments apply to the coreductive radical in place of the nilpotent radical. Now n(g) is pronilpotent by definition and ncored (g) is nilpotent by Theorem 7.67. Then by the Equivalence Theorem for Nilpotency of Connected Pro-Lie Group 10.36, N (G) and ncored (N ) are pronilpotent pro-Lie groups. Since the pro-Lie groups N (G) and Ncored (G) may be identified with (n(g)), respectively, (ncored (g)), they are simply connected, and thus by Theorem 8.5, they are isomorphic to (n(g), ∗), respectively, (ncored (g), ∗). (ii) By Definition 10.40, N (G) = expG n(g); now expG n(g) = A(n(g)) and by Definition 9.9, this equals im in(g) = πG ((n(g))) where (n(g)) is again identified  namely, the unique one having n(g) as with a closed normal subgroup of (g) = G,  by (i) above. Thus N (G) = πG (N (G))  follows. The Lie algebra, and that is N(G) same arguments apply to the coreductive radical in place of nilradical. From Theorem 6.6 (i) we know that L(πG ) is an isomorphism of pro-Lie algebras, and thus maps the unique largest pronilpotent ideal isomorphically onto the largest pronilpotent ideal, and the coreductive radical to the coreductive radical. (iii) Let G be a pro-Lie group. By definition, N (G) and Ncored (G) are connected closed subgroups of G and thus are connected pro-Lie groups by the Closed Subgroup Theorem for Pro-Lie Groups 3.36. By (ii) we know L(N (G)) = n(g), respectively, L(Ncored (G)) = ncored (g). Since n(g) and ncored (g) are pronilpotent pro-Lie algebras, N(G) and Ncored (G) are pronilpotent pro-Lie groups by the Equivalence Theorem for Nilpotency 10.36. Since the subgroups N (G) and Ncored (G) are characteristic in the normal subgroup G0 , they are normal in G. Theorem 10.42. If G is a pro-Lie group, then the nilradical N (G) is the largest connected transfinitely topologically nilpotent normal subgroup and is a closed connected characteristic subgroup of G such that L(N (G)) = n(g). Proof. Let N be a transfinitely topologically nilpotent connected normal subgroup. Then N is transfinitely topologically nilpotent by 10.11 (v) and N is a connected normal

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def

subgroup. Then n = L(N) is a pronilpotent pro-Lie algebra by the Equivalence Theorem 10.36. Since N is normal, n is an ideal by the Correspondence Theorem of Subgroups and Subalgebras 9.22 (v). By Definition 7.65 and Theorem 7.66, n(g) is the unique largest topologically transfinitely nilpotent subgroup. Therefore n ⊆ n(g). Then N ⊆ N = expG n ⊆ expG n(g) = N (G). This shows that N (G) contains all connected normal pronilpotent subgroups. Since the connected normal subgroup N(G) is pronilpotent by Lemma 10.41 (iii), it is one of these. Hence N (G) is the unique largest connected normal pronilpotent subgroup. Thus it is preserved under all automorphisms of topological groups. Hence N (G) is characteristic. The remainder of the theorem follows from Lemma 10.41. Let G be the Lie group of all real 2 × 2 matrices , r s def e (s; r) = , r, s ∈ R. 0 1 Define f : G → R, f ((s; r)) = r. Then f is a quotient morphism. The nilradical N(G) is the set of all (s; 0), s ∈ R, that is, the kernel of f . The nilradical N (R) is R. Thus f (N (G)) = {0} = R = N(R). This very simple example shows, that the nilradical does not behave too well with respect to quotient morphisms. We have seen in Theorem 10.25 and will see in Theorem 10.43 below that the radical and the coreductive radical behave better in this regard. Theorem 10.43. If G is a connected pro-Lie group, then the coreductive radical Ncored (G) is the smallest connected closed normal subgroup N such that G/N is reductive. In particular, G/Ncored (G) and G/N (G) are reductive. It is a closed connected characteristic subgroup of G such that L(Ncored (G)) = ncored (g). If f : G → H is a quotient morphism of connected pro-Lie groups, then f (Ncored (G)) = Ncored (H ). Proof. Let N be a closed connected normal subgroup such that G/N is reductive. def

Then n = L(N ) is a closed ideal of g by the Correspondence Theorem of Subgroups and Subalgebras 9.22 (v). We know that g/n ∼ = L(G/N ) by Corollary 4.21 (i). Since G/N is reductive, g/n ∼ L(G/N ) is reductive by Theorem 10.29 (i). Now ncored (g) = is the smallest ideal j of g such that g/j is reductive by Definition 7.65. Hence ncored (g) ⊆ n. Then Lemma 10.41 (i) and Corollary 4.22 (i) allow us to conclude Ncored (G) = expG ncored (g) ⊆ expG n = N . This shows that the ideal Ncored (G) is contained in all connected normal pronilpotent subgroups N such that G/N is reductive, and since it is itself one of those N, it is the unique smallest ideal among these N. In particular, the coreductive radical Ncored (G) is preserved under all automorphisms of topological groups. Hence it is characteristic. The claim on the homomorphic images of the coreductive radicals is proved similarly to the corresponding proof of 10.25, as follows. Let f : G → H be a quotient morphism of connected pro-Lie groups. Then L(f ) : g → h, g = L(G)

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and h = L(H ), is surjective by Corollary 4.21. Now according to 7.67 we have ncored (g) = [g, r(g)]. As we argued in the proof of 10.25, L(f )(r(g)) = r(h). Thus L(f )[g, r(g)] = [h, r(h)], and it follows that L(f )(ncored (g)) ⊆ ncored (h), and [h, r(h)] ⊆ L(f )([g, r(g)]) ⊆ L(f )(ncored (g)), whence ncored (h) [g, r(h)] ⊆ f (ncored (g)). Therefore, L(f )(ncored (g)) = ncored (h). Because of expH L(f ) = f  expG and Ncored (G) = expG ncored (g), we get Ncored (H ) = expH ncored (h) = expH L(f )(ncored (g)) ⊆ expH L(f )(ncored (g)) ⊆ f (expG ncored (g)) ⊆ f (expG ncored (g)) = f (expG ncored (g)) = f (expG ncored (g)) = f (expG ncored (g)) = f (Ncored (G)). We further obtain f (expG ncored (g) = expH L(f )(ncored (g)) ⊆ expH ncored (h) ⊆ Ncored (H ), whence f (Ncored (G)) ⊆ Ncored (H ). Thus f (Ncored (G)) = Ncored (H ), which we had to show. The remainder of the theorem follows from Lemma 10.41. Example 10.44. Let let H = R2 × R denote the Heisenberg group defined by the multiplication (v, r) ∗ (w, s) = (v + w, r + s + 21 det(v, w)) with det(v, w) = v1 w2 − v2 w1 . Then the commutator group (and coreductive radical) of H is {0} × R and this is also the center of H . If we let G = H and H = H /Z(H), and if we let f : G → H be the quotient morphism, then f (Ncored (G)) = {1}. We let Zp denote the p-adic completion of Z and SSp the p-adic solenoid (Zp × R)/D, D = {(−n, n) : n ∈ Z}. Let z  → z∗ : Zp → SS p , be the embedding given by z∗ = (z, 0) + D. Then we get a quotient morphism q : SSp → SO(2) with kernel Z∗p , giving us an automorphic action of SSp on R2 defined by z · v = q(z)(v). So we set M = R2 q SSp , multiplication being (v1 , z1 )+(v2 , z2 ) = (v1 +z1 ·v2 , z1 +z2 ). Thus M is a 3-dimensional locally compact connected metabelian group whose commutator group (and coreductive radical) is R2 × {0} and whose center is {0} × Z∗p . Now let G = M × H. Then G is a 6-dimensional connected locally compact metabelian group such that Ncored (G) = [G, G] = (R2 × {0}) × ({0} × R) ∼ = R3 . The subgroup D = {((0, n∗ ), (0, −n)) : n ∈ Z} is discrete and central. Set H = G/D and let f : G → H denote the quotient morphism. Then H contains the central subdef def group S = (({0} × Z∗p ) × ({0} × R))/D ∼ = SSp . Let T = {((v, n∗ ), (0, n)) : v ∈ R2 , ∼ R2 . The commutator subgroup of H is f ([G, G]) = f (Ncored (G)) n ∈ Z}/D = whose closure is S + T = ((R2 × Z∗p ) × ({0} × R))/D ∼ = R2 × SSp . Thus f (Ncored (G)) = Ncored (H ) but f (Ncored (G)) = Ncored (H ).

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Proposition 10.45. For any simply connected pro-Lie group G, the coreductive radical Ncored (G) agrees with ([G, G] ∩ R(G))0 . Proof. First we note from Proposition 9.26 that L([G, G]) = [g, g]. Now L([G, G] ∩ R(G)) = L([G, G]) ∩ L(R(G)) = [g, g] ∩ r(g) = ncored (g) in view of Lemma 10.24 (i) and Theorem 7.67. But then by Corollary 4.22 we have ([G, G] ∩ R(G))0 = expG L([G, G] ∩ R(G)) = ncored (g) = Ncored (G). Lemma 10.46. For any connected Lie group G the coreductive radical Ncored (G) agrees with [G, R(G)]. Proof. For finite-dimensional groups, [G, R(G)] is an analytic subgroup with Lie algebra [g, r(g)] = ncored (g). That is, [G, R(G)] = expG ncored (g). By Definition 10.40, Ncored (G) = expG ncored (g) = [G, R(G)]. Exercise E10.9. Verify the following result. Proposition. Let A = expG a, B = expG b, be analytic subgroups of a finitedimensional Lie group G, where a = L(A), b = L(b) for subalgebras a, b of g = L(G). Assume that b is an ideal. Then the group [A, B] generated by all commutators aba −1 b−1 , a ∈ A, b ∈ B is an analytic subgroup with Lie algebra [a, b]. [Hint. This is not exactly trivial. Modify the procedure given in [102, Lemma 5.57ff.] through Theorem 5.60 and generalize it appropriately. The fact that [102] deals with linear Lie groups there is immaterial.] Proposition 10.47. For any connected pro-Lie group G, the coreductive radical ˙ ∩ R(G). Ncored (G) agrees with [G, R(G)] ⊆ G Proof. Let N ∈ N (G). Then from 10.45 we know [G, R(G)]N/N = [G, R(G)]N/N = [G/N, R(G/N)] = Ncored (G/N ) = Ncored (G)N/N = Ncored (G)N /N.  This means [G, R(G)]N = Ncored (G)N. Now N ∈N (G) Ncored (G)N is the set of def

points of adherence of the filter basis B = {Ncored (G)N : N ∈ N (G)}, where a point is a point of adherence of a filter basis of sets if every neighborhood meets each set of the filter basis. The closed set Ncored (G) clearly consists of such points of adherence. Conversely, if g is a point of adherence of B, let U be an identity neighborhood of G. Find an identity neighborhood V such that V V ⊆ U . Now let N ∈ N (G) be such that N ⊆ V . Then Ncored (G)N ∩ gV = Ø, that is, there are points x ∈ Ncored (G), n ∈ N , and v ∈ V such that xn = gv. Then x = gvn−1 ∈ gV V ⊆ gU . Thus every neighborhood  of g meets Ncored (G), and since Ncored (G) is closed, g ∈ Ncored (G) follows. Thus N ∈N (G) Ncored (G)N = Ncored (G). In the same vein one establishes that the points of adherence of the filter basis {[G, R(G)]N : N ∈ N (G)} are the points of [G, R(G)]. Thus [G, R(G)] = Ncored (G) follows.

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The Structure of Reductive Pro-Lie Groups We saw that every connected pro-Lie group is the extension of a pronilpotent characteristic subgroup Ncored (G) and a reductive group G/Ncored (G). We have information on pronilpotent connected pro-Lie groups through our knowledge of abelian pro-Lie groups in Chapter 5 and Lemma 10.41. We need to have information on reductive groups now. Recall from Definition 10.27 that a connected pro-Lie group G is reductive iff R(G) = Z(G)0 . The following includes a summary of what we know at this point, notably from 10.29 through 10.34: Characterisation of Reductive Pro-Lie Groups Theorem 10.48. Let G be a connected pro-Lie group. Then the following statements are equivalent: (i) G is reductive. (ii) g is reductive. (iii) g = z(g) ⊕ g˙ for a unique semisimple pro-Lie algebra g˙ , obtained as the closed commutator subalgebra of g. (iv) g = a ⊕ r for a central closed subalgebra a ⊆ z(g) and a closed reductive def subalgebra r = L(A(˙g)). (v) G = AS = AS for a closed connected central subgroup A and a semisimple minimal analytic subgroup S. (vi) G = Z(G)S for a semisimple minimal analytic subgroup S, that is, S = A(s) for some semisimple subalgebra (indeed ideal) s. (vii) G = Z(G)S for the minimal analytic subgroup S = A(˙g), and g˙ is semisimple. Proof. (i) ⇔ (ii): See Theorem 10.29 (i). (ii) ⇔ (iii): See Theorem 7.27 and Corollary 7.28 def

(iii) ⇒ (iv): We let S = A(˙g); then S is a minimal analytic semisimple subgroup, def

and the closed connected subgroup S is reductive by Proposition 10.32. Thus r = L(S) is reductive by (i) ⇔ (ii) and thus r = z(r)⊕˙r = z(r)⊕˙g. We have z(r) ⊆ z(˙g, g) = z(g). Every closed vector subspace of a weakly complete topological vector space is a direct factor, and thus there is a closed vector subspace a ⊆ z(g) such that z(g) = a ⊕ z(r). Then g = a ⊕ z(r) ⊕ g˙ = a ⊕ r. (iv) ⇒ (v): We know that (g) = (a) · (r) and that this product is direct (algebraically and topologically) since  implements an equivalence of categories proLieAlg → proSimpConLieGr and thus preserves finite products. def

Let A = πG ((a)) = expG a and S = πG ((r)) = A(r), where L(S) = r. Then expG (a ⊕ r) ⊆ (expG a)(expG r) ⊆ AS and since expG g is dense in G by Corollary 4.22 (i) we have G = AS. (v) ⇒ (vi): By Theorem 10.32 (v), S = Z(S)S. Since A is central, and G = AS = AS we have Z(S) ⊆ Z(G) and thus AS ⊆ Z(G)S, and so Z(G)S is dense in G. Recall

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that the second center Z2 (G) of a group G is that characteristic subgroup containing Z(G) for which Z2 (G)/Z(G) = Z(G/Z(G)). The quotient group G/Z(G) has a dense semisimple analytic subgroup SZ(G)/Z(G). Then it is reductive and satisfies G/Z(G) = Z(G/Z(G))SZ(G)/Z(G) = Z2 (G)S/Z(G) by Theorem 10.32. Thus Z(G)S = G = Z2 (G)S. We claim Z2 (G) = Z(G) and follow the lead of the proof of 10.32 (v). For this purpose let N ∈ N (G). Then G/N is a connected reductive Lie group with an analytic semisimple subgroup SN/N such that G/N = Z(G/N)(SN/N ) = Z2 (G/N )(SN/N ), in particular, Z2 (G/N ) ⊆ Z(G/N )(SN/N ). Now SN/N is reductive and of the form Z(SN/N )0 (SN/N ). Then Z(SN/N )0 ⊆ Z(G/N )0 . Thus Z(G/N )0 (SN/N ) = Z(G/N)0 Z(SN/N )0 (SN/N ) = Z(G/N )0 SN/N is a reductive analytic group and its closure is reductive. (See Exercise E10.10 below.) The class 2 nilpotent normal subgroup Z2 (G/N ) therefore is contained in a reductive group, whose maximal solvable normal subgroup is its center. Thus Z2 (G/N ) is abelian and so Z2 (G/N ) = Z(G/N). In the proof of 10.32 (v) this allowed us to conclude that Z2 (G) = Z(G). Therefore G = Z(G)S. (vi) ⇒ (i): The maximal connected normal solvable subgroup R(G) is Z(G)0 . Thus G is reductive. (vi) ⇒ (vii): Since (vi) and (iii) are equivalent, the semisimple ideal s is contained in the unique largest semisimple ideal g˙ . Therefore G = Z(G)A(s) ⊆ Z(G)A(˙g) ⊆ G. (vii) ⇒ (vi): This is trivial. Exercise E10.10. Prove the following Proposition. Let G be a finite-dimensional Lie group and H a dense analytic subgroup such that h = L(H ) = a ⊕ s for an abelian subalgebra a and a semisimple finite-dimensional Lie algebra s. Then G is reductive and g = z ⊕ s for an abelian ideal z containing a. [Hint. Since H is dense in G then g = h = s by Theorem 9.22. It follows that g is the def

direct sum of the radical r(g) and the ideal s. Since a is central in h, so A = expG a is central in H and then also in H = G. Hence a ⊆ z(g) ⊆ r(G). Now by Theorem 7.67 we have ncored (g) = [g, r(g)] = [g, g] ∩ r(g) = s ∩ r(g) = {0}. Thus r(g) is central, and this means that g and therefore G is reductive.] Since we are dealing with finite-dimensional Lie groups, we remark that the algedef

braic commutator group G is the analytic subgroup S = expG g  = expG s (see for instance [102, Theorem 5.60]), and S is semisimple. Example 10.33 shows that S need not be closed. Let s be a semisimple closed subalgebra of the pro-Lie algebra L(G) of a pro-Lie  group G. Then there is an isomorphism α : s → j ∈J sj with a family of simple finite-dimensional Lie algebras sj . For each k ∈ J there is a morphism "  s if j = k, σk : sk → sj , σk (s) = (sj )j ∈J , sj = 1 otherwise. j ∈J

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We know that H = A(s) is a reductive pro-Lie group. Definition 10.49. An analytic subgroup S of H is called an atom of the semisimple analytic subgroup A(s) if there is an isomorphism α : s → j ∈J sj and an index j ∈ J such that S = A(α −1 (σj (sj )). In this case L(S) = α −1 (σ (sj )) ∼ = sj . An atom may or may not be closed, as we have seen in examples like 10.26 (b), 10.30, or 10.33. Let s be a reductive real finite-dimensional Lie algebra and s = k⊕a⊕n its Iwasawa decomposition (see [78] or [197]). Then k is a maximally compactly embedded subalgebra, that is, ead k is a compact subgroup of gl(s). (This concept is extensively explored in [84].) The compact Lie algebra k may or may not have a nontrivial center z(k). Lemma 10.50. For a semisimple finite-dimensional real Lie algebra s, the following statements are equivalent: z(k) = {0}. Z((k)) is infinite. Z((s)) is infinite. Z((s)) ∼ = F × Zn for n > 0 and some finite abelian group F . There is a finite-dimensional connected pro-Lie group S such that L(S) = s and A(s) = expS s is a dense analytic subgroup of S different from S. (vi) There is a connected pro-Lie group S such that L(S) = s and πS :  S → S is not a quotient morphism. (vii) There is a connected semisimple pro-Lie group H such that H = A(s) such that H = A(s). (i) (ii) (iii) (iv) (v)

Proof. (i) ⇔ (ii) ⇔ (iii): Exercise E10.11. (iii) ⇔ (iv): The center of a semisimple Lie group such as (s) is a finitely generated abelian group. (For a compact connected Lie group K the fundamental group π1 (G)  → K, is finitely which is isomorphic to the kernel of the universal covering πK : K generated (see for instance [102, Proposition 5.75, or Corollary 6.16, Theorem 6.18]), and the fundamental group of (s) is that of a maximal compact subgroup). Thus the equivalence of (iii) and (vi) follows from the Fundamental Theorem of Finitely Generated Abelian Groups. (See for instance [102, Theorem A1.11].) (iv) ⇒ (v): This is a special case of Proposition 10.31. (v) ⇔ (vi) is clear from the fact that  S = (s) and A(s) = πS ((s)). (v) ⇒ (vii) is trivial. (vii) ⇒ (iii): We are in the situation of Theorem 10.32 with a one-element family {s} of simple Lie algebras. We have L(H ) = z(h) ⊕ s. Since H is a pro-Lie group by hypothesis; from Corollary 4.21 (ii), we obtain limN ∈N (H ) L(N ) = {0}. Since s is a Lie algebra, we find a zero neighborhood U of s not containing any vector spaces; then z(h) ⊕ U is a zero neighborhood of h in which z(h) is the unique largest vector

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subspace. Hence we find an N ∈ S such that L(N ) ⊆ z(h). Then N0 ⊆ Z(H ); as A(s) is a semisimple analytic subgroup, N ∩ A(s) is central whence N ⊆ Z(S). In the def

reductive Lie group H /N the analytic subgroup A = AH /N ((s + n)/n) = A(s)N/N is dense and if we choose N small enough so that n = z(h), then A = H and L(A) ∼ = s. Since A is a homomorphic image of A(s), if Z(A) is infinite, then Z(A(s)) is infinite, whence Z((s)) is infinite, since there is a surjective homomorphism of topological groups (s) → A(s), and (s) is a simply connected semisimple Lie group. Thus, in order to simplify notation, we replace H by H /N and rename everything; thus we assume that H is a reductive Lie group with a dense semisimple analytic subgroup A = H and we have to show that Z(A) = Z(H ) ∩ A is infinite. Suppose def

that this was not the case, that is, that F = Z(H )∩A is finite. Then H /F is a reductive Lie group in which the semisimple analytic subgroup A/F is dense while remaining different from H /F ; since A/F is center-free, Z(H )/F is the center Z(H /F ). The Lie group H satisfies H = Z(H )A by Theorem 10.32 (v). Thus H /F is algebraically the direct product of H /F and A/F . We simplify notation and rename H /F to be H . Let L be the Lie group on the underlying group A such that L → A is the continuous identity map inducing an isomorphism of Lie algebras. (See Proposition 9.6 (iii).) Then (z, a) → za : Z(H )0 × L → H is a bijective morphism of connected Lie groups inducing on the Lie algebra level the isomorphism z(h)×s → z(h)⊕s = h. It is therefore open and thus is an isomorphism. Hence L → A is an isomorphism and A is a Lie group. Hence it is a closed subgroup of H . This entails that A = A = H , and this is a contradiction. This contradiction finally proves the claim and completes the proof of the lemma. Exercise E10.11. Prove the equivalence of (i), (ii) and (iii) in Lemma 10.50. [Hint. If a compact Lie group K is semisimple then its center is finite. (See for instance [102, Theorem 5.76].) For a compact group R(K) = Z(K)0 , and this subgroup is  the universal covering group is compact. Conclude that singleton iff z(k) = {0} iff K (i) ⇔ (ii). For Z((k)) = Z((s)) note that (s) = (k + a + n) is homeomorphic to (k) × (a) × (n) (see [78, p. 270, Theorem 5.1]; see also [197]), and that (a) isomorphic to a vector group, and (n) is homeomorphic to n (see Theorem 8.13). Thus (k), (k) × (a) × (n) and (g) have the same center, namely π1 (Ad(G)) = π1 (G/Z(G)).] Definition 10.51. A finite-dimensional semisimple real Lie algebra s is said to be of unbounded type, if the equivalent conditions of Lemma 10.50 are satisfied, otherwise it is said to be of bounded type. If S is an atom of a semisimple analytic subgroup A(s) of a pro-Lie group G, then it is said to be of unbounded type if L(S) is of unbounded type. Otherwise it is said to be of bounded type. The analytic subgroup A(s) is said to be of unbounded type if it has an atom of unbounded type, otherwise it is said to be of bounded type. Typically, sl(2R) is of unbounded type.

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10 The Global Structure of Connected Pro-Lie Groups

It is discussed in [84] that the semisimple Lie algebras of unbounded type are exactly the ones which support convex closed generating pointed cones which are invariant under all inner automorphisms. Recall from Definition 9.9 that a subalgebra s of the pro-Lie algebra g of a pro-Lie group G defines, by the universal property of (s) a morphism is = is,G : (s) → G such that, if we write s = L((s)) which we may, L(is ) : s → g is just the inclusion morphism and thus has zero kernel. The corestriction αs : (s) → A(s) of is onto the minimal analytic subgroup A(s) = A(s, G) of G with Lie algebra s has a central totally disconnected kernel D = ker is (see Theorem 4.20 and Proposition 4.23) and thus induces a bijective morphism αs : (s)/D → A(s). If s is semisimple, then we can write s =  j ∈J sj with simple Lie algebras sj by Corollary 7.29. Hence we may write (s) = j ∈J Sj for Sj =  (sj ), a simply connected simple Lie group for each j ∈ J . Thus D ⊆ Z((s)) = j ∈J Z(Sj ). Therefore Z((s)/D) ∼ = Z((s))/D and  αs yields a bijective morphism 

αs

Z(Sj )/D −−−→ Z(A(s)).

(†)

j ∈J

Each Z(Sj ) is isomorphic to Znj × Fj for some nonnegative integer nj and some finite def  abelian group Fj . Hence Z = j ∈J Z(Sj ) is isomorphic to ZI × C for a set I and the unique maximal compact subgroup comp(Z) = {0} × C which is a product of cyclic groups. We are dealing with a surjective morphism f : Z → K onto a compact group. Now f induces a surjection f  : Z/ comp(Z) → K/f (comp(Z)); if that is a quotient morphism so is f and vice versa. The question therefore is as follows: Problem P10.2. Is a surjective morphism f : ZI → K onto a compact group open for any set I ? The results in Theorem 4.1 and in Chapter 5 do not answer this question. If I is countable, then ZI is a Polish group and an Open Mapping Theorem applies in this case and gives an affirmative answer. The Open Mapping Theorem for Pro-Lie Groups does not apply since ZI is never almost connected for I nonempty. Theorem 10.52. Let G be a pro-Lie group and let s be a semisimple pro-Lie subalgebra def of g = L(G) defining a minimal analytic subgroup A(s, G) with Lie algebra s. (a) If Z(A(s, G)) is compact, then A(s, G) is closed. (b) The following statements are equivalent: (i) s is of bounded type. (ii) Whenever s ⊆ L(G) for a pro-Lie group L(G) then A(s, G) is closed in G. In other words, s is of bounded type if and only if A(s, G)) is closed in all pro-Lie groups G. Proof. (a) follows from Theorem 10.32 (vii).

The Structure of Reductive Pro-Lie Groups

(b) We may write s =



457

sj for a family of finite-dimensional simple real Lie  def algebras. We write Sj = (sj ). Then S = (s) = j ∈J Sj . (ii) ⇒ (i): We assume that (i) is false and show that (ii) fails. Thus we assume that (i) is of unbounded type. By Definition 10.51, there is at least one index k ∈ J such that sk is of unbounded type. By Lemma 10.50, there is a connected pro-Lie group k such  that L(k ) = sk and that A(sk ) is dense in k but not equal to k . Now let G = j ∈J Gj where " k if j = k, Gj = Sj if j = k.  Then L(G) ∼ = j ∈J L(Gj ) where " L(k ) ⊇ sk if j = k, L(Gj ) = L(Sj ) = sj if j = k. j ∈J

 We may identify a with a subalgebra of L(G). Then A(s) = j ∈J A(sj ) and since A(sk ) is not closed in k , the minimal analytic group A(s) is not closed in G.  (i) ⇒  (ii): By 10.32 (vi), A(s) = πG (S) for a closed semisimple subgroup of G ∼ and S = j ∈J Sj , Sj = (sj ). By (i) we assume that all sj are of bounded type. If sj is of bounded  type then Z(Sj ) is finite by Lemma 10.50 and Definition 10.51. Then Z(S) ∼ = j ∈J Z(Sj ) is compact. If D = ker πG |S then D ⊆ Z(S) and thus Z(S/D) = Z(S)/D is compact. Now πG induces a continuous algebraic isomorphism S/D → A(s) and thus Z(A(s)) is a continuous bijective homomorphic image of Z(S/D) and thus is compact and therefore closed. Then (a) shows that A(s) is closed. Proposition 10.53. (i) Let G be a topological group. Then a maximal connected abelian subgroup C is closed. (ii) If G is a connected pro-Lie group, then a maximal connected abelian subgroup C has a maximal abelian subalgebra L(C) of L(G). If a is a maximal abelian subalgebra of L(G), then the subgroup expG a = A(a) is a maximal connected abelian subgroup. If C is a closed connected subgroup of G then C is maximal connected abelian iff L(C) is maximal abelian in L(G). (iii) Any maximal connected abelian subgroup C of a pro-Lie group G has a maximal compact subgroup comp(C) and G is isomorphic algebraically and topologically to the direct sum of a weakly complete vector group and comp(G). (iv) If G is a connected semisimple pro-Lie group, then a maximal connected abelian subgroup C is isomorphic to RI × TJ for sets I and J that is, the maximal compact subgroup comp(C) of C is a torus, that is, a product of circles. Proof. (i) This an immediate consequence of the observation that the closure of a connected abelian subgroup is connected and abelian. (ii) If C is a closed abelian subgroup of a pro-Lie group G, then L(C) is a closed abelian subalgebra of C. (See Theorem 9.22 (v) and Corollary 9.24.). Let a be an

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10 The Global Structure of Connected Pro-Lie Groups

abelian subalgebra containing L(C). Assume that C is a maximal abelian connected subgroup; we must show a = L(C). Since a is abelian we assume a to be closed. Then expG a is an abelian (minimal) analytic subgroup, and A = expG a is a closed connected abelian subgroup. Now we recall Corollary 4.22 and compute C = expG L(C) ⊆ expG a = A. Since C was maximal connected abelian, we get A = C. Then L(C) ⊆ a ⊆ L(A) = L(C), yielding a = L(C), as asserted. Thus L(C) is maximal abelian if C is maximal connected abelian. Now let a be a maximal abelian subalgebra of L(G) and set C = expG a. Then C is a closed connected abelian subalgebra. Let A be a closed connected abelian subgroup of G containing C. We must show A = C. Now A is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35, and L(A) is an abelian closed subalgebra of L(G) by 9.24, and a ⊆ L(C) ⊆ L(A) by 9.22. The maximality of a implies L(A) = a and thus C = expG a = expG L(A) = A (by 4.22). The last assertion of (ii) follows from these facts in view of the fact that by 4.22, C = expG L(C). (iii) This is an immediate consequence of the Vector Group Splitting Theorem for Connected Abelian Pro-Lie Groups 5.12. (iv) Now let G be a connected semisimple pro-Lie group and C a maximal connected abelian subgroup. By Theorem 10.29 and Corollary 7.29, L(G) ∼ = j ∈J sj for a family of simple finite-dimensional Lie algebras sj . The image c of L(C) in the product is a maximal abelian subalgebra  by (ii) above. Let cj = pr j c; then cj is an abelian closed subalgebra of sj and c ⊆ j ∈J cj and this last product is abelian. The maximality of c  now implies c = j ∈J cj

Postscript We recall from Chapter 7, that every pro-Lie algebra g is a semidirect sum r(g) ⊕ s of a maximal pro-solvable ideal r(g) and a semisimple subalgebra which is a product of finite-dimensional simple Lie algebras. In Chapter 8 we were able to translate this into a fairly satisfactory structure theorem on simply connected pro-Lie groups. Namely, every simply connected pro-Lie group G is a semidirect product R(G)  S of a closed normal subgroup R(G)  whose Lie algebra is the radical of L(G), and a subgroup S which is a product j ∈J Sj of a family of simply connected simple finite-dimensional Lie groups Sj . Also R(G) is homeomorphic to RI for some set I . As a variation of this theme we found in any pro-Lie algebra g a smallest pronilpotent ideal ncored (g) such that  g/ ncored (g) is a reductive Lie algebra which we know to be of the form RI × j ∈J sj for a family of finite-dimensional simple Lie algebras sj . This gives us for every simply connected pro-Lie group G a normal subgroup Ncored (G) which is isomorphic to (ncored (g), ∗), its Lie algebra equipped with the Baker–Campbell– Dynkin–Hausdorff multiplication ∗; moreover we  know that G/Ncored (G) is a simply I connected reductive group isomorphic to R × j ∈J Sj for the same family of simply

Postscript

459

connected simple Lie groups Sj we had before. This is a fairly satisfactory state of affairs for the structure of simply connected pro-Lie groups. So now we turn to the nonsimply connected case. In Chapter 6 we discovered, that for every connected pro-Lie group G we obtain functorially a simply connected  = (L(G)) and a morphism πG : G  → G with convincing universal pro-Lie group G properties; we observed in Theorem 8.21 that for any pro-Lie group G that has a universal covering group in the topological sense, πG is the universal covering morphism. This applies, in particular, to any connected Lie group. In order to proceed further we need a better understanding of the structure of the radical and the coreductive radical in the case of connected but not necessarily simply connected pro-Lie groups. In particular, the following are the key questions: (A) Is the radical R(G) in any way a “solvable” topological group? (B) Is the coreductive radical Ncored (G) in any way a “nilpotent” group? (C) How do we define reductive and semisimple pro-Lie groups? What is their structure if they are not simply connected? Even in the case of finite-dimensional Lie groups this topic is not devoid of surprises; so surely in our context we had to take a careful look. Regarding questions (A) and (B), solvability and nilpotency, we had to face a great variety of possible definitions; the complications we face on the pro-Lie group level are even greater than those we faced on the pro-Lie algebra level in Chapter 7. The topic may seem tedious and pedestrian, but one has to penetrate the jungle. In Theorem 10.18 it is shown that eight sensible concepts of solvability coincide for pro-Lie groups. So in our structure theorem for a simply connected pro-Lie group G we now know that the radical R(G) is solvable in any of these reasonable meanings. In particular, in descending the algebraic or topological commutator series transfinitely, it suffices to go to the first infinite ordinal, namely the countable ordinal ω represented by the well-ordered set of natural numbers. One question is certainly appropriate for a Lie theory of solvable pro-Lie groups: (A ) How do commutator series on the group level compare with the commutator series on the Lie algebra level? Theorem 10.20 says that this works as well for the topological commutator series as might be expected. Once that much of a theory of connected solvable pro-Lie groups is developed, one finds satisfactory information on the radical R(G), up to a point. Theorem 10.25 combines group theory and Lie algebra theory in the spirit of Lie theory by specifying that, for a pro-Lie group G, the radical R(G) is the largest connected transfinitely topologically solvable normal subgroup and is a closed connected characteristic subgroup of G such that L(R(G)) = r(g). Now that we understand the radical it is appropriate to call a connected pro-Lie group semisimple if its radical is singleton, and reductive, if it is central.

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10 The Global Structure of Connected Pro-Lie Groups

At this stage we are ready for the first structure theorem on semisimple and reductive pro-Lie groups: see 10.29. This structure theorem exposes certain aspects which are familiar from classical Lie group theory, and other aspects which are unfamiliar due to several complications that arise in pro-Lie group theory: Firstly, quotients may be incomplete and therefore fail to be pro-Lie groups as we saw in Chapter 4, and the minimal analytic subgroup belonging to the Lie algebra of the group may be properly smaller than the group itself. Yet Theorem 10.29 keeps these phenomena under control and paves the way for a theorem that describes the structure of the closure of a semisimple analytic subgroup of a pro-Lie group: Theorem 10.32. Results on nilpotency which are analogous to the ones on solvability are obtained in Theorem 10.36. This theorem is not entirely parallel to Theorem 10.18. We do not know whether a transfinitely nilpotent pro-Lie group is pronilpotent. On the other hand, the topological descending series is related to the topological descending series of the Lie algebra in a way analogous to the results for the closed commutator series expressed in Theorem 10.20. Moreover, the coreductive radical Ncored (G) of a connected pro-Lie group is well understood after our Theorem 10.43. Since G/Ncored (G) is a reductive pro-Lie group, this is the place to resume the thread of answers to Question (C) on the structure of semisimple and reductive groups that we started with Theorems 10.29 and 10.32; in Theorem 10.48 we complete a structure theorem on reductive pro-Lie groups. If G is a pro-Lie group with Lie algebra g and if s is a semisimple closed subalgebra of g, then we obtain a unique minimal analytic subgroup A(s; G) of G whose Lie  algebra is s ∼ = j ∈J sj . But what is the structure of the connected pro-Lie subgroup H = A(s; G)? In Theorem 10.52 (a) we show that H = A(s; G) if the center of A(s; G) is compact. We recognize that the nature of the analytic subgroup A(s; G) depends much on the structure of the finitely generated center Z((sj )) of the simply connected simple group (sj ). Things are generally under control, if for all j this group is finite. We shall pursue the construction of examples in the next chapter, when we resume the thread of the discussion of reductive pro-Lie groups in the context of splitting theorems. Then we shall see examples showing a great diversity of groups with curious properties which can be constructed when atoms of unbounded type are involved. What we learn from the structure theory of semisimple and reductive pro-Lie groups is that considerable complications in the global structure theory arise whenever elements of infinite order are present in the center of the simply connected pro-Lie group (s) generated by a semisimple Lie algebra.

Chapter 11

Splitting Theorems for Pro-Lie Groups

Splitting theorems say that certain topological groups are direct or semidirect products of suitable subgroups, or at least that they come close to decomposing in this fashion. We have seen examples such as Theorem 11.1 (= 5.19). Assume that G is an abelian pro-Lie group with a closed subgroup G1 that is isomorphic to a product copies of R and of T. Then G1 is a homomorphic retract of G, that is, a direct summand algebraically and topologically. So G ∼  = G1 × G/G1 . The Vector Group Splitting Theorem 5.20 was the core structural result on abelian pro-Lie groups, and is used throughout the book. Of course, Chapter 7 on pro-Lie algebras is full of splitting theorems in the context of reductive modules. For pro-Lie algebras we had the Levi–Mal’cev Splitting Theorem. Theorem 11.2 (= 7.52). Every pro-Lie algebra is a semidirect sum, algebraically and topologically, of the radical r(g) and some semisimple subalgebra s. If s1 and s2 are two such summands, then there is an inner automorphism α such that s2 = α(s1 ). Every simply connected pro-Lie group G is the semidirect product, algebraically and topologically, of its radical R(G) and a subalgebra S, and two such algebras are conjugate under inner automorphisms. The radical is topologically the direct product of the nilradical N (G) which is isomorphic to its Lie algebra (n(g), ∗) endowed with the Campbell–Baker–Hausdorff–Dynkin multiplication and a weakly complete topological vector space (8.3, 8.13, 8.14). Every compact connected group G is a homomorphic image of a direct product of its (central) radical R(G) and its (semisimple) commutator subgroup G and is a semidirect product of G and an abelian subgroup isomorphic to G/G ([102, Theorem 9.24, Theorem 9.39]).  Theorem 11.3 (= 9.44). Every finite-dimensional pro-Lie group G is a homomorphic image of a direct product of a compact totally disconnected metric central subgroup of G and a simply connected Lie group L modulo a discrete subgroup and the homomorphism implements an isomorphism L(L) → L(G).  In this chapter we shall provide more splitting theorems in the same spirit. Prerequisites. We need the general theory of pro-Lie groups and pro-Lie algebras up through Chapter 10.

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11 Splitting Theorems for Pro-Lie Groups

Splitting Reductive Groups Semidirectly We have recalled in Theorem 11.1 that in an abelian pro-Lie group, a subgroup which is a product of a family of real lines and a family of circles will always split as a direct factor, algebraically and topologically. In Proposition 10.52 (iv) we noted that a maximal abelian subgroup of a connected semisimple pro-Lie group is always of this form. We shall use these facts to prove a remarkable splitting theorem for certain connected reductive pro-Lie groups. For this purpose we do need some preparations. Lemma 11.4. Assume that N and H are abelian subgroups of a topological group such that N ∼ = RI × TJ and [N, H ] = {1}. If H is a pro-Lie group, then the inclusion map i : N ∩ H → N extends to a morphism H → N . Proof. The groups RI × TJ are relative injectives for the class of embeddings in the category AbproLieGr of abelian pro-Lie groups by Theorem 5.19. We say that a function f : H → N between topological groups is a 1-cocycle if there is a continuous action (h, n)  → h · n : H × N → N such that every n  → h · n is an automorphism of n and such that the following functional equation is satisfied:   f (h1 h2 ) = h1 · f (h2 ) f (h1 ). (In Exercise E1.5 we discussed this concept with the reverse order of factors on the right side; this is a matter of convenience for the algorithms in which it is used, and if N is abelian, the difference evaporates.) An action of the type we need here is also called an automorphic action. If the action is constant and N is abelian, the 1-cocycle is a homomorphism. We have encountered cocycles in Chapter 1 in the discussion of semidirect products. Let Z 1 (H, N ) denote the set of all cocycles f : H → N . If N and H are subgroups of G and H is in the normalizer of N , acting on N under inner 1 automorphisms, let j : N ∩ H → N be the inclusion morphism and let ZN ∩H (H, N ) denote the set of cocycles f : H → N with f |(N ∩ H ) = j . Using this notation we formulate the group theoretical background of our discussion in the following proposition. Lemma 11.5. (i) Assume that G is a topological group, N a normal subgroup, and H a closed subgroup such that G = NH . The group H acts on N automorphically via h · n = hnh−1 . The following conditions are equivalent: (1) There is a subgroup A such that the function ν : N  A → G,

ν(n, a) = na,

is a bijective morphism of topological groups such that ν −1 |H : H → N  A is continuous. In particular, G is algebraically a semidirect product of N and A. (2) The inclusion function N ∩ H → N extends to a 1-cocycle f : H → N with respect to the action of H on N .

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(ii) Also, the following conditions are equivalent and imply the conditions (1) and (2) above: (3) There is a closed subgroup A such that the function ν : N  A → G,

ν(n, a) = na,

is an isomorphism of topological groups, that is, G is a semidirect product of N and A. (4) The inclusion function j : N ∩ H → N extends to a 1-cocycle f : H → N with respect to the action of H on N and the function μ : N  H → G,

μ(n, h) = nh,

is an open surjective morphism of topological groups, that is, G ∼ = (N  H )/D, D = {(h−1 , h) : h ∈ N ∩ H } ∼ = N ∩ H. 1 (iii) Let ZN ∩H (H, N ) and C(N ) denote the sets of cocycles extending the identity of N ∩ H and the set of cofactors (that is semidirect factors complementary to N ), respectively. Then the function 1  : ZN ∩H (H, N ) → C(N ),

(f ) = {f (h)−1 h | h ∈ H },

is a bijection. Proof. (i) First we show (1) ⇒ (2). Every g ∈ G decomposes uniquely and continuously into a product na with n ∈ N and a ∈ A. In particular, each h ∈ H defines a unique element f (h) ∈ N and a ϕ(h) ∈ A such that (ϕ(h), f (h)) = ν −1 (h), that is, h = f (h)ϕ(h). Thus f : H → N is a continuous function. If h ∈ N ∩ H , then ϕ(h) = 1 and  thus f (h) = h. Also,  f (h1 h2 )ϕ(h1 h2 ) = h1 h2 = h1 f (h2 )ϕ(h2 ) = h1 · f (h2 ) h1 ϕ(h2 ) = h1 · f (h2 ) f (h1 )ϕ(h1 )ϕ(h2 ). Then f is the desired cocycle. Next we prove (2) ⇒ (1). Assume that f : H → N is a 1-cocycle with respect to the action of H on N and agreeing with the identity on N ∩ H . We define p : H → G by p(h) = f (h)−1 h. Then p is continuous. We note p(h1 h2 ) = #  $−1 h1 · f (h2 ) f (h1 ) h1 h2 = f (h1 )−1 h1 f (h2 )−1 h−1 1 h1 h2 = p(h1 )p(h2 ). Thus p is a morphism of topological groups having N ∩ H in its kernel. The surjective morphism of topological groups μ : N  H → G, μ(n, h) = nh has the kernel def

K = {(h−1 , h) : h ∈ N ∩ H }, and there is a bijective morphism of topological groups μ1 : (N  H )/K → G. The morphism p  pr H : N  H → G vanishes on ker μ and thus induces a morphism p : (N  H )/K → G giving us a homomorphism of groups P : G → G, P = p   μ−1 1 , and if g ∈ H then P (g) = p(g), if g ∈ N then P (g) = 1. Thus no matter how an element g is represented as a product nh with n ∈ N and h ∈ H we have P (g) = p(h) = f (h)−1 h. We set A = p(H ) = P (H ). Note that P 2 (nh) = P (f (h)−1 h) = P (f (h)−1 )P (h) = 1 · P (h) = P (n)P (h) = P (nh) and thus P 2 = P , that is, P is an idempotent endomorphism of G. For g = nh we have 1 = P (g) = f (h)−1 h iff h = f (h); since f (h) ∈ N this equation holds only if

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11 Splitting Theorems for Pro-Lie Groups

h ∈ N ∩ H , and if h ∈ N ∩ H , then f (h) = h by hypothesis. Thus ker P = N . Therefore, NA = NH = G and N ∩ A = {1}. The bijective morphism ν : N  A → G, ν(n, a) = na, has the inverse function g  → (gP (g)−1 , P (g)), and if h ∈ H , then ν −1 (h) = (hp(h)−1 , p(h))(h(f (h)−1 h)−1 , p(h)) = (f (h), f (h)−1 h), and so this inverse is continuous when restricted to H . (ii) We note that P is continuous iff ν is an isomorphism of topological groups iff μ : N  H → G is open. Under these circumstances, A is a closed subgroup. 1 −1 (iii) Let f ∈ ZN ∩H (H, N ). We saw that the set A = {f (h) h | h ∈ H } is a cofactor of N and that every cofactor arises in this way. Hence the function  is well 1 defined and is surjective. In order to see its injectivity, let fj ∈ ZN ∩H (H, N ), j = 1, 2 be two cocycles such that (f1 ) = (f2 ). Write A = (f1 ). Define the morphisms ϕj : H → A by h = fj (h)ϕj (h). Since the product G = N A is semidirect, we have a projection p : G → A such that every element g ∈ G is uniquely written as g = np(g) with n ∈ N . Accordingly, ϕ1 (h) = p(h) = ϕ2 (h) for h ∈ H . It follows that f1 = f2 . This completes the proof. If G = NH and H centralizes N, then f : H → N is a cocycle iff h  → f (h)−1 is a morphism; so the issue is whether the inclusion morphism of N ∩ H → N extends to a morphism H → N; this is an important special case as we shall see presently. Note that N ∩ H is central in H . We define HomD (H, N ) to be the set of all morphisms f : H → N which agree on D with the inclusion j : D → N. If it happens that H and N commute elementwise and the action of H on N is by inner automorphisms, and if H happens to be commutative, then for f ∈ Z 1 (H, N ) we have f (xy) = f (yx) = (yf (x)y −1 )f (y) = f (x)f (y). In particular, in these 1 (H, N ) = Hom (H, N ). circumstances, ZD D The simplest case in which 11.5 applies is that of an abelian pro-Lie group. Lemma 11.6. Let N and H be abelian subgroups of a topological group such that N is a group isomorphic to RI × TJ , for sets I and J , and N and H commute elementwise. If H is a pro-Lie group, then there exists a closed subgroup A ⊆ N H such that (n, h)  → nh : N × A → NA is a bijective morphism of topological groups. In particular, N H is algebraically a direct product and equals N A. Moreover, if the surjective morphism (n, h)  → nh : N × H → N T is open, then A is closed and (n, h)  → nh : N × A → NA is an isomorphism of topological groups, that is, NH is a direct product of N and A. Proof. By the preceding Lemma 11.5 it suffices to extend the inclusion morphism N ∩ H → N to a homomorphism H → N . Since H and N are pro-Lie groups, and since N ∼ = RI × TJ for some sets I and J this is accomplished by Lemma 11.4 above. Recall Definition 10.51 where it was explained when a semisimple pro-Lie algebra is of bounded type.

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465

Lemma 11.7. Let G be a connected reductive pro-Lie group and assume that the closed ˙ and the morphism commutator algebra s = g˙ is of bounded type. Then G = Z(G)0 G ˙ → G, μG : Z(G)0 × G

μG (z, g) = zg,

is open, that is, is a quotient morphism inducing an isomorphism L(μG ) : z(g) × g˙ → g,

L(μG )(z, x) = z + x,

on the Lie algebra level. Proof. Since g˙ is of bounded type, A(s) is closed by Theorem 10.52 and therefore agrees ˙ by Theorem 9.26. Since g˙ is of bounded type, with the closed commutator subgroup G ˙ Z(G) is compact. (Compare the proof of 10.52.) Since quotients of pro-Lie groups modulo compact subgroups are pro-Lie groups, we may readily form the quotient group ˙ ∩G ˙ = Z(G) ˙ without changing the of G modulo the compact central subgroup Z(G) ˙ = {1} and that G ˙ is center-free. general situation. Thus we may assume that Z(G) ∩ G ˙ By 10.48 (vi) we have G = Z(G)G. We consider the completion (G/Z(G))∗ of the quotient G/Z(G) and the injective ˙ → (G/Z(G))∗ , ψ(g) = gZ(G) = Z(G)g. Then L(ψ) is an isomormorphism ψ : G phism on the pro-Lie algebra level between pro-Lie algebras isomorphic to g. ˙ Let us ˙ is center-free and is therewrite s for L((G/Z(G))∗ ); then A(s, (G/Z(G))∗ ) = ψ(G) fore closed by Theorem 10.29 (iii)(d) and thus is complete. It follows that G/Z(G) is a center-free pro-Lie group with Lie algebra g. ˙ By the Sandwich Theorem 10.29 (iii)(c) we have a commutative diagram ψ ˙ G −−−−→ G/Z(G) ⏐ ⏐ ⏐ ⏐ f F   −−−→ j ∈J Sj /Z(Sj ) − j ∈J Sj /Z(Sj ) id

with quotient morphisms and a bijective morphism ψ. It follows that ψ is an isomorphism. Therefore, quot ψ −1 ˙ pr G˙ : G −−−−→ G/Z(G) −−−−→ G is a morphism and thus it follows readily that ˙ → G, μG : Z(G) × G

μG (z, g) = zg

is an isomorphism of pro-Lie groups. In particular this shows that Z(G) = Z(G)0 , since G is connected. We are now ready for a proof of the splitting theorem which we announced for this subsection.

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Splitting Theorem for Reductive Pro-Lie Groups Theorem 11.8. Let s be a Levi summand of the pro-Lie algebra g of a connected proLie group G, and let A(s) = expG g be the unique minimal analytic subgroup with Lie algebra s, and denote its center by Z. Consider the following statements: (a) The semisimple pro-Lie algebra g/r(g) is of bounded type. (b) Z is compact and contained in a maximal connected compact abelian subgroup of A(s) which is a torus. (c) A(s) is closed and has a maximal connected abelian subgroup isomorphic to RI × TJ . Then (a) ⇒ (b) ⇒ (c). Further, if (c) holds, then the following statements are equivalent: (i) G is reductive. ˙ is the minimal analytic subgroup A(˙g) for (ii) The closed commutator subgroup G the semisimple ideal s = g˙ of g = L(G) and there is a closed connected abelian ˙ under inner automorphisms such that ι(a)(g) = subgroup A of G acting on G aga −1 and such that ˙ ι A → G, ν: G

ν(g, a) = ga

is an isomorphism of pro-Lie groups. Proof. By the Levi–Mal’cev Theorem 7.52, g = r(g) ⊕ s, and thus s ∼ = g/r(g). Thus (a) implies that s is of bounded type. Then Z is compact as we saw in the  def proof of 10.52 (i) ⇒ (ii). The group S = (s) is isomorphic to  a product j ∈J Sj , Sj = (sj ) of simply connected simple Lie groups and Z(S) ∼ = j ∈J Z(Sj ). Since Z(Sj ) is contained in every maximal compact subgroup Kj (see for instance [78] or [197]), we know that Z(Sj ) is contained in every maximal compact connected abelian subgroup C; every such group is a maximal connected abelian subgroup of S. The kernel D of the morphism πG |S : S → A(s) is contained in Z(S). Thus πG |S induces an isomorphism of C/D onto a compact maximal connected abelian subgroup of A(s). By Proposition 10.53 (iv), C is a torus. Every homomorphic image of a torus is a torus (since dually, every subgroup of a free abelian group is free). Thus A(s) has a maximal connected abelian subgroup which is a torus, that is, it is isomorphic to TJ . Thus (b) holds. Assume (b) is true. Then Z is closed and so A(s) is closed by 10.52 (a). A torus is trivially isomorphic to RI × TJ with I = Ø. Thus (c) follows. Now we assume (c). (i) ⇒ (ii): If G is reductive then r(g) = z(g) and g = z(g) ⊕ s, where s = g ∼ = def g/r(g). Thus S = A(s) is a closed normal subgroup. Since G = Z(G)A(s) = A(s)Z(G) by 10.32. Let N be a maximal connected abelian subgroup of A(s) which is isomorphic to RI × TJ . Then we obtain a maximal connected abelian subgroup in the form of N H with H = Z(G)0 . Now we apply Lemma 11.6 and conclude the

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existence of an abelian subgroup A such that N A = N H and N ∩ A = {1}. Then SA = SN A = SNH = SH = G. We have to show that S ∩ A = {1}. Now a ∈ S ∩ A implies a ∈ S ∩ NA = S ∩ NH . Hence a = nh ∈ A with n ∈ N ⊆ S and h ∈ H . Thus h = n−1 a ∈ H ∩ S ⊆ N by hypothesis. Hence a = nh ∈ N ∩ A = {1}. Since μG : Z(G)0 × S → G is a quotient morphism by Lemma 11.7, it follows from Lemma 11.6 that ν is an isomorphism of topological groups. ˙ is semisimple, it is the unique (ii) ⇒ (i): Since the closed commutator group G Levi factor. Since the coreductive radical Ncored (G) is contained in the intersection ˙ by Theorem 10.47 and is, therefore, singleton in the present situation, G is R(G) ∩ G reductive. Corollary 11.9 (Uniqueness in 11.8). Assume that Condition (ii) of Theorem 11.8 is ˙ denote the set of cofactors of G ˙ in G. Then the function satisfied. Let C(G) ˙  ) → C(G), ˙  : HomD (Z(G)0 , G

(f ) = {f (z)−1 z | z ∈ Z},

is a bijection. Proof. (i) This is a consequence of Lemma 11.5 (iii) Exercise E11.1. Prove the following variations of this theme: Lemma A. (i) Let G be a connected simple real Lie group such that g has only one conjugacy class of Cartan subalgebras. (This is the case if g is a compact Lie algebra or the underlying real Lie algebra of a complex Lie algebra.) Then all maximal connected abelian subgroups of G are conjugate. (ii) Let G be a connected semisimple pro-Lie group such that all simple factors of g have only one conjugacy class of Cartan subalgebras. Then any two maximal connected abelian subalgebras of G are conjugate. Proposition B. Assume that the condition (a) in Theorem 11.8 is satisfied and that in addition each simple factor of g/r(g) has only one conjugacy class of Cartan sub˙ denote the set of conjugacy classes of C(G). ˙ algebras. Let Cconj (G) Let N be a ˙ ˙ maximal connected abelian subgroup of G and i : N → G the inclusion morphism, ˙ be the induced and let HomD (Z(G)0 , i) : HomD (Z(G)0 , N) → HomD (Z(G)0 , G) injection. Then the composition HomD (Z0 (G),i)  cls ˙ −− ˙ −− ˙ → C(G) → Cconj (G) HomD (Z0 (G), N ) −−−−−−−−→ HomD (Z0 (G), G)

˙ is conjugate to one that is generated by an is surjective. Thus every cofactor for G f ∈ HomD (Z(G)0 , N). [Hint. A (i). The maximal connected abelian subgroups of G are the exponential images of the Cartan subgroups h of g which are conjugate under inner automorphisms. Use expG ead x h = expG Ad(g)h = g(expG h)g −1 .   A (ii). Write g = j ∈J gj and consider the subalgebras h = j ∈J hj . The formula in (i) persists in the current situation.

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˙ such B. Let A ∈ Cconj (G ). By Corollary 11.9 there is an f ∈ HomD (Z(G)0 , G) ˙ and that (f ) = A. The image f (Z(G)0 ) is a connected abelian subgroup of G therefore, by A (ii), there is a g ∈ G such that gf (Z(G)0 )g −1 ⊆ N . Then the function g · f given by (g · f )(z) = gf (z)g −1 belongs to HomD (Z(G)0 , N ) and (g · f ) = {gf (z)−1 g −1 z | z ∈ Z(G)0 } = g{f (z)−1 z | z ∈ Z(G)0 }g −1 = gAg −1 ∈ cls(A) in view of the centrality of Z(G)0 .] We use a simple lemma for constructing examples: Lemma 11.10. Let P and Q be topological groups and P1 and Q1 closed central subgroups of P and Q, respectively. Let f : P1 → Q1 be an injective morphism of topological groups and let  = {(p, f (p)) : p ∈ P1 }. Then  is a closed central subgroup of P × Q and the factor group G = P ×Q  has the following two normal P ×f (P1 ) P1 ×Q ∗ ∗ and Q =  . subgroups: P =  Then the following conditions are satisfied: (i) G = P ∗ Q∗ , and P ∗ ∩ Q∗ = G/Q∗ = G/P ∗ =

and

P∗ ∼ = P ∗ ∩ Q∗

P1 × f (P1 ) ∼ = f (P1 ),  P ×Q ∼ = P /P1 , P1 × Q P ×Q ∼ Q , = P × f (P1 ) f (P1 ) P × f (P1 ) ∼ P , = P1 × f (P1 ) P1

P1 × Q ∼ Q Q∗ ∼ . = = ∗ ∗ P ∩Q P1 × f (P1 ) f (P1 ) (ii) The functions p → (p, 1) :P1 → P ∗ ∩ Q∗ , p  → (p, 1) :P → P ∗ , and q  → (1, q) :Q → Q∗ are bijective morphisms of topological groups, and the last one is an isomorphism. G G ∗ ∗ ∗ ∗ (iii) π : P ∗G ∩Q∗ → Q∗ × P ∗ , π(g(P ∩ Q )) = (gQ , gP ) is an isomorphism of topological groups (iv) There is a strict exact sequence of abelian topological groups 0 → Q → G → P /P1 → 0, where Q → G is an embedding and G → P /P1 a quotient morphism.

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Proof. The graph of a continuous function between Hausdorff spaces is always closed. Hence  is a closed subgroup of P1 ×Q and since it is contained in the central subgroup P1 ×Q1 of P ×Q is central and hence normal. Thus G = P ×Q  is Hausdorff topological group. We note that (P × {1}) = P × f (P1 ) and ({1} × Q) = P1 × Q; the intersection of these two groups is P1 × f (P1 ). The function ϕ : P1 × Q → P1 × Q, ϕ(p1 , q) = (p1 , q + f (p1 )) is an automorphism mapping P1 × {0} to  and thus P ×f (P ) ×Q ∼ P1 ×Q ∼ Q∗ = P1 = P1 ×{0} = Q yielding the isomorphism 1  1 ∼ = f (P1 ). It is clear ∗ ∗ that P , respectively, Q are the natural homomorphic images of P × {1}, respectively,

{1} × Q in G, whence G = P ∗ Q∗ and P ∗ ∩ Q∗ = P1 ×f(P1 ) . All assertions in (i) and (ii) then are straightforward group theory. Regarding (iii) it is clear that π is a bijective morphism. That it is in fact an isomorphism follows from ∗ ∗ Q . the facts that P ∗P∩Q∗ ∼ = QG∗ ∼ = PP1 and P ∗Q∩Q∗ ∼ = PG∗ ∼ = f (P 1) Assertion (iv) is a reformulation of the preceding results. We emphasize once more that f (P1 ) need not be closed in Q and that f need not implement an isomorphism onto its image. Neither of the factor groups G/P ∗ nor G/(P ∗ ∩ Q∗ ) need be Hausdorff. The universal “Hausdorffisation” of G/P ∗ is G/P ∗ . We should point out to the category theory minded reader that Lemma 11.10 is a pushout construction. Indeed, for the data given, let α : P1 → P be the inclusion morphism, β : P1 → Q the morphism given by β(p) = f (p)−1 (well defined because of the centrality of Q1 in Q), let σ : P → G be given by σ (p) = (p, 1), and τ : Q → G by τ (q) = (1, q), then P⏐1 ⏐ β Q

α

−−−−→ P ⏐ ⏐σ  −−−−→ G τ

is a pushout diagram. One illuminating example is the following: Take P√= P1 = R, Q = Q1 = T2 for T = R/Z, and let f : R → T2 , f (r) = (r + Z, a · r + Z). Then G ∼ = T2 , ∗ ∗ ∗ ∗ ∗ 2 ∗ ∼ ∼ P = P ∩ Q = f (R), Q = G, G/P = T /f (R) and G/Q = {0}. This example shows that none of the bijective morphisms in the lemma need be isomorphisms. A more canonical example that is familiar to workers in topological groups is as follows; it is explicitly treated in [102], Exercise E1.11 following Definition 1.30, complete with picture: Take P = R, P1 = Z, Q = Q1 = Zp the additive group of p-adic integers (see Example 1.20 (A)(i)), and let f : Z  → Zp be the canonical injection given by f (z) = (z + pn Z)n∈N ∈ limn Z/pn Z ⊆ n∈N Z/pn Z. Then G is the p-adic solenoid SSp (see Example 1.20 (A)(ii)), P ∗ = P ∗ ∩ Q∗ = (R × f (Z))/ is the dense one-parameter subgroup of SSp and Q∗ ∼ = Q = Zp is a compact subgroup ∼ P /P of SSp such that G/Q∗ ∼ = R / Z T . A detailed theory of finite-dimensional = = 1 pro-Lie groups subsuming this example was given in this book in Chapter 9, 9.43, 9.44.

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We now emulate this classical example in the context of abelian pro-Lie groups; later we shall embed this example into a connected reductive pro-Lie group, and this is the reason we present it here. Let denote the power ZN but equipped with the discrete topology and not the product topology. As in Chapter 4 in Example 4.9, we consider the compact, connected, and strongly locally connected (see Definition 4.8) abelian character group  , call it A, and let Aa = expA L(A) be the arc component of the identity. The exponential function expA : L(A) → A has the kernel K(A) ∼ = Hom( , Z) while L(A) ∼ = Hom( , R) and the exponential function gives a corestriction expA : L(A) → Ga which happens to be a quotient morphism by Proposition 4.10 so that it induces an isomorphism of topological groups L(A)/K(A) ∼ = Aa . The prodiscrete group K(A) is algebraically isomorphic to the free abelian group with countably many generators Z(N) (see Corollary 4.11; this goes back to the fact that Z is a “slender” abelian group (see [49, pp. 51ff., notably, p. 60, Corollary 2.4]). The group K(A) is an interesting example of a pro-Lie group itself as we pointed out in Proposition 5.2. The inclusion function Z → R induces the inclusions function Hom( , Z) → Hom( , R) = L(A). We recalled from Chapter 4 that there is a strict exact sequence expA

0 → Hom( , Z) → L(A) −−−−→ Aa → 0, where the corestriction expA : L(A) → Aa of the exponential function to its image is a quotient morphism onto a proto-Lie group which is not a pro-Lie group; the completion of Aa is A. The inclusion Z(N) → induces the restriction morphism Hom( , Z) → Hom(Z(N) , Z) ∼ = ZN and since Z is slender ([49, p. 51ff.]) every morphism ZN → Z is uniquely determined def

by its action on Z(N) and thus we have an injection f : K(A) = Hom( , Z) → ZN . This injection is not an embedding, because if it were, K(A) would be a Polish countable group and would therefore be discrete by the Baire Category Theorem; but, on the contrary, it is not discrete (see Proposition 5.2). The image f (K(A)) is none other than (Z(N) )p ⊆ ZN , the countably generated free abelian group with the induced topology from the product. It is important to keep the three topologies on Z(N) apart: the discrete one, the one transported from K(A) via f , and the one induced from ZN , each  = (L(A)) and L(A). one properly coarser than the previous one. We may identify A  We now apply Lemma 11.10 with P = A, P1 = K(A), Q = Q1 = ZN , and with  × ZN is an abelian pro-Lie group and  = {(k, f (k)) : the injection f above. Then A k ∈ K(A)} is a closed totally disconnected subgroup and so we can form the factor group  × ZN def A . G =  The connected subgroup P ∗ =

 (K(A)) A×f 

 is a bijective homomorphic image of A

and it is not closed but arcwise connected. Its closure P ∗ is

 (K(A)) A×f 

=

 ZN A× 

= G.

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Thus G, having a dense arcwise connected subgroup is a connected abelian protoLie group. The automorphism ϕ of the group K(A) × ZN , ϕ(k, z) = (k, z + f (k)) def K(A)×ZN 

maps K(A) × {0} onto , and induces an isomorphism from H = K(A)×ZN ∼ N = Z . Also,

onto

K(A)×{0}

 × ZN 4 K(A) × ZN A ∼ ∼  = A/K(A) = Aa .   Hence G is a connected abelian pro-Lie group embedded into an exact sequence G/H =

q

0 → ZN → G −−−→ Aa → 0. Since L(H ) = {0} we have an isomorphism L(q) : L(G) → L(Aa ) = L(A) ∼ = RR . Let us summarize this example as follows, recalling that denotes the power ZN with the discrete topology: Example 11.11. Let A =  , and let Aa be the arc component of 0 in A. There is a connected abelian proto-Lie group G with a Lie algebra g isomorphic to Hom( , R), and there is a closed subgroup H of G isomorphic to such that G/H ∼ = Aa , that is, the quotient G/H is a connected arcwise and locally arcwise connected abelian protoLie group whose completion is the compact connected and strongly locally connected group A =  . The minimal analytic subgroup A(g) with Lie algebra g is a faithful homomorphic image of g, and G = A(g)H .  × ZN . The group G is an incomplete quotient of the pro-Lie group A Exercise E11.2. Prove the following assertions: The completion G of G in Example 11.11 satisfies L(G) = g and A(g, G) = A(g, G). There is a strict exact sequence 0 → ZN → G → A → 0 In the light of this proposition we should recall from the Vector Group Splitting Lemma 5.12 for Connected Pro-Lie Groups that G ∼ = V × comp(G) for a weakly complete topological vector space V and the unique maximal compact subgroup of G. We shall now use the preceding Lemma 11.10 and ingredients we have prepared in the course of this book to construct a somewhat bizarre reductive connected pro-Lie group. We shall first discuss the construction and then summarize the salient features in Example 11.12 below. The idea is to embed Example 11.11 into the center of a reductive group. def

Let S = Sl(2, R) be the group of 2 × 2-matrices of determinant 1 and s = sl(2, R) the Lie algebra of all 2 × 2-matrices with trace 0. Let  S = (s) be the universal covering of S and πS :  S → S the covering morphism. We identify the Lie algebra L( S) with s via the natural isomorphism ηs : s → L((s)) so that πS  expS˜ = expS . We consider the injective morphisms , 0 1 ι : R → s, ι(r) = π r , −1 0

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11 Splitting Theorems for Pro-Lie Groups

 j : Z → R and notice that exp S ι  j : Z → S is one of the two possible injective morphisms with image Z( S). We write z = ι(Z) ⊆ s. N N  The power (s ) = S is a simply connected (Polish) semisimple pro-Lie group. The power sN may be considered as the Lie algebra of  S N with componentwise exS N and thus an ponential function. We have an injection (expS˜  ι  j )N : ZN →  injection f = (expS˜  ι  j )N  ϕ : K(A) →  SN, which is not an embedding but has the uncountable center Z( S)N as range.  We now apply Lemma 11.10 with P = A, P1 = K(A), Q =  S N , Q1 = Z( S)N , and N   with the injection f above. Then A× S is a reductive simply connected pro-Lie group and  = {(k, f (k)) : k ∈ K(A)} is a closed central totally disconnected subgroup and so we can form the factor group ×  A SN .  By Theorem 4.1, this factor group is a connected proto-Lie group. The central con (K(A))  and it is not is a bijective homomorphic image of A nected subgroup P ∗ = A×f  closed but is arcwise connected. It is the minimal analytic subgroup A(L(A), G) with  (K(A))   S)N = A×Z( = Z(G), and this is a copy Lie algebra L(A). Its closure P ∗ is A×f   of the group constructed in Example 11.11 Hence Z(G) is a connected abelian pro-Lie group embedded into an exact sequence def

G =

0 → ZN → Z(G) → Aa → 0. S ˙ = K(A)× is isomorphic to Q =  SN. The closed commutator subgroup Q∗ = G  K(A)×f (K(A)) ∼ ( N ) ∗ ∗ The intersection P ∩ Q is = f (K(A)) ∼ = (Z )p . We compute the  factor groups N

 ×  SN ∼ A G∼ A ∼ = = = Ga , ˙ K(A) G K(A) ×  SN ×   A SN SN G ∼ ∼ . = =  × f (K(A)) P∗ f (K(A)) A ˙ is an incomplete abelian proto-Lie group whose completion is the compact Thus G/G connected strongly locally connected abelian group A. In particular, this shows that G cannot be a pro-Lie group, because if it were, then by Theorem 4.28 (i) the quotient ˙ would have to be a pro-Lie group. The factor group  G/G S N /f (K(A)) contains the factor group Z( S)N /f (K((A)) ∼ = ZN /f (K(A)) ∼ = ZN /(ZN )p . This last group is not Hausdorff, and thus not unexpectedly, G/P ∗ is not Hausdorff; its Hausdorffisation is G/Z(G) ∼ = PSl(2, R)N . A reductive proto-Lie group is a proto-Lie group whose Lie algebra is a reductive pro-Lie algebra. We now summarize the salient features of this discussion as follows:

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Example 11.12. There exists a connected reductive proto-Lie group G with the following properties: ˙ is isomorphic to  (i) The closed commutator group G S N where  S is the universal covering group of Sl(2, R). ˙ is an arcwise connected and arcwise locally connected (ii) The factor group G/G abelian proto-Lie group Aa whose completion is the connected, strongly locally  N. connected compact abelian group A = Z (iii) There is an analytic subgroup Z of G such that L(Z) = L(A) = Hom(ZN , R) ∼ = RR , and that Z is dense in the center Z(G) which is a connected abelian pro-Lie group, and the exponential function expZ : L(Z) → Z is bijective. There is an exact sequence 0 → ZN → Z(G) → Aa → 0. ˙ and Z ∩ G ˙ is isomorphic to the proto-discrete countably generated (iv) G = Z G, ( N ) ˙ free group (Z )p whose completion is ZN . Notably, G = Z(G)G. ˙ is isomorphic to the direct product The non-Hausdorff group G/(Z ∩ G) ˙ ˙ where D is a countable dense subgroup of Z(G). Aa × G/D N   (v) G is a factor group of the reductive pro-Lie group A× S modulo a closed central subgroup isomorphic to ZN . In particular, L(G) ∼ = (R×sl(2))N . = L(A)×sl(2)N ∼ def

def

(vi) The completion G = GN (G) is a reductive Polish pro-Lie group with g = L(G) = L(G) such that there is a strict exact sequence ˙ S N → G → A → 1. 1→G=

˙ ˙ ∼ S N and R(G) = A(z(g), G) = A(z(g), G) = Z(G) = We have G = G =  ˙ Z(G). Thus R(G)G = R(G)A(˙g, G) = G = G. ˙ Here the closed commutator group G is far from splitting semidirectly in G such as it was described in the Splitting Theorem for Reductive Pro-Lie Groups 11.8. Also, the reductive pro-Lie group G is an example of a connected pro-Lie group with Lie algebra h = r(h) + s for a Levi summand s such that H = R(H )A(s, H ). We shall have more to say about this later (see 12.62 ff.).

Vector Group Splitting in Noncommutative Groups The example of the 3-dimensional Heisenberg group, that is, the group of all matrices ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 x z ⎝0 1 y ⎠ : x, y, z ∈ R ⎭ ⎩ 0 0 1

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shows that normal vector subgroups (here the one-dimensional center and commutator subgroup) do not split. However, what we wish to show here is that a weakly complete normal vector subgroup V of a pro-Lie group G does split whenever the factor group G/V is compact. We cite a lemma which we proved as Theorem 5.71 in [102] and which we adjust for our present purposes. Lemma 11.13. Let N be a normal weakly complete vector subgroup of a topological group G such that the following three conditions are satisfied: def

(i) H = G/N is compact. (ii) The quotient morphism q : G → H has a local cross section, that is, there is an open identity neighborhood V of H and a continuous map s : V → G such that q(s(h)) = h for all h ∈ V . Then G contains a compact subgroup K such that the function (n, k)  → nk : N × K → G is a homeomorphism. Proof. This is a special case of the result in [102, Theorem 5.71]. We draw the following conclusion: Corollary 11.14. Let G be a pro-Lie group with a normal weakly complete vector subgroup N such that G/N is a finite-dimensional compact Lie group. Then G is the semidirect product of N and a compact subgroup K. Proof. By Theorem 4.22 (iv) we have a local cross-section and so the hypotheses of Lemma 11.13 are satisfied and the lemma yields the assertion. With the aid of this result we can now prove the splitting theorem that we announced at the beginning of the subsection. It is convenient for us to agree to the following convention: If N is a normal subgroup and H is a subgroup of a topological group G, then we shall say that G is the semidirect product of N and H iff the function (n, h)  → nh : N × H → G is a homeomorphism. We shall abbreviate this statement by writing G = N  H. The Vector Group Splitting Theorem for Compact Quotients: Existence Theorem 11.15. Let G be a pro-Lie group with a normal weakly complete vector subgroup N such that G/N is compact. Then G has a compact subgroup K such that G = N  K.

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Proof. Case 1. dim N < ∞. The filter base {N ∩ M : M ∈ N (G)} converges to 1 and so, since N is finite-dimensional, there is an M ∈ N (G), such that N ∩M = {1}. By the Closed Subgroup Theorem 1.34 (iv), n  → nM : N → N M/M is an isomorphism, and so NM/M is a weakly complete vector subgroup of G/M. Hence by Corollary 11.14 there is a compact subgroup K/M of G/M such that G/M is a semidirect product of NM/M and K/M. Thus G = NMK = NK and N ∩ K ⊆ N ∩ M = {1}. The group G is an extension of the locally compact group N by the compact group G/N and is therefore locally compact. Hence all sufficiently small members of N (G) are compact, and we may assume that M is compact. Then K is an extension of the compact group M by the compact group K/M and is therefore compact. Then N × K is σ -compact and G is a Baire space. Hence the morphism (n, k)  → nk : N  K → G is an isomorphism by the Open Mapping Theorem for Locally Compact Groups. Therefore the assertion is true in Case 1. (This is the “classical” case.) Case 2. N arbitrary. Let M ∈ N (G). Then N/(N ∩ M)0 is a finite-dimensional vector group and thus by Case 1 there is a subgroup K of G containing (N ∩ M)0 such that K/(N ∩ M)0 is compact and G/(N ∩ M)0 is a semidirect product of N/(N ∩ M)0 and K/(N ∩ M)0 , and thus G = NK and N ∩ K = N ∩ M0 . Let K be the set of closed subgroups K of G such that (aK ) N ∩ K is a vector subgroup normal in G, and (bK ) The function fK : K/(N ∩ K) → G/N, fK (k(N ∩ K)) = kN is an isomorphism of topological groups. Containment ⊆ makes K a partially ordered set; the set of filters  in this poset contains a maximal element M by Zorn’s Lemma. Now let C = M. Then C is a closed vector subgroup of N which is normal in G and we claim that fC : C/(N ∩ C) → G/N is an isomorphism. Indeed, notice that because of (bK ) we have natural isomorphisms ∼ =

limK∈K fK

C ∗ = lim K/(N ∩ K) −−−−−−→ lim G/N −−−→ G/N. def

K∈M

K∈M

We have a morphism ϕ : C/(C ∩ N) → C ∗ , ϕ(c(C ∩ N )) = (c(N ∩ K))K∈K . The factor group N/(N ∩ C) is a weakly complete vector group. The filter basis / . N N ∩K :K∈M in N ∩C N ∩C intersects in the singleton set containing the identity, hence converges to the identity by Lemma 5.18 (b). Therefore, if (cK (K ∩ N))K∈K ∈ C ∗ , then (cK (C ∩ N ))K∈K is a Cauchy sequence in N/(C ∩ N) and thus converges to an element c(C ∩ N ) such that c ∈ (K ∩ N) for all K ∈ K. This shows that ϕ is in fact an isomorphism proving that fC is an isomorphism. This establishes the claim. Therefore C ∈ K. Now we claim that N ∩ C = {1}. Suppose that this is not the case. Then there is an M ∈ N (G) such that def

DM = (N ∩ C ∩ M)0 = N ∩ C.

(∗)

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Now G/M and therefore (N ∩ C)/(N ∩ C ∩ M) ∼ = (N ∩ C)M/M are Lie groups, by the Theorem on Closed Subgroups of Rn (see [102, Theorem A1.12]). Hence dim(N ∩ C)/DM = dim(N ∩ C)/(N ∩ C ∩ M)0 < ∞. Applying Case 1 to C/DM with the vector subgroup (N ∩ C)/DM gives a closed subgroup L ⊆ C such that, firstly, L ∩ (N ∩ C) = N ∩ L = DM = (M ∩ C ∩ N )0 a normal subgroup of G since N ∩ C and M are normal and the identity component of a topological group is characteristic, and that, secondly, C/DM is the semidirect product of (N ∩ C)/DM and L/DM . This means that (aL ) and (bL ) are satisfied. Then L ∈ Kand M ∪ {L} is a filter. By the maximality of M, this implies L ∈ M and thus C = M ⊆ L. Since L ⊆ C this implies L = C. But now DM = N ∩ L = N ∩ C.

(∗∗)

But (∗) and (∗∗) contradict each other. This contradiction proves our claim that N ∩ C = {1}. But now (bC ) shows that fC : C → G/N , fC (c) = cN is an isomorphism of topological vector spaces, and therefore that G is a semidirect product of N and C. This splitting theorem entails some other, more general splitting theorems which we describe now. Corollary 11.16. Let N be a normal subgroup of a pro-Lie group G such that G/N is compact connected and that N is a connected abelian pro-Lie group. Then N is a direct product of a weakly complete vector subgroup V of N and a central compact connected subgroup comp(N ); and G contains a compact subgroup K such that G = N K and N ∩ K is a totally disconnected subgroup of comp(N ). Proof. The characteristic subgroup comp(N ) is a compact normal abelian subgroup of G; it is central in G (see Exercise E12.7 in the next chapter). The Vector Group Splitting Theorem for Connected Abelian Pro-Lie groups 5.12 proves the first assertion. Now apply 11.15 to G/ comp(N ) and find a closed subgroup C of G containing comp(N) such that G/ comp(N ) is a semidirect product of N/ comp(N ) and the compact group C/ comp(N ). Then C is a compact connected group with the compact central subgroup comp(N ). Now we find a compact connected normal subgroup K of C such that C = comp(N ) · K and that comp(N ) ∩ K is totally disconnected (see [102, Theorem 9.77]). Then K is the required group. If G = U(n), n ≥ 2, then Z(G) = e2π R · 1n , 1n being the n × n unit matrix, is a circle group, but Z(G) does not split, yet meets SU(n) in the cyclic subgroup e2π Z/n · 1n . In this regard, Corollary 11.16 cannot be improved to yield a complete splitting of N, due to the presence of the compact connected subgroup comp(N ). Splitting of Simply Connected Prosolvable Groups: Existence Corollary 11.17. Let G be a pro-Lie group with a normal subgroup N such that N is simply connected prosolvable and G/N is compact. Then G has a compact subgroup K such that G = N  K.

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Proof. Claim 1. The corollary holds when N is topologically solvable. We prove this claim by induction with respect to the solvable length n, N = N ((0)) ⊇ N ((1)) ⊇ · · · ⊇ N ((n)) = {1}, N ((n−1)) = {1} (see Definition 10.8). We prove by induction the following claim: For all m = 0, . . . , n there exists a subgroup Km such that N ((m)) ⊆ Km

and

G

=

N ((m))

N N ((m))



Km . N ((m))

(∗m )

For m = 0 this is trivially satisfied with KM = G. Assume that the claim is true for m < n. From (∗n ) we get G = NKm

and

N ∩ Km = N ((m)) .

Since (∗m ) implies Km /N ((m+1)) ∼ Km ∼ G/N ((m)) ∼ = = G/N = N ((m)) N ((m)) /N ((m+1)) N/N ((m)) def

we know that G∗ = G/N ((m+1)) is a pro-Lie group by Theorem 4.28 (i) with a closed def

normal subgroup N ∗ = N ((m)) /N ((m+1)) such that G∗ /N ∗ is compact. From Corollary 8.16 we know that N ((1)) and N/N ((1)) are simply connected. Continuing inductively we derive that all abelian pro-Lie groups N ((m)) /N ((m+1)) are simply connected and thus are weakly complete vector groups by 8.13 (or the Vector Group Splitting Lemma 5.12 together with the fact that a compact abelian group is never simply connected: see also [102, Theorem 9.29]). Thus N ∗ is a weakly complete vector group. Now the Vector Group Splitting Theorem for Compact Quotients 11.15 applies to G∗ and gives us a compact subgroup K ∗ such that G∗ = N ∗  K ∗ . The full inverse image Km+1 of K ∗ in G contains N ((m+1)) and is such that Km+1 /N ((m+1)) is compact and N ((m+1)) ⊆ Km+1

and

G N ((m+1))

=

N N ((m+1))



Km+1 . N ((m+1))

(∗m+1 ) def

This completes the induction. In particular, taking m = n we get a subgroup K = Kn such that G = N  K. (∗n ) This proves Claim 1. Claim 2. The assertion of the corollary holds for all prosolvable N . From Claim 1 we obtain a sequence of subgroups {Km : m = 0, 1, 2, . . . }, each containing N ((m))) with a compact factor group Km /N ((m)) such that the morphism fm : Km /N ((m)) → G/N , fm (kN ((m)) ) = kN, is an isomorphism and G N ((m))

=

N N ((m))



Km . N ((m))

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11 Splitting Theorems for Pro-Lie Groups

def  ((n)) : n = 0, 1, 2, . . . } converges Now we define K = ∞ n=0 Kn . The sequence {N to 1 since N is a prosolvable pro-Lie group. The natural map k  → kG((n)) : K → limn∈N Kn /G((n)) is an isomorphism, and the map

(fn )n∈N : lim Kn /G((n)) → lim G/N ∼ = G/N n∈N

n∈N

is an isomorphism as well. This means that k  → kN : K → G/N is an isomorphism and that says exactly that G = N  K. ˙ = Recall that for a prosolvable connected pro-Lie group we have ncored (G) = G [[1]] =G and that this group is contained in the nilradical N (G). Recall further that a topological group is called almost connected if G/G0 is compact for the identity component G0 of G.

G((1))

Corollary 11.18 (Structure of Prosolvable Pro-Lie Groups). Let G be an almost connected pro-Lie group such that G0 is prosolvable and assume that N (G0 ) is simply connected. Then there is a normal closed simply connected subgroup V containing N(G0 ) and a compact subgroup K such that G = V  K. Proof. By Theorem 4.28 (i), the factor group G0 /N (G0 ), as a homomorphic image ˙ is a connected abelian pro-Lie group A. The group G/N(G0 ) acts under of G/G, inner automorphisms on A; but since A is abelian, it is in fact the compact group def

 = G/G0 that acts on A. Now by 5.14 (ii), A is a direct product of an -invariant weakly complete vector subgroup V /N(G0 ) and a characteristic compact subgroup C/N(G0 ). The factor group V /N(G0 ) is a weakly complete vector group isomorphic to RJ for some set J and thus is strictly exponential. By Proposition 8.10, the quotient morphism V → V /N(G0 ) has a continuous cross section. Hence V is homeomorphic to N(G0 ) × RJ , and N(G0 ) is simply connected by hypothesis. Hence V is simply connected. As a closed subgroup of a prosolvable pro-Lie group it is prosolvable. Now G/V ∼ = (G/N(G0 ))/(V /N (G0 )) ∼ = C/N (G0 ). Now Corollary 11.17 on the Splitting of Simply Connected Prosolvable Groups applies to C and shows C = N (G0 )  K for a compact group K isomorphic to C/N (G0 ) ∼ = G/V . So we have G = V C = V · N(G) · K = V C and V ∩ K ⊆ V ∩ C ∩ K = N (G) ∩ K = {1}. Since K is compact, this shows G = V  K. This applies in a lucid fashion to a basic structure theorem for any connected prosolvable group G.

The Structure of Pronilpotent and Prosolvable Groups But before we can exploit the splitting theorems, we need to have more precise information on pronilpotent groups.

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Lemma 11.19. Let G be a connected nilpotent Lie group. Then the center Z(G) is connected. Proof. By Theorem 8.5 (or the simpler finite-dimensional equivalent), the simply  may be identified with (g, ∗) and the universal covering connected covering group G morphism is expG : g → G where g is given the Campbell–Baker–Hausdorff–Dynkin and multiplication. In g we have X ∗ Y ∗ (−X) = ead X Y (see Proposition 5∞ 1 9.2), n in the thus X ∈ Z(g, ∗) iff ead X = idg . Let L(z) = log(1 + z) = z n=1 n ring of formal power series over Q, and L((exp z) − 1) = z. Then by 9.2 (or its simpler finite-dimensional equivalent), we have ad X = L(ead X − idg ), and thus X ∈ Z(g, ∗) iff ad X = 0, that is, iff X ∈ z(g). Thus Z(g, ∗) = z(g) is connected. def Since L(D) = ker L(πG ) = {0}, by 4.23, the group D = ker expG is a totally disconnected subgroup (and indeed a discrete subgroup in the finite-dimensional case), hence is a central subgroup of (g, ∗). That is, D ⊆ z(g). Now g = exp X ∈ Z(G) iff 0 = comm(X, Y ) = X ∗ Y ∗ (−X) ∗ (−Y ) ∈ D for all Y ∈ g. Since comm(X × g) is connected, and contains 0 while D is totally disconnected, we conclude g ∈ Z(G) iff X ∈ z(g), and thus, since G = im expG , Z(G) = exp z(g) ⊆ Z(G)0 .



Recall that for a topological group G we denote by comp(G) the union of all compact subgroups. Proposition 11.20. Let G be a connected pronilpotent pro-Lie group. Then (i) its center Z(G) is connected, (ii) comp(G) ⊆ Z(G), and (iii) Z(G) ∼ = RI × comp(G), where I is a suitable set, and where comp(G) is a compact connected abelian group. Proof. (i) Let N ∈ N (G) and let ZN = {g ∈ G : comm(g × G) ⊆ N } be the full inverse image of Z(G/N) in G. Then N ⊆ ZNand ZN /N = Z(G/N ). If g ∈ ZN then  comm(g, x) ∈ N for  all x ∈ G. Thus g ∈ N ∈N (G) ZN implies comm(g, x) ∈ N (G) = {1}. Thus N ∈N (G) ZN ⊆ Z(G). The reverse containment is obvious.  Thus Z(G) = N ∈N (G) ZN ∼ = limN ∈N (G) ZN /N . All groups ZN /N = Z(G/N ) are connected by Lemma 11.19. From Theorem 9.7 (ii) it now follows that their projective limit Z(G) is connected as well. (ii) A compact connected subgroup T of G is abelian as a compact connected pronilpotent group (see [102, Chapter 9]: By Proposition 9.4 of [102], a compact connected group C satisfies C  = C  ; thus C is prosolvable if and only if C  = {1} if and only if it is abelian). If g = exp Z ∈ G satisfies g n = exp n · Z = 1, then exp n · Z ∈ Z(G), and so Z ∈ z(g) by the preceding section and so g ∈ Z(G) and indeed the circle group exp R.Z is central. Thus (∗∗) every finite cyclic subgroup and every circle subgroup of G is central.

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Let us momentarily assume that G is a Lie group. Then, firstly, G = exp g, and so (∗) implies that Z(G) is connected. Secondly, any compact connected subgroup T is generated by the union of the circle groups it contains, and so T ∈ Z(G) follows. Since every finite cyclic subgroup is central, all finite subgroups are central. By Dong Hoon Lee’s Theorem 6.74 in [102] a compact Lie group is a product of its identity component and a finite group. Hence a compact subgroup of G is central. Now let G be an arbitrary connected pronilpotent pro-Lie group and C a compact subgroup of G. If N ∈ N (G), then CN/N is a compact  Lie subgroup of G/N and thus is central, that is, comm(C × G) ⊆ N. Since N (G) = {1}, we conclude comm(C × G) = {1}, and therefore C ⊆ Z(G). (iii) The group Z(G) is a connected pro-Lie group by (i) and contains the union comp(G) of all compact subgroups of G. Then the Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups 5.12 applies to Z(G) and shows that Z(G) is the direct product of a weakly complete vector group and a unique maximal compact connected subgroup of Z(G) which, according to (ii) is comp(G). According to the preceding Proposition 12.61 for a pronilpotent group G we may write comp(G) = MaxK(G) We recall that for a pronilpotent pro-Lie algebra g the group (g) may be identified with (g, ∗), the underlying space of g with the pro-Lie group operation ∗ implemented by the Baker–Campbell–Hausdorff–Dynkin multiplication on g. Lemma 11.21. If A1 is a connected abelian group and A2 a compact abelian group, and if B is a topological abelian group whose characters separate the points, then any continuous Z-bilinear map β : A1 × A2 → B is zero.  χ : B → T, be a character of B. Then χ  β : A1 × A2 → T is a Proof. Let χ ∈ B, continuous Z-bilinear map into the circle group. Then we define a function β ∗ : A1 → 2 = Hom(A2 , T) is that of uniform 2 by β ∗ (a1 )(a2 ) = χβ(a1 , a2 ). The topology of A A ∗ convergence. We claim that β is continuous. (Exercise E11.3 below.) Since A2 is 2 is discrete. Since A1 is connected, β ∗ (A1 ) = {0}. Therefore χ  β = 0 compact, A  Since the characters of B separate the points, β = 0 follows. for all χ ∈ B. Exercise E11.3. Prove the following assertion: Let A1 and A2 be abelian topological groups and assume that A2 is compact. Let β : A1 ×A2 → B be a continuous Z-bilinear function into an abelian topological group B, endow the set Hom(A2 , B) of continuous morphisms with the topology of uniform convergence. Then the function β ∗ : A1 → Hom(A2 , B), β ∗ (a1 )(a2 ) = β(a1 , a2 ) is continuous. def [Hint. The basic zero neighborhoods of Hom(A2 , B) are of the form W ∗ = {f ∈ Hom(A2 , B) : f (A2 ) ⊆ W } as W ranges through the zero neighborhoods of B. Since β({0} × A2 ) = {0}, for each x ∈ A2 there is a zero neighborhood Ux of A1 and a neighborhood Vx of x in A2 such that β(U x × Vx ) ⊆ W . Using  the compactness of A2 , find x1 , . . . , xn ∈ A2 such that A2 ⊆ nj=1 Vxj . Set U nj=1 Uxj and observe that β(U × A2 ) ⊆ W . Then β ∗ (U ) ⊆ W ∗ .]

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Lemma 11.22. Let T be a topological group with a central subgroup B that is contained in a pro-Lie subgroup K such that K/B is compact. If comm(T × K) ⊆ B, T /T˙ is connected, and B = B0 comp(B), then K is central and satisfies K ⊆ B · comp(K) ⊆ B · comp(T ) = comp(T ) · B. Proof. We recall some simple commutator calculations for t, t1 , t2 ∈ T and k, k1 , k2 ∈ K: comm(t1 t2 , k) = t1 t2 kt2−1 t1−1 k −1 = t1 comm(t2 , k)kt1−1 k −1 = comm(t1 , k) comm(t2 , k), comm(t, k1 k2 ) = tk1 k2 t −1 k2−1 k1−1 = tk1 t −1 comm(t, k2 )k2 = comm(t, k1 ) comm(t, k2 ), due to the fact that comm(t2 , k) and comm(t, k2 ) are central. Therefore and

comm(·, k) : T → B, comm(t, ·) : K → B

are morphisms of topological groups with an abelian range; therefore the first factors through T /T˙ while the second factors, firstly, through K/B since B is central, and then, secondly, through (K/B)/(K/B) ˙ = (K/B)/(K  B/B) ∼ = K/K  B. So there  ˙ is a continuous Z-bilinear function β : T /T × K/K B → B, β(t T˙ , kK  B) = [t, k].  By hypothesis, T /T˙ is connected, and K/K B, as a homomorphic image of K/B, is compact. Since B is an abelian pro-Lie group, the characters of B separate the points by Exercise E5.5 following Theorem 5.36. Thus Lemma 11.21 applies with A1 = T /T˙ and A2 = K/K  B and shows that β is constant. This means that [T , K] = {1}, that is, K ⊆ Z(T ). In particular, K is abelian. Since the group B satisfies B = B0 · comp(B), it is a direct product V comp(B) ∼ = V × comp(B) of a weakly complete vector group V and the subgroup comp(B) by Theorem 5.12 (v). The abelian pro-Lie group K is a direct product V H ∼ = V × H of V and a complementary factor H by Theorem 5.19. Since the projection of comp(B) into V along H is a union of compact subgroups of V and is therefore singleton, we have comp(B) ⊆ H . Then H /K ∼ = (V × H )/(V × K) ∼ = K/B is compact and so H is compact and agrees with comp(K). Thus K = V comp(K) ⊆ B comp(K) ⊆ B comp(T ). Let us notice that by Theorem 5.20 (v) the condition B = B0 comp(B) is tantamount to saying that B/B0 is a union of compact subgroups, and that this condition is satisfied if B is almost connected, and certainly if it is connected. Exercise E11.4. Prove the following result. Proposition. Let T be a connected topological group and K be a normal subgroup containing a central almost connected pro-Lie subgroup B of T such that K/B is compact and central in T /B. Then K is central in T . [Hint. Apply Lemma 11.22.]

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Recall that for a topological group G we have defined G[[0]] = G and have set inductively G[[n+1]] = [G, G[[n]] ]. Then for each n the factor group G/G[[n]] is nilpotent, and if G is pronilpotent and connected, comp(G/G[[n]] ) is a compact connected abelian subgroup. Hence we may define Cn to be that closed subgroup of G which contains G[[n]] and satisfies Cn /G[[n]] = comp(G/G[[n]] ). ˙ for n = 1, 2, . . . . Lemma 11.23. C1 = Cn G[[1]] = Cn G Proof. Let pn : (G/G[[n+1]] ) → (G/G[[n]] ), pn (gG[[n+1]] ) = gG[[n]] , be the quotient morphism. Then (Cn+1 nG[[n]] /G[[n]] ) = pn (comp(G/G[[n+1]] )) ⊆ comp(G/G[[n]] ) = Cn /G[[n]] . So Cn+1 G[[n]] ⊆ Cn . We claim that the reverse containment holds: We apply the preceding Lemma 11.22 with T = G/G[[n+1]] , B = G[[n]] /G[[n+1]] , and K = Cn /G[[n+1]] and deduce Cn+1 G[[n]] Cn+1 G[[n]] Cn = K ⊆ B comp(H ) = · = . G[[n]] G[[n+1]] G[[n+1]] G[[n+1]] So Cn ⊆ Cn+1 G[[n]] and this was our claim. Therefore equality holds and we have C1 = C2 G[[1]] . Assume that Cn−1 = Cn G[[1]] has been established, then Cn = (Cn+1 G[[n]] )G[[1]] = Cn+1 G[[1]] which is what we had to show. ˙ denote its Lemma 11.24. Let G be a connected pronilpotent pro-Lie group and let G ˙ = (comp(G) · G)/ ˙ G. ˙ closed commutator subgroup of G. Then comp(G/G) Proof. The containment ⊇ is clear, we have to show the reverse containment. Using ˙ the notation of the preceding lemma, this means C1 = comp(G) · G. Firstly, we claim ∞ def  C = Cn = comp(G). n=1

We recall from 11.20 that comp(G) is a compact connected central subgroup of G. Then comp(G) · G[[n]] /G[[n]] ⊆ comp(G/G[[n]] ) = Cn /G[[n]] . Hence comp(G) ⊆ Cn for all n and thus comp(G) ⊆ C. To see the converse we first recall limn G[[n]] = 1 (indeed let N ∈ N (G) and let U be an open identity neighborhood of G satisfying U N = U such that U/N contains no subgroups other than the identity, then G/N being nilpotent and G[[n]] N/N ⊆ (G/N )[[n]] = {1} for large enough n gives G[[n]] ⊆ U which establishes the claim). Since pn : G/G[[n+1]] → G/G[[n]] maps Cn+1 /G[[n+1]] = comp(G/G[[n+1]] ) into comp(G/G[[n]] ) = Cn /G[[n]] , the Cn /G[[n]] form a projective system and we can form the limit L = lim(. . . Cn /G[[n]] ←− Cn+1 /G[[n+1]] . . . ) ⊆ lim G/G[[n]] = G{G[[n]] :n∈N} ∼ = G. n

/G[[n]]

comp(G/G[[n]] )

All factor groups Cn = are compact by 11.20. Hence L is compact as a projective limit of compact groups, whence L is compact, and it contains limn C/G[[n]] . Let us identify limn G/G[n]] and G in the obvious way. Then

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483

L ⊆ comp(G). Since comp(G)G[[n]] /G[[n]] ⊆ comp(G/G[[n]] ) = Cn /G[[n]] and so comp(G) ⊆ Cn and thus comp(G) ∼ = lim comp(G)G[[n]] /G[[n]] ⊆ lim Cn /G[[n]] = L. n

n

Hence, without identification, comp(G) ⊆ L and so L = comp(G). Now let (cn G[[n]] )n∈N ∈ L. Then m ≤ n implies cn ∈ G[[m]] . Thus, since limn G[[n]] = 1, the sequence {cn G[[n]] : n ∈ N} is  a Cauchy filter and thus has a limit c in the complete  [[m]] = CG[[m]] for all pro-Lie group G. We have c ∈ n∈N Cn G[[n]] ⊆ n∈N Cn G m ∈ N. Since limm G[[m]] = 1 we conclude c ∈ C. Hence L = limn CG[[n]] /G[[n]] = C with our identification. Therefore, C = comp(G). Since G[[n]] ⊆ G[[1]] ∩ Cn and Cn /G[[n]] comp(G/G[[n]] ) is compact we know that ˙ ∩ Cn ) is compact and so the function Cn /(G ˙ ∩ Cn )  → c G ˙ : c(G

˙ Cn Cn G ˙ = C1 /G → ˙ ∩ Cn ˙ G G

is an isomorphism of topological groups in view of 11.23. Therefore, ∼ =

comp(G) = C −−→ lim n

∼ Cn C1 = ˙ = C1 /G. −−→ lim ˙ ˙ n G ∩ Cn G

˙ = C1 , and this is what had to be shown. In other words comp(G)G According to a definition preceding Theorem 9.50, we call a topological group compactly simple, if it has no compact normal subgroup except the trivial one. Lemma 11.25. Let G be a compactly simple connected pronilpotent pro-Lie group. ˙ ∼ Then G/G = RI for some set I . ˙ are connected, G/G ˙ is a pro-Lie group by Theorem 4.28 (i) and Proof. Since G and G ˙ But thus is isomorphic to RI × C for a compact connected abelian group C ∼ = (G/G). Lemma 11.24 now shows that C = {1} because comp(G) = {1} by hypothesis. Lemma 11.26. Let G be a compactly simple connected pronilpotent pro-Lie group and M a connected normal subgroup. If C is any subgroup of G containing M such that C/M is compact, then C = M. Proof. By 4.28 (i) again G/M is a pronilpotent connected pro-Lie group to which Proposition 11.20 applies. Hence the subgroup comp(G/M) is compact connected central, and it contains C/M. Let H ⊇ M be that connected closed subgroup for which H /M = comp(G/M). Since comm(H × G) ⊆ M we have H˙ ⊆ M. Since G is compactly simple and comp(H ) is characteristic in H , the subgroup H is compactly simple as well. Thus Lemma 11.24 applies to H and shows H /H˙ is a weakly complete vector group. The subgroup M/H˙ is connected and closed, and so H /M ∼ = (H /H˙ )/(M/H˙ ) is a vector group on the one hand and a compact group on the other. This implies H = M and thus C = M.

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11 Splitting Theorems for Pro-Lie Groups

Simple Connectivity of Pronilpotent Pro-Lie Groups Theorem 11.27. For a connected pronilpotent pro-Lie group G, the following statements are equivalent: (i) (ii) (iii) (iv)

G has no compact subgroups. G is compactly simple. G has no compact connected normal subgroups G is simply connected and expG : (g, ∗) → G is an isomorphism of topological groups.

Proof. Clearly, (iv) ⇒ (i) ⇒ (ii) ⇒ (iii). So we have to show (iii) ⇒ (iv). By Theorem 8.15, we have to verify that exp is bijective. We claim it is injective: Let X ∈ ker exp. Then exp R · X is a homomorphic image of R/Z and therefore is a compact connected group. Since G has no compact subgroups, exp R · X = {1} implying R · X ⊆ ker exp and so X = 0. Hence exp is injective. Now let N ∈ N (G). The finite-dimensional factor group G/N0 is compactly simple and so does not contain any nonsingleton compact normal subgroups by Lemma 11.26.  → G is surjective. Then Proposition 9.40 shows that πG : G If we consider the proof of Theorem 11.27 and everything that went into it we have to admit that it is remarkably complex. It is therefore worthwhile to observe that the structure of the Lie algebra of a pronilpotent pro-Lie algebra is somewhat easier to describe than the corresponding situation on the group level. Typically, an abelian direct factor can be split as is indicated in the following remark: Exercise E11.5. Prove the following observations: Remark A. Let g be a pronilpotent pro-Lie algebra and let g˙ = [g, g]. Let v be a closed vector space complement of z(g) ∩ g˙ in z(g) so that z(g) = v ⊕ (z(g) ∩ g˙ ) according to Theorem 7.7 (iv). Then any vector space complement w of z(g) + g˙ yields a closed ideal h of g such that g = h ⊕ v,

z(g) = z(h) ⊕ v,

˙ z(h) ⊆ h.

[Hint. The sum z(g) + g˙ is closed by Lemma A2.12 (c). Let w be a vector space complement according to Theorem 7.7 (iv). Then g = v ⊕ w ⊕ g˙ . We set h = w ⊕ g˙ ; then h is closed by A2.12 (c) again and [g, h] ⊆ g˙ ⊆ h. Thus h is a closed ideal, and g = h ⊕ v. Clearly, z(g) = z(h) + v. Further, g˙ = h˙ and thus v ⊕ z(h) = z(g) = ˙ and thus z(h) = z(g) ∩ h.] ˙ v ⊕ (z(g) ∩ h) Remark B. Let G be a connected pronilpotent pro-Lie group and g = L(G) its Lie algebra. According to Chapter 9 there are minimal analytic subgroups A(v) and A(h) with v and h as in Remark A above. The first one is central, the second one has a pronilpotent proto-Lie group topology, and their product A(v)A(h) is dense in G. What can we say about the closures of these groups?

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Now we have all the ingredients to prove the following structure theorem which belongs to the type of “almost” splitting theorems. Structure of Almost Connected Prosolvable Pro-Lie Groups Theorem 11.28. Let G be an almost connected pro-Lie group whose identity compodef nent is prosolvable. Let C = comp(Z(G)0 ) = KZ(G)0 be the unique largest compact connected central subgroup, clearly contained in the nilradical N (G0 ). Then there is a maximal compact subgroup K which is abelian containing C, and there is a connected closed normal subgroup V containing N(G0 ) such that V /C is simply connected and G/C = V /C  K/C. There is a compact subgroup M of G such that G = V M and V ∩ M is totally disconnected central. G    V   N(G)  0   C    {1} def

 

compact, splits mod V ∩ M ∼ = (L(V /N (G0 )), +) splits topologically over N (G0 )



∼ = (L(N (G0 ))/C, ∗) = comp(Z(G))0 = KZ(G)0  compact central

Proof. Let C = comp(Z(G)0 ) = KZ(G)0 be the unique largest compact central connected subgroup which exists by Theorem 9.50 and is clearly contained in N (G0 ). Since C is central, the factor group N(G0 )/C is the nilradical of G0 /C and does not contain any compact connected normal subgroup by 9.50 (i). By Theorem 11.27 we know that N(G0 )/C is simply connected. Now we can apply Corollary 11.18 to G/C and obtain a compact subgroup K containing C and a connected closed subgroup V containing N(G0 ) such that V /C is simply connected and G/C = V /C  K/C. Thus G = V K and V ∩K = C. From Corollary 11.16 we get a connected compact subgroup M such that K = CM and C ∩ M is totally disconnected. Then G = V CM = V M and V ∩ K ⊆ V ∩ C ∩ K = C ∩ K. It remains to show that K is maximal compact. Let K ⊆ K1 for a compact subgroup. Then G/C = V /C  K/C implies K1 /C = (V /C ∩ K1 /C)  K/C. But V /C is simply connected and V /C ∩ K1 /C is compact; since (V /C)/(N (G0 )/C) ∼ = V /N(G0 ) is a vector group, it has no nontrivial compact subgroups. So (V ∩ K1 )/C ⊆ N(G0 )/C. But N(G0 )/C is simply connected pronilpotent and thus does not contain any nontrivial compact subgroup by 11.27. Thus (V ∩ K1 )/C = C/C and thus K1 = K. The proof of the theorem is completed.

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The theorem says that a maximal compact subgroup K is divided into a part “way down”, namely KZ(G)0 and a part “way up”, namely M meeting V in a thin set, that is a totally disconnected compact central set. Since the vector group complement in the Vector Space Splitting Theorem 5.20 for Abelian Pro-Lie Groups is not uniquely determined, the normal subgroup V is not unique. In Remark 5.14 it was explained what, in the abelian case, was the precise reason for the nonuniqueness of a vector group complement. When a near splitting is observed, like the one of M in Theorem 11.28, one has a “Sandwich Theorem” like the following: Exercise E11.6. Prove the following theorem: Theorem 11.28.A. (Sandwich Theorem for Prosolvable Groups.) Assume the hypotheses and the notation of Theorem 11.28. Let the morphism α : M → Aut(V ) be defined by the restriction of the action by inner automorphisms as α(m)(v) = mvm−1 . There are quotient morphisms p : V α M → G, p(v, m) = vm, q : G → V /(C ∩ M)  M/(C ∩ M). The kernels of both are isomorphic to the compact totally disconnected central subgroup C ∩ M. [Hint. We assume the statement and proof of Theorem 11.28. The function p is a continuous morphism. By Theorem 11.28 it is surjective. Its kernel is the set of all (v, m) ∈ V × M such that vm = 1, that is v = m−1 ∈ V ∩ M = C ∩ M, so ker p = {(m−1 , m) : m ∈ C ∩ M} and m  → (m−1 , m) : C ∩ M → ker p. Since C ∩ M is compact, p is a quotient morphism. The function q is the quotient morphism G → G/(C ∩ M) and G/(C ∩ M) = (V /(C ∩ M)) · (M/(C ∩ M)) while (V /(C ∩ M)) ∩ (M/(C ∩ M)) = {C ∩ M}.] Exercise E11.7. Prove the following generalisation of Theorem 11.28 leading us outside the domain of prosolvable pro-Lie groups. We say that a pro-Lie group G is almost prosolvable if the quotient group G/R(G) is compact for the radical R(G). Since R(G) ⊆ G0 , an almost prosolvable pro-Lie group is almost connected. Theorem 11.28.B. (Structure Theorem of Almost Prosolvable Pro-Lie Groups.) Let G be an almost prosolvable pro-Lie group and let C = KZ(G)0 be the unique largest compact connected central subgroup. Then there is a maximal compact subgroup K, obviously containing C, and a normal subgroup V containing N (G) such that V /C is simply connected prosolvable and G/C = (V /C)  K/C. There is a compact connected subgroup M of K such that G = V M and V ∩M = C∩M is compact totally disconnected central.

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[Hint. G/N(G) is reductive with (G/N (G))/R(G/N (G)) ∼ = G/R(G) being compact. Such a group is a direct product of a weakly complete vector subgroup and a compact group. Let V be the full inverse image of the vector group factor in G. Then V /C is prosolvable simply connected and (G/C)/(V /C) ∼ = G/V is compact. Now proceed as in the proof of Theorem 11.28.] Simple Connectivity of Prosolvable Lie Groups Corollary 11.29. Let G be a connected prosolvable pro-Lie group. Then the following statements are equivalent: (i) G does not contain any nontrivial compact subgroup. (ii) G is simply connected. (iii) G is homeomorphic to (L(N (G)), ∗) × (L(G/N (G)), +). If these conditions are satisfied, then G = expG g = A(g, G). Proof. For the equivalence of (ii) and (iii) see 8.13 and 10.18. (ii) ⇒ (i): Assume G is simply connected and assume that C is a compact subgroup. ˙ is a weakly complete vector group by 8.13. Hence C G/ ˙ G ˙ is singleton, that Then G/G ˙ = ncored (G). But ncored (G) is simply connected as well, and so C = {1} is, C ⊆ G by 11.27. (i) ⇒ (ii): Conversely, assume G has no compact subgroups. Then KZ(G)0 = {1} and so N(G) is simply connected by 11.27. Since G has no compact subgroups in the terminology of 11.28 we have K = {1} and so G = V is simply connected. The last assertion is an immediate consequence of Corollary 8.17 in view of 9.10 (iii). The Lie group G = C  R, (c, r)(d, s) = (c + e2π ir d, r + s) is solvable and homeomorphic to R3 while G = expG g.

Conjugacy Theorems We have seen in the Levi–Mal’cev Splitting Theorem for Pro-Lie Algebras in Theorems 7.52 and 7.77 and the Splitting Theorem for Reductive Groups 11.8 and 11.9 that for semidirect decompositions we obtain, as a rule, information on the conjugacy of the factors complementary to the normal factors. This is also true for the Vector Group Splitting Theorem for Compact Quotients 11.15 as we shall see now. In this theorem we encounter a pro-Lie group G = N  K with a normal subgroup N which is a weakly complete vector group N , that is, a subgroup for which expN = expG |n : n → N , n = L(N ) implements an isomorphism of abelian topological groups and with a compact subgroup K. The adjoint representation of K on n gives us an automorphic action of K on n making n into a K-module such that for k ∈ K and

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X ∈ N we have exp(k · X) = k(exp X)k −1 ,

Ad(k)(X) = k · X,

giving us the morphism Ad : K → Aut(n). Every g ∈ G provides a compact subgroup gKg −1 for which G = N  (gKg −1 ). Thus with one semidirect cofactor K we get a whole conjugacy class in the set C(N ) of semidirect cofactors for N in G, and we see that G acts on C(N ) under conjugation. Since g = nk for a unique pair (n, k) ∈ N × K we note gKg −1 = nKn−1 , so the G-orbits of C(N ) are in fact the N-orbits under the induced action. In the discussion preceding 11.5 we recalled the concept of a cocycle. We now consider the additive version n of N and let G act on n via g · X = Ad(g)(X). Now, for any subgroup K of G, we recognize that a 1-cocycle f : K → n is a continuous function satisfying (∀k1 , k2 ∈ K)

f (k1 k2 ) = k1 · f (k2 ) + f (k1 ) = f (k1 ) + k1 · f (k2 ).

The vector space under pointwise addition and scalar multiplication of all such cocycles is denoted Z 1 (K, n). A function f : K → n is called a coboundary, if there is an element X ∈ n such that f (k) = k · X − X. The vector space of all coboundaries is written B 1 (K, n). We compute f (k1 k2 ) = k1 k2 ·X −X = k1 ·(k2 ·X −X)+(k1 ·X −X) = k1 ·f (k2 )+f (k1 ). Thus every coboundary is a cocycle: B 1 (K, n) ⊆ Z 1 (K, n). The quotient group is written H 1 (K, n) and called the first cohomology group of K with coefficients in n, but we shall not make further use of this fact but would be amiss not to mention it here. In Lemma 11.5 we proved a special case relevant for the present situation: Proposition 11.30. Let G be a pro-Lie group and N a normal weakly complete vector group such that G = N  K for a closed subgroup K. Then (i) the function ! : Z 1 (K, n) → C(N ),

!(f ) = {exp(−f (k))k | k ∈ K}

is a bijection, and (ii) ! maps B 1 (K, n) onto the conjugacy class of K in C(N ). Proof. The proof of (i) is an immediate consequence of 11.5 (iii). For a proof of (ii) it suffices to recall {exp(−f (k))k : k ∈ K} = !(f ) is a conjugate nKn−1 , n ∈ N iff for each k ∈ K we have exp(−f (k))k = nkn−1 , that is exp f (k) = knk −1 n−1 . If we set n = exp X, this means exp f (k) = exp k · X(exp X)−1 = exp(k · X − X) and this is equivalent to f (k) = k · X − X since expN : n → N is bijective. However, if H is any compact subgroup of K in G = N  K, as in the Splitting Theorem 11.15, a new element enters. Let f : H → n be a cocycle. This is tantamount to having a compact subgroup K2 = (expG f (h), h) : h ∈ H , and every compact subgroup K2 satisfies K2 ∩ N = {1} and is of this form. Then normalized Haar

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def = integral on H provides an element Y = h∈H = f (h) dh in the complete locally convex vector space n, and by invariance we have h∈H f (kh) dh = Y for all k ∈ H . If, for a fixed k, we integrate the functional equation f (kh) = k · f (h) + f (k), we obtain Y = k · Y + f (k) and thus f (k) = k · X − X for X = −Y . Thus f ∈ B 1 (H, n). But then the function F : K → n given by F (k) = k · X − X is a coboundary F ∈ B 1 (K, n) extending f , and so {(expG F (k), k) : k ∈ K} is a semidirect cofactor of N containing K2 . Thus we get the following concomitant of the Splitting Theorem 11.15.

The Vector Group Splitting Theorem for Compact Quotients: Conjugacy Theorem 11.31. Let G be a pro-Lie group with a normal weakly complete vector subgroup N such that G/N is compact. Then there are two semidirect cofactors K1 and K2 for N , such that G = N  K1 = N  K2 are conjugate under an inner automorphism implemented by an element of N. More specifically, if K1 is a semidirect cofactor and K2 is any compact subgroup of G, then there is an inner automorphism ϕ implemented by an element of N such that K2 ⊆ ϕ(K1 ). Proof. By the preceding remarks, we have Z 1 (K1 , n) = B 1 (K1 , n). Then Proposition 11.30 proves the first assertion. Now let K2 be a compact subgroup such that N ∩ K2 = {1}. Let H be the projection of K2 into K1 along N. Then there is an f ∈ B 1 (H, n) such that K2 = {(expG f (h), h) : h ∈ H }. By the preceding remarks, f extends to a coboundary def

F ∈ B 1 (K1 , n). Then K2∗ = {(F (k), k) : k ∈ K1 } is a semidirect cofactor of N containing N2 . By the first part of the proof, K2∗ is conjugate to K1 under an element of N. This completes the proof of the theorem. Splitting Simply Connected Prosolvable Groups: Conjugacy Corollary 11.32. Let G be a pro-Lie group with a simply connected prosolvable normal subgroup N such that G/N is compact, and assume that K1 and K2 are two semidirect cofactors for N, that is, closed subgroups satisfying G = N  K1 = N  K2 . Then K1 and K2 are conjugate under an inner automorphism implemented by an element of N . Specifically, if K1 is a semidirect cofactor of N and K2 is any compact subgroup of G, then there is an inner automorphism ϕ implemented by an element of N such that K2 ⊆ ϕ(K1 ). Proof. (i) By Theorem 11.31 there is an element n1 such that (n1 N ((1)) ) that is,

K2 N ((1)) K1 N ((1)) ((1)) −1 (n N ) ⊆ , 1 N ((1)) N ((1)) ((1)) . n1 K2 n−1 1 ⊆ K1 N

(∗1 )

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Continuing inductively, we find elements nk ∈ N ((k)) such that (nk nk−1 . . . n1 )K2 (nk nk−1 . . . n1 )−1 ⊆ K1 N ((k)) .

(∗k )

Let xk = nk nk−1 . . . n1 , then xk+1 xk−1 = nk+1 ∈ N ((k+1)) . The filter basis {N ((k)) : k ∈ N} converges to 1 since N is prosolvable. Then (xk )k∈N is a Cauchy sequence and thus converges to an element x ∈ N. Then for all k ∈ N we get from (∗k ), upon passing to the limit on the left hand side, xK2 x −1 ⊆ K1 N ((k)) . Since limk∈N N ((k)) = 1 we finally get xK2 x −1 ⊆ K1 . Now assume that K2 is a semidirect cofactor as well. Then, since inner automorphisms of G permute semidirect cofactors, xK1 x −1 is a semidirect cofactor, contained in the semidirect cofactor K2 . Hence equality follows and K2 is a conjugate of K1 under an inner automorphism implemented by an element x of N. Remark 11.33. The maximal compact subgroups K in the Structure Theorem 11.28 of Connected Prosolvable Groups are conjugate under inner automorphisms. Proof. This is now a straightforward exercise from 11.28 and 11.32. Exercise E11.8. Write down the details of the proof of Remark 11.33.

Postscript A theorem about a topological group G which asserts that a normal subgroup N is, under suitable circumstances, a semidirect factor, is called a splitting theorem. Sometimes a splitting of a group exists locally only, that is a covering group of the given group is split, and the given group is therefore a quotient of a semidirect product modulo a discrete central subgroup. In this fashion, splittings arise in the environment of connected Lie groups from clean splittings on the Lie algebra level. This chapter is concerned with genuine splitting theorems on the group level. We discuss one splitting theorem in the context of reductive groups that vastly generalizes a splitting theorem known for compact groups, and we discuss a group of splitting theorems that deal with pro-Lie groups that are compact modulo their radical. These two categories of splitting theorems have different roots, as we shall see. Let us first comment on the reductive situation. We do know from Chapter 8 that a simply connected reductive group is a product of a possible large family of Lie groups which are either isomorphic to R or are simply connected simple Lie groups. This, of course is a splitting itself, but it is one that arises from the splitting theorems of reductive pro-Lie algebras and the fact that the categories of pro-Lie algebras and simply connected pro-Lie groups are equivalent.

Postscript

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In the absence of simple connectivity, we have a variety of examples of connected reductive groups that show what a great variety of groups with curious properties can be constructed if there are simple atoms (that is, simple factors in the Lie algebra) which are of the unbounded type; a very representative example is the case that in the Lie algebra is a product of Lie algebras isomorphic to sl(2, R), that is s = sl(2, R)J . In this case, Z((sl(2, R))) ∼ = Z and the remarkable properties of the abelian groups Z((s)) ∼ = Z J cause the structure theory of H = A(s; G) to be so complex. So what we have learned from the structure theory of semisimple and reductive pro-Lie groups in the preceding Chapter 10 is that serious complications in the global structure theory arise whenever elements of infinite order are present in the center of the simply connected pro-Lie group (s) generated by a semisimple Lie algebra. On the other hand, if all Z((sj )) are finite, we obtain the remarkable Splitting Theorem 11.8. A corollary we may formulate, in view of Theorem 8.14, as follows: If G is a connected reductive pro-Lie group such that the center of the semisimple  is compact then the closed commutator subgroup splits, that is, factor S of G ˙  A. there is a closed abelian subgroup A of G such that G ∼ =G If G is compact, this reduces to the Borel–Scheerer–Hofmann Splitting Theorem (see [102, Theorem 9.39]). Our present generalisation appears to be a new result even for finite-dimensional Lie groups. While the first part of the chapter is devoted to reductive pro-Lie groups, the second one deals with prosolvable pro-Lie groups and, a bit more generally, with those which are compact modulo the radical R(G). Indeed we prove the generalisation to pro-Lie groups of some classical splitting theorems for normal vector subgroups and simply connected solvable Lie groups in locally compact almost connected groups. These were discussed in [108], but their proofs require some extra attention in the much bigger category of locally compact pro-Lie groups. At the root of these theorems, in the last evaluations, is invariant integration on compact groups. The grandfather theorem of all of these theorems is as follows (11.15): Let G be a pro-Lie group, and N a closed normal weakly complete vector subgroup such that G/N is compact. Then N splits, that is, there is a compact subgroup C such that μ : N  C → G, μ(n, c) = nc is an isomorphism of topological groups. This allows us to give a significant structure theorem on connected prosolvable groups. We recall that every pro-Lie group has a radical which is exactly such a group (11.28). Let G be an almost connected pro-Lie group such that G0 is prosolvable, then there is a (not necessarily unique) closed subgroup V containing the nilradical N(G) which is simply connected modulo the unique largest compact connected central subgroup C and there is a compact connected group K such that G = V K and V ∩ K is a totally disconnected compact subgroup of the center. This theorem cannot be improved in any obvious direction as various examples show. It shows that compact subgroups in connected solvable pro-Lie groups have a

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“low lying” compact connected central portion C and a “high flying” nonnormal part K complementing a subgroup V which is simply connected modulo C. The splitting Theorems 11.15 and 11.17 are accompanied by the conjugacy Theorems 11.31 and 11.32 which perhaps one expects to hold from the experience with more classical situations.

Chapter 12

Procompact Subalgebras of Pro-Lie Algebras and Compact Subgroups of Pro-Lie Groups

In Chapter 5 we saw that a connected abelian pro-Lie group G contains a unique largest compact subgroup comp(G) and indeed that G is isomorphic to a direct product of a weakly complete vector group and comp(G). In Chapters 10 and 11 it emerged that in the case of reductive connected pro-Lie algebras the chances of identifying maximal compact subgroups and maximal compact normal subgroups are not bad; in particular if g/r(g) is of bounded type as in Theorem 11.8, then G is a semidirect product of a normal closed semisimple pro-Lie group S with L(S) ∼ = g/r(g), and an abelian connected pro-Lie group A. In Chapter 11 we encountered useful splitting theorems by which compact factor groups of a pro-Lie group modulo simply connected solvable normal subgroups split as a semidirect cofactor. Obviously, we need a general theory of compact subgroups of pro-Lie groups. In the spirit of the Lie theory of pro-Lie groups we begin with the Lie algebra theory of compact groups and subgroups of pro-Lie groups. Later we use the insights so obtained for locating compact subgroups in connected pro-Lie groups. Prerequisites. We shall use Chapter 7 on the theory of pro-Lie algebras and their modules heavily. It is helpful if the reader has encountered the structure theory of compact groups, although we shall develop essential aspects from scratch but using information accumulated in this book. For instance, later in the section on pro-Lie algebras we shall use inner automorphisms as provided in the earlier parts of Chapter 9. It is natural that, at this stage, many of the results presented in Chapters 6 through 11 should now enter into the discussions of this chapter. In the proof of Corollary 12.87 characterizing local compactness we shall invoke Yamabe’s Theorem (see [145], [206], [207]).

Procompact Modules and Lie Algebras We shall first deal with the algebraic equivalence of the compactness of a pro-Lie group G as a property of its pro-Lie algebra g = L(G). In this discussion we shall concentrate on pro-Lie algebras g and their g-modules in the spirit of what we have expounded in Chapter 7. We begin by considering an arbitrary Lie algebra L without the specification of any additional algebraic or topological properties. Definition 12.1. (i) Let L be a Lie algebra and let V be an L-module. Then V is called a pre-Hilbert L-module if V is a real vector space with an inner product (•|•), that is,

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a symmetric positive bilinear form, such that (∀x ∈ L, v, w ∈ V )

(x · v|w) = −(v|x · w).

(1)

It is called a Hilbert L-module if, √ in addition, V is a complete topological vector space with respect to the norm v2 = (v|v). (ii) V is called a compact L-module if V can be given an inner product relative to which it is a Hilbert L-module, and dim V < ∞. (iii) An L-module V is called procompact if V is profinite-dimensional and if all finite-dimensional quotient modules are compact L-modules. (iv) An L-module V is called an algebraically locally compact L-module if V is locally finite-dimensional and if all finite-dimensional submodules are compact Lmodules. The terminology of a “compact” L-module formulated in (ii) arises from the adjoint module of a compact Lie group; for details see for instance [102, Proposition 6.2ff]. The labelling of an algebraic property of a locally finite-dimensional module as “locally compact” even though it has nothing to do with the topological property of local compactness is not worse than calling certain finite-dimensional Lie algebras compact, and this practice is well established. Proposition 12.2. Assume that V is a Hilbert L-module. Let δ : V → V  the isomorphism defined by δ(v)(w) = (v|w). Then δ is an isomorphism from the L-module V to its dual module V  . In particular, every compact L-module is isomorphic to its topological dual. Proof. Let δ(x · v)(w) = (x · v|w) = −(v|x · w) = −δ(v)(x · w)) = x · δ(v)(w) by (1). Thus δ(x · v) = x · δ(v) and this says that δ is a morphism of L-modules according to Definition 7.1 (1). Proposition 12.3. For a profinite-dimensional L-module V over a Lie algebra L the following conditions are equivalent: (i) V is a procompact L-module. (ii) V  is an algebraically locally compact L-module. Proof. Exercise E12.1. Exercise E12.1. Prove Proposition 12.3. [Hint. Apply duality according to Theorem 7.11 (v) and Proposition 12.2.] We say that two vector subspaces E1 and E2 of a pre-Hilbert space are orthogonal, 5 in symbols E1 ⊥ E2 , if (v1 |v2 ) = 0 for all v1 ∈ E1 , v2 ∈ E2 . A sum j ∈J Ej of vector subspaces in a pre-Hilbert space is called an orthogonal sum if Ej ⊥ Ek whenever j = k. By the duality statement of Proposition 12.3 we know the structure of any profinitedimensional L-module V as soon as we know the structure of its algebraically locally

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compact dual V  . The structure of the latter, however, is completely clarified in the following satisfactory structure theorem. We shall use of the Axiom of Choice. The Structure of Algebraically Locally Compact Modules Theorem 12.4. (i) An algebraically locally compact L-module is a pre-Hilbert Lmodule which is an orthogonal direct sum of compact submodules. (ii) Any L-submodule of an algebraically locally compact L-module is algebraically locally compact and is an orthogonally direct summand. (iii) Any L-module homomorphic image of an algebraically locally compact Lmodule is algebraically locally compact. (iv) For each algebraically locally compact L-module E there is a Hilbert L-module  in which E is dense in the Hilbert space norm. E Proof. (i) Let F be a submodule of the algebraically locally compact L-module E. We consider the set P of triples (F, B, F) where F is a submodule of E containing F and where B is an invariant scalar product on F , 5 and F is an orthogonal family of finite-di5 F and F = {M : M ∈ F, M ⊆ F }. mensional submodules of F such that F = We partially order P by saying (F1 , B1 , F1 ) ≤ (F2 , B2 , F2 ) if and only if the following conditions are satisfied: (a) F1 ⊆ F2 , (b) B2 |(F1 × F1 ) = B1 , and (c) F1 ⊆  F2 . If T = {(Fj , Bj , Fj ) | j ∈ J } is a totally ordered subset of P we set F = j ∈J Fj and define B : F × F → R as follows: Let v, w ∈ F . Then there is a j such that v, w ∈ Fj . If we have also a k ∈ J with v, w ∈ Fk , then one of the two spaces contains the other, say, Fk does. Then Bk (v, w) = Bj (v, w) and we can unambiguously define B(v, w) = Bj (v, w). It is straightforward to observe  that B is symmetric, positive 5 definite, and invariant. Furthermore we set F = j ∈J Fj . We note that F = F is an orthogonal sum. Thus (F, B, F) belongs to P and is an upper bound of the tower T . Hence P is inductive and so we have a maximal element (M, B, M) in P . We claim M = E; a proof of this claim will finish the proof. Suppose M = E. Then there is an element x ∈ E \ M. Since E is algebraically locally compact, there is a compact submodule F with an invariant scalar product C containing x. Thus F ⊆ M. The submodule M ∩ F has a C-orthogonal complement G in F which, def

by the invariance of C, is a submodule. Now M = M ⊕ G is a module which is properly larger than M, and we define a function B : M × M → R on it by setting B(m1 ⊕ g1 , m2 ⊕ g2 ) = B(m1 , m2 ) + C(g1 , g2 ). It is easily seen that B is an invariant 5 scalar product (Exercise E7.7). Moreover, we set M = M ∪ {G}. Then M = M and this sum is orthogonal. Thus (M, B, M) is a member of P that is properly bigger than the maximal element (M, B, M), and that is a contradiction. This proves our claim. (ii) Let E be an algebraically locally compact L-module and F a submodule. Then F is the sum of finite-dimensional submodules and every such is a compact module since E is algebraically locally compact. Hence F is algebraically locally compact. In the proof of (i) we constructed a scalar 5product B on E and an orthogonal 5family F of compact submodules such that F = {M : M ∈ F, M ⊆ F } and E = F. Then F

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is the 5 disjoint union of the set of M ∈ F, M ⊆ F and the set of M ∈ F, M ⊥ F . Let F  = {M : M ∈ F, M ⊥ F }. Then E is the orthogonal direct sum of F and F  . (iii) Let E be an algebraically locally compact L-module and F a submodule. Then E/F is an L-module. By (ii) we find for F an orthogonal complement F ⊥ . Then F ⊥ and E/F are isomorphic as L-modules. Since F ⊥ is algebraically locally compact by (ii), the quotient module E/F is algebraically locally compact. (iv) We leave the proof of this assertion as an exercise. Exercise E12.2. (a) Verify that, in the proof of Theorem 12.4 (i), B is an invariant scalar product on M. (b) Prove assertion (iv) in Theorem 12.4. (c) Prove the following observations: Lemma A. A compact L-module is an orthogonal direct sum of finite-dimensional simple submodules. Lemma B. Let V be a semisimple finite-dimensional L-module V and let πV : L → gl(V ) be the representation given by πV (x)(v) = x · v. This applies to compact Lmodules. Assume that for each x ∈ L the endomorphism πV (x) is nilpotent. Then L is a zero module, that is L · V = {0}. [Hint. (b): From Part5 (i) of Theorem 12.4 we know that E has an inner product (•|•) such that E = j ∈J Ej is the orthogonal direct sum of finite-dimensional  L-modules Ej . Define E to be the Hilbert space directsum j ∈J Ej of the family of L-modules Ej . Formally, this is the vector subspace of j ∈J Ej of all J -tuples (xj )j ∈J such that the family of real numbers (xj 2 )j ∈J is summable; we verify that this is a vec5  by ((xj )j ∈J |(yj )j ∈J ) = tor subspace. We define an inner product on E j ∈J (xj |yj ); the sum exists by the Cauchy–Schwarz inequality. Since E is an orthogonal direct sum of the Ej , there is a linear isometry from E to the vector subspace of all tuples (xj )j ∈J with xj = 0 for all but a finite number of the indices j ∈ J . This vector space is  and the module operation extends in the obvious way as inherited from the dense in E  product module j ∈J Ej . (c): Proof of Lemma A. Let V be a compact L module according to 12.1 (ii). The orthogonal complement of a submodule is a submodule. Thus V is a semisimple L-module according to 7.15 (ii). Apply Theorem 7.16 to the special case of finite dimensions. (The Axiom of Choice is not needed here.) Proof of Lemma B. Since V is a compact L module it is a finite direct sum of simple compact modules. For a proof that V is a zero module we may therefore assume that V is simple and nonzero. By the Theorem of Engel (see for instance [16, §4, no 2, Théorème 1]), there is a nonzero v ∈ V such that L · v = {0}. Since V is simple, V = R · v and L · V = {0}.] Since property (iv) is just a sidelight in our discourse, we shall not pursue this further here. Corollary 12.5. Every algebraically locally compact L-module E is a semisimple L-module.

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Proof. By the Structure Theorem of Algebraically Locally Compact Modules 12.4, E is a direct sum of finite-dimensional compact submodules; each compact L-module is a direct orthogonal sum of finite-dimensional simple L modules by Exercise E12.2 (c). Hence E is an orthogonal direct sum of simple submodules. Then by Theorem 7.16, E is a semisimple L-module. By way of duality, we now can instantly formulate the following results on procompact L-modules. The Structure of Procompact Modules Theorem 12.6. (i) A procompact L-module is a direct product of compact simple L-modules and is a semisimple L-module. (ii) A continuous homomorphic image of a procompact L-module is procompact. (iii) A closed submodule of a procompact L-module is procompact. Proof. (i) and (iii) follow from 12.5 via duality as does the assertion that a quotient module of a procompact L-module is procompact. But then (ii) follows from Theorem A2.12 in Appendix 2 on morphisms between weakly complete vector spaces. Before we exploit the Structure Theorem further, let us recall a definition which is a group theoretical analog of what we discussed in Chapter 7 for L-modules for a Lie algebra. (Compare Definition 7.1.) Definition 12.7. (i) Let G be a group and E a vector space. Then E is a G-module if there is a map (g, v)  → g · v : G × E → E satisfying 1 · v = v and gh · v = g · (h · v) for all g, h ∈ L and v ∈ E. A function f : E1 → E2 between G-modules is said to be a morphism of G-modules if it is linear and satisfies (∀g ∈ L, v ∈ E1 )

f (g · v) = g · f (v).

(ii) If G is a group and V is a topological vector space, then V is said to be a continuous G-module if the underlying vector space is a G-module in the sense of (i) above and the functions v  → g · v : V → V are continuous for all g ∈ G. (iii) If G is a topological group, then a topological vector space V is said to be a topological G-module if it is a continuous G-module and the maps g  → g · v : L → V are continuous for all v ∈ V ; in other words, if (x, v)  → x · v : L × V → V is continuous in each variable separately. (iv) If, in the circumstances of (iii), the function (g, v)  → g · v : G × V → V is continuous, then V is called a jointly topological G-module If the underlying topological space V is a Baire space (for instance if E is a Fréchet space, that is, a completely metrizable space) and G is locally compact, then a topological G-module is a jointly topological G-module (see for instance [102, Theorem 2.3]).

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The groups we consider in this book frequently fail to be locally compact, and weakly complete topological vector spaces are Baire spaces only if they are first countable. Let V be a topological vector space and give the group Aut(V ) of automorphisms of the topological vector space V the topology of pointwise convergence, that is, the subspace topology of the inclusion Aut(V ) ⊆ V V . Then this topology is also called the strong operator topology. Exercise E12.3. Let V be a topological G-module. Define π : G → Aut(V ) ⊆ V V by π(g)(v) = g · v. Then π is well defined, that is, π(g) is an automorphism and homeomorphism of V , and π is a continuous group homomorphism. The Compact Group Associated with a Procompact Module Theorem 12.8. Let V be a procompact L-module. Then there is a compact connected group GV ⊆ Aut(V ) such that V is a jointly topological GV -module and there is a homomorphism of Lie algebras λ : L → L(GV ) such that GV = expGV λ(L), (∀x ∈ L, v ∈ V ) x · v = limh→0,h =0 h1 ((expGV λ(h · x)) · v − v), (∀x ∈ g, w ∈ V ) (expGV λ(x))(w) = w + xV (w) + 2!1 · xV2 (w) + · · · ∈ V , and a closed vector subspace W of V is an L-module if and only if it is a GV -module.  Proof. By Theorem 12.6 we may write V = j ∈J Vj with compact (hence finite-dimensional) simple L modules. We let (•|•)j denote an L-invariant inner product which exists since Vj is compact. Let πj : L → gl(Vj ) be the representation defined by πj (x)(vj ) = x · vj . We let ø(Vj ) be the group  of all skew-symmetric endomorphisms. Then πj (L) ⊆ ø(Vj ). We define π : L → j ∈J ø(Vj ) by π(x) = (πj (x))j ∈J and get a morphism of Lie algebras of L into a semisimple pro-Lie algebra. We let SO(Vj ) denote the special orthogonal group of (Vj , (•|•)). We have L(SO(Vj )) = ø(Vj ) with the standard exponential function expj : ø(Vj ) → SO(Vj ), expj X = 1 + X + 1/2!X 2 + · · · . def  We have an obvious homomorphism embedding ϕ : K = j ∈J SO(Vj ) → Aut(V ) given by ϕ((gj )j ∈J )((vj )j ∈J ) = (gj (vj ))j ∈J . We thus make V into a topological K-module via k · v = ϕ(k)(v). By the remark following Definition 12.7, V is in fact a jointly  topological K-module. Clearly, K is a compact group with the Lie algebra k = j ∈J ø(Vj ) for the exponential function expK ((Xj )j ∈J ) = (expj Xj )j ∈J ). Fi(i) (ii) (iii) (iv)

def

nally we set GV = expK π(L) ≤ K. Then GV is a compact group as a closed subgroup of a compact group; since the compact group GV contains the dense arcwise connected subgroup expK π(L), it is connected. The K-module structure of V makes V into a jointly topological GV -module. Clearly π(L) ⊆ L(GV ); let λ : L → L(GV ) be the corestriction of the morphism π . The commutative diagram λ

incl

L −−−−→ L(G ⏐ V ) −−−−→ ⏐k ⏐exp expGV ⏐   K GV −−−−→ K incl

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illustrates the situation. We have proved the assertions of Theorem 12.8 through statement (i). For a proof of (ii) we note that for each j ∈ J we know that for the finite-dimensional Lie group SO(Vj ) we have (∀xj ∈ ø(Vj ), vj ∈ Vj )

xj · vj =

lim

h→0,h =0

1 ((expj h · xj ) · vj − vj ). h

It follows at once that (∀x  ∈ k, v ∈ V )

x·v =

lim

h→0,h =0

1 ((expK h · x  )) · v − v). h

= λ(h · x) we obtain the proof ofassertion (ii). Taking x  = λ(x) and noting h · λ(x) (iii) Let x = (Xj )j ∈J ∈ L(K) = j ∈J ø(Vj ), and w = (vj )j ∈J ∈ j ∈J Vj = V . Then (expK x)(w) = ((expj Xj )(vj ))j ∈J = (vj + Xj (vj ) + Xj2 (vj ) + · · · )j ∈J 1 = w + xV (w) + · xV2 (w) + · · · ∈ V . 2! For x ∈ L(GV ) ⊆ L(K) this is what was asserted. (iv) Let W a closed vector subspace of V . We have L · W ⊆ W iff π(L)(W ) ⊆ W iff λ(L)(W ) ⊆ W . If this is satisfied, it then follows that (expGV λ(x))(w) = w + x(w) + 2!1 · x 2 (w) + · · · ∈ W for all x ∈ L and w ∈ W by (iii) above. Since GV is topologically generated by the elements expGV λ(x), this implies GV · W ⊆ W . Conversely, if this condition is satisfied, then we have expGV λ(h · x) ∈ GV and thus x ·w = limh→0,h =0 h1 ((expGV λ(h·x))·w −w) ∈ W for x ∈ L and w ∈ W . Therefore L · W ⊆ W . This completes the proof of the theorem. Corollary 12.9. Let L = L1 + L2 be a Lie algebra with two subalgebras L1 and L2 satisfying [L1 , L2 ] ⊆ L2 . If V is a profinite-dimensional L-module such that V is a procompact Lj -module for each of j = 1 and j = 2, then V is a procompact L-module. Proof. Let Gj ⊆ Aut(V ) be the subgroups associated with Lj , respectively, according to Theorem 12.8. Let V  be the dual L-module and E a finite-dimensional L-submodule. Assume that π : L → gl(E) is the representation associated with the L-module structure of E. There are representations γj : Gj → Gl(E) such that γj (Gj ) = expGl(E) π(Lj ). Let gj = exp π(xj ) for xj ∈ Lj . Then g1 g2 g1−1 = exp Ad(g1 )π(x2 ) where Ad(g1 )π(x2 ) = ead π(x1 ) π(x2 ) = π(x2 ) + π [x1 , x2 ] + 1/2! ad π(x1 )π [x1 , x2 ] + · · · ⊆ π(L2 )

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since [L1 , L2 ] ⊆ L2 and π(L2 ) is finite-dimensional and therefore closed. Consequently Ad(g1 )π(L2 ) ⊆ π(L2 ) and so g1 g2 g1−1 ∈ γ2 (G2 ). The group generated by def

gj is dense in γj (Gj ). Thus γ1 (G1 ) normalizes γ2 (G2 ), and thus  = γ1 (G1 )γ2 (G2 ) is a compact subgroup of Gl(E). Hence E supports a -invariant inner product and thus an γj (Gj ) invariant inner product for both j = 1 and j = 2. This inner product, consequently is simultaneously L1 - and L2 -invariant and thus is L-invariant. Hence V  is an algebraically locally compact L- module and thus V is a procompact L-module as was asserted. The hypothesis [L1 , L2 ] ⊆ L2 may be rephrased by saying that L2 is assumed to be an ideal of L = L1 + L2 .

Procompact Lie Algebras and Compactly Embedded Lie Subalgebras of Pro-Lie Algebras We now apply our knowledge of “compactness” and L-modules to the case of a proLie algebra g and the adjoint and coadjoint modules. Then the adjoint module gad is a profinite-dimensional g-module and the coadjoint module gcoad is a locally finite-dimensional g-module dual to g. If k is any Lie subalgebra of g, then the restriction of the adjoint action of g on gad to k makes gad a profinite-dimensional k-module and the restriction of the coadjoint action of g on gcoad to k makes gcoad into a locally finitedimensional k-module which is dual to the k-module gad . For the following definition we recall from Definition 12.1 (iii) the concept of a procompact L-module. Definition 12.10. Let g be a pro-Lie algebra and k a Lie subalgebra. Then k is said to be compactly embedded into g if the adjoint module gad is a procompact k-module. If g is compactly embedded into itself, then g is said to be a procompact pro-Lie algebra Obviously, a closed compactly embedded subalgebra is procompact in its own right. A noncentral one-dimensional subalgebra of the three-dimensional Heisenberg algebra is a compact (hence procompact) Lie algebra which is not compactly embedded into the Heisenberg algebra. In the Heisenberg algebra, every 2-dimensional vector subspace containing the 1-dimensional center and commutator subalgebra is a maximal abelian subalgebra and is also an ideal. This shows that maximal abelian subalgebras and ideals are not unique. It also shows that the analytic subgroup belonging to a maximal abelian subalgebra may not be compact in a Lie group having the Heisenberg algebra as Lie algebra. This example shows that maximal procompact subalgebras need not lead us to maximal compact subgroups. We need to consider the ambient pro-Lie algebra as well. Indeed this is what we have to do in the finite-dimensional situation, too, and this is where the concept of compactly embedded subalgebras comes in. The Heisenberg

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algebra is a good simple test for the viability of conjectures on procompact subalgebras on the one hand and compactly embedded ones on the other. Proposition 12.11. Let g be a pro-Lie algebra and k a Lie subalgebra. Then the following conditions are equivalent. (i) k is compactly embedded into g. (ii) The coadjoint module gcoad of gad is an algebraically locally compact k-module. If these conditions are satisfied, then gad is a semisimple k-module under the adjoint action. Proof. This is an immediate consequence of Proposition 12.3 with L = k, V = gad and V  = gcoad . The last assertion follows from Theorem 12.6 (i). As a first major step we present the structure of procompact pro-Lie algebras. The Structure Theorem of Procompact Lie Algebras Theorem 12.12. (A) Let g be a pro-Lie algebra. Then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

g is procompact. The coadjoint g-module gcoad is an algebraically locally compact g-module. The coadjoint g-module gcoad is a direct sum of simple compact g-modules. g is a direct product of simple compact Lie algebras and copies of R. g is a direct product of its center z(g) and its commutator algebra g and g is a product of simple compact Lie algebras. In particular, for a procompact Lie-algebra g, the radical r(g) agrees with its center z(g).

(B) A closed subalgebra of a procompact pro-Lie algebra is procompact. (C) The image of a procompact pro-Lie algebra under a continuous morphism of Lie algebras is procompact. (D) A product of any family of procompact pro-Lie algebras is procompact. (E) A closed procompact semisimple subalgebra k of a pro-Lie algebra g is compactly embedded in g. (F) g/r(g) is procompact iff g/ ncored (g) is procompact. (G) If V is a profinite-dimensional k-module for a semisimple procompact pro-Lie algebra k, then V is a procompact k-module. (H) Assume that a subalgebra g of a pro-Lie algebra h is the sum of two subalgebras g1 and g2 such that, firstly, g1 and g2 are compactly embedded in h, and, secondly, [g1 , g2 ] ⊆ g2 . Then g is compactly embedded in h. Proof. First we prove (A). (i) ⇔ (ii): This follows from Proposition 12.3.

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(ii) ⇒ (iii): This follows from Theorem 12.5. (iii) ⇔ (iv) is a consequence of duality. (iv) ⇒ (i): If (iv) holds, then the filter basis P of partial products with finite-dimensional quotients converges to zero and thus is cofinal in the filter basis  of all ideals with finite-dimensional quotients (see 2.17). Since g/i is a compact Lie algebra for each i ∈ P , it follows that g is a procompact pro-Lie algebra. (iv) ⇔ (v): A compact Lie algebra which is simple as an adjoint module over itself is either isomorphic to R or to a nonabelian simple compact Lie algebra s. Thus (iv) is equivalent to  (iv ) g = RI × j ∈J sj for some set I and for a family of simple compact nonabelian Lie algebras sj .  I  In this statement,  clearly z(g) = R × {0} and g ⊆ {0} × j ∈J sj . Thus it remains to be shown that j ∈J sj agrees with its own commutator algebra. But this is an immediate consequence of Theorem 7.27. (B) and (C)are direct consequences of Corollary 12.6 (iii) and (ii). (D) If g = k∈K gk for a family of procompact subalgebras gk , then by (A), each gk is a direct product of simple compact finite-dimensional Lie algebras or copies of R; accordingly, g is a product of simple compact finite-dimensional Lie algebras and copies of R. Hence g is procompact by (A) above.  (E) Let k be a closed procompact subalgebra of g. Then by (A) (iv), k ∼ = j ∈J sj for a family of simple compact Lie algebras sj . Let V ⊆ gcoad be a finite-dimensional g submodule; by Definition 12.1 (iv) and Proposition 12.11 it suffices to show that V is a compact k module. Let π : g → gl(V ) be the representation associated with the module V (thatis, π(x)(ω), y = −ω, [x, y]). Then the image π(k) is a homomorphic image of j ∈J sj in a finite-dimensional Lie algebra and is therefore a finite product of simple compact Lie algebras, that is, it is a semisimple compact Lie algebra. The assertion then is a consequence of the following Lemma. Let V be a finite-dimensional k-module for a semisimple compact Lie algebra k. Then V is a compact k-module. Proof of the lemma. We have to show that V supports a k-invariant inner product (•|•). Let π : k → gl(V ) be the representation associated with the k-module structure of V . Then π(k) is a Lie subalgebra of gl(V ) which is a semisimple compact Lie algebra. It is therefore no loss of generality to assume that k ⊆ gl(V ). The analytic def

subgroup K = exp k ⊆ gl(V ) has the Lie algebra L(K) = k (compare 9.9ff.; but the present claim refers to finite-dimensional Lie theory only, see for instance [102, Theorem 5.52]). Then the underlying Lie group of K is compact (see for instance [102, Theorem 6.6 with m = 0]) and therefore K is compact in the subgroup topology. Then by Weyl’s Unitary Trick, there is a K-invariant inner product (•|•) on V (see for instance [102, Theorem 2.10]). This inner product is k-invariant. (See for instance [102, Exercise E6.1 following Proposition 6.2]). This completes the proof of the lemma and thereby the proof of Assertion (E) of the theorem.

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def (F) Since h = g/ ncored (g) is reductive h ∼ = z(h) ⊕ h where h is a semisimple pro-Lie algebra, that is, a product of simple finite-dimensional Lie algebras. Then h is procompact iff h is procompact. On the other hand, h ∼ = g/r(g). So g/ ncored (g) is procompact iff g/r(g) is procompact. (G) Let E be a finite-dimensional submodule of the dual g-module V  . We must show that E has an invariant inner scalar product (•|•). Let π : k → gl(E) be the representation associated with the k-module E. Since dim gl(E) is finite, π(k) is a finite-dimensional semisimple compact Lie subalgebra of gl(E). The analytic subgroup G of Gl(E) generated by π(k) has the compact semisimple Lie algebra π(k) as Lie algebra L(G). By [102, Theorem 6.6 (iv)] there is a compact simply connected Lie group S with L(S) = L(G), and so by [102, Corollary A2.26] there is a morphism f : S → G which implements a local isomorphism and thus is a quotient morphism. Hence G is a compact subgroup of Gl(E). By Weyl’s Trick (see for instance [102, Theorem 2.10]), there is an inner product (•|•) relative to which all elements of G are orthogonal transformations of E. But then this inner product is π(k)-invariant (see for instance [102, Proposition 6.2]). But this means exactly that E is a compact k-module as asserted. (H) We apply Corollary 12.9 with V = h, L = g, L1 = g1 and L2 = g2 and we obtain the assertion.

Regarding Assertions (B) and (E) we recall that any closed abelian subalgebra of a pro-Lie algebra is a procompact pro-Lie algebra in its own right but there are many examples of abelian subalgebras in Lie algebras which are not compactly embedded (for instance in the Heisenberg algebra or in the 2-dimensional nonabelian Lie algebra). Proposition 12.13. Let ϕ : g1 → g2 be a quotient morphism of pro-Lie algebras. If k is a compactly embedded subalgebra of g1 , then ϕ(k) is compactly embedded in g2 . Proof. The weakly complete topological vector space g2 is a k-module via X · Y = [ϕ(X), Y ], X ∈ k, Y ∈ g2 . Then ϕ is a continuous k-module homomorphism. Since g1 is a procompact k-module by Corollary 12.6 (ii), g2 is a procompact k-module and that is tantamount to saying that g2 is a procompact ϕ(k)-module. Thus ϕ(k) is a compactly embedded subalgebra of g2 . Proposition 12.14. Let g be a pro-Lie algebra. Then the following conclusions hold: (i) If k is a compactly embedded subalgebra of g, then z(g) + k is compactly embedded. (ii) The center z(g) of g is compactly embedded. The closure of a compactly embedded subalgebra is compactly embedded. (iii) A compactly embedded subalgebra k contained in a pronilpotent ideal n is contained in the center z(g). In particular, a pronilpotent compactly embedded ideal is central. (iv) Any procompact subalgebra k of a prosolvable pro-Lie algebra g is abelian. In particular, a compactly embedded subalgebra of a prosolvable pro-Lie algebra is abelian.

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Proof. We prove (i). Let V be a finite-dimensional compact k-submodule of gcoad . The annihilator V ⊥ ⊆ gad of all elements of g annihilated by all functionals in V is a def

cofinite-dimensional closed k-submodule. The set a = {x '∈ g : [x,&V ⊥ ]' ⊆ V ⊥ } &is a closed subalgebra of g, since x, y ∈ a and v ∈ V ⊥ implies [x, y], v = x, [y, v] − ' & y, [x, v] ∈ V ⊥ . Clearly z(g) is contained in a, and thus z(g) + k ⊆ a. The vector subspace V of gcoad is an a-module and then a z(g) + k-module. Let sk(E) ⊆ gl(E) denote the Lie subalgebra of all endomorphisms ϕ of E satisfying (ϕ(v)|w) + (v|ϕ(w)) = 0 for all v, w ∈ E. The dual a-module of E is gad /E ⊥ . Let πE : a → gl(E) denote the representation defined by πE (x)(v), y + E ⊥  = −v, [x, y] + E ⊥  for x ∈ a, y ∈ g, v ∈ E. If z ∈ z(g), then [z + x, y] = [x, y] and thus πE (z + x) = πE (x). Since πE (x) ∈ sk(E) as E is a compact k-module we have πE (z(g) + k) ⊆ sk(E), and thus E is a compact z(g) + k)-module. Since E was an arbitrary compact k-submodule of gcoad we see that gcoad is an algebraically locally compact z(g) + k-module. This shows (i) by Proposition 12.3 applied with V = gad and L = k. (ii) is an immediate consequence of (i). (iii) Let j ∈  (g), then (k + j)/j is compactly embedded in g/j by Proposition 12.13, and it is contained in a nilpotent ideal (n + j)/j of g/j since n is pronilpotent. Let x ∈ k. def

Then x ∈ n and [x, g] ⊆ n. Then ν = adg/j (x + j ) is a nilpotent endomorphism of the vector space g/j , as ν m+1 (g/j) ⊆ ν m ((n + j)/j) = {0} for m large enough. Then Lemma B of Exercise E12.2 (c) applies and  shows that g/j is a zero k-module. So [k, g] ⊆ j for all j ∈  (g), and thus [k, g] ⊆ j∈ (g) j = {0}. This means that k ⊆ z(g). (iv) Let g be a prosolvable pro-Lie algebra and k a procompact subalgebra. By Theorem 12.14 (A), k is a direct product of simple compact Lie algebras. By Theorem 7.53, g does not contain any finite-dimensional simple Lie algebras. Hence k is singleton, and k is abelian. Now assume that k is a compactly embedded subalgebra. Then k is compactly embedded by (ii) above. Hence k is a procompact Lie algebra and thus is abelian by what we just saw.

Maximal Compactly Embedded Subalgebras of Pro-LieAlgebras It is not immediately clear that an arbitrary pro-Lie algebra should contain any nontrivial compactly embedded subalgebra, and indeed the two-dimensional nonabelian real Lie algebra contains no nonzero compactly embedded subalgebra. However, the following theorem secures the existence of compactly embedded subalgebras to the best extent possible. Maximal Compactly Embedded Subalgebras: Existence Theorem 12.15. Let g be a pro-Lie algebra. Then (i) every compactly embedded subalgebra of g is contained in a maximal compactly embedded subalgebra, and

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505

(ii) every compactly embedded abelian subalgebra of g is contained in a maximal compactly embedded abelian subalgebra. Proof. Let F denote the set of all finite-dimensional g-submodules of the coadjoint g-module gcoad . For each E ∈ F we have a representation πE : g → gl(E) such that π(X)(ω) = X · ω for X ∈ g and ω ∈ E. Now let h be a compactly embedded subalgebra [respectively, a compactly embedded abelian subalgebra] of g and let K denote the set of compactly embedded subalgebras [respectively, compactly embedded abelian subalgebras] of g containing h. Let k ∈ K; since gcoad is an algebraically locally compact k-module, E is a compact k-module (see Definition 7.81 (ii)). We shall now prove the claim of the theorem via Zorn’s Lemma. Therefore let C be a chainin K, that is, a subset which is totally ordered with respect to ⊆, and define c = C. Then πE (C) = {πE (k) : k ∈ C} is a chainof subspaces   vector  in the finite-dimensional vector spaces gl(E), and πE (c) = πE C = πE (C). A finite-dimensional vector space satisfies the ascending chain condition for vector subspaces. Hence  there is a kE∈ C such that kE ⊆ k ∈ C implies πE (kE ) = πE (k). Thus πE (c) = πE (C) = kE ⊆k∈C πE (k) = πE (kE ). Thus E is a compact cmodule. Since E ∈ F was arbitrary, every element of gcoad is contained in a compact c-submodule of gcoad . Hence the c-module gcoad is algebraically locally compact. By Proposition 7.88, c is compactly embedded in g. Thus (K, ⊆) is inductive. Hence this partially ordered set has maximal elements, as asserted. Proposition 12.16. Let g be a pro-Lie algebra. Then: (i) A maximal compactly embedded subalgebra (respectively, maximal compactly embedded abelian subalgebra) m of g is closed and contains the center. (ii) If n(g) denotes the nilradical of g, that is, the largest nilpotent ideal of g, then m ∩ n(g) = z(g) for any maximal compactly embedded subalgebra (respectively, any maximal compactly embedded abelian subalgebra) m. (iii) If m and k are compactly embedded subalgebras and one of the two normalizes the other, that is, [m, k] is contained in m or in k, and if m is maximal compactly embedded, then k ⊆ m. Proof. We begin by noting that the alternative conclusions in square brackets pertaining to maximal compactly embedded abelian subalgebras follow immediately in the same way as the other assertions; therefore we will not particularly refer to the abelian case. (i) is an immediate consequence of Proposition 12.14 (1) Proof of (ii). We note that (i) shows that z(g) ⊆ m ∩ n(g). Moreover, m ∩ n(g) is a compactly embedded subalgebra contained in the pronilpotent ideal n(g). Then by 12.14 (iii) we have m ∩ n(g) ⊆ z(g) and therefore equality holds. Proof of (iii). Since both m and k are compactly embedded in g and one of the two subalgebras normalizes the other, by Theorem 12.12 (H), the sum m + k is a compactly embedded subalgebra which contains m. Thus if m is maximal compactly embedded, then m + k = m which means k ⊆ m.

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Exercise E12.4. Prove the following Proposition. Let g be a pro-Lie algebra. (i) If k is a maximal compactly embedded subalgebra, and t is a maximal compactly embedded abelian subalgebra of k, then t is a compactly embedded abelian subalgebra of g. (ii) Let t be a maximal compactly embedded abelian subalgebra of g. Then there is a maximal compactly embedded subalgebra k of g containing t and t is a maximal compactly embedded abelian subalgebra of k. [Hint. (i) Since g is procompact k-module it is a procompact h-module for any subalgebra of k. This applies to h = t. (ii) If t ⊆ t1 ⊆ k where t1 is an abelian subalgebra. Then t1 is compactly embedded in g by (i); since t was maximal compactly embedded in g we conclude t = t1 , and thus t is maximal compactly embedded abelian in k.] We use a lemma from Bourbaki [19, Chap. VII, §2, no 3, Proposition 10] which works for us without dimensional restriction. The relevance is that every (maximal) compactly embedded abelian subalgebra a of a pro-Lie algebra satisfies the hypotheses of the subsequent theorem and shows that it is contained in a Cartan subalgebra; we discussed Cartan subalgebras of pro-Lie algebras in Chapter 7 from 7.81 on. Theorem 12.17. Let g be a pro-Lie algebra and a a subalgebra satisfying the following two conditions: (i) a is abelian. (ii) g is a semisimple a-module under the adjoint action. Then a is contained in a Cartan subalgebra of g; in fact the nonempty set C(a, g) of Cartan subalgebras of g containing a is the set of Cartan subalgebras of z(a, g), the centralizer of a in g. Proof. The centralizer z(a, g) is closed and thus is a pro-Lie algebra in its own right. By Theorem 7.93, there is a Cartan subalgebra h of z(a, g). Since a is central in z(a, g) we have a ⊆ h by Exercise E7.17. Let n(h, g) be the normalizer of h in g. Then [a, n(h, g)] ⊆ [h, n(h, g)] ⊆ h.

(∗)

Since g is a semisimple a-module, there is an a-submodule v of n(h, g) such that n(h, g) = h ⊕ v. Now [a, v] ⊆ [a, n] ⊆ h by (∗) and [a, v] ⊆ v since V is an amodule. So [a, v] ⊆ h ∩ v = {0}. Thus v ⊆ z(a, g) and so n(h, g) = h + v ⊆ z(a, g). Thus n(h, g) = n(h, z(a, g)) = h, by Definition 7.88, since h is a Cartan subalgebra of z(a, g). But then again by Definition 7.88, h is a Cartan subalgebra of g. Corollary 12.18. Let g be a pro-Lie algebra and a a maximal compactly embedded abelian subalgebra of g. Then a is contained in a Cartan subalgebra of g and indeed the set C(a, g) of all Cartan subalgebras containing a is exactly the set of all Cartan subalgebras of z(a, g), the centralizer of a in g.

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Proof. Since a is compactly embedded in g, the adjoint module gad is a semisimple a-module by Proposition 12.11 (i). The corollary then follows from Theorem 12.17.

We recall from Theorem 7.101 that in a pro-solvable Lie algebra all Cartan subalgebras the  1 are  In conjugate.  3-dimensional simple Lie algebra sl(2, R) the subalgebras 0 0 1 R · 0 −1 and R · −1 0 are two Cartan subalgebras; the second one is compactly embedded, the first one is not. So they certainly cannot be conjugate. Corollary 12.19. Let g be a pro-Lie algebra. Then the following conclusions hold. (i) Assume that all Cartan subalgebras are conjugate. Then all maximal compactly embedded abelian subalgebras are conjugate. (ii) In a prosolvable pro-Lie algebra all maximal compactly embedded subalgebras are conjugate under inner automorphisms. Proof. (i) Let a1 and a2 be maximal compactly embedded subalgebras. By Corollary 12.18 there exist Cartan subalgebras h1 and h2 such that aj ⊆ hj , j = 1, 2. By hypothesis, there is an inner automorphism ϕ of g such that ϕ(h1 ) = h2 . Then both ϕ(a1 ) and a2 are maximal compactly embedded abelian subalgebras of g, and both are central in h2 by Corollary 12.18. Hence ϕ(a1 ) and a2 centralize each other and thus by Proposition 12.16 (iii) we have ϕ(a1 ) = a2 and this proves the corollary. (ii) By Theorem 7.101, all Cartan subalgebras are conjugate under inner automorphisms in a prosolvable pro-Lie algebra. The assertion now follows from (i).

Conjugacy of Maximal Compactly Embedded Subalgebras We are now working toward a theorem which says that two maximal compactly embedded subalgebras of a pro-Lie algebra are conjugate. The first problem here is to express clearly what conjugacy is supposed to mean. In Theorem 7.77 we proved that two Levi summands of a pro-Lie algebra are conjugate under a “special automorphism”, that is an automorphism ϕ = ead x with an element x ∈ ncored (g); the existence of such automorphisms was discussed in Corollary 7.74 and Definition 7.75. In Chapter 9 we generalized this concept by showing that every element x of a pro-Lie algebra g defined an element ead x ∈ Aut(g), and we introduced the group Inn(g) of inner automorphisms which is algebraically generated by {ead x : x ∈ g}. (See Definition 9.3.) If g = L(G) for a pro-Lie group G, then ead x = Ad(expG x) for all x ∈ g. Conjugacy in a pro-Lie algebra g is meant to be conjugacy with respect to automorphisms from Inn(g). Proposition 12.20. Let g be a pro-Lie algebra, j a closed ideal and h a closed subalgebra. Then the following conclusions hold:

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(i) Any inner automorphism ϕ ∈ Inn(g) leaves j invariant and induces an inner automorphism ϕg/j ∈ Inn(g/j) via ϕg/j (X + j) = ϕ(X) + j. (ii) ϕ  → ϕg/j : Inn(g) → Inn(g/j) is a surjective morphism of groups. (iii) For each ϕ ∈ Inn(h) there is a ϕh ∈ Inn(g) such that ϕh |h = ϕ. Proof. (i) Since j is a closed ideal, for each X ∈ g and Y ∈ j we have ead X Y = Y + [X, Y ] + 2!1 [X, [X, Y ]] + · · · ∈ j. Thus j is invariant under each inner automorphisms. (ii) Each inner automorphism of g/j is a finite product of automorphisms of the form eadg/j (X+j) . This automorphism is induced on g/j by the inner automorphism eadg X of g. (iii) Each ϕ ∈ Inn(h) is a finite product of inner automorphisms of the form eadh X with X ∈ h. But each of these is the restriction of the inner automorphism eadg X of g to h. In the following we shall utilize information on finite-dimensional semisimple Lie algebras. Lemma 12.21. Let g be a finite-dimensional real reductive Lie algebra and k1 and k2 two maximal compactly embedded subalgebras. Then (i) there is an inner automorphism ϕ of G such that ϕ(k1 ) = k2 . (ii) If t1 and t2 are two maximal compactly embedded abelian subalgebras of g then there is an inner automorphism ϕ such that ϕ(t1 ) = t2 . (iii) If t is compactly embedded Cartan subalgebra then there is a unique compactly embedded subalgebra k such that t ⊆ k. (iv) If t is a maximal compactly embedded abelian subalgebra such that z(t, g) = t, then t is contained in more than one maximal compactly embedded subalgebra of g. Proof. (i) This assertion is the Lie algebra version of [78, p. 183, Theorem 7.2]. (ii) By Theorem 12.15, for each j = 1, 2, the subalgebra tj is contained in a maximal compactly embedded subalgebra kj . (In the finite-dimensional case, the full power of Theorem 12.15 is not required.) Also, for j = 1, 2, the subalgebra tj is a Cartan subalgebra of kj . By (i) above, k1 and k2 are conjugate under an inner automorphism of g. However, in a compact Lie algebra, the Cartan subalgebras are conjugate (see for instance [102, Theorem 6.27] or [78, p. 248, Theorem 6.4 (iii)]). Claim (ii) follows. (iii) and (iv) These points are discussed in Appendix A3, Proposition A3.6ff. Lemma 12.22. Two maximal compactly embedded subalgebras of a reductive pro-Lie algebra are conjugate under inner automorphisms. The same is true for maximal compactly embedded abelian subalgebras. Every compactly embedded Cartan subalgebra of a reductive pro-Lie algebra is contained in unique compactly embedded subalgebra.  Proof. Assume that g is reductive, that is, g = z(g) ⊕ j ∈J sj for a family of simple ideals sj . Let k be a maximal compactly embedded subalgebra. Then z(g) ⊆ k by Proposition 12.20 (i); let kj = pr j (k) be the projection into sj . Then kj is compactly

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 embedded into the simple Lie algebra sj by Proposition 12.13 k ⊆ z(g) ⊕ j ∈J kj . The subalgebra k j is contained in a maximal compactly embedded subalgebra mj , and then z(g) ⊕ j ∈J mj is compactly embedded and the maximality of k implies  k = z(g) ⊕ j ∈J mj . Then mj = kj and  k = z(g) ⊕ kj . j ∈J

Assume that h = z(g) ⊕



hj

j ∈J

is a second compactly embedded subalgebra; then by Lemma 12.20 (i), there is an inner def  automorphism ϕj ∈ Inn(sj ) such that ϕj (hj ) = kj . Now we set ϕ = j ∈J ϕj ∈ def  Inn(s) where s = j ∈J sj , ϕ(z + (sj )j ∈J ) = z + (ϕj (sj ))j ∈J . Then we obtain ϕ(h) ⊆ k. Since ϕ is an inner automorphism, the claim is proved. An identical proof using Lemma 12.20 (ii) in place of 12.20 (i) takes care of the conjugacy of two maximal compactly embedded abelian subalgebras. Finally let t be a maximal compactly embedded abelian subalgebra of g. Then it is of the form t = z(g)× j ∈J tj for some maximal compactly embedded abelian subalgebra of sj . Let k and k (1) be maximal compactly embedded subalgebras containing t. Then  k = z(g) × kj , j ∈J

k

(1)

= z(g) ×



(1)

kj ,

j ∈J (1)

for suitable compactly embedded subalgebras kj and kj of sj containing tj . Then (1) Lemma 12.21 (iii) implies kj = kj for each j ∈ J and thus k(1) = k. Exercise E12.5. Verify the details of the proof of the abelian case of Lemma 12.22. In particular, Lemma 12.22 entails the following remark. Lemma 12.23. The group Inn(g) operates transitively on the set C, respectively, Ca of all subalgebras c of g containing r(g) for which c/r(g) is maximal compactly embedded in g/r(g), respectively, maximal compactly embedded abelian in g/r(g). def

Proof. We write r = r(g). Let mj , j = 1, 2, be subalgebras of g containing r such that def

m∗j = mj /r are maximal compactly embedded subalgebras of g/r. By Lemma 12.22 we find an inner automorphism  of g/r such that (m∗1 ) = m∗2 . By Lemma 12.20 (ii) there is an inner automorphism ϕ of g such that (X + r) = ϕ(X) + r. It then follows that ϕ(m1 ) = m2 . The abelian case is completely analogous.

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Theorem 12.24. Two maximal compactly embedded abelian subalgebras of a pro-Lie algebra g are conjugate under an inner automorphism. Proof. Let a1 and a2 be maximal compactly embedded abelian subalgebras of g. Then the quotients (aj + r(g))/r(g), j = 1, 2, are compactly embedded abelian in g/r(g) by Proposition 12.13. Thus by Theorem 12.15 there are subalgebras bj , j = 1, 2, such that aj ⊆ bj and (bj + r(g))/r(g), j = 1, 2 are maximal compactly embedded abelian subalgebras of g/r(g). By Lemma 12.23 there is an inner automorphism  of g/r(g) such that ((b1 + r(g))/r(g)) = (b2 + r(g))/r(g). Then by Proposition 12.20 (ii) there is an inner automorphism ϕ of g such that ϕ(b1 ) = b2 . Thus ϕ(a1 ) and a2 ⊆ b2 are two maximal compactly embedded abelian subalgebras of the prosovable pro-Lie algebra b2 . By Corollary 12.19 (ii), ϕ(a1 ) and a2 are conjugate in b2 under an inner automorphism and so, a fortiori, they are conjugate in g under an inner automorphism (see 12.20 (iii)). This proves the claim. From Theorem 7.52 (i) we know that every c ∈ C contains a Levi summand s, and since s ∼ = g/r(g) we know that s is procompact, and from 12.12 (E) we infer that s is compactly embedded in c. Lemma 12.25. Assume that g is the direct sum of r(g) and s for a unique Levi summand (according to Theorem 7.77 (ii)), and assume that s is procompact. Then a compactly embedded subalgebra k is maximal iff it is of the form k = m ⊕ s with a maximal compactly embedded subalgebra m of r(g). def

Proof. If m is a maximal compactly embedded subalgebra of r(g), then k = m ⊕ s is compactly embedded by Theorem 12.12 (H), and if k ⊆ k1 then k1 = m1 ⊕ s for m ≤ m1 ≤ r(g); if k1 is compactly embedded in g then m1 is compactly embedded in r(g), since r(g) ∼ = g/s is a homomorphic image of g and m1 is the homomorphic image of k1 (see Proposition 12.15). But then m1 = m by the maximality of m. Thus k1 = k. Therefore k is maximal. Conversely, let k be maximal compactly embedded in g. Let m0 be the projection of k into r(g), then m0 is compactly embedded into r(g) by Proposition 12.13. Let m be maximal compactly embedded in r(g) containing m0 by Theorem 12.14. Since [r(g), s] = {0}, the subalgebra m is in fact compactly embedded into g. Then m ⊕ s is compactly embedded in g by Theorem 12.12 (H). Now k ⊆ m ⊕ s and k is maximal, hence k = m ⊕ s. Lemma 12.26. Assume that g/r(g) is procompact and let k be a maximal compactly embedded subalgebra containing a Levi summand s. Then k = m ⊕ s where m is a maximal compactly embedded subalgebra of z(s, g) and where z(s, g) denotes the centralizer of s in g. def

Proof. Since s is semisimple we have z(s) = z(s, g) ∩ s = {0}. Set gs = z(s, g) ⊕ s. Then z(s, g) = r(gs ) and gs is the direct sum of its radical r(gs ) and its unique procompact Levi summand gs (see Theorem 7.77 (ii)). Then by Lemma 12.25, all

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maximal compactly embedded subalgebras k of g containing s are of the form k = m⊕s with a maximal compactly embedded subalgebra m of z(s, g). Maximal Compactly Embedded Subalgebras: Conjugacy Theorem 12.27. Let g be a pro-Lie algebra. (i) If k1 and k2 are two maximal compactly embedded subalgebras of g, then there is an inner automorphism ϕ of g such that ϕ(k1 ) = k2 . (ii) If t1 and t2 are two maximal compactly embedded abelian subalgebras of g, then there is an inner automorphism ϕ of g such that ϕ(t1 ) = t2 . (iii) If t is a compactly embedded Cartan subalgebra of g then there is a compactly embedded subalgebra k of g containing t and k is unique modulo the coreductive radical ncored (g), that is, if k1 is a maximal compactly embedded subalgebra containing t, then there is an x ∈ ncored (g) such that ead x k1 = k. In particular, if g is reductive, then k is unique. Moreover, there is an inner automorphism ϕ of g such that ϕ(k1 ) = k and ϕ(t) = t. Proof. Part (ii) is Theorem 12.24. So we have to prove (i). By Proposition 12.13, for r = r(g), the quotients (kj + r)/r, j = 1, 2 are compactly embedded in g/r. Hence each of kj , j = 1, 2 is contained in some cj ∈ C. By Lemma 12.23, there is an inner automorphism of g mapping c1 onto c2 . Thus as far as conjugacy of k1 and k2 under inner automorphisms is concerned, we may just as well assume that g ∈ C, that is, g/r(g) is procompact. Let s ∼ = g/r(g) be a Levi summand. By Theorem 12.12 (E), s is compactly embedded and thus is contained in a maximal compactly embedded subalgebra k in view of Theorem 12.14. By Lemma 12.25, k = m ⊕ s where m is a maximal compactly embedded subalgebra of the prosolvable subalgebra z(s, g). If we can show that k1 is conjugate to k, then the same will hold for k2 showing that k1 and k2 are conjugate. Thus there is no loss in assuming k2 = k = m ⊕ s, s = k , and k1 ⊆ s, and g = r(g) + s. Let a1 and a be maximal compactly embedded abelian subalgebras contained in k1 and k, respectively. Then a1 and a are conjugate by Theorem 7.24. Thus we may actually assume without losing generality that a1 = a, that is, that a is a maximal compactly embedded abelian subalgebra which is contained both in k1 and k. Since k = m ⊕ s and m is central in k we have m ⊆ a. Also, s ∼ = g/r(g) = (k+r(g))/r(g) ∼ k/(k∩r(g)). Thus k∩r(g) = r(k) = m ⊆ a∩r(g) ⊆ k∩(g). = Therefore m = a ∩ r(g). Then k1 /m ∼ = k1 and k/m ∼ = k = s. So m = r(k1 ) = z(k1 ) and m = r(k) = z(k) since k1 and k are procompact. Thus both k1 and k are contained in z(m, g) and so we may assume g = z(m, g), that is, that m ⊆ z(g). By Theorem 7.77 (iii) the semisimple closed subalgebra k1 is contained in some Levi complements s1 . By Theorem 7.77, there is a ψ ∈ Inn(g) such that ψ(k1 ) ⊆ s. Since m is central we have ψ(m) = m. Thus we may assume that k1 ⊆ k. But then k1 = m ⊕ k1 ⊆ m ⊕ k = k. This final reduction completes the proof of part (i) of the theorem. (iii) Let t be a maximal compactly embedded subalgebra of g. By 12.15, there is at least one maximal compactly embedded subalgebra k of g containing t. Assume

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that k and k1 are compactly embedded subalgebras containing t. Let r = r(g) be an abbreviation for the radical of g. Since k = k˙ (= [k, k]) is semisimple, it is contained in a Levi summand s of g by Theorem 7.77 (iii). Since a maximal compactly embedded subalgebra of s is contained in a maximal compactly embedded subalgebra, after conjugation if necessary, this may be considered to be k, and thus we may assume that k is maximal compactly embedded in s. Then k is of the form k = z(k) ⊕ k . The projection of z(k) along r into s commutes elementwise with k and thus has to be singleton, as k is maximal compactly embedded in s. Therefore z(k) ⊆ r. Now z(k) ⊆ t and t = z(k) ⊕ a with a suitable abelian subalgebra a of k . Since t is maximal compactly embedded abelian in g and therefore also in k, it follows that a is maximal compactly embedded abelian in k . But then k1 , containing t = z(k) ⊕ a is of the form k1 = z(k) ⊕ k1 , where a ⊆ k1 , and a is maximal compactly embedded abelian in k1 . From Proposition 12.21 (iii) we know that the projection of k1 into s along r agrees with k . Thus r ⊕ k = r ⊕ k1 , and we may assume that s = k and g = r ⊕ k . Since both k and k1 are contained in z(z(k), g), we may replace g by z(m, g) and assume thereafter that z(k) = z(g). Since z(g) ⊆ ncored (g) we may concentrate on ncored (g) ⊕k and simplify notation further by assuming that r(g) = ncored (g), that is g = ncored (g) ⊕s, k = z(g) ⊕ s, k1 = z(g) ⊕ s1 , t = z(g) ⊕ a ⊆ k ∩ k1 .

s = k , s1 = k1 ,

Now by Theorem 7.77 (i) there is an element x ∈ ncored (g) such that ead x s1 = s. Thus ead x k1 = k. By Part (ii) of the theorem applied to s and by Proposition 12.20 (iii) there is an inner automorphism ψ of g restricting to an inner automorphism ψ|s of s such def

that ψ  ead x (a) = a. Setting ϕ = ψ  ead x completes the proof of Assertion (iii). Theorem 12.27 (iii) cannot be much improved, even in the case of finite-dimensional Lie groups. This is illustrated by the following example. Example 12.28. (a) We let g be the semidirect product of euclidean 3-space R3 and the rotation algebra so(3): def

g = R3 ⊕ so(3),

[(v, X), (w, Y )] = (X(w) − Y (v), [X, Y ]).

Equivalently, we might consider ⎧ ⎛ ⎞⎫ x ⎬ ⎨, X v g= : X ∈ so(3), v = ⎝y ⎠ . ⎩ 0 0 ⎭ z The group belonging to this Lie algebra is ⎧ ⎛ ⎞⎫ x ⎬ ⎨, R v G= : R ∈ SO(3), v = ⎝y ⎠ . ⎩ 0 1 ⎭ z

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Let us consider S = (v, 0) and T = (0, Y ). Then [S, T ] = (−Y (v), 0) and [S, [S, Accordingly, ead S T = T + [S, T ] = (−Y (v), Y ). Now take T ]] = (0, 0).0 −1  0 0 v = 0 , Y0 = 1 0 0 . Set k = {0} × so(3) and t = {0} × R · Y0 ⊆ k. Then 1

0 0 0

Y0 (v) = 0 and thus ead S t = t. If we set k1 = ead S k = {(−Y (v), Y ) : Y ∈ so(3)}, then k and k1 are maximal compactly embedded subalgebras of g and t = k ∩ k1 is a maximal compactly embedded abelian subalgebra of g. (b) Let g be sl(2, C) considered as a 6-dimensional real Lie algebra. Then  0  def is a maximal compactly embedded abelian subalgebra, and its cent = Ri · 01 −1  0   0  . For each t ∈ R, set θ (t) = ad 0t −t . Then eθ (t) su(2) tralizer z(t, g) is C · 01 −1 is a maximal compactly embedded subalgebra of g containing t for each t, providing many different maximal compactly embedded subalgebras of g containing t. For more precise information see Appendix 3, Proposition A3.9. Exercise E12.6. Prove the following consequences of Theorem 12.27. Corollary. If k is any compactly embedded subalgebra, respectively, an abelian compactly embedded subalgebra, and m is a maximal compactly embedded subalgebra, respectively, a maximal compactly embedded abelian subalgebra of a pro-Lie algebra g, then there is an inner automorphism ϕ such that ϕ(k) ⊆ m. In order to derive the next corollary, we note the following lemma. Lemma 12.29. Let g be a pro-Lie algebra, let g˙ = [g, g], and let m be a vector subspace of g. Then (∀x ∈ g) g˙ + ead x m = g˙ + m. ' & Proof. Let x ∈ g and y ∈ m. Then ead x y = y + [x, y] + 2!1 x, [x, y] + · · · ⊆ y + g˙ . Thus e± ad x m ⊆ g˙ + m and so also m ⊆ e∓ ad x (˙g + m) = g˙ + e∓ ad x m. The asserted equality of the ideals g˙ + m and g˙ + ead x m follows. Corollary 12.30. If m is a maximal compactly embedded subalgebra, respectively, def maximal compactly embedded abelian subalgebra, then gc = g˙ + m is an ideal of g which is independent of the choice of m and is invariant under all automorphisms of g. Proof. Let m1 and m2 be maximal compactly embedded subalgebras. By Theorem 12.27 we have ead x m1 = m2 for some x ∈ g. By Lemma 12.29, this implies that g˙ + m2 = g˙ + ead x m1 = g˙ + m1 . (∗) def

Set gc = g˙ + m = ncored (g) +m where m is any maximal compactly embedded subalgebra of g. If ϕ ∈ Aut(g), then ϕ(˙g) = g˙ and ϕ(m) is a maximal compactly embedded subalgebra. By (∗) this implies ϕ(gc ) = gc . Since the vector space gc contains [g, g], it is an ideal.

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We observe that this corollary is particularly significant for prosolvable Lie algebras g. It provides a characteristic ideal g˙ + m that contains ncored (g) = g˙ and is often properly contained in g. Corollary 12.31. Let  be the set of pairs (a, k) where k is a maximal compactly embedded subalgebra of a pro-Lie algebra g and a is a maximal compactly embedded abelian subalgebra of g contained in k. Let Inn(g) act on  via ϕ ·(a, k) = (ϕ(a), ϕ(k)). Then the action is transitive. Proof. Let (a1 , k1 ), (a2 , k2 ) ∈ . Then by Theorem 12.27, there is a ϕ ∈ Inn(g) such that ϕ(k1 ) = k2 . Then ϕ(a1 ) and a2 are maximal compactly embedded abelian subalgebras of k2 . Thus by Theorem 12.27 again, there is a ψ ∈ Inn(k) such that ψ(ϕ(a1 )) = a2 . By Proposition 12.20 (iii), we may consider ψ as an inner automorphism of g leaving k fixed as a whole. Then (ψϕ) · (a1 , k1 ) = (a2 , k2 ). We now sharpen a proposition we proved in Exercise E12.4 following Proposition 12.16. Proposition 12.32. Let g be a pro-Lie algebra. If k is a maximal compactly embedded subalgebra, and t is a maximal compactly embedded abelian subalgebra of k, then t is a maximal compactly embedded abelian subalgebra of g. Proof. If t ⊆ a where a is a maximal compactly embedded abelian subalgebra of g according to Theorem 12.15, then a is contained in some maximal compactly embedded subalgebra k1 of g again by Theorem 12.15. Then (a, k1 ) ∈  according to Corollary 12.30. There is an inner automorphism ϕ of g such that ϕ(k) = k1 by Theorem 12.27. Then ϕ(t) is a maximal compactly embedded abelian subalgebra of ϕ(k) = k1 . By Part (ii) of the proposition in Exercise E12.4, a is maximal compactly embedded abelian in k1 . By assumption on t, we know that ϕ(t) is maximal compactly embedded abelian in k1 . So there is an inner automorphism ψ of g inducing an inner automorphism of k1 such that ψ(a) = ϕ(t). But then as an image of a under an inner automorphism of g, the subalgebra ϕ(t) is maximal compactly embedded abelian in g and then, by the same token, t is maximal compactly embedded in g as was asserted. Lemma 12.33. For a closed subalgebra h of a pro-Lie algebra g, the following conditions are equivalent: (i) (ii) (iii) (iv) (v) (vi)

(∀x ∈ g) [x, h] ⊆ h, that is, h is an ideal of g. h is invariant under Inn(g). (∀x ∈ g) ead x h ⊆ h. (∀x ∈ g, j ∈  (g)) eadg/j (x+j) (h + j)/j ⊆ (h + j)/j. (∀x ∈ g, j ∈  (g)) [(x + j), (h + j)/j] ⊆ (h + j)/j. (∀x ∈ g, j ∈  (g)) [x, h] ⊆ h + j

Proof. (i) ⇒ (ii): By (i), h is invariant under ad x for all x, hence under all ead x and thus under all finite products of these.

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That (ii) ⇒ (iii) is trivial, and (iii) implies (iv) by passing to the quotient g/j. Since g/j is a finite-dimensional Lie algebra, (iv) ⇔ (v). The  implication (v) ⇒ (vi) is immediate. But condition (vi) implies (∀x ∈ g) [x, h] ⊆ j∈ (g) (h + j) = h, and this proves (i). Exercise E12.7. Prove the following Lemma. Let C be a connected topological group and N a compact connected abelian normal subgroup. Then N is central. [Hint. The (topological) automorphism group Aut(N ) of a compact connected abelian group N is totally disconnected: see for instance [102, Proposition 9.85]. If N is a normal subgroup of a connected topological group C, then g  → Ig : C → Aut(N ), Ig x = gxg −1 , is continuous and therefore, having a connected domain and totally disconnected range, is constant. Hence N is central in C.] Recall that for any subalgebra a of a Lie algebra b we write a for the algebraic commutator algebra [a, a], the linear span of all brackets [X, Y ], X, Y ∈ a. The Largest Compactly Embedded Ideal of a Pro-Lie Algebra Corollary 12.34. Let g be a pro-Lie algebra and r = r(g) its radical. Then (i) there is a unique largest compactly embedded ideal m(g), and (ii) there is a unique largest compactly embedded abelian ideal, namely, the center z(m(g)) = z(g). (iii) m(g) ∩ r(g) = z(g) = z(m(g)) and m(g) = m(g) ∩ s for each Levi summand s of g. Further m(g), z(g), and m(g) are invariant under all automorphisms of g. Proof. (i) We let K denote the set of all maximal compactly embedded subalgebras. def  K. Then m(g) is a compactly embedded subalgebra which is invariant Set m(g) = under all automorphisms of g. This applies to all inner automorphisms of the form ead X , X ∈ g. Hence by Lemma 12.31, m(g) is an ideal. If k is a compactly embedded ideal, then by Theorem 12.15, there is a maximal compactly embedded subalgebra c containing k. If ϕ ∈ Inn(g) then by Lemma 12.31 we have  k = ϕ(k) ⊆ ϕ(c). By Theorem 12.27 we have K = {ϕ(c) : ϕ ∈ Inn(g)}. Thus k ⊆ ϕ∈Inn(g) ϕ(c) = m(g). So m(g) is the unique maximal compactly embedded ideal. (ii) A completely analogous procedure applies to the unique largest compactly embedded abelian ideal a(g). The relation a(g) ⊆ m(g) is trivial. Then a(g) ⊆ r(m(g)) = z(m(k)). Since z(m(k)) is a compactly embedded abelian ideal, being invariant under inner automorphisms of g, we have z(m(k)) ⊆ a(g). Thus a(g) is the center z(m(g)) of m(g). def If j ∈  (g) then a∗ = (a(g) + j)/j is a compactly embedded ideal of the finite-didef

mensional Lie algebra g∗ = g/j. Then ead a is a compact connected abelian normal ∗

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subgroup of the inner automorphism group of g∗ and thus also of its closure in Aut(g∗ ). Since Aut(g∗ ) is a Lie group, so is this closure, and by Exercise E12.7 above, a compact connected abelian normal subgroup in a connected Lie group  is central. It follows that a∗ is central in g∗ . Thus (∀x ∈ g) [x, a(g)] ∈ j. Since  (g) = {0} it follows that (∀x ∈ g) [x, a(g)] = {0}, that is, a(g) is central. (iii) In (ii) we saw z(m(g)) = z(g), and by the definition of the radical as the unique largest prosolvable ideal, we clearly have z(g) ⊆ r(g). Thus z(g) ⊆ m(g) ∩ r(g). For the reverse containment we observe firstly that m(g) ∩ r(g) is a compactly embedded ideal since m(g) is compactly embedded and the intersection of two ideals is an ideal. Since r(g) is prosolvable, so is m(g) ∩ r(g). Thus m(g) ∩ r(g) ⊆ r(m(g)). As the pro-Lie algebra m(g) is compactly embedded in g, it is, in particular, procompact. Hence r(m(g)) = z(m(g)) by Theorem 12.12 (A). Thus m(g) ∩ r(g) ⊆ z(m(g)) = z(g) follows. Since m(g) is a procompact Lie algebra, by Theorem 12.12 (A)(v), the algebraic commutator algebra m(g) is a closed characteristic subalgebra and by Theorem 7.77 (ii) is the unique Levi summand of m(g). Since m(g) is a characteristic ideal of m(g) it is a semisimple ideal of g and thus is contained in every Levi summand s of g by Theorem 7.77 (A)(iv). Therefore, m(g) ⊆ m(g) ∩ s. For proving the reverse containment, let s be a Levi summand of g. Then m(g)∩s is a semisimple subalgebra of m(g) and thus is contained in the unique Levi summand m(g) of m(g) by Theorem 7.77. Thus m(g) ∩ s ⊆ m(g) follows. In the following corollary we keep the notation of Theorem 12.34. By Theo rem 12.12 (v), the compactly embedded ideal m(g) is of the form m(g) = z(g) ⊕ m(g) ,  ∼ where m(g) = j ∈J sj for a family of compact simple Lie algebras sj . Corollary 12.35. The quotient algebra g/m(g) has no nondegenerate compactly embedded semisimple ideal. Proof. Let s be a closed ideal of g containing m(g) such that s/m(g) is a compactly embedded semisimple ideal of g/m(g). Then z(g) is the radical of s, and thus by Theorem 7.77 (ii), s = z(g) ⊕ s ∗ with a unique Levi summand s∗ such that s∗ = m(g) ⊕ s# where s# ∼ = s/m(g). This quotient, being compactly embedded in g/m(g) is a procompact semisimple pro-Lie algebra, and thus there is a family of compact simple Lie # # algebras sj with j ∈ J # with  an index set J disjoint from J∗ such that s is isomorphic  ∗ ∼ to j ∈J # sj . Then s = j ∈J ∪J # sj ; and this shows that s = s is a procompact ideal of g. Hence s is a compactly embedded ideal of g by Theorem 12.12 (E). The maximality of m(g) then shows s∗ = s  ⊆ m(g) which entails s ⊆ m(g) and so s = m(g). This corollary is a little asymmetric, but this is in the nature of things: In the threedimensional Heisenberg algebra g, the center z(g) is the unique maximal compactly embedded ideal m(g), whereas the quotient algebra g/m(g) is abelian, thus compact. So it cannot be asserted that g/m(g) does not contain nondegenerate compactly embedded ideals.

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Compact Connected Groups In the following theorem one of four equivalent statements uses the concept of a projective compact connected group for which we refer to [102], Definition 9.75 and Theorem 9.76, and Theorem 8.78 through Proposition 8.81. Procompact Lie algebras were introduced in Definitions 12.1 (iii) and 12.9. For simply connected compact groups we refer the reader to Theorem 9.29 in [102]. Group Theoretical Characterisation of Procompact Pro-Lie Algebras Theorem 12.36. For a pro-Lie algebra g the following statements are equivalent. (i) g is a procompact Lie algebra. (ii) There is a simply connected topological group G of the form RI × S for some set I and a simply connected compact group S such that g ∼ = L(G). (iii) There is a compact connected group G such that L(G) ∼ = g. (iv) There is  a unique projective compact connected group G such that L(G) ∼ = g. (v) (g) = i∈I Si for a family of Lie groups Si each of which is either isomorphic to R or else is a compact simply connected Lie group. In these circumstances, if L((g)) is identified with g as is possible by Theorem 6.4, then exp(g) g = (g). If H is a pro-Lie group with Lie algebra h containing g, then expH g is an analytic subgroup of H , indeed the minimal analytic subgroup with Lie algebra g. Proof. (i) ⇔ (ii): A simply connected compact group S is isomorphic to a product  j ∈J Sj of simply connected simple compact groups: see [102, Theorem 9.29]. If s is a simple compact Lie algebra, then there is a unique simply connected simple compact Lie group Ssuch that s = L(S). (See [102, Theorem 6.6].) Thus (ii) holds iff g∼ = RJ × j ∈J sj for a family of simple compact Lie algebras sj . By 6.35 this is equivalent to (i).  (ii) ⇔ (v): Since a simply connected compact group is isomorphic to a product j ∈J Sj of a family of simply connected compact Lie groups, the asserted equivalence is immediate from the properties of the functor . (See Theorem 6.6 (vi) and Theorem 8.15.) (ii) ⇒ (iii): The groups RJ and RJ /ZJ = TJ have the same Lie algebra RJ . Then G = TJ × S is a compact group with L(G) ∼ = g. (iii) ⇔ (iv): The implication (iv) ⇒ (iii) is trivial; the implication (iii) ⇒ (iv) follows from [102, Theorem 9.76 (i) and (iii)]. (iv) ⇒ (i): ByProposition 9.74 of [102], a projective compact connected group is of the form C × j ∈J Sj with simply connected simple compact Lie groups Sj and a compact connected abelian group C with character group of the form Q(J ) (see [102,   R) ∼ Theorem A1.42]). Then L(C) = Hom(R, C) ∼ = Hom(C, = Hom(Q(J ) , R) ∼ = J J ∼ ∼ Hom(Q, R) = R (see [102, Proposition 7.40]). Thus g = L(C × j ∈J Sj ) ∼ =  RJ × j ∈J sj for simple compact Lie algebras sj . By 6.35 this is equivalent to (i).

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 By (v) we may write (g) = i∈I Si for a family of Lie groups Si each of which satisfies expSi L(Sj ) = Sj (see [102, Theorem 6.30 (26)]). The functor L preserves products, and since the exponential function is calculated componentwise, it follows that exp(g) g = (g). Now let H be a pro-Lie group with Lie algebra h such that g ⊆ h. Then by Theorem 6.5 there is a morphism f : (g) → H such that the morphism of pro-Lie algebras L(f ) : L((g)) → h maps L((g)) onto g. Thus by definition def

G = f ((g)) is the minimal analytic subgroup A(g; H ) of H with Lie algebra g (see Definition 9.9), and G = f ((g)) = f (exp(g) L((g)) = expH L(f )(L((g))) = expH g. This permits us to formulate a new (and at this stage shorter) proof of one of the core structure theorems on compact groups. The proof here is more algebraic than the one given in [102] for Theorem 9.19 and for Theorem 9.23. Corollary 12.37 (The Structure Theorem of Semisimple Compact Connected Groups and the Levi–Mal’cev Structure Theorem for Connected Compact Groups). Each compact connected group G is a quotient group  modulo a central totally disconnected compact subgroup of the product Z0 (G) × j ∈J Sj , where the Sj are simply connected simple Lie groups and where Z0 (G) is the identity component of the center of G. Proof. Let G be a compact group. From the preceding Theorem 12.17 we know that  sj . L(G) = g ∼ = z(g) ⊕ [g, g] ∼ = RJ × j ∈J

Now expG z(g) = Z0 (G) and G = expG [g, g]. Hence G = Z0 (G)G and Z0 (G) ∩ G modulo a totally G is totally disconnected. Hence G is a quotient of Z0 (G) ×   disconnected central subgroup. It remains to show that L(G ) = j ∈J sj implies that  G is a quotient of the group j ∈J Sj modulo a central subgroup. The details are left as an exercise (Exercise E12.8) Exercise E12.8. Work out the details of the proof of Corollary 12.37. [Hint. Consult [102], from Theorem 9.2 through Corollary 9.25.] The second major structure theorem on compact connected groups is the Borel– Scheerer–Hofmann Splitting Theorem of Compact Connected Groups (see [102, Theorem 9.39]). We record that it is a consequence of the structure theory of pro-Lie groups as well. It follows immediately from Theorem 10.58. Recalling that the algebraic commutator group G of a compact connected group is closed ([102, Theorem 9.2]) we see that it is also a consequence of our results in Chapters 10 and 11 as we note now. Corollary 12.38. Let G be a compact connected group. Then there is a closed connected abelian subgroup A of G such that (g, a)  → ga : G ι A → G, is an isomorphism of compact groups.

ι(a)(g) = aga −1 ,

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Proof. The Lie algebra L(G ) = g is of bounded type (see Definition 10.51). Hence the conclusion of Theorem 11.8 applies and proves the assertion.

Compact Subgroups We are now beginning to look for maximal compact subgroups of a pro-Lie group, if there are any. Remember, the additive group of p-adic rationals (see Example 1.20 (A)(i)) is a nondiscrete locally compact but noncompact abelian group which is a union of an ascending chain of compact (open) subgroups; thus there is no maximal compact subgroup in such a pro-Lie group. Of course, there are simple discrete abelian  ∞ 1 examples: The groups Z(p∞ ) = n=1 p n Z /Z and n=1 Z/mn Z (for a family of positive integers mn ) are countably infinite torsion groups which are the union of ascending towers of finite groups. It is therefore not a priori clear whether, for instance, connected pro-Lie groups have maximal compact subgroups at all. def

Lemma 12.39. Let G be a pro-Lie group and H a compact subgroup. Then h = L(H ) def is compactly embedded in g = L(G). def

Proof. Let j ∈  (g); then V = g/j is a finite-dimensional G-module under the adjoint action (see Chapter 2, Definition 2.27 through Proposition 2.30, and Chapter 8, Proposition 8.1), that is, g · (x + j) = Ad(g)(x) + j. Let πV : G → Gl(V ) be the corresponding representation. Since H is a compact subgroup, by Weyl’s Unitary Trick (see for instance [102, Theorem 2.10]) we have an inner product on V making V into a real Hilbert space such that π(H ) ⊆ O(V ). We have a morphism of pro-Lie algebras L(π) : L(G) = g → L(Gl(V )) = gl(V ) and L(π )(h) = L(π )(L(H )) ⊆ so(V ). Thus V is a compact h-module in the sense of Definition 12.1 (ii). This shows that gad is a procompact h-module. Then by Definition 12.10, h is a compactly embedded subalgebra of g. We have used on earlier occasions the adjoint representation AdG : G → Aut(g) as it was introduced in Chapter 2 (see 2.27–2.30) and in Chapter 8 (see 8.1). This allows us to give more details of the situation described in Lemma 12.39. Lemma 12.40. Let G be a pro-Lie group with Lie algebra g and H an analytic subgroup def with Lie algebra h. Assume that K = Ad(H ) is compact in Aut(g). Then h is compactly embedded. Proof. g is a K-module for the compact group K. Hence g is a procompact L(K)module. We have ad h ⊆ L(H ) since t  → et·ad x : R → K is a one parameter subgroup of K for each x ∈ h. Hence h is a compactly embedded subalgebra of g. Proposition 12.41. Let G be a pro-Lie group and H a compact subgroup with Lie def algebra h = L(H ). Then eadg h = Ad(H ) ⊆ Aut(g) is a compact subgroup and h is compactly embedded.

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Proof. From [102, Theorem 9.60] we know that expG h = expH h is the arc component Ha of the identity in H and is therefore a group. By 8.1(1) we have eadg h = AdG (expG h) ⊆ AdG (H ). By Proposition 2.28, AdG (H ) is a compact subgroup of Aut(g) in the topology of pointwise convergence which acts jointly continuously on g, that is, (h, X)  → Ad(h)(X) : H × g → g is continuous. By [102, Theorem 9.60 and its proof], the image expG h = expH h is Ha = H0  Aa where A is an abelian subgroup isomorphic to H0 /H0 ; but Aa is dense in A by [102, Theorem 7.71]. Hence Ha is dense in H . Thus eadg h = Ad(H ) is a compact subgroup of Aut(g), contained, of course, in Ad(G).

Exercise E12.9. Use 12.41 to provide a second proof of Lemma 12.39, saying that h is a compactly embedded subalgebra of g. def

Now we assume that G is a pro-Lie group with Lie algebra g = L(G). If h is a compactly generated subalgebra of g, then the adjoint module gad is a procompact h-module V by assumption. By Theorem 12.8 on the Compact Group Associated with a Procompact Module, there is a compact connected group GV ⊆ Aut(V ) and a morphism of Lie algebras λ : h → L(GV ) such that GV = expGV λ(h) and (∀x ∈ h, y ∈ g) [x, y] = limh→0,h =0 h1 ((expGV λ(h · x))(y) − y). We continue to denote the adjoint representation of G on g by Ad : G → Aut(g). Proposition 12.42. Let G be a pro-Lie group and h a compactly embedded subalgebra of g = L(G). Assume that H is an analytic subgroup of G with Lie algebra h, and denote by V the weakly complete topological vector space g considered as a procompact h-module under the adjoint action. Then Ad(H ) ⊆ GV ⊆ Ad(g) ⊆ Aut(g), and Ad(H ) = Ad(H ). In particular, Ad(H ) is a compact group and agrees with ead h . Proof. If x ∈ h, then Ad(expG x) = eadg x ∈ Ad(G) so that ead x y = y + [x, y] + 1/2! · (ad x)2 (y) + · · · ∈ g by the definition of ead x . On the other hand, we have (expGV λ(x))(y) = y + [x, y] + 1/2! · (ad x)2 (y) + · · · ∈ g by Theorem 12.8 (iii). Thus Ad(expG x) = expGV λ(x), that is, e{•}  ad = Ad  expG = expGV λ. Since expH h is dense in H and thus in H , we get Ad(H ) ⊆ Ad(H ) ⊆ expGV λ(h) = GV by Theorem 12.8 (i). Since GV is compact, Ad(H ) is compact. It remains to show that ead h = Ad(H ). Since h is compactly embedded and so is procompact in its own right, by Theorem 12.36, exp h is the minimal analytic subgroup of G with Lie algebra h, and since H is an analytic subgroup with Lie algebra h, we have exp h ⊆ H and ead h = Ad(exp h) ⊆ Ad(H ). We notice H = exp h (see Proposition 9.11), whence ead h = Ad(exp h) = Ad(exp h) = Ad(H ) = Ad(H ). Now the proposition is proved. Corollary 12.43. Let h be a compactly embedded subalgebra of g = L(G) for some pro-Lie group G. Let H be any analytic subgroup with L(H ) = h. Then L(H ) is a compactly embedded subalgebra.

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Proof. This is a consequence of 12.40 and 12.42. Corollary 12.44. Let h be a maximal compactly embedded subalgebra of the Lie algebra g of a pro-Lie group G. Then for any analytic subgroup H of G with L(H ) = h one has L(H ) = h. In other words, among the analytic subgroups with Lie algebra h there is a closed one which is the unique largest one. Proof. This is an immediate consequence of 12.43. Theorem 12.45. Let h be a closed subalgebra of the Lie algebra g of a pro-Lie group G. Then the following statements are equivalent: (i) h is compactly embedded into g. (ii) Ad(exp h) = ead h is a compact subgroup of Aut(g). (iii) Ad(exp h) = ead h  is a compact subset of Aut(g). Proof. (i) ⇒ (ii): This follows from 12.36 and 12.42. (ii) ⇒ (iii): Since ead h is a subgroup of Aut(g) by (ii) we have ead h  = ead h and thus (iii) follows from (ii). (iii) ⇒ (i): This follows from Lemma 12.40 applied to H = A(h; G).

Potentially Compact Pro-Lie Groups Definition 12.46. (i) A connected pro-Lie group G is called potentially compact, if its Lie algebra g is a procompact pro-Lie algebra. (ii) A subgroup H of a pro-Lie group G is called compactly embedded, if Ad(H ) is compact in Aut(g) (with respect to the topology of pointwise convergence). Corollary 12.47. Let H be an analytic subgroup of a pro-Lie group G and h the Lie algebra of H inside the Lie algebra g of G. Then the following conditions are equivalent: (i) H is compactly embedded in G. (ii) h is compactly embedded in g. Proof. (i) ⇒ (ii): We know H ⊆ exp h from Chapter 9 and Ad(exp h) = ead h from Chapter 2. Hence Ad(H ) ⊆ ead h  ⊆ Ad(H ), that is Ad(H ) = ead h .

(∗)

Now Ad(H ) is compact by (i) and Definition 12.46. Thus (ii) follows from (∗) and Theorem 12.45. (ii) ⇒ (i): This is a consequence of Theorem 12.45 and (∗).

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For an analytic subgroup H of G therefore to be compactly embedded into G is the same as for its Lie algebra h to be compactly embedded into g. The Structure Theorem of Procompact Lie Algebras 12.12 and the Vector Group Splitting Lemma 5.12 for Connected Abelian Pro-Lie Groups permit us to determine the structure of potentially compact connected pro-Lie groups. Characterisation of Potentially Compact Connected Pro-Lie Groups Theorem 12.48. Let G be a connected pro-Lie group. Then the following conditions are equivalent. (i) G is potentially compact. (ii) G contains a closed weakly complete central vector subgroup V and a maximal compact subgroup C which is characteristic, such that the function μ : V × C → G, μ(v, c) = vc is an isomorphism of topological groups. (iii) There is a morphism f : G → K into a compact group with dense image such that L(f ) : L(G) → L(K) is an isomorphism. If these conditions are satisfied, then G = Z0 (G)G , where Z0 (G), the identity component of the center is isomorphic to V × comp(Z0 (G)), and where the algebraic commutator group G is a semisimple compact connected characteristic subgroup. Proof. (i) ⇒ (ii): By Theorem 12.12 g = z(g) ⊕ g and g is a compact semisimple pro-Lie algebra, indeed a product of finite-dimensional simple ones, while z(g) is a weakly complete vector group. Thus g is reductive. By Theorem 10.25, the radical R(G) of G satisfies L((R(G))) = r(g) = z(g) and thus R(G) = Z(G)0 by Proposition 9.23. The group (g ) is simply connected compact by Theorem 12.36 (v), and thus the minimal analytic subgroup A(g , G) of G with Lie algebra g , as a homomorphic image of (g ) by Proposition 9.10 is a compact subgroup. By 9.10, (g ) may be considered as a closed subgroup of (g) and indeed all of (g ) consists of commutators by Gotô’s Commutator Subgroup Theorem for Compact Connected Groups (see [102, Theorem 9.2]). Accordingly, A(g , G) = exp g consists of commutators. By Theorem 9.26, A(g ) = A(g ) = G ⊇ G ⊇ A(g ), and thus the algebraic commutator subgroup G is compact and has the Lie algebra g . Thus G = Z(G)0 G (see Theorem 10.48). By the Vector Group Splitting Lemma 5.12 for Connected Abelian Pro-Lie Groups we know that Z(G)0 = V comp(Z(G)0 ) for a suitable vector group complement V ∼ = RJ for some set J , and (v, z)  → vz : V × comp(Z(G)0 ) → Z(G)0 is an isomorphism of topological groups. The compact subgroup comp(Z(G)0 )G is contained in comp(G), and V comp(Z(G)0 )G = Z(G)0 G = G. Thus comp(G) = (V ∩ comp(G)) comp(Z(G)0 )G ; since V does not contain nonsingleton compact subgroups we have V ∩ comp(G) = {1}, and so comp(G) = Z(G)0 G and this is the unique maximal compact characteristic subgroup C. It follows that μ : V × C → G, μ(v, c) = vc, is a bijective morphism of compact groups. We

Potentially Compact Pro-Lie Groups ∼ =

523 ∼ =

notice that the sequence of morphisms G → G/G → Z(G)0 /(G ∩ Z(G)0 ) → V × comp(Z(G)0 )/G ∩ Z(G)0 → V yields the continuity of the projection of G onto V , and thus μ is an isomorphism. (ii) ⇒ (iii): Let G = RJ × C with a compact connected group C according  be the Pontryagin to (ii) and consider the discrete topology on Q. Let i : R → Q ) is an dual morphism of the inclusion map Q → R. Then L(i) : R = L(R) → L(Q isomorphism (see [102, Chapters 7 and 8, Example 8.31]). Since L preserves products, i J × idC : G → QJ × C is a dense morphism such that L(i J × idC ) = L(i)J × idL(C) is an isomorphism of pro-Lie algebras. (iii) ⇒ (i): By (iii), there is a compact group K with L(G) ∼ = L(K), and so g = L(G) is procompact by 12.36. Thus the equivalence of (i), (ii) and (iii) is established, and the proof yielded, in addition the information stated in the last paragraph of the theorem. Exercise E12.10. Verify the details of the proof of the last paragraph of Theorem 12.48. For instance, if g is the Heisenberg algebra spanned by X, Y , and [X, Y ] with [X, Y ] being central, and if G = (h, ∗) for X ∗ Y = X + Y + 21 [X, Y ], then the subgroup def

N = (R · X + R · [X, Y ], ∗) is potentially compact by Theorem 12.36 as L(N ) is abelian. But N is not compactly embedded in G. Recall that a subgroup of a topological group is called fully characteristic if it is mapped into itself by each continuous endomorphism. Corollary 12.49. Each potentially compact pro-Lie group G contains a unique maximal compact subgroup C and C is fully characteristic. Proof. By Theorem 12.48, comp(G) (see Definition 5.4) is a subgroup which is closed and characteristic. Corollary 12.50. For a connected pro-Lie group G the following conditions are equivalent: (i) G is potentially compact. (ii) The Bohr compactification morphism γG : G → α(G) is injective. Proof. Exercise. Exercise E12.11. Prove Corollary 12.50. [Hint. (i) ⇒ (ii): By Theorem 12.48 condition (i) implies that there is an injective morphism from G into a compact group. The universal property of α(G) yields the injectivity of γG . (ii) ⇒ (i): This is a consequence of Theorem 12.48.] Proposition 12.51. If H is a compactly embedded connected subgroup of a pro-Lie group G, then H is a compactly embedded hence procompact connected pro-Lie group, in particular, a closed analytic subgroup. In particular, any maximal compactly embedded connected subgroup is closed.

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Proof. Let H be a compactly embedded connected subgroup of G. Then Ad(H ) is compact by Definition 12.46 (ii). Now Ad(H ) = Ad(H ), and so H is compactly embedded into G by Definition 12.47 (ii) and is a pro-Lie group by the Closed Subgroup Theorem 3.34 and is, in particular, an analytic subgroup of G. Now L(H ) is compactly embedded into g by Corollary 12.47. Then h is a procompact pro-Lie algebra by a comment following Definition 12.10. Hence H is a potentially compact pro-Lie group by Definition 12.46 (i). The remainder is straightforward. Proposition 12.52. Let G be a connected pro-Lie group and h a closed subalgebra of g = L(G). Then h is maximal compactly embedded in g, respectively, maximal compactly embedded abelian in g if and only if there is a maximal compactly embedded connected subgroup H of G, respectively, a maximal compactly embedded connected abelian subgroup H of G such that L(H ) = h. Proof. Assume that h is maximal compactly embedded. Then by Corollary 12.44, there is a closed connected subgroup H with L(H ) = h. By Corollary 12.47, H is compactly embedded in G. We claim that H is maximal compactly embedded connected in G. Indeed, if H ⊆ K and K is compactly embedded, then K is compactly embedded connected as well and we may now assume that K is compactly embedded connected closed. It is then a pro-Lie group and thus a closed analytic subgroup. Then its Lie algebra L(K) is compactly embedded by Corollary 12.47. Since h ⊆ L(K) and h is maximal, we conclude h = L(K). Then K = exp L(k) = exp h = H . This proves the claim. Conversely assume that h = L(H ) for some maximal compactly embedded closed connected subgroup H of G. Then H is connected and compactly embedded by Definition 12.46. Now H is a closed connected pro-Lie and thus analytic subgroup of G. By the maximality of H we conclude H = H , that is, H is closed. Then h is compactly embedded by Corollary 12.47; we claim that h is maximal with respect to this property. Let k be a compactly embedded closed subalgebra containing h. Define K = A(k, G). Corollary 12.47 shows that K is compactly embedded. Then K is a compactly embedded closed analytic subgroup of G and then h ⊆ k = L(K) ⊆ L(K). Now H = exp h ⊆ exp L(K) = K. The maximality of H now shows H = K and thus h ⊆ L(K) ⊆ L(K) = h and therefore h = k showing the asserted maximality of h. The abelian case is completely analogous.

The Conjugacy of Maximal Compact Connected Subgroups We proved the Conjugacy Theorem for maximal compactly embedded subalgebras (12.27). Thus the group G of inner automorphisms of g acts transitively on the set of maximal compactly embedded subalgebras of g.

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Maximal Compactly Embedded Connected Subgroups: Existence and Conjugacy Theorem 12.53. Each connected pro-Lie group G contains maximal compactly embedded connected subgroups, respectively, maximal compactly embedded connected abelian subgroups, and these are conjugate in G under inner automorphisms. Proof. Existence. By Theorem 12.15, g has maximal compactly embedded subalgebras. Then by Corollary 12.47, G has maximal compactly embedded connected subgroups. Conjugacy. Let K1 and K2 be two compactly embedded connected subgroups of G. def

Then by Proposition 12.51, K1 and K2 are closed analytic subgroups and k1 = L(K1 ) and k2 = L(K2 ) are maximal compactly embedded subalgebras by Proposition 12.52. Hence there is an inner automorphism ϕ ∈ Inn(g) such that ϕ(k1 ) = k2 by Theorem 12. By Definition 9.3 there are elements x1 , . . . , xn ∈ g such that ϕ = ead x1 . . . ead xn . We set gk = expG xk , k = 1, . . . , n. Define Igk ∈ Inn(G), the group of inner automorphisms of G by Igk (h) = gk hgk−1 . Define g = g1 . . . g2 . Then Ig = Ig1 . . . Ign . Further, Igk (expG x) = expG (Ad(gk )x) = expG (Ad(expG xk )x) = expG (ead xk x). It follows that Ig (expG x) = expG (ϕ(x)). Therefore Ig (K1 ) = Ig (expG k1 ) = expG k2  = K2 . This proves the conjugacy of maximal compactly embedded connected subgroups of G. The abelian case is completely analogous.

Maximal Compact Connected Subgroups: Existence and Conjugacy Corollary 12.54. Each connected pro-Lie group G contains maximal compact connected subgroups, respectively, maximal compact connected abelian subgroups. Maximal compact connected subgroups, respectively, maximal compact connected abelian subgroups, are conjugate in G under inner automorphisms. More specifically, every compact connected, respectively, compact connected abelian subgroup is contained in a maximal compact connected, respectively, maximal compact connected abelian subgroup. Proof. Step 1. If K be a compact connected subgroup of G, then there is a maximal compactly embedded connected subgroup H of G such that K ⊆ comp(H ). Since the subgroup K is compact, it is closed, as it is connected it is a closed analytic subgroup with a Lie algebra k. Then k is a compactly embedded closed subalgebra of g by Corollary 12.47. By Theorem 12.15, there is a maximal compactly embedded subalgebra h of g containing k. Then h = L(H ) for some maximal compactly embedded connected subgroup H of G by 12.52. Then K = exp k ⊆ exp h = H . Now H is potentially compact, and thus H has a unique largest compact subgroup comp(H ) of H (see Theorem 12.48). Then K ⊆ comp(H ). Step 2. Let K1 and K2 be two compact connected subgroups and assume that K2 is maximal compact connected. Then there is a g ∈ G such that Ig (K1 ) ⊆ K2 , where Ig (x) = gxg −1 , and K2 = comp(H2 ) for a maximal compactly embedded

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connected subgroup H2 of G. By Step 1 we find two maximal compactly embedded connected subgroups H1 and H2 such that Kj ⊆ comp(Hj ), j = 1, 2. Since K2 is maximal, K2 = comp(H2 ). By Theorem 12.53, there is a g ∈ G such that Ig (H1 ) = H2 . Since comp(Hj ) is a fully characteristic subgroup of Hj we conclude Ig (K1 ) ⊆ Ig (comp(H1 )) = comp(H2 ) = K2 . Step 3. For a maximal compactly embedded connected subgroup H , the subgroup comp(H ) is maximal compact connected in G. Let K be a compact connected subgroup of G containing comp(H ). Then by Step 1 there is a maximal compactly embedded subgroup H2 of G such that K ⊆ comp(H2 ). Then by Step 2, we find a g ∈ G such that comp(H2 ) = Ig (comp(H )). Then comp(H ) ⊆ K ⊆ comp(H2 ) = Ig (comp(H )). Now let N ∈ N (G). Then comp(H )N/N ⊆ IgN comp(H )N/N for a compact connected Lie subgroup comp(H )N/N of the Lie group G/N and an automorphism IgN of G/N. This automorphism induces an injective endomorphism of L(comp(H )N/N ) which is an automorphism since an injective endomorphism of a finite-dimensional vector space is an automorphism. Hence comp(H )N = g comp(G)g −1 N . Taking intersections over all N ∈  (G) yields comp(H ) = g comp(G)g −1 and thus K = comp(H ). Thus comp(H ) is maximal as asserted. Now the existence of maximal compact connected normal subgroups is guaranteed by Step 3, and their conjugacy by Step 2. The abelian case follows analogously. Later we shall prove that maximal compact subgroups exist and that they are connected (see Theorem 12.77 below). Exercise E12.12. Formulate an alternative proof for the abelian version of Corollary 12.54, using the information that all maximal connected abelian subgroups of a compact group are conjugate under inner automorphisms. [Hint. [102, Theorem 9.32]. Every compact connected abelian subgroup T is contained in some maximal compact connected group K. If T1 and T2 are maximal compact connected abelian groups, conjugate T1 into a maximal compact connected group K2 containing T2 . Then apply the theorem cited.] Lemma 12.55. Every totally disconnected normal subgroup of a connected group is central. Proof. Let N be a totally disconnected normal subgroup of a connected topological group G, then for each n ∈ N, the continuous function x  → xnx −1 n−1 : G → N from a connected to a totally disconnected space is necessarily constant, and it maps 1 to 1. This means that N ⊆ Z(G). (See e.g. [102, Lemma 6.13 and Proposition A4.27]).

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The Largest Compactly Embedded Connected Normal Subgroup of a Pro-Lie Group Theorem 12.56. Let G be a connected pro-Lie group and R(G) its radical. Let Z(G)0 = V C denote the direct product decomposition of the identity component of the center into a vector group factor V and the maximal compact subgroup C = comp(Z(G)0 ) according to the Vector Group Splitting Lemma 5.12 for Connected Abelian Pro-Lie Groups. Then (i) there is a unique maximal compactly embedded connected normal subgroup MaxCE(G), and (ii) L(MaxCE(G)) = m(g), the algebraic commutator subgroup MaxCE(G) is compact and agrees with exp m(g) , and MaxCE(G) = Z(G)0 exp m(g) . There is a quotient morphism V ×C×MaxCE(G) → MaxCE(G), given by (v, z, c)  → vzc whose kernel is isomorphic to C ∩ MaxCE(G) . (iii) (MaxCE(G) ∩ R(G))0 = Z(G)0 = Z(MaxCE(G))0 and MaxCE(G) = (MaxCE(g) ∩ S)0 for each Levi factor S of G. (iv) There is a unique maximal compact connected normal subgroup MaxC(G) and MaxC(G) = comp(MaxCE(G)) = C MaxCE(G) . (v) The factor group G/ MaxC(G) has no nontrivial compact connected normal subgroups. Proof. (i) Let G be a connected pro-Lie group and g its Lie algebra. Then by Coroldef

lary 12.34 there is a unique largest compactly embedded ideal m(g). Let MaxCE(G) = A(m(g), g) be the closure of the minimal analytic subgroup with Lie algebra m(g). By Proposition 9.20 (a), the normalizer N (A(m(g), G) is closed and has the Lie algebra n(m(g), g) = g. Thus this normalizer agrees with G and hence A(m(g), G) is normal in G. (This can also be concluded from Theorem 9.22.) But then its closure MaxCE(G) is normal as well. Since MaxCE(G) is a closed connected subgroup of G it is an analytic subgroup; its Lie algebra is an ideal by Theorem 9.22 (v) and is a closed ideal of g containing m(g). Now m(g) is compactly embedded and thus A(m(g), g) is compactly embedded by Corollary 12.47. Hence MaxCE(G) is compactly embedded by Proposition 12.51. Then L(MaxCE(G)) is compactly embedded in g by Corollary 12.47. Since m(g) ⊆ L(MaxCE(G)) and since m(g) is the largest compactly embedded ideal, we conclude m(g) = L(MaxCE(G)). From the fact that m(g) is the maximal compactly embedded ideal of g it follows straightforwardly that MaxCE(G) is maximal compactly embedded connected normal. (ii) The first assertion is implicit in the proof of (i). For the second we recall that m(g) = z(g) ⊕ m(g) , whence exp m(G) = (exp z(g))(exp m(g) ). Now exp z(g) = Z(G)0 by Proposition 9.23 (ii). Further, m(g) is a direct product of a family of finitedimensional simple compact Lie algebras by Theorem 12.12. Therefore (m(g) ) is a product of finite-dimensional, simple and simply connected compact Lie groups (by Theorem 12.36 (v)) and therefore is algebraically perfect, that is, agrees with

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its own algebraic commutator subgroup. Also, this product is compact and satisfies (m(g) ) = exp(m(g) ) m(g) . Hence the image of ηm(g) : (m(g) ) → expG m(g)  def

is K = expG m(g) . This group, therefore, is a compact connected algebraically perfect group, and is contained in the algebraic commutator subgroup MaxCE(G) . In any topological group, the product of a closed subset and a compact subset is closed. Therefore, exp m(g) = exp z(g) exp m(g) = Z(G)0 K ⊆ MaxCE(G). From L(MaxCE(G)) = m(g) we conclude equality, and therefore MaxCE(G) = Z(G)0 K. Hence MaxCE(G)/K is abelian, and therefore MaxCE(G) ⊆ K. It follows that K = MaxCE(G) , and the first part of assertion (ii) is proved. The function (v, c)  → vc : V × C → Z(G)0 is an isomorphism. It is clear that μ : V × C × K → MaxCE(G),

μ(v, c, d) = vcd

is a surjective morphism implementing on each factor an inclusion morphism. The compact subgroup V ∩ CK of the vector group V is degenerate. Hence v −1 = cd ∈ V ∩ CK only if v = 0 and c = d −1 . The kernel of μ therefore is the set {(0, d −1 , d) : d ∈ K ∩ C}. The function d  → (0, d −1 , d) : K ∩ C → ker μ is a continuous bijective morphism from a compact group onto ker μ and is, therefore an isomorphism of topological groups. Hence ker μ is compact and μ is a proper (perfect) morphism by Proposition 1.23 and is therefore a quotient morphism. (iii) In view of (ii) we clearly have Z(G)0 ⊆ MaxCE(G) ∩ R(G). By (ii) we know that MaxCE(G) = Z(G)0 K, K = exp m(g) . Let x = MaxCE(G) ∩ R(G). Then x = zk ∈ R(G) where z ∈ Z(G)0 and k ∈ K. Then k = z−1 x ∈ R(G) ∩ K ⊆ Z(K). The totally disconnected center of the characteristic subgroup K is normal in G and hence central in G by Lemma 12.55. So x = zk ∈ Z(G). It follows that Z(G)0 ⊆ MaxCE(G) ∩ R(G) ⊆ Z(G). By definition of the identity component in a topological group, Z(G)0 = (MaxCE(G) ∩ R(G))0 follows. From (ii) we know that MaxCE(G) = K and so L(MaxCE(G) ) = L(K) = m(g) . From Corollary 12.34 we recall m(g) = m(g) ∩ s for every Levi summand. The Levi factors of G are the analytic subgroups S satisfying L(S) = s for some Levi summand s. Accordingly K = MaxCE(G) = exp m(g) = exp(m(g) ∩ s) ⊆ exp m(g) ∩ exp s = MaxCE(G) ∩ S. We have K = (MaxCE(G) ∩ S)0 iff {1} = (MaxCE(G)/K ∩ S/K)0 . Thus it is no loss of generality to assume that K = {1} because otherwise we factor K and treat G/K. Now MaxCE(G) = Z(G)0 and assertion (ii) reduces to the claim that Z(G)0 ∩ S is totally disconnected  for all Levi factors S. The group (s) is a product j ∈J Sj of finite-dimensional simply connected simple Lie groups. Let is : (s) →  G be the morphism defined by s ⊆ L(G) (see Definition 9.9). Let Z((s)) = j ∈J Z(Sj ). Then is−1 (Z(G)0 ∩ S) = is−1 (Z(G)0 ) is a closed subgroup P ⊆ Z((s)) and is therefore a procountable group (see Definition 4.32) satisfying is (P ) = Z(G)0 ∩ S. By Lemma 4.33, Z(G)0 ∩ S is totally disconnected and this is what we had to show. (iv) From (ii) we easily see that comp(MaxCE(G)) = comp(Z(G)0 ) MaxCE(G) .

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We set MaxC(G) = comp(MaxCE(G)). As a characteristic subgroup of the normal subgroup MaxCE(G) this connected subgroup is normal in G. Every compact connected normal subgroup N is compactly embedded and thus is contained in MaxCE(G). Therefore, by the compactness of N we have N ⊆ comp(MaxCE(G)), and since N is connected we have N ⊆ MaxC(G). Hence MaxC(G) contains every compact connected normal subgroup of G. (v) Let K be a subgroup of G containing MaxC(G) such that K/ MaxC(G) is a compact connected normal subgroup of G/ MaxC(G). Then the subgroup K is normal in G; it is compact as a compact extension of a compact group, and it is connected as a connected extension of a connected group. But MaxC(G) is the unique maximal compact connected normal subgroup of G and therefore contains K. Thus K = MaxC(G) and K/ MaxC(G) is singleton. Corollary 12.57. Let G be a connected pro-Lie group. Then: (i) G contains a largest compactly embedded connected abelian normal subgroup which is central, namely, Z(MaxCE(G))0 = Z(G)0 , the identity component of the center of MaxCE(G) and of G itself. (ii) Similarly, G contains a unique largest compact connected abelian normal subgroup which is also central, namely, Z(MaxC(G)0 ) = comp(Z(MaxCE(G))0 ) = comp(Z(G)0 ). (iii) The factor group G/ comp(Z(G)0 ) has no nontrivial compact connected central subgroups. Proof. (i) Since Z(MaxCE(G)) is a characteristic closed connected subgroup of MaxCE(G), it is normal. If A is any compactly embedded connected abelian normal subgroup of G, then it is contained in MaxCE(G). But by Theorem 12.48, MaxCE(G) = Z(MaxCE(G))0 G , and the quotient morphism MaxCE(G) → G /(G ∩ Z(MaxCE(G))0 ) maps A onto a connected normal abelian subgroup of a compact semisimple group and thus to the singleton subgroup. This means A ⊆ Z(MaxCE(G))0 . Thus Z(MaxCE(G))0 is the unique largest compactly embedded connected normal subgroup of G. We have L(Z(MaxCE(G))0 ) = z(m(g)) by Proposition 9.23 and Theorem 12.50; likewise L(Z(G)) = z(g). Now z(m(g)) = z(g) by Corollary 12.34 (ii). Therefore Z(MaxCE(G))0 = exp z(m(g)) ⊆ exp z(g) = Z(G)0 . But z(g) is compactly embedded in g by 12.34 (ii) and then Z(G)0 is compactly embedded in G by Corollary 12.47. Hence Z(G)0 ⊆ Z(MaxCE(G))0 , and equality follows. (ii) The proof of (ii) and (iii) is Exercise 12.13 below. Exercise E12.13. Prove the following.

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Every connected pro-Lie group G contains a unique largest compact connected abelian normal subgroup which is also central, namely, Z0 (MaxC(G)) = comp(Z0 (MaxCE(G))) = comp(Z0 (G)). The factor group G/(comp(Z(G)0 )) contains no compact connected central subgroups other than the identity. [Hint. Verify that Z0 (comp(MaxCE(G))) = comp(Z0 (MaxCE(G))). Then use Z0 (MaxCE(G)) = Z0 (G) to have all equalities. Show that Z0 (MaxC(G)) is the unique largest compact connected abelian normal subgroup. For a proof of the second part follow the proof of 12.56 (v).] Recall the three-dimensional Heisenberg group N = (R · X + R · Y + R · Z, ∗), [X, Y ] = Z, x ∗ y = x + y + 21 · [x, y]. Let G = N/Z · Z. Then Z(G) = MaxC(G) = C(G) = R · Z/Z · Z ∼ = R/Z = T. But G/ MaxC(G) ∼ = R2 is abelian and therefore procompact; that is C(G/C(G)) = G/C(G) = R2 = {0}. Thus, in general, G/C(G) and G/ MaxC(G) may have compactly embedded normal subgroups. Set Gn = Sl(n, C), then Z(Gn ) = {e2π im/n En : m = 0, . . . , n − 1} ∼ = Z/nZ. def ∞ Therefore G = n=2 Gn is a pro-Lie ∞ group without nontrivial compact connected normal subgroup. But Z(G) ∼ = n=2 Z/nZ, and this is a nonsingleton compact normal subgroup. In other words, compact normal subgroups need not be trivial in pro-Lie groups without compact connected normal subgroups. However, we remind the reader that in Theorem 9.50 we proved that a connected pro-Lie group always contains a largest compact abelian and consequently central subgroup KZ(G), and G/ KZ(G) does not contain any compact central subgroups other than the identity. Accordingly, KZ(G)0 is the unique largest compact connected central subgroup. Therefore Corollary 12.57 (ii) recovers a fact we already knew since Chapter 9. This result, however, suffices to sharpen the “compact” portion of the Largest Compact Normal Connected Subgroup of a Pro-Lie Group Theorem 12.56. For easy reference we reformulate a definition that we mentioned already just before Theorem 9.50 and Lemma 11.25. Definition 12.58. A group G will be called compactly simple, if it is a topological group in which every compact normal subgroup is singleton. Largest Compact Normal Subgroup of a Pro-Lie Group Theorem 12.59. Every connected pro-Lie group G has a unique largest compact normal subgroup MaxK(G) and G/ MaxK(G) is a compactly simple connected proLie group. Proof. Let G be a connected pro-Lie group. By Theorem 12.56 (iv, v) there is a unique largest normal compact connected subgroup MaxC(G). By Theorem 4.28 (i) or (iii),

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def

H = G/ MaxC(G) is a pro-Lie group. Then by Theorem 9.50 H has a unique largest compact central subgroup KZ(H ) of H . We define MaxK(G) to be the full inverse image of KZ(H ) under the quotient morphism G → H and claim that MaxK(G) is the desired largest compact normal subgroup of G. Firstly, as MaxC(G) and MaxK(G)/ MaxC(G) are compact, MaxK(G) is compact. Secondly, let N be any compact normal subgroup of G; we claim that N ⊆ MaxK(G). The groups MaxC(G) and N are compact normal subgroups, and so def

M = N · MaxC(G) is a compact normal subgroup containing MaxC(G). By Theorem 12.56 (v), the factor group H = G/ MaxC(G) has no compact connected normal subgroups. The identity component M0 of M contains the connected subgroup MaxC(G) and being characteristic, is normal in G; by the maximality property of MaxC(G) we have M0 ⊆ MaxC(G) and therefore M0 = MaxC(G). Hence M/ MaxC(G) is a totally disconnected normal subgroup of the connected topological group H and is therefore central in H by Lemma 12.55. By Theorem 4.28 (i) or (iii), H = G/ MaxC(G) is a pro-Lie group. Hence the compact central subgroup M/ MaxC(G) of the connected pro-Lie group H is contained in the unique largest compact central subgroup KZ(H ) of H according to Theorem 9.50 and therefore N ⊆ M ⊆ MaxK(G) as KZ(H ) = MaxK(G)/ MaxC(G). This proves our claim. Thirdly, let N be a subgroup of G containing MaxK(G) such that N/ MaxK(G) is def

a compact normal subgroup of K = G/ MaxK(G). Then the subgroup N is normal in G; it is compact as a compact extension of a compact group. But MaxK(G) is the unique maximal compact normal subgroup of G and therefore contains N. Thus N = MaxK(G) and N/ MaxK(G) is singleton. By Theorem 4.28 (iii), the factor group G/ MaxK(G) is a pro-Lie group since MaxK(G) is compact. According to this theorem, every connected pro-Lie group G is the extension of a compact group MaxK(G) by a connected compactly simple pro-Lie group. One would therefore like to learn as much as possible about the structure of connected compactly simple pro-Lie groups. In view of the Yamabe Theorem ([206], [207]), that every locally compact almost connected group is a pro-Lie group and all sufficiently small members of N (G) are compact for a locally compact pro-Lie group G we see that an  almost connected locally compact group is compactly simple only if it is a compactly simple Lie group. In Theorem 11.27 we saw that a pronilpotent connected pro-Lie group is compactly simple iff it is simply connected. This fails for prosolvable groups even in the case of Lie groups as the group of motions of the euclidean plain shows. By in Corollary 11.29 we saw that a prosolvable connected pro-Lie group has no nondegenerate compact subgroups iff it is simply connected.

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Recall from Chapter 10 that each connected pro-Lie group G has a unique largest connected closed connected pronilpotent normal subgroup N (G), its nilradical. The nilradical contains Z(G)0 , the identity component of the center of G. Let us summarize some of our recent results in the following form. Proposition 12.60. Let G be a connected pro-Lie group without nontrivial compact central subgroups. Then the following conclusions hold. (i) Every compact connected normal subgroup is semisimple and center-free. (ii) The center is an abelian pro-Lie group which is isomorphic to Z(G)0 ×H , where the identity component Z(G)0 of the center Z(G) is a vector group isomorphic to RI for some set I and where H is a totally disconnected subgroup of Z(G) which contains no compact subgroups. (iii) The nilradical N (G), that is, the largest pronilpotent connected closed normal subgroup, is simply connected and thus is isomorphic to (n(g), ∗) where n(g) is the nilradical of g = L(G). (iv) The radical R(G) is a semidirect product V  K of a simply connected normal subgroup V containing N (G) and a compact connected group. Proof. (i) The center of a compact normal subgroup is contained in KZ(G) and is, therefore trivial. The claim then follows from the Levi–Mal’cev Theorem for Compact Connected Groups 12.37. (ii) This was proved implicitly in Theorem 9.50 (ii). (iii) The group KZ(N (G)) is a characteristic subgroup of N (G) and is therefore normal in G and thus is contained in KZ(G). Hence (iii) is an immediate consequence of Theorem 11.27. (iv) follows from Theorem 11.28.

The Analytic Subgroups Having a Full Lie Algebra We have the ingredients for significant results on the unique minimal analytic subgroup A(g, G) = expG g with full Lie algebra g in any pro-Lie group G. These results allow generalisations that are relevant now in the context of dense analytic subgroups. Proposition 12.61. Let G be a connected prosolvable pro-Lie group such that G/ KZ(G) is simply connected. Then KZ(G)A(g, G) = G. def

Proof. From Theorem 9.50 (i) we know that H = G/ KZ(G) is compactly simple. Then Corollary 11.29 shows H = A(h, H ). If q : G → H is the quotient morphism, then q(A(g, G) = A(h, H ) = H by 9.10 (iii) and 4.22 (iii). Then it follows that KZ(G)A(g, G) = (ker q)A(g, G) = G.

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Notice that all simply connected pronilpotent pro-Lie groups satisfy the hypothesis of Proposition 12.61. For the result in 12.61 on connected prosolvable pro-Lie groups there is a parallel result on connected reductive pro-Lie groups. Proposition 12.62. Let G be a connected reductive pro-Lie group. Then KZ(G)A(g, G) = G. Proof. We may factor KZ(G) and assume that Z(G) is compactly simple and show that A(g, G) = G. We know that R(G) = Z(G)0 is a weakly complete vector group by the Vector Group Splitting Lemma 5.12. Thus R(G) = exp(r(g)) ⊆ A(g, G). The factor group H = G/R(G) is semisimple. Let N be the full inverse image of Z(H ) in G. Then N/R(G) is totally disconnected and central in H . Thus N is nilpotent of class ≤ 2 and [N, N] ⊆ R(G) ⊆ Z(G). We let K be the full inverse image in N of comp(Z(H )). Note that K/R(G) is compact by 9.50 and totally disconnected as a subgroup of N/R(G). Now we apply Lemma 11.22 with G in place of T , the radical R(G) in place of B, and K as is. The conclusion is that K ⊆ R(G) · comp(K). Now R(G) ∩ comp(K) = {1}, since R(G) is a vector group. Therefore K is algebraically the direct product of R(G) and comp(K). But comp(K) is characteristic in K, hence normal in G. As a totally disconnected subgroup of G, it is central. Since KZ(G) was assumed to be trivial, it follows that comp(K) is trivial and that therefore K = R(G); that is, KZ(H ) is singleton. Now Theorem 10.32 (vi)(c) applies and shows that A(h, H ) = H . But since R(G) ⊆ A(g, G) we have A(h, H ) = A(g, G)/R(G) and therefore A(g, G) = G as asserted. There is not too much leeway in combining the two propositions as the CenterFree Embedding Lemma 9.41 and its Corollary 9.42 show. Indeed, let us pursue this for an illustration of the present situation by looking at an example which was first d be the character group of the additive introduced after Example 9.42. So let K = Q group of rational numbers given the discrete topology, then 9.42 demonstrates the existence of a metabelian pro-Lie group CQ π K, where π(χ )((cq )q∈Q ) = (χ (q))q∈Q , χ ∈ K. This group is center-free; its nilradical N (G) is CQ × {0}, a weakly complete topological vector space which is certainly simply connected, and we have A(g, G) = d ) is identified with Hom(Qd , R) and expK with the morphism CQ ×expK R, where L(Q Hom(Qd , R) → Hom(Qd , R/Z) according to [102, Theorem 7.66]. Every morphism f : Qd → R is uniquely determined by a real number r so that f (q) = qr and, d is given by (expK f )(q) = qr + Z ∈ T = R/Z. In this accordingly, expK f ∈ Q def

fashion we identify L(K) with R. The quotient Q → Q/Z induces an embedding  =  Q Z(p ∞ (for q /Z = Hom((Q/Z)d , T) → Hom(Qd , T) = K. But Q/Z =  p prime 1 ∞ Z(p ) = p∞ · Z/Z, see [102, Definition A1.30 ff.]).So  = p prime Zp , where Zp is the additive group of p-adic integers, and  is in fact the zero-dimensional Bohr

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compactification of Z (see Chapter 1, Examples after Theorem 1.41). It is known that K =  expK R (see [102, Theorem 8.20]). Therefore, if we set D = {0} ×  ⊆ G, then G = DA(g, G) and in place of D we cannot have any central compact abelian group, because G has a trivial center. Of course, KZ(G) is central in both Proposition 12.61 and 12.62. Definition 12.63. Let G be a connected pro-Lie group, let q : G → G/R(G) denote def

the quotient map and define Q(G) = q −1 (KZ(G/R(G))). Call Q(G) the extended radical of G. Proposition 12.64. For a connected pro-Lie group G, the extended radical Q(G) satisfies the following conditions: (i) Q(G) is a prosolvable characteristic subgroup of G whose identity component is the radical R(G). (ii) Q(G) contains a closed subgroup V that is normal in Q(G), contains the nilradical N (G) of G and is simply connected modulo KZ(G)0 . (iii) Q(G) contains a maximal compact abelian subgroup K such that Q(G) = V K and V ∩ K = KZ(G)0 . (iv) K contains KZ(G) and is contained in a compact connected abelian subgroup C of G. (v) There is a totally disconnected compact subgroup D of K such that K = A(L(K), K)D and Q(R) = R(G)D. Proof. (i) Since R(G) is a characteristic subgroup of G and KZ(G/R(G)) is a characteristic subgroup of G/R(G), the subgroup Q(G) is characteristic in G. As R(G) is prosolvable and Q(G)/R(G) = KZ(S) is abelian, Q(G) is prosolvable. Since R(G) is connected and Q(G)/R(G) = KZ(G/R(G)) is totally disconnected, R(G) = Q(G)0 . It follows that Q(G) is an almost connected prosolvable pro-Lie group, and N(Q(G)0 ) = N (G). (ii) By the Structure Theorem on Almost Connected Prosolvable Pro-Lie Groups (11.28) we find a normal subgroup V of Q(G) containing N (G) such that V / KZ(G)0 is simply connected. (iii) Likewise, we find a compact maximal subgroup K of Q(G) such that Q(G) = V K and V ∩ K = KZ(R(G))0 . (iv) Since KZ(G)R(G)/R(G) ⊆ KZ(S) we have KZ(G) ⊆ Q(G), and so by the maximality of K we have KZ(G) ⊆ K. By Theorem 10.28, the factor group G/R(G) is a semisimple pro-Lie group. Now we apply 10.29 (iii)(f) to G/R(G) and find a compact connected abelian group B1 containing KZ(G/R(G)) whose full inverse image in G we call B; then B/R(G) = B1 . Now by 11.28 we find a maximal compact subgroup C of B which is connected and of course abelian. From 11.32 it follows that we may assume K ⊆ C. (v) From [102, Theorem 8.20 (i)] it follows that there is a totally disconnected compact subgroup of K such that K = A(L(K), K)D = K0 D (see also [102, Theorem 9.41]). Now, using (ii), we get G = V K ⊆ R(G)K = R(H )K0 D = R(G)D.

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Supplementing the Minimal Analytic Subgroup Generated by the Full Lie Algebra Theorem 12.65. Let G be a connected pro-Lie group and K a maximal compact subgroup of the extended radical Q(R). Then G = K · A(g, G). There is a totally disconnected compact subgroup D of K such that G = D · A(g, G) and Q(G) = R(G)D. Moreover, there is a compact connected abelian subgroup C of G containing K. def

def

Proof. Write S = G/R(G) and s = L(S). The quotient morphism q maps A(g, G) surjectively onto A(s, S) and S = KZ(S) · A(s, S) by Proposition 12.62. Therefore G = Q(G) · A(g, G).

(∗)

From (∗) and Q(G) = V K in 12.63 (iii) we obtain G = Q(G) · A(g, G) = V K · A(g, G) = KV · A(g, G).

(∗∗)

Now V / KZ(G)0 is simply connected solvable and KZ(G0 ) = KZ(V ). Then Proposition 12.61 implies V = KZ(G)0 · A(L(V ), V ). From (∗∗) we now get G = KV A(g, G) = K KZ(G)0 · A(L(V ), V )A(g, G) = K · A(g, G). Finally, let D be as in 12.64 (v). Then Q(G) = R(G)D and G = A(g, G) · K = A(g, G) · A(L(K), K)D = A(g, G) · D, and notice that 12.64 (iv) proves the last assertion This result has a number of immediate consequences. Corollary 12.66. Let G be a connected pro-Lie group. (i) The abstract group G/A(g, G) is abelian. (ii) The algebraic commutator group of G is contained in A(g, G). (iii) Algebraically, G is generated by divisible subgroups. (iv) G has no normal subgroups of finite index whatsoever. Proof. (i) There is an isomorphism of groups K/(K ∩ A(g, G)) ∼ = G/A(g, G), and K is abelian. (ii) is a consequence of (i). (iii) By 12.65 we have G = C · expG g, where C is a compact connected abelian group. Every such is divisible by [102, Corollary 8.5]. Every subgroup expG R · X, X ∈ g is divisible. Hence G is algebraically generated by divisible abelian subgroups. (iv) If f : G → F is an algebraic homomorphism into a finite group and D is any divisible subgroup, then f (D) = {1}. Therefore (iii) implies f (G) = {1} and the assertion follows.

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12 Compact Subgroups of Pro-Lie Groups

Conclusion (iv) was used in an essential fashion in the proof of the Open Mapping Theorem 9.60. Conclusion (i), for what it is worth, could have been sharpened to say that the abelian group G/A(g, G) is an algebraic homomorphic image of a compact totally disconnected abelian group. In fact, for what it is worth, the abelian group G/A(g, G) is an algebraic homomorphic image of a compact totally disconnected abelian group. For the next one we need a lemma by which a subgroup K of a connected pro-Lie  = (g). group G yields an automorphic action of K on G Since  : proLieGr → proSimpConLieGr is a functor we have a morphism of  given by α(g) = Ig where Ig (h) = ghg −1 . The pro-Lie groups α : G → Aut(G)  group G is defined as (g). Therefore we can give a more detailed discussion of α as follows. Recall the adjoint representation Ad : G → Aut(g) of G on g = L(G) discussed in 2.27ff. and used frequently in the entire Lie theory of pro-Lie groups. Since  : proLieAlg → proSimpConLieGr and L : proSimpConLieGr → proLieAlg are functors implementing an equivalence of categories, for each pro-Lie algebra g we have an isomorphism ιg : Aut(g) → Aut((g)), ιG (ϕ) = (ϕ) such that, if, as usual, g is identified with L((g)), we have L(ιG (ϕ)) = ϕ and ιG (L(λ)) = λ. Recall that X ∈ g is a one parameter subgroup X : R → G and that (Ad(g)X)(r) = gX(r)g −1 for g ∈ G, X ∈ g, r ∈ R. Thus we define  α : G → Aut(G),

α = ιG  Ad,

(L(α(g))X)(r) = gX(r)g −1 .

Then action (g, X)  → Ad(g)(X) : G × g → g is continuous by Proposition 2.28. The →G  is continuous next lemma states that in fact the action (g, x)  → α(g)(x) : G × G as well. Lemma 12.67. Let G be a pro-Lie group. Then there is an automorphic action →G  such that g·x = α(g)(x) and that πG (g·x) = gπG (x)g −1 , (g, x)  → g.x : G×G that is, the following diagram is commutative:  (g,x)→g·x G× ⏐ G −−−−−−→ ⏐ idG ×πG  G × G −−−−−−−→ (g,h) →ghg −1

 G ⏐ ⏐π  G G.

The following conditions hold: →G  is continuous. (i) The action (g, x)  → g · x : G × G  α G with multiplication (ii) There is a well defined semidirect product G (x1 , g1 )(x2 , g2 ) = (x1 (g1 · x2 ), g1 g2 )  α G → G given by δ(x, g) = πG (x)g, and there is a surjective morphism δ : G  whose kernel contains exactly the pairs (x −1 , πG (x)), x ∈ G.

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537

Proof. The existence of the action is clear from the definition of α. We shall call the →G  the lifted action of G on G.  action μG : G × G =G <0 we may assume that G is connected and we recall from Chapter 6, (i) Since G   → G/N  Let λN : G  be the limit morphisms. We shall that G = limN ∈N (G) G/N.   prove continuity of μG : G × G → G by proving continuity of λN  μK,N for all N ∈ N (G). We consider the commutative diagram  G× ⏐G ⏐ quot ×λN   G/N × ⏐ G/N ⏐ id ×πG/N  G/N × G/N

μG

−−−−→ −−−−→ μG/N

−−−−−−−−−−→

(kN,gN ) →kgk −1 N

 G ⏐ ⏐λ  N  G/N ⏐ ⏐πG/N  G/N.

 . If this action is continuous, then where μG/N is the lifted action of G/N on G/N λN  μG is continuous as desired. Thus Claim (ii) is true if it is true for Lie groups. So for the remainder of the proof of (i) we assume that G is a connected Lie group. In the commuting diagram (g,X) →Ad(g)(X)

G⏐ × g −−−−−−−−−→ ⏐ idG × expG˜   G×G −−−−→ μG

g ⏐ ⏐exp ˜  G  G,

 the top horizontal and the two vertical maps are continuous, and since G and thus G is a Lie group, the map expG  is a local homeomorphism at 1. Specifically, there is  such that an open neighborhood U of 0 in g and an open neighborhood V of 1 in G expG |U : U → V is a homeomorphism. Thus all maps in the following commutative  diagram are continuous: G× U ⏐ −1 idG ×(expG˜ |U ) ⏐ G×V

(g,X) →Ad(g)(X)

−−−−−−−−−→ −−−−−→

μG |(G×V )

g ⏐ ⏐exp ˜  G  G.

Thus μG is continuous at all points of {g} × V . Since μG is an automorphic action it  is connected, we have V  = G.  Thus is continuous at all points of {g} × V . As G  μG is continuous in all points (g, x) of G × G. This completes the proof of (ii).  α G is now clear, and the re(ii) The construction of the semidirect product G maining assertions on δ are straightforwardly verified. We now return to Theorem 12.65. By Lemma 12.67, we have a morphism of groups  such that πG (α(k)(x)) = kπG (x)k −1 , allowing us to introduce the α : K → Aut(G)  α K. semidirect product G Recall that we use the words quotient morphism and open surjective morphism synonymously. For pro-Lie groups, quotient morphisms with prodiscrete kernels are the next best thing to a covering morphism we can expect to have in this environment.

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The Resolution Theorem for Connected Pro-Lie Groups Corollary 12.68. Let G be a connected pro-Lie group. Then there is a totally disconnected compact subgroup D of the extended radical and a quotient morphism of pro-Lie groups  α D → G, δ(x, k) = πG (x)k. δ: G The morphism

−1 x  → (x −1 , πG (x)) : πG (D) → ker δ

is an isomorphism of prodiscrete groups. Proof. Let D be as in Theorem 12.65 and construct the semidirect product and δ as in Lemma 12.67. The surjectivity of δ then follows from Theorem 12.65. The assertion about ker δ follows straightforwardly from 12.67 and the fact that x  → (x −1 , πG (x)) is clearly invertible. Since D is a compact totally disconnected group and G is connected, the domain of δ is almost connected. Thus the Open Mapping Theorem 9.60 applies and shows that δ is a quotient morphism. We retain the notation of Theorem 12.65. Corollary 12.69. Let G be a connected pro-Lie group and let g = r(g) + s be a Levi–Mal’cev decomposition. Then G = Q(G) · A(s, G). Proof. We have Q(G) = KV = KA(L(V ), V ) ⊆ KA(r(g), R(G)) ⊆ Q(G). Note A(r(g), R(G)) = expR(G) r(g) = expG r(g) = A(r(g), G). The universal mor = (g) → G has the image A(g, G). From Theorem 8.15 and Theophism πG : G rem 6.11 we get (g) = (r(g))(s). (∗) Since A(r(g), G) = πG ((r(g))) and A(s, G) = πG ((s)) from (∗) we obtain A(g, G) = A(r(g), R(G))A(s, G).

(∗∗)

Thus Q(G) · A(s, G) = KA(r(g), R(G)) · A(s, G) = K · A(g, G) = G from Theorem 12.65. This proves the assertion. One might surmise that for a connected pro-Lie group G whose Lie algebra g = L(G) has a Levi–Mal’cev decomposition r(g) + s we have the relation G = R(G)A(s, G). However, this conjecture is false, as Example 11.12 (vi) shows. In fact, it is false for reductive groups and in the case that A(s, G) is closed and isomorphic to  S N where S = Sl(2, R). The following is the common generalisation of Propositions 12.61 and 12.62: Theorem 12.70. Let G be a connected pro-Lie group and let g = r(g) + s be a Levi decomposition of its Lie algebra. Assume the following hypotheses:

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539

(i) [r(g), s] = {0}, and (ii) R(G)/ KZ(G)0 is simply connected. Then G = KZ(G) · A(g, G). Proof. After factoring KZ(G) and renaming we may assume again that KZ(G) = {1} and show that G = A(g, G). From Theorem 12.65 we know G = D · A(g, G)

(∗)

for a totally disconnected compact subgroup D of the extended radical Q(G). Since R(G)/ KZ(G)0 is simply connected by (ii) and KZ(G)0 ⊆ K, we know that K ∩ R(G) = KZ(G)0 . Now D ∩ R(G) = {1} and so Q(G) = R(G)  D. Also, D is central in G modulo R(G), that is, if [G, D] is the closed subgroup generated by comm(G × D), then [G, D] ⊆ R(G).

(0)

We proceed in two steps. def

Step 1. Assume that R(G) is pronilpotent, that is, R(G) = N (G). Let Nk = N [[k]] . We set G1 = G/N1 . Then G1 is reductive since g1 = (n(g)/n1 )⊕(s+n1 )/n1 by (i). In particular N(G1 ) = Z(G1 )0 , and Q(G1 ) = N(G1 )  D1 , where D1 is the isomorphic copy of D in G1 , is in fact a direct product. We apply Lemma 11.22 with G1 in place of T , N(G1 ) = N0 /N1 in place of B and Q(G1 ) in place of K and get that Q(G1 ) is central in G1 . It follows that [G, D] ⊆ N1 . (1) Now let G2 = G/N2 . Then n1 /n2 is central in n(g)/n2 and n(g) commutes with s by (i). Hence n1 /n2 is central in g/n2 and thus N1 /N2 is central in G/N2 . Now we apply Lemma 11.22 with G2 in place of T , N1 /N2 in place of B and Q(G2 ) in place of K and get that Q(G2 ) is central in G2 . It follows that [G, D] ⊆ N2 .

(2)

We have clearly presented the first steps of an induction which shows that [G, D] ⊆ Nk (k)  for all k = 1, 2, . . . and thus [G, D] ⊆ ∞ k=1 Nk = {1}. Thus D is a compact central subgroup and is therefore contained in KZ(G). Since KZ(G) = {1}, we have G = A(g, G) by (∗) and Q(G) = R(G) in this case. ˙ the Step 2. Let R(G) be simply connected but otherwise arbitrary. Consider G, ˙ = (G ˙ ∩ closed commutator subgroup of G. By Proposition 10.45, we have R(G) R(G))0 = Ncored (G) ⊆ N (G). Since N(g) is simply connected, the closed connected characteristic subgroup Ncored (G) is simply connected as well and so Ncored (G) = ˙ is characteristic in G, ˙ it follows that A(ncored (g), G) ∼ = (ncored (g), ∗). Since Z(G)

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12 Compact Subgroups of Pro-Lie Groups

˙ ⊆ KZ(G) = {1}. Similarly KZ(R(G)) = {1}. Now by Step 1 we have KZ(G) ˙ = A(L(G), ˙ G). By Proposition 12.62 we know that R(G) = A(r(g), G) and G ˙ Therefore g = r(g) + L(G). ˙ G) = R(G)G. ˙ A(g, G) = A(r(g), G)A(L(G), In view of Theorem 12.65 we recognize ˙ = Q(G)G ˙ G = DA(g, G) = DR(G)G

(∗)

for a compact totally disconnected subgroup of the extended radical Q(G) = R(G)D. Since R(G) is simply connected, D ∩ R(G) = {1}. Thus Q(G) = R(G)  D. Define  = DNcored (G)/Ncored (G) to be the isomorphic image of D in the factor group Q(G)/Ncored (G). Then Q(G)/Ncored (G) = R(G)/Ncored (G)  ,

(∗∗)

and by (∗) there is a surjective morphism ˙ ϕ : Q(G)/Ncored (G) → G/G,

˙ ϕ(xNcored (x)) = G/G

˙ The automorphism Ad(g) of g leaves r(g) and of pro-Lie groups. Let g ∈ Q(G) ∩ G. s invariant, and since gR(G) is central in G/R(G) by the definition of Q(G) we know ˙ = R(G) ˙ by Step 1 above. that Ad(g) fixes s elementwise. This implies that g ∈ Q(G) ˙ = Q(G) ∩ G ˙ = ˙ = (G ˙ ∩ R(G))0 ⊆ R(G). We have shown (R(G)  D) ∩ G But R(G) ˙ ˙ ˙ ˙ 0, R(G)∩ G. Thus the kernel of ϕ is (R(G)∩ G)/Ncored (G) = (R(G)∩ G)/(R(G)∩ G) ˙ a prodiscrete group. Since G/G is abelian, so is the algebraically isomorphic group ˙ Q(G)/(R(G) ∩ G). Thus the adjoint representation of Q(G)/R(G) ∼ = D ∼ =  ∼ ˙ ˙ on L(Q(G)/(R(G) ∩ G)) = L(Q(G)/(R(G) ∩ G)0 ) is trivial. This implies that Q(G)/Ncored (G) is an abelian pro-Lie group, isomorphic to the direct product of the weakly complete vector group R(G)/Ncored (G) and the compact totally disconnected group D. The factor group Q(G)/Ncored (G) Q(G) ∼ = ˙ ˙ ˙ 0 R(G) ∩ G (R(G) ∩ G)/(R(G) ∩ G) is an abelian proto-Lie group, which is isomorphic to the direct product of the connected ˙ and the compact abelian group D. The abelian proto-Lie group R(G)/(R(G) ∩ G) surjective morphism ϕ induces a bijective morphism :

R(G) ˙ × D → G/G ˙ R(G) ∩ G

˙ (see Theorem 4.28 (i)). A subgroup P of an onto the abelian pro-Lie group G/G abelian group C is said to be pure, if the existence of an element c ∈ C and a natural number n such that cn ∈ P implies the existence of a p ∈ P such that p n = cn . (See for instance [102, Appendix 1, Definition A1.22.]) Since the direct factor {1} × D

The Analytic Subgroups Having a Full Lie Algebra

541

def

of the domain is a pure subgroup, its image P = ({1} × D) is a pure compact subgroup of the range, contained in the unique maximal compact connected subgroup def ˙ Since C is a direct factor by Lemma 5.12, it is a pure subgroup C = comp(G/G). ˙ of G/G, and so P is pure in C. As a compact connected abelian group, C is divisible  is torsion-free (see [102, Chapter 8, Corollary 8.5]). Thus P , being pure in a since C divisible abelian group is divisible itself. Hence, as a compact divisible abelian group, P is connected ([102, Corollary 8.5]). On the other hand, being isomorphic to the totally disconnected group D, the group P is also totally disconnected. Therefore, P is singleton. Hence D is singleton. Then (*) shows that G = A(g, G) and this is what had to be shown. The set-up of Theorem 12.70 allows an immediate application of our Open Mapping Theorem 9.60. Indeed, in the circumstances of Theorem 12.70, we get a surjective morphism f : KZ(G) × G → G, f (z, x) = zx.  is an almost connected pro-Lie group. Thus Since KZ(G)0 is compact, KZ(G) × G the Open Mapping Theorem 9.60 applies and proves Corollary 12.71. Under the hypotheses of Theorem 12.70, there is a quotient morphism  → G. f : KZ(G) × G It is easy to calculate its kernel and to improve it in one direction, because the compact abelian group KZ(G) has a totally disconnected compact subgroup D such that KZ(G) = D KZ(G)a for the identity arc component KZ(G)a ⊆ A(g, G) of KZ(G).  → G whose The restriction of f to D ×KZ(G) still yields a quotient morphism D × G kernel is isomorphic to D ∩ G, and this is a compact totally disconnected group. However, we wish to generalize the preceding results in such a fashion that an immediate application of the Open Mapping Theorem is no longer possible; it will be applied, however, in the course of the proof of the final result. The preceding results motivate the following definition: Definition 12.72. A connected pro-Lie group G will be called centrally supplemented if  G = Z(G)A(g, G) = Z(G)πG (G). From Theorem 12.70 we know that all connected pro-Lie groups satisfying the following hypotheses are centrally supplemented: (i) [r(g), s] = {0}, and (ii) R(G)/ KZ(G)0 is simply connected. This includes all reductive connected pro-Lie groups and all prosolvable pro-Lie groups which are simply connected modulo the maximal connected central compact subgroup. Example 9.42 gives us plenty center-free metabelian connected pro-Lie groups G in which A(g, G) is a proper subgroup and which, therefore, are not centrally supplemented.

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12 Compact Subgroups of Pro-Lie Groups

We shall now prove a Resolution Theorem for centrally supplemented groups. We prepare for its proof by stating and proving the following lemma: Lemma 12.73. Let G and H be topological groups, N ⊆ G and M ⊆ H be closed normal subgroups, and ϕ : G → H a morphism of topological groups, satisfying the following hypotheses: def

(i) ϕ|N : N → M = ϕ(N) is a quotient morphism, that is, an open map. (ii) The morphism  : G/N → H /M, (gN ) = ϕ(g)M, implements a local isomorphism. (iii) The quotient morphism q : G → G/N admits a local cross section. ϕ

−−−−→

G /   

G  H N → N

N

−−−−→

 .H   

ϕ|N

M.

Then ϕ is a quotient morphism. Proof. The surjectivity of ϕ|N and  implies the surjectivity of ϕ. We have to show that ϕ is open. Denote the quotient morphism G → G/N by p and let U be an open identity neighborhood of G/N and σ : U → G a continuous map such that p  σ : U → G/N is the inclusion map. Then the filter of identity neighborhoods of G has a basis consisting of neighborhoods of the form W σ (U  ) where W is an open identity neighborhood of N and U  is an open identity neighborhood of G/N contained in U . def Now let V = (U ) ⊆ H /M; since the restriction of  to a sufficiently small identity neighborhood of G is a homeomorphism onto its image, we may assume that |U : U → V is a homeomorphism. Define τ : V → H by τ = ϕ  σ  (|U )−1 and let q : H → H /M denote the quotient morphism. Then we have a commutative diagram U ⏐ ⏐ σ G ⏐ ⏐ p G/N

|U

−−−−→ H /M ⏐ ⏐τ  ϕ −−−−→ H ⏐ ⏐q  −−−−→ H /M 

such that the vertical compositions are the inclusion maps. Now we apply ϕ to one of the neighborhoods W σ (U  ) and obtain ϕ(W σ (U  )) = ϕ(W )ϕ(σ (U  )) = ϕ(W )τ ((U  )). Now by Hypothesis (i), ϕ(W ) is open in M; by Hypothesis (ii), (U  ) is open in H /M. Since τ is a local cross section, this implies that ϕ(W ) · τ ((U  )) is open in H . Thus ϕ maps arbitrarily small identity neighborhoods to identity neighborhoods and thus is open.

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Theorem 12.74 (The Resolution Theorem for Centrally Supplemented Pro-Lie Groups). Let G be a connected centrally supplemented pro-Lie group. Then there is a prodiscrete central subgroup D of G such that there is a quotient morphism  × D → G, δ: G

δ(x, d) = πG (x)d

inducing an isomorphism L(δ) of Lie algebras, and the kernel ker δ is prodiscrete and isomorphic to π −1 (D) ∩ A(g, G). Proof. First we apply the Resolution Theorem of Abelian Pro-Lie Groups (5.46) to the center Z = Z(G) of G. Writing Z(G) = V × H with a complete vector subgroup V and a complement H containing the compact subgroup comp(Z(G)) = KZ(G) (see Theorem 9.50). Now let D be a cotorus subgroup of H (see Definition 5.42). Then  × D → Z(G), δZ : Z

δZ (X, x) = (expG X)x,

is a quotient morphism inducing an isomorphism of Lie algebras and having the prodiscrete kernel {(x, πZ (x)−1 : πZ (x) ∈ D} ∼ = πZ−1 D ∼ = πZ−1 (D ∩ comp(Z0 )).  and the central subgroup D, commute elementwise, the As A(g, G) = πG (G) function δ is a morphism of topological groups, and the hypothesis of G to be centrally supplemented shows that G = DA(g, G) and thus that δ is surjective. The morphism −1 (D) → ker δ x  → (x −1 , πG (x)) : πG −1 is an isomorphism, causing the kernel of δ to be isomorphic to πG (D) = π −1 (D ∩ −1 im π) = πG (D ∩ A(g, G)). Our task is now to show that δ is an open map. We intend to apply Lemma 12.73 to  × D → G by δ(x, d) = do that. Abbreviate the center Z(G) of G by Z, define δ : G πG (x)d, and consider the diagram

×D G /    × D Z

δ

−−−−→  ˜ G×D →G Z ˜ Z×D δZ

−−−−→

 .G    Z.

 × D)) = where  is the induced morphism of the quotients given by ((x, d)(Z  is in the center of πG (G)  = πG (xd)Z. Notice that  is well defined since πG (Z)  ⊆ Z. Define ! : G/  Z  → G/Z, A(g, G), which is dense in G, whence πG (Z)  = πG (x)Z and α : G/  Z  → (G  × D)/(Z  × D), α(x Z)  = (x, 1)(Z  × D). !(x Z) Then α is an isomorphism, and we have a commutative diagram  Z  G/ ⏐ ⏐ α  × D G × D/Z

!

−−−−→ −−−−→ 

G/Z ⏐ ⏐id  G/Z G/Z.

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12 Compact Subgroups of Pro-Lie Groups

Now we claim that ! is an isomorphism. Once that is proved,  is an isomorphism and Lemma 12.72 applies to show that δ is a quotient map. Thus in order to complete the proof it remains to show that !:

 G G → ,  Z Z

 = πG (x)Z !(x Z)

is an isomorphism.  and thus ! is surjective. Firstly, since G is centrally supplemented, G = ZπG (G)  = 1, that is πG (x) ∈ Z; in other words, x ∈ π −1 (Z). Then Now assume !(x Z) G  × {x}) ∈ ker πG = P (G); but the Poincaré group ker πG of G is prodiscrete comm(G and thus is totally disconnected, while comm(G × {x}) is connected and contains 1.  = Z.  Thus ! is bijective. The Open Hence comm(G × {x}) = {1} and so x ∈ Z(G) Mapping Theorem for Connected Pro-Lie Groups 9.60 applies and proves this claim.

Maximal Compact Subgroups of Connected Pro-Lie Groups One of the applications we are making of (a special case of) the Resolution Theorems is that we can prove a fact on maximal compact subgroups of connected pro-Lie groups that is overdue. By Theorem 12.54 we know that maximal compact connected groups exist and are conjugate, and by Theorem 12.59 we know that a unique maximal compact normal subgroup exists. Yet up to this point we do not know whether maximal compact subgroups exist. In this section we clarify this question in the affirmative by even showing, that maximal compact subgroups are connected (and hence are conjugate by Corollary 12.54). We should begin with a warning. Most of the existence and conjugacy theorems for maximal compact connected subgroups or maximal compactly embedded connected subgroups have analogs for abelian subgroups. This is not the case for what we are proving in this section. Indeed, SO(3) contains the maximal abelian subgroup of 4 elements ⎧⎛ ⎫ ⎞ ⎨ d1 0 0 ⎬ ⎝ 0 d2 0 ⎠ : dn ∈ {1, −1}, d1 d2 d3 = 1 . ⎩ ⎭ 0 0 d3 Thus maximal compact abelian subgroups of Lie groups (and so a fortiori of pro-Lie groups) need not be connected. (See [102, Corollary 6.33 and Exercise E6.10 following Corollary 6.33].) We shall see that the problem of maximal compact subgroups in connected proLie groups reduces fairly quickly to connected compactly simple reductive groups. In order to understand, that even in the case of Lie groups there are some surprises, let us consider and keep in mind the following example

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545

 R) be the simply connected covering group of Sl(2, R) Example 12.75. Let S = Sl(2, def

and z one of the two generators of its cyclic center Z(S). Let  = {(−n, zn ) ∈ R × S : n ∈ Z} be a cyclic subgroup of the center R × Z(S) of R × S. We let X ∈ sl(2, R) 0 π and identify L(S) with sl(2, R) so that π be −π Sl(2,R) : S → Sl(2, R) induces the 0 identity on the Lie algebra level. Then exp SX ∈ {z, z−1 }, say expS X = z. Now we let def G = (R ×S)/ and set C = {(−r, expS r ·X) : r ∈ R} ⊆ G. Then C ∼ = R/Z and C is a maximal compact subgroup of G. The radical R(G) = (R ×Z(S))/ is isomorphic to R; the unique Levi complement (Z × S)/ is isomorphic to S, and R(G) ∩ S = (Z × Z(S))/ ∼ = Z while R(G) ∩ S ∩ C = {1}. We have L(G) = R × sl(2, R) and L(C) = R · (−1, X) is a subalgebra which is not an ideal. In particular, the maximal compact subgroups of G fail to meet the center and the Levi complement nontrivially. Lemma 12.76. Let G be a reductive connected pro-Lie group in which {1} is the only compact normal subgroup, and assume that C is a compact subgroup. Then there is a maximal compact subgroup C # that is connected and contains C. Proof. Let G be a reductive compactly simple connected pro-Lie group. Then  → G is surjective by 12.62 KZ(G) = {1} and therefore the universal morphism πG : G and then a quotient morphism by the Open Mapping Theorem 9.60. Recall that the  (G). Poincaré group P (G) of G is the prodiscrete kernel ker πG , and thus G ∼ = G/P  We claim that any compact normal subgroup N of the universal group G is totally disconnected central and contained in the Poincaré group P (G); indeed if N is any  then πG (N ) is a compact normal subgroup of G, since compact normal subgroup of G, πG is surjective. Since G is compactly simple, πG (N ) is singleton and thus N ⊆ P (G) and N is totally disconnected as claimed. Therefore, by Theorem 8.14, we may write   = RI × G Sj j ∈J

with a family of noncompact simple simply connected Lie groups Sj . For each j there is a maximal compactly embedded subgroup Kj of Sj = Kj Aj Nj (in terms of an Iwasawa decomposition of Sj ), and Kj is simply connected. Then Kj is the direct product of Z(Kj )0 ∼ = Rnj and the semisimple simply  connected commutator  subgroup  Kj . Since Z(Sj ) ⊆ Kj , we have P (G) ⊆ RI × j ∈J Z(Sj ) ⊆ RI × j ∈J Z(Kj ).   Since we have R I ∼ = j ∈J Z(Kj )0 for a suitable index set I  , we may define a connected subgroup      def = RI ∪I × Kj ⊇ RI ∪I × Z(Kj ) ⊇ P (G) K j ∈J

j ∈J

and a connected quotient group  ∼  (G), K ⊆ G. K = πG (K) = K/P     def = We claim that K is a pro-Lie group. Set B j ∈J Aj Nj . Then G = K B, and ×B  → G  is a homeomorphism. Therefore b  → the function (k, b)  → kb : K

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 → P (G)B/P  (G) is a homeomorphism, and so G = KB, B = πG (B),  P (G)b : B where (k, b)  → kb : K × B → G is a homeomorphism. By assumption G is a proLie group, and K is a closed subgroup as a continuous retract. Hence K is a complete topological group as a closed subgroup of a complete topological group and thus is a :K  → K of πG is the universal pro-Lie group as claimed. Now the restriction πG |K 0 ∼ morphism πK . Since G has no compact central subgroups, πG maps Z(G) = RI I ∼ isomorphically onto Z(G)0 = R and we observe that Z(G)0 ⊆ K From Theorem 5.19 we get Z(G) = Z(G)0 ×D with a prodiscrete central subgroup D, and since G is compactly simple, comp(D) = comp(Z(G)) = KZ(G) = {1}. So D is compact-free. The factor group G/D is a reductive group with a connected center isomorphic to Z(G)0 and thus is of the form  def Sj /Z(Sj ). G∗ = RI × j ∈J

Let p : G → G∗ be the quotient map. Then we have a sandwich situation p πG  −− → G −−→ G∗ , where p  πG = πG∗ . G Now let C be a compact subgroup of G; we claim that we find a compact connected subgroup C # of G containing C. Now p(C) is a compact subgroup of G∗ , projecting triviallyinto RI along the center-free complementary factor and therefore being contained in j ∈J Sj /Z(Sj ). Its projection p(G)j into Sj /Z(SJ ) is contained in a maximal compact subgroup; these are all conjugate and Kj /Z(Sj ) is one of them. We may just as well assume that the Kj were selected in such a way that p(G)j ⊆ Kj /Z(Sj ). It follows from this and the surjectivity  −1 −1 −1 −1  Therefore (C) = πG p (p(C)) = πG (p(C)) ⊆ RI × j ∈J Kj = K. of p that πG ∗ −1  K = πG (K) ⊇ πG (πG (C)) = C as πG is surjective and C is a compact subgroup of K. Now we have the following sandwich situation  p|K πK I  K −−−−→ K   −−−−→∗ R × j ∈J Kj /Z(Sj ) ⏐ ⏐ ⏐ incl⏐ incl⏐ ⏐incl πK−1 (C) −−−−→ ∗ πK |C

C

−−−−→ p|C

p(C),

in which all horizontal maps are quotient morphisms.  = RI ∪I  × K  . (Recall that Moreover, we know that C is compact in K and that K by Gotô’s Theorem the commutator subgroup of a compact connected group is closed; def  ) is a see for instance [102, Theorem 9.2ff.].) The homomorphic image K  = πK (K  compact connected normal subgroup of K, and so K C is a compact subgroup of K containing C. Hence there is no harm in assuming K  C = C, that is K  ⊆ C. Since K  is connected, we even have K  ⊆ C0 . This implies that C0 is normal in K and that K/C0 is abelian. By Theorem 4.28 (i), K/C0 is a pro-Lie group. By Lemma 5.12, the factor group K/C0 is isomorphic to RA × M for a cardinal A and a compact connected group M = comp(K/C0 ) and C/C0 ⊆ comp(K/C0 ) since comp(K/C0 ) is the unique largest compact subgroup of K/C0 . Let C # be the full inverse image of

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comp(K/C0 ) in K. Then C # is an extension of a compact connected group by a compact connected group and is therefore compact connected. Since C # /C0 = comp(K/C0 ) is maximal compact in K/C0 , the group C # is maximal compact in K. Since all compact subgroups, such as C are contained in K or a conjugate, C # is maximal compact in G. By what we just saw, C ⊆ C # , and that completes the proof. Connectedness of Maximal Compact Subgroups Theorem 12.77. Every compact subgroup of a pro-Lie group is contained in a maximal one. Each maximal compact subgroup is connected. Two maximal compact subgroups are conjugate. Proof. Let G be a connected pro-Lie group. By Theorem 12.59, G has a unique largest compact normal subgroup MaxK(G) so that G/ MaxK(G) has no nonsingleton compact normal subgroup. If C is any compact subgroup, then C MaxK(G) is a compact subgroup containing C. In looking for a maximal compact subgroup we may therefore consider only compact subgroups containing MaxK(G). Therefore it is no loss of generality to assume from here on out that {1} is the only compact normal subgroup. We have to show that G has maximal compact subgroups C and that C = C0 for all of these. The conjugacy then follows from Corollary 12.54. Since G is compactly simple, so is N (G), the nilradical. Then N (G) is simply connected by Proposition 12.60 (iii), and G/N (G) is reductive by Theorem 10.43. Now let C be a compact subgroup of G. We claim that C is maximal compact in G iff its homomorphic image N (G)C/N (G) is maximal compact in G/N (G). First let C be maximal compact in G. Let N (G)C ⊆ M ⊆ G be a subgroup such that M/N(G) is compact. Now by Theorem 11.17, there is a compact subgroup K of M such that M = N (G)  K. From Theorem 11.32 it follows that a conjugate of K contains C, and we may assume without loss of generality that C ⊆ K. But since C was maximal compact C = K and thus N (G)C/N (G) = M/N (G) which establishes that CN(G)/N(G) is maximal compact in G/N (G). But conversely, if CN (G)/N (G) is maximal compact in G/N (G), let M be a compact subgroup of G containing C. Then MN(G)/N(G) is a compact subgroup of G/N (G) containing CN (G)/N (G). By the maximality of the latter, we get CN (G) = MN (G). Since MN (G) is a semidirect product and C ⊂ M, we may conclude C = M, as desired. Thus G has maximal compact subgroups if and only if G/N (G) has maximal compact subgroups. Now C is connected if and only if N (G)C/N (G) is connected since N (G)C = N (G)  C. Thus it is no loss of generality if we now assume that G is reductive and show that G has maximal compact subgroups and these are connected. Every compact subgroup C of G may be enlarged by forming C MaxK(G) unless MaxK(G) ⊆ C. Thus there is no loss in generality to pass to the quotient group G/ MaxK(G) which is compactly simple. Thus we may and shall assume that G does not have any compact connected nontrivial normal subgroups. But then the preceding Lemma 12.76 showed that every compact subgroup of G is contained in a maximal compact connected subgroup.

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Exercise E12.14. Prove the following results: Theorem. Let G be a connected pro-Lie group. 1. G contains maximal compact abelian connected subgroups and any two of these are conjugate. 2. If g ∈ G and g is compact (see Theorem 5.3), then there is a compact connected abelian subgroup C of G containing g. 3. G = SO(3) contains a maximal compact abelian subgroup A isomorphic to Z(2)2 which, accordingly, is not contained in a connected abelian subgroup. [Hint. (i) Let M be a maximal compact subgroup of G, By 12.77, M is connected. M is the union of all of its maximal compact connected abelian subgroups T and all of these are conjugate in M (see for instance the Maximal Pro-Torus Theorem [102, 9.32]). (ii) Let g ∈ G be as in (ii); then there is a maximal compact subgroup M of G such that g ⊆ M by 12.77. Then by the Maximal Pro-Torus Theorem there is a maximal compact connected abelian subgroup T of M such that g ∈ T . (iii) Take ⎧⎛ ⎫ ⎞ ⎨ a 0 0 ⎬ ⎝0 b 0⎠ : a, b, c ∈ {1, −1}, abc = 1 . ⎩ ⎭ 0 0 c (See for instance [102, Corollary 6.33 and Exercise E6.10].)] What remains is the question how a maximal compact connected subgroup K of a connected pro-Lie group G determines the topological structure of G in terms of that of K which is very well known and documented. We begin with the case of a reductive, compactly simple group G and return to Lemma 12.76, its hypotheses and its notation. We would like to say more about the way that a maximal compact subgroup C is located in K and how K is located in G. Let us first reformulate Lemma 12.74 in a form that will recur several times. Lemma 12.78. Let G be a connected pro-Lie group satisfying the hypotheses of Lemma 12.76. Then there are a maximal compact subgroup C and simply connected pronilpotent subalgebras nk , k ≤ 3 such that (c, X1 , X2 , X3 )  → c expG X1 · expG X2 · expG X3 : C × n1 × n2 × n3 → G is a homeomorphism. def

Proof. We define n1 = L(V ). From Theorem 8.13 applied to B we know that N2 = def

N(B) and N3 = B/N2 give us pronilpotent simply connected pro-Lie groups, N3 in fact being abelian, such that (X2 , X3 )  → expB X2 · expB X3 : n2 × n3 → B is a homeomorphism. With this information we get the lemma from 12.76.

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Lemma 12.79. Let G be a pro-Lie group and K a closed normal subgroup such that G/K is simply connected pronilpotent. Then (i) g contains a vector subspace v such that (z, X)  → z expG X : K × v → G is a homeomorphism. (ii) If K is also compact, then the product KZ(G)0 expG v is a closed connected pronilpotent subgroup and is the nilradical N (G) of G. (iii) If G is a normal subgroup of H and C is a maximal compact subgroup of H such that H = CG and C ∩ G = K, then (c, X) → c expG X : C × v → H is a homeomorphism. Proof. (i) Every closed vector subspace of a weakly complete topological vector space is a direct summand (see Theorem A2.11 (i) in Appendix 2). Thus we can write g = L(K) ⊕ v. The pronilpotent simply connected group G/K is compactly simple (see Theorem 11.27). Therefore G/K ∼ = (g/L(K), ∗) by Theorem 8.5 and so G/K is strictly exponential. Then Assertion (i) follows from Proposition 8.10 and its proof. (ii) The hypotheses of the lemma yield additional information: Since G/K is connected, and K is compact, G0 K = G, and therefore G is almost connected. Since G/K is pronilpotent, all semisimple summands of g have to be contained in k and thus g = r(g) + k by the Levi–Mal’cev Theorem 7.52 (i). The analytic subgroup expG (r(g) + k) is dense in G0 and is contained in expG r(g)exp k = R(G)K0 and this is a closed group. So we get G0 = R(G)K, that is, G is almost prosolvable. By the Structure Theorem of Almost Connected Almost Prosolvable Pro-Lie Groups 11.28.B of E11.7, there is a maximal compact subgroup C, clearly containing the normal compact subgroup K, and a closed prosolvable subgroup V containing KZ(G)0 , def

def

def

such that G∗ = G/ KZ(G)0 = V ∗  C ∗ , V ∗ = V / KZ(G)0 , C ∗ = C/ KZ(G)0 . Since G/K is connected pronilpotent, so is G/K KZ(G) and V ∗ is a closed connected subgroup thereof. Hence V ∗ and then V are pronilpotent, and V is a connected normal subgroup of G and so is contained in the nilradical N (G). Therefore, V = N (G). Hence G = N(G)C. Now G/K = N (G)C/K is simply connected pronilpotent; this implies that C/K is singleton and thus C = K. So G = N (G)K, and G/(N (G) ∩ K) is the direct product of subgroups N (G)/(N (G) ∩ K) and K/(N (G) ∩ K). Since K is compact, n(N(G) ∩ K)  → nK : N ((N (G) ∩ K) → N (G)/K is an isomorphism and since N(G)/K is simply connected, N (G) ∩ K is connected and agrees with KZ(G)0 . Therefore, KZ(G)0 expG v = N (G). (iii) The function (c, X)  → c expG X : C × v → H is a homeomorphism since firstly, the function Kg  → Cg : G/K → H /C is a homeomorphism, so C expG X → expG X : H /C → H is a cross-section, since K expG X → expG X : G/K → G is a cross-section.

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This allows us to prove the following Induction Lemma. Lemma 12.80. Let G be a connected pro-Lie group, N a normal subgroup, and C a maximal compact subgroup of G. Assume that the following hypotheses are satisfied. Hypothesis 1. The pro-Lie algebra L(G/N ) contains p closed vector subspaces v∗k , def

k ≤ p such that, with G∗ = G/N, C ∗ = CN/N (c∗ , X1∗ , . . . , Xp )  → c∗ expG∗ X1∗ . . . expG∗ Xp∗ : C ∗ × v∗1 × · · · × vp ∗ → G∗ is a homeomorphism. def Hypothesis 2. K = KZ(N ) = KZ(G)0 = C ∩ N and N/(C ∩ N ) is connected pronilpotent. Hypothesis 3. Let q : G → G∗ denote the quotient map. Then q −1 KZ(G∗ ) expG v∗k is a connected pro-Lie subgroup for k = 1, . . . , p. Then we have the following Conclusion. There are vector subspaces vk ⊆ g = L(G), k = 0, 1, . . . , p, such that (c, X0 , X1 , . . . , Xp )  → c expG X1 . . . expG Xp : C × v0 × · · · × vp → G is a homeomorphism and KZ(G)0 expG vk is a pronilpotent pro-Lie subgroup for k = 0, 1, . . . , p Proof. Set N0 = N, and for k = 1, . . . , p define Nk = q −1 KZ(G∗ )0 expG∗ v∗k . Then by Hypotheses 2 and 3, Nk /N are simply connected pronilpotent pro-Lie groups for k = 0, . . . , p. Since K is a compact normal subgroup of G we have Nk ∩ C = K, and by Lemma 12.79, we find a vector subspace v0 of L(CN ) so that m : C × v0 → CN,

m(c, X0 ) = c expG X0 ,

(∗)

is a homeomorphism and that K expG v0 is a pronilpotent subgroup, and by assumption K = KZ(G) and so KZ(G)0 expG v0 is a pronilpotent subgroup. By Lemma 12.79 (iii) again we find vector subgroups vk of Nk , k = 1, . . . , p such that (n, Xk )  → n expG Xk : N × vk → Nk is a homeomorphism and N expG vk is a pro-Lie group. Now we note that CN/N = C expG v0 , expG∗ vk = (N expG vk )/N, whence G∗ = C ∗ expG∗ v∗1 . . . expG∗ v∗p implies G = CN · N expG v1 . . . N · expG vp = C expG v0 expG v1 . . . expG vp . Thus the function ε : C × v0 × v1 × · · · × vp → G, ε(c, X0 , X1 , . . . , Xp ) = c expG X0 expG X1 . . . expG Xp ,

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is surjective and we must show that it is a homeomorphism. We have ker q = N ⊆ CN. The function μ : A × B → G, μ(a, b) = ab, A = CN = C expG v0 , B = expG v1 . . . expG vk is surjective and the quotient map q : G → G/N = G∗ maps C to C/N = C ∗ under the quotient morphism, B to def

B ∗ = expG∗ v∗1 . . . expG∗ v∗p homeomorphically, and there is a commutative diagram C × v⏐0 × B ⏐ q|C×idv0 × idB  C ∗ × v0 × B ∗

−−−−→

m×idB

μ

m ×idB

μ

A −−−−→ ⏐×B ⏐ ⏐ ⏐ q|A × q|B −−∗−−→ A∗ × B ∗ −−−− → ∗

G ⏐ ⏐q  G∗ ,

where μ∗ , defined by μ∗ (a ∗ , b∗ ) = a ∗ b∗ is a homeomorphism. Now define ϕ : G → A and ψ : G → B as follows: First set ψ = (q|B)−1 pr B ∗ (μ∗ )−1  q : G → expG v0 , def

and then define ϕ0 (g) = gψ(g)−1 for each g ∈ G. This defines a function ϕ0 : G → G such that ϕ0 (g)ψ(g) = g for all g. But from the commuting diagram we obtain q(ϕ0 (g)) = q(g)q(ψ(g))−1 . Every g ∗ ∈ G∗ is uniquely of the form g ∗ = μ ∗ (a ∗ , b∗ ) and then pr B ∗ (μ∗)−1 (q(g)) = pr B ∗ (a ∗ , b∗ ) = b∗ = q(b) for a unique b ∈ B. Now ε is surjective, and so there is an a ∈ A and a b ∈ B such that ab = g; then q(b ) = q(b) follows whence b = b and then gψ(g)−1 = (ab)b−1 = a ∈ A. Therefore, ϕ0 (g) ∈ A and we define ϕ : G → A by ϕ(g) = ϕ0 (g) and since we just saw that g  → (ϕ(g), ψ) : G → A × B is surjective, μ(ϕ(g), ψ(g)) = g implies that it is μ−1 . Thus μ is a homeomorphism, and this implies that ε is a homeomorphism.

Splitting a Maximal Compact Subgroup Topologically Theorem 12.81. Let G be a connected pro-Lie group. Then G has a maximal compact group C which is connected, and all other maximal compact subgroups are conjugate to C. Also, for some p ≤ 4 there are vector subspaces vk ⊆ g, k ≤ p, such that the function (c, X1 , . . . , Xp )  → c expG X1 . . . expG Xp : C × v1 × · · · × vp → G is a homeomorphism. Moreover, all N (G) expG vk , k ≤ p, are prosolvable subgroups. Finally, let Ca be the arc component of the identity in C. Then Ca × v1 × · · · × vp is mapped homeomorphically onto the arc component Ga of the identity of G, and Ga = A(g, G). Proof. From Theorem 12.77 we know that maximal compact subgroups exist, that are all connected, and that are conjugate to each other under inner automorphisms. We proceed in steps.

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Step 1. Let G be a reductive connected pro-Lie group and assume that C is a maximal compact subgroup. We continue the notation we have used in Lemma 12.76. Unlike in 12.76 we allow the presence of compact factors. Therefore we have    = RI × Sj Kj × G j ∈Jc

j ∈J

with two disjoint index sets Jc and J such that all Kj , j ∈ Jc are simply connected compact, and all Sj , j ∈ J are noncompact simple simply connected. Hence  Sj has an Iwasawa decomposition Kj Aj Nj . Let us write J ∗ = Jc ∪ J and set B = j ∈J Aj Nj ,   ∼ nj  = RI × so that G j ∈J ∗ Kj × B. Recall that Kj = Vj × Kj , Vj = R , where     = RI × nj may be zero and will be if j ∈ Jc . We set K j ∈J ∗ Vj × j ∈J ∗ Kj .    = The commutator subgroup K j ∈J ∗ Kj is compact and simply connected, and    the identity component of the center is RI ∪I , where RI ∼ = j ∈J ∗ Vj . The Poincaré ∼  and agrees with P (K), K = πG (K)  = subgroup P (G) = ker πG is contained in K  def  (K). The group B = K/P j ∈J Aj Nj is simply connected, since it is topologically a  direct factor of G and is prosolvable since every Aj Nj in the Iwasawa decomposition    contains is solvable. The nilradical N(B) j ∈J Nj . By Theorem 8.13, B is homeo  morphic in a natural way to (n(b), ∗)× B/N(B) and thus is homeomorphic to a weakly = K B,  then this is topologically complete topological vector space. If we write G  isomorphically onto  then πG maps B a direct product and P (G) = P (K) ⊆ K, def ∼ ∼    B = πG (B), and G = G/P (G) = KB = K × B, topologically. It remains to   ∼ clarify the topological structure of K. Since L(K) = L(K) = RI ∪I × k and this is a procompact pro-Lie algebra, K is a potentially compact connected pro-Lie group (see 12.46). Then by Theorem 12.48 K contains a maximal central vector subgroup def

V and a unique maximal compact connected subgroup C = comp(K) such that (c, v)  → cv : C × V → K is an isomorphism. Thus G contains subgroups V , B and C such that (c, v, b) → cvb : C × V × B → G is a homeomorphism. Thus we let v1 = L(V ), v2 = n(b), v3 any vector space direct summand complementary to n(b) ∈ b and the conclusion holds with p = 3 and all vk being subalgebras. Step 2. Let G be a connected pro-Lie group for which the nilradical N (G) is simply connected and let C be a maximal compact group. Then G/N (G) is a connected reductive pro-Lie group by Theorem 4.28 (i) and it satisfies the conclusion of Step 1. Now we apply the Induction Lemma 12.80 with K = {1} and N = N (G). It yields v0 = n(g) and vector spaces v1 , v2 , and v3 such that (c, X0 , . . . , X3 )  → c expG X0 . . . expG X3 : C × v0 × · · · × v3 → G is a homeomorphism. The connected groups N (G) expG vk , k = 1, 2, 3 are prosolvable. def Step 3. The general case: We know that G∗ = G/ KZ(G)0 satisfies the conclusions since the assertion is true if KZ(G)0 , the unique largest compact connected central

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subgroup, is singleton. Then by Theorem 9.50, KZ(G∗ )0 = {1} and so Step 2 applies to G∗ and yields at most four vector subspaces vk of g∗ = g/L(KZ(G))∗ such that (c∗ , X1∗ , . . . , Xp∗ )  → c∗ expG∗ X1 . . . expG∗ Xp : C ∗ × v1 × · · · × vp → G∗

(∗)

is a homeomorphism and that all N(G∗ ) expG∗ vk are prosolvable subgroups. Each of the full inverse images Gp of N(G∗ ) expG∗ v∗k satisfies the hypotheses of Lemma 12.79 (i). So L(Gp ) contains a vector subspace vk such that the function (c, X1 , . . . , Xp )  → c expG X1 . . . expG Xp : C × v1 × · · · × vp → G is a homeomorphism. In view of what has been proved above, the final remark follows, firstly, from the fact that Ca × v1 × · · · × vp is the arc component of (1, 0, . . . , 0) ∈ C × v1 × · · · × vp , secondly, from Ca = expC L(C) according to [102, 9.60 (i)], and thirdly, from Ga = expC L(C) expG v1 . . . expG vp ⊆ expG L(G) = A(g, G) ⊆ Ga . We have accurate information on Ca through [102, 8.30 and 9.30]. The preceding theorem applies, in particular, to any locally compact connected group. Topological Splitting of Connected Pro-Lie Groups Corollary 12.82. (i) Each connected pro-Lie group, respectively, connected locally compact group, is homeomorphic to a product of a compact connected semisimple group, a compact connected abelian group, and a family, respectively, finite family, of copies of R. (ii) Each connected pro-Lie, respectively, connected locally compact, group is homeomorphic to a product of a compact connected semisimple group and a connected abelian pro-Lie group, respectively, locally compact connected abelian group. Proof. First we recall that every locally compact connected group is a pro-Lie group (see [145] or [129]), and that a weakly complete vector group RJ is locally compact iff J is finite (see A2.2 (iv)). These facts explain the locally compact portion of the corollary. (i) This is a consequence of the preceding Theorem 12.81 and the Borel–Scheerer– Hofmann Theorem (see [102, Theorem 9.39], and see also Theorem 11.8 of this book). (ii) follows from (i) via the Vector Group Splitting Lemma 5.12. The following corollaries, of course apply to locally compact connected groups as well. Corollary 12.83. Each connected pro-Lie group has a compact connected group as deformation retract. The entire algebraic topology, homotopy, homology, cohomology is supported by one and hence any of the maximal compact subgroups.

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Proof. This is an immediate consequence of the preceding corollary. Corollary 12.84. Let G be a connected pro-Lie group and let C be a maximal compact subgroup. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

G is simply connected. C is simply connected. C is semisimple and simply connected.  C∼ = j ∈J Sj for a family of simple and simply connected compact Lie groups.

Proof. By Theorem 12.81, there is a set I such that G and C × RI are homeomorphic. A product is simply connected iff all of its factors are simply connected. Thus C × RI is simply connected iff C is simply connected. Therefore (i) and (ii) are equivalent. As mentioned in the proof of 12.82, C = C   C/C  by the Borel–Scheerer– Hofmann Splitting Theorem. A compact connected abelian group is simply connected if and only if it is singleton (see [102, Theorem 9.29]). Thus a compact connected group is simply connected iff it is semisimple and indeed is a product of simply connected simple compact Lie groups ([102, Theorem 9.29]). Thus (ii), (iii) and (iv) are equivalent. We return to the formulation of Theorem 12.81 and recall that for any connected pro-Lie group G there is a number p ≤ 4 and there are vector subspaces vk , k ≤ p, of g such that the function (c, X1 , . . . , Xp )  → c expG X1 . . . expG Xp : C × v1 × · · · × vp → G is a homeomorphism. If we set v = v1 × · · · × vp , then v is a weakly complete topological vector space and thus (by Theorem A2.9) is isomorphic to RJ for a set J whose cardinality is an invariant of G which we call the topological dimension of v; it agrees with the linear dimension of the vector space v , the topological dual of v. The def

subset M = expG v1 . . . expG vp ⊆ G is homeomorphic to v, and (c, m)  → cm : C × M → G is a homeomorphism. Definition 12.85. In the notation of Theorem 12.81 and the one introduced in the preceding remarks, the subset M ⊆ G is called a manifold factor, and if greater precision is asked for, a vector space manifold factor. We write def

dim M = card J and call this cardinal the dimension of the manifold factor. For more details on dimension, we refer to [103]. def Let G/C = {Cg : g ∈ G} denote the orbit space for the left action of the maximal compact subgroup C by multiplication on G.

Maximal Compact Subgroups of Connected Pro-Lie Groups

555

The Existence of Manifold Factors Corollary 12.86. Let G be a connected pro-Lie group and C a maximal compact subgroup. Then: (i) There is a manifold factor M and (c, m) → cm : C × M → G is a homeomorphism. (ii) The function σ : M → G/C,

σ (m) = Cm

is a homeomorphism. (iii) G is homeomorphic to C × G/C. Proof. (i) is a direct consequence of Theorem 12.81. Proof of (ii): We denote the homeomorphism of (i) by μ : C × M → G. We let C act on the left on C × M by c · (c , m) = (cc , m). Then μ is an equivariant homeomorphism, that is, c · μ(x) = μ(c · x) for all x = (c , m) ∈ C × M. Evidently, the function s : M → (C × M)/C,

s(m) = C · (1, m) = C × {m}

is a homeomorphism. The equivariant homeomorphism μ induces a homeomorphism μ∗ : (C × M)/C → G/C. Since σ = μ∗  s, assertion (ii) follows. (iii) is an immediate consequence of (i) and (ii). In the light of this corollary we attach to a connected pro-Lie group a cardinal dim G/C = dim M = dim v = card J . We call this cardinal the manifold rank of G. By Corollary 12.86, two connected pro-Lie groups G1 and G2 are homeomorphic if and only if they have homeomorphic maximal compact subgroups C1 , respectively, C2 and the same manifold rank. These aspects allow us to present a very convincing way of characterizing the local compactness of a connected pro-Lie group. We recall from Theorem 12.59 that every connected pro-Lie group G has a unique largest compact normal subgroup MaxK(G). Characterisation of Local Compactness Corollary 12.87. Let G be a connected pro-Lie group. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

G is locally compact. The manifold rank dim G/C of G is finite. The vector space codimension of L(C) in L(G) is finite. The factor group G/ MaxK(G) modulo the unique maximal compact normal subgroup is a Lie group.

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Proof. (i) ⇔ (ii): By Corollary 12.87, G is homeomorphic to C × G/C. This product is locally compact iff G/C is locally compact. Now G/C is homeomorphic to RJ for a set J with card J = dim G/C. But RJ is locally compact iff J is finite. (iii) ⇒ (ii): If L(C) has finite codimension, dim vk < ∞ for each k = 1, . . . , p by the proof of Theorem 12.81. So dim G/C = dim v = dim v1 + · · · + dim vp < ∞. (i) ⇒ (iv): The factor group of a locally compact group is locally compact. By Theorem 12.59, G/ MaxK(G) has no compact normal subgroup. By Yamabe’s Theorem (see [145], [206], [207]) a locally compact group without nontrivial compact normal subgroups is a Lie group. (iv) ⇒ (iii): Every maximal compact subgroup C contains MaxK(G) and the factor group C/ MaxK(G) is a closed subgroup of the Lie group G/ MaxK(G) and thus dim L(G/ MaxK(G))/L(C/ MaxK(G)) < ∞. By Theorem 4.20 we have L(G/ MaxK(G)) ∼ = L(G)/L(MaxK(G)) and Therefore

L(C/ MaxK(G)) ∼ = L(C)/L(MaxK(G)). L(G) ∼ L(G)/L(MaxK(G)) ∼ L(G/ MaxK(G)) = = L(C) L(C)/L(MaxK(G)) L(C/ MaxK(G))

is finite-dimensional.

An Alternative Open Mapping Theorem We deal often with an Open Mapping situation, i.e., a surjective morphism f : G → H between topological groups of which we would like to know that it is an open mapping, equivalent to a quotient map. We always refer to the example H = R with the natural topology, G = Rd with the discrete topology, f = idR . This is a nonopen bijective morphism between abelian Lie groups. We have the classical Open Mapping Theorem for surjective morphisms between locally compact groups whose domain is a countable union of compact sets, and we have a rather different Open Mapping Theorem for surjective morphisms between pro-Lie groups whose domain is almost connected (Theorem 9.60). Related information is to be found in Corollary 4.22. But here is a lemma that we have not yet formulated: Theorem 12.88 (Alternative Open Mapping Theorem). Let f : G → H be a surjective morphism between pro-Lie groups and assume that (i) G/ ker f is a pro-Lie group, (ii) H is connected, (iii) L(f ) : L(G) → L(H ) is surjective. Then f is open.

An Alternative Open Mapping Theorem

557

Proof. The morphism f factors through a bijective morphism F : G/ ker f → H , whose domain is a pro-Lie group by (i) and whose range is connected by (ii). Also, L(f ) factors through the quotient L(G) → L(G)/L(ker f ) ∼ = L(G)/ ker L(f ) (see Theorem 4.20), and so L(F ) : L(G/ ker f ) → L(H ) is bijective and hence an isomorphism. Thus we may assume that f : G → H is bijective and (iii ) L(f ) : L(G) → L(H ) is an isomorphism. →H  is an isomorphism. Secondly, we claim that f It follows firstly, that f: G maps the maximal compact connected subgroups of G bijectively onto the maximal compact connected subgroups of H and if M is a maximal compact connected subgroup of G then f |M : M → f (M) is an isomorphism. Indeed, since L(f ) is an isomorphism of pro-Lie algebras, it maps the maximal compactly embedded subalgebras of g = L(G) onto those of h = L(H ). By Proposition 12.52, the maximal compactly embedded connected subgroups of G, respectively, H are exactly those closed analytic subgroups, whose Lie algebras are maximal compactly embedded into g, respectively h. Let M be a closed analytic subgroup of G, then M is maximal compactly embedded iff m = L(M) is maximally compactly embedded in g and M = expG m, and that is the case if and only if L(f )(m) is maximally compactly embedded in h iff expH L(f )m = f (M) is maximal compactly embedded in H . m ⏐ expM ⏐  M

L(f )|m

−−−−→ −−−−→ f |M

L(f⏐)m ⏐exp  f (M) f (M).

Each maximal compactly embedded connected subgroup of G, respectively, of H contains a unique characteristic maximal compact connected subgroup by Theorem 12.48; see also Theorem 12.54 and its proof. Thus if M is maximally compactly embedded in G, then comp(M) is a maximal compact connected subgroup of G and every such is of the form comp(M) for a maximal compactly embedded connected subgroup M of G. Then f maps comp(M) isomorphically onto comp(f (M)). By Theorem 12.65 and Corollary 12.68 there is a totally disconnected compact subgroup  of H act such that there is a quotient morphism δ : H    → H, ing automorphically on H δ(x, d) = πH (x)d. Now  is contained in some maximal compact and therefore connected subgroup of H according to Theorem 12.77. Every such is of the form f (comp(M)) for a suitable M and thus f −1 () is a compact subgroup  of comp(M), ∼  and  ∼ mapped isomorphically onto  by f . Since G =H = , clearly  acts auto  morphically on G. Define ω : G   → G by ω(x, w) = πG (x)w. Then we have a commutative diagram 

f ×f |  −  G −−−→ H ⏐ ⏐ ⏐ ⏐ ω δ G −−−−→ H. f

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The vertical maps are morphisms between pro-Lie groups, and δ is surjective by Corollary 12.68; therefore, since the horizontal functions are bijective, ω is surjective    and H    are almost connected since G  as well. The isomorphic groups G  are connected and  is compact totally disconnected. Thus the Open Mapping and H Theorem 9.60 applies to both vertical maps and shows that they are open. But this implies that f = δ  (f× f |)  ω−1 is open as asserted. Notice that the proof of the Alternative Open Mapping Theorem 12.88 uses the Open Mapping Theorem 9.60. Moreover, it uses a good deal of structural information we have accumulated in this chapter. Corollary 12.89. Let G and H be two pro-Lie groups of which at least one is assumed to be connected. Let f : G → H be a bijective morphism. Then the following statements are equivalent: (i) f is an isomorphism. (ii) L(f ) : L(G) → L(H ) is an isomorphism. Proof. Theorem 12.88 proves (ii) ⇒ (i). The implication (i) ⇒ (ii) was proved in Corollary 4.22 (ii). This corollary is a small testimony to the functioning of the Lie theory of pro-Lie groups. We now give a sample application of the Alternative Open Mapping Theorem 12.88 to a result on the structure of connected pro-Lie groups, in whose proof it is used.

On the Center of a Connected Pro-Lie Group Theorem 12.90. The center of a connected pro-Lie group is contained in some connected abelian subgroup. Proof. Let G be a connected pro-Lie group and Z(G) its center. (a) Assume that the assertion is true for G in which compact connected central subgroups are trivial. We claim that then the assertion is true in general. Indeed let G be arbitrary. Then G/ KZ(G) has no nontrivial compact central subgroups by Theodef

rem 9.50 and H = G/ KZ(G)0 has the totally disconnected subgroup KZ(G)/ KZ(G)0 as unique largest compact central subgroup. Therefore H has no compact connected central subgroups other than the singleton one. Thus by our assumption there is a connected closed abelian subgroup A of H such that Z(H ) ⊆ A. If p : G → H dedef

notes the quotient morphism, we set B = p−1 (A). Since p(Z(G)) ⊆ Z(H ) we have Z(G) ⊆ B and B is a connected closed subgroup of G which is nilpotent of class ≤ 2. Now the center Z(B) of B is connected by Lemma 11.19. Since Z(G) ⊆ Z(B) we have found a connected abelian subgroup containing Z(G).

On the Center of a Connected Pro-Lie Group

559

Thus we may assume from here on without losing generality that G has no nondegenerate compact connected central subgroups. Equivalently, the nilradical N (G) is simply connected by Theorem 11.27. From Theorem 9.90 (ii) we know that the center Z(G) is a direct product of a weakly complete vector group Z(G)0 and a prodiscrete subgroup . (b) We claim that the assertion is true if G is semisimple. In the proof of 12.77 we have already argued that a connected semisimple pro-Lie group G has maximal compactly embedded subgroups K and that these are connected and conjugate. Since the center is compactly embedded it is contained in one (hence each) of them, say K. But then Z(G) ⊆ Z(K). Now K ∼ = RJ × C for the unique maximal compact subgroup C J and so Z(K) ∼ = R × Z(C). But the center of a compact connected group C is contained in each maximal pro-torus T of C. Hence Z(G) ⊆ Z(K) ∼ = RJ × Z(C) ⊆ RJ × T . This proves the assertion in the case that G is semisimple. (c) Assume that the assertion is true for prosolvable groups. We claim that then it is true in general. Set S = G/R(G) and let p : G → S again be the quotient morphism. According to (b) we find a connected closed abelian subgroup A containing Z(S). Then def

p(Z(G)) ⊆ Z(S) ⊆ A and hence Z(G) ⊆ B = p −1 (A). Now B is a closed subgroup of G and thus is a pro-Lie group by the Closed Subgroup Theorem 3.35; moreover, it is the extension of the connected prosolvable group R(G) by the connected abelian group A and thus is a connected prosolvable pro-Lie group. By our assumption we find a connected closed abelian subgroup containing Z(B) and thus, a fortiori, Z(G). (d) Thus we may and will assume from here on, that G is prosolvable and has a simply connected nilradical N (G). We know that Z(G) is the direct product of Z(G)0 = N(G) ∩ Z(G) and a prodiscrete subgroup : Z(G) = (N (G) ∩ Z(G)) ×  = Z(G)0 × . def

The quotient Q = G/ is a proto-Lie group by Theorem 4.1. The quotient morphism q : G → Q induces an isomorphism L(q) : L(G) → L(Q) by Theorem 4.20, and the def

completion H = Q is a pro-Lie group such that the inclusion i : Q → H induces an isomorphism L(i) : L(Q) → L(H ). We set p = i  q : G → H . Then L(p) : g → h is an isomorphism, and there is no harm in identifying g and h. Let T be a maximal compact subgroup of H . Since a prosolvable compact connected group is abelian, T is abelian. By Theorem 12.77, T is connected, and by Theorem 11.28 and our assumption KZ(G)0 = {1}, there is a simply connected normal subgroup V containing N (H ) such that H = V  T . It follows that g = v ⊕ t semidirectly with a prosolvable ideal v and a compactly embedded subalgebra t. Again Z(H ) is a direct product of Z(H )0 and a prodiscrete D. Indeed p −1 (D) is a totally disconnected normal subgroup of the connected group G and is therefore central, that is, p −1 (D) ⊆ Z(G) = Z(G)0  and so D ∩ p(G) = p(p−1 (D)) ⊆ p(Z(G)0 ) ⊆ Z(H )0 . This shows D ∩ p(G) = {1}. Since L(p) is an isomorphism, p(G) is an analytic subgroup whose Lie algebra is h = g. In particular, it contains expH h and therefore V = expH v by Theorem 8.13. Therefore, Z(H ) ∩ p(G) = Z(H )0 D ∩ p(G) = Z(H )0 by the modular law; in

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particular, Z(H ) ∩ V = Z(H )0 = N (H ) ∩ Z(H ). The simply connected universal group T of the compact connected abelian groups T is the additive group of t and the universal map πT : t → T is the exponential map. =H  = (g) is of the form V α t, where The simply connected universal group G α(X)(v) = (expT X)v(expT −X) and πH (v, X) = v expG X. Accordingly, P (H ) = ker πH ∼ = ker expT . The following diagram may serve as an orientation: id

˜ G  −−−− → V  G ⏐ ⏐α t ⏐ ⏐π πG   H G −−−−→ H.

p

 ⊆ Z(H ) ∩ p(G) = Z(H )0 . Once again, the center Z(G)  We note that πH (Z(G))  0 and some prodiscrete group E. Let (v, X) ∈ E; then is a direct product of Z(G)  0 . This shows that (v, expT X) ∈ Z(H )0 and so X ∈ ker expT = P (H ) and v ∈ Z(G)  = Z(G)  0 × P (G). Z(G) def

Set W = expG v, then πG (V × {1}) = W and p(W ) = V = πH (V × {1}). It follows that p|W : W → V is an isomorphism. If z ∈ W ∩ Z(G), then p(z) ∈ V ∩ Z(H ) = N(H ), whence g ∈ N (G) ∩ Z(G) = Z(G)0 . def

Let A = (p−1 (T )). Since L(p) is an isomorphism, L(A) = t and A0 = expG t by Corollary 4.22 (i). Then p(A0 ) is a closed subset of p(G) and is therefore of the form p(G) ∩ C with a closed subset C of T . It contains p(expG t) = expH t and thus T = expH t ⊆ C, whence C = T and thus p(A0 ) = p(G) ∩ T whence A0  = p−1 (T ) = A. Therefore, A is abelian. Let g ∈ G. Then p(g) = vt with v ∈ V and t ∈ T . Let w = (p|W )−1 (v) ∈ W ; then p(w) = v. So t = p(w−1 g) giving us w −1 g ∈ p−1 (T ) = A, that is g = wa for some a ∈ A. Thus G = W A. If g ∈ W ∩ A, then p(g) ∈ V ∩ T = {1}, and so g ∈ ∩W ⊆ ∩(Z(G)∩W ) = ∩Z(G)0 = {1}. Therefore W ∩A = {1}. Accordingly, the morphism μ : W  A → G, μ(w, a) = wa is a bijective morphism of pro-Lie groups with a connected range, inducing an isomorphism L(μ) : L(W  A) → L(G). idV ×L(p)|L(A) id  −−− −→ V ⏐L(A) −−−−−−−−→ G ⏐ ⏐ ⏐ πG  idV × expA  W A −−−−→ G −−−−→ μ

p

V ⏐ t ⏐id × exp  V T V  T.

Now the Alternative Open Mapping Theorem 12.88 applies to the bijective morphism μ and shows that it is an isomorphism. Therefore, since G is connected, A is connected. So  = p−1 (1) ⊆ p−1 (T ) = A and Z(G) = Z(G)0  = Z(G)0 A ⊆ Z(G)0 A. This proves that Z(G) is contained in the connected abelian group Z(G)0 A. This completes the proof of the theorem.

Postscript

561

Postscript In this chapter, amongst other significant results, we are able to describe completely the topological structure of connected pro-Lie groups. A study of compact subgroups of pro-Lie groups naturally begins with the adjoint action of the group on its Lie algebra, a pro-Lie algebra. This is a special case of the representation of a pro-Lie group on a weakly complete topological vector space or, of a G-module V for a pro-Lie group G and a weakly complete topological vector space V . If K is a compact subgroup of G, then we have in a natural fashion a K-module V but also an L(K)-module V , and that is our point of departure. Indeed considerable information on L-modules for Lie algebras L were amassed in Chapter 7. If L is an arbitrary Lie algebra (to start with) and V an L-module, we say that V is a pre-Hilbert module if there is a scalar product (positive definite bilinear form) (· | ·) on V such that (∀x ∈ L, v, w ∈ V )

(x · v|w) = −(v|x · w).

(1)

We say that V is a compact L-module, if it is finite-dimensional and supports the structure of a pre-Hilbert module. And we say that V is an algebraically locally compact L-module if every element is contained in a finite-dimensional submodule carrying the structure of a pre-Hilbert module. Our starting point is a structure theorem for algebraically locally compact L-modules which requires the Axiom of Choice: Indeed if V is an algebraically locally compact module, then we can endow V with a scalar product such that V is the orthogonal direct sum of finite-dimensional compact modules (Theorem 12.4). The point is that this dualizes and yields a structure theorem for a class of modules that are relevant in our very own context. Namely, we say that an Lmodule is a procompact L-module, if it is a weakly complete topological vector space and the cofinite-dimensional modules W form a filter basis converging to 0 in such a way that all factor modules V /W are compact L-modules. Then we derive by duality from the result above that any procompact L-module is a direct product of compact simple L-modules. Every such module is semisimple. that is, every closed submodule is a direct module summand algebraically and topologically. (See 12.6) A set-up like this cries out for some compact group emerging out of nowhere and acting on such a module. We do produce such a group (in Theorem 12.8) which, technically speaking, acts on a procompact L-module in a fashion compatible with all the data. This group action is actually used in some of the applications we are making, for instance in the proof of Corollary 12.9. We proceed instantly to apply this sort of module theory to the problems at hand: Indeed we say that a subalgebra k of a pro-Lie algebra g is compactly embedded into g if the adjoint module gad is a procompact k-module, and we say also that g itself is a procompact pro-Lie algebra if g is compactly embedded into itself. We have made sure by pointing to the relevant examples of even low-dimensional Lie algebras what pitfalls we have to avoid: Any abelian closed subalgebra is a procompact Lie algebra in its own right but may fail grossly to be compactly embedded. The structure of procompact pro-Lie algebras is completely and satisfactorily elucidated (Theorem 12.12). After these preliminaries we proceed to the main theme of

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this chapter as long as we travel on the pro-Lie algebra level (the group level afterwards posing its own additional problems): The study of maximal compactly embedded subalgebras (and maximal compactly embedded abelian subalgebras). The Axiom of Choice permits a relatively quick proof of their existence (12.15). However, the issue of their conjugacy is all but simple. It took considerable preparation, including our discussion of Cartan subalgebras and their conjugacy theorems in Chapter 7, yet in the end it emerges that two maximal compactly embedded subalgebras of a pro-Lie algebra are conjugate under inner automorphisms (Theorem 12.27). It is important that we are able to derive a theorem on the existence of a unique maximal compactly embedded ideal m(g) of a pro-Lie algebra (Corollary 12.34), giving us another in the comparatively long sequence of characteristic ideals of a pro-Lie algebra g, such as for instance z(g), r(g), n(g), ncored (g), g˙ , and so on. The point of this theory is to apply it to an analysis of compact subgroups of a proLie group, in other words to utilize Lie algebra results for producing group theoretical insights. As a first and still simple step, there is a group theoretical characterisation of the Lie algebraic property of g to be procompact (12.36), and this in turn is tested to yield basic results on the structure theory of compact connected groups whose proofs are embedded here in a more general setting but which, given the present theory, emerge very quickly (see 12.37 and 12.38). The Lie algebra L(H ) of a compact subgroup H of a pro-Lie group G is compactly embedded in L(G); the reverse direction is more complicated: If h is a compactly embedded subalgebra of L(G) and if H is an analytic subgroup such that L(H ) = h, then Ad(H ) is a compact group and agrees with ead h . (See Theorem 12.45.) So we do have a bridge between compact subgroups of a pro-Lie group and compactly embedded Lie subalgebras. In the end, when the dust clears we have the following result (12.54, 12.56): Each connected pro-Lie group G contains maximal compact connected subgroups, respectively, maximal compact connected abelian subgroups. Maximal compact connected subgroups, respectively, maximal compact connected abelian subgroups, are conjugate in G under inner automorphisms. Also, there is a unique maximal compact connected normal subgroup MaxC(G) and the factor group G/ MaxC(G) has no nontrivial compact connected normal subgroups. It is then easy to deduce that there is a largest compact connected abelian normal subgroup which is central, namely, Z(MaxCE(G)0 ) = comp(Z(G)0 ). Indeed we saw in Chapter 9 (see 9.50) that a connected pro-Lie group always contains a largest compact normal abelian and consequently central subgroup KZ(G), and G/ KZ(G) does not contain any compact central subgroups other than the identity. Bringing this result into the picture allows us to prove that every compact connected pro-Lie group G has a unique largest compact normal subgroup MaxK(G) and G/ MaxK(G) is a connected pro-Lie group having no nontrivial compact normal subgroups.

Postscript

563

In Chapter 11 we had already observed the nontrivial fact that the absence of nontrivial compact subgroups in a connected prosolvable Lie group G is equivalent to simple connectivity. The absence of compact subgroups is a condition in the domain of topological algebras whereas simple connectivity is a purely topological condition. Since analytic subgroups were first introduced in Chapter 9 we have frequently encountered analytic subgroups of a pro-Lie group G, we have observed the analytic subgroup A(g, G) = expG g, the unique minimal analytic subgroup whose Lie algebra is g. This is clearly a characteristic subgroup which is dense in G and is perhaps the most important such, although even the theory of connected compact abelian groups shows us that there may be many dense analytic subgroups. Indeed a compact connected group G has a dense one parameter subgroup (that is, a one-dimensional analytic  has a cardinality not exceeding 2ℵ0 . We saw in Chapter 9 subgroup) if and only if G in 9.32ff. that whenever A(h, G) is dense in G then g˙ ⊆ h, that is g/h is an abelian algebra. However, we were not able to show in general that G/A(h, G) is an abelian group. In this chapter we are finally able to show that the analytic subgroup can be supplemented by compact abelian groups in the following fashion: Let G be a connected pro-Lie group and K a maximal compact subgroup of the extended radical Q(R). Then G = K · A(g, G). There is a totally disconnected compact subgroup D of K such that G = D · A(g, G) and Q(G) = R(G)D. There is a compact connected abelian subgroup C of G such that G = C ·A(g, G). Here the extended radical Q(G) of G plays a surprisingly significant role: It is defined as that subgroup containing the radical R(G) such that Q(G)/R(G) = KZ(G/R(G)). In particular, it follows that the abstract group G/A(g, G) is abelian, and indeed is an algebraic homomorphic image of a compact totally disconnected subgroup of the extended radical. One notices that A(g, G) is normal while, in general, K and D are not normal. This unfortunately cannot be helped. If K, respectively, D were normal in the connected group G, then they would have to be central, and we had in Chapter 9 a construction of center-free metabelian groups with proper dense analytic subgroups A(g, G). On the other hand, for a relatively wide class of connected pro-Lie groups G which nevertheless excludes these center-free examples, we have a result that is significantly better: Let G be a connected pro-Lie group and let g = r(g) + s be a Levi decomposition of its Lie algebra. Assume that [r(g), s] = {0}, and R(G)/ KZ(G)0 is simply connected. Then G = KZ(G) · A(g, G). Here both factors are characteristic. If one is satisfied with having a central (instead of characteristic) subgroup as a compact supplement one can improve the conclusion by saying

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there is a compact totally disconnected central subgroup D such that G = D · A(g, G). The results listed here allow us to improve on the universal covering morphism  → G. If G is a Lie group, π is a covering morphism. In the case of a pro-Lie πG : G group which is not a Lie group, this is not the case any more as we have observed on numerous occasions beginning in Chapter 6. As a rule π is not even surjective. Here we prove that the compact totally disconnected abelian subgroup D of the extended  → G,  d · x = α(d)(x) radical allows an automorphic action (d, x)  → d · x : D × G  Accordingly one can form the semidirect product via a morphism α : D → Aut(G).  α D. Recall from Chapter 8 that G  is an exponentially generated, arcwise connected G and simply connected group which is, topologically a product of copies of R and of simple simply connected Lie groups Sj . On the other hand, D is a compact totally disconnected abelian group; these are exactly the character groups of abelian torsion groups (see [102, Corollary 8.5]). We show in the Resolution Theorem for Connected Pro-Lie Group 12.68 that there is an open surjective morphism  α D → G δ: G which induces an isomorphism of Lie algebras and, accordingly, has a prodiscrete kernel. In 12.74, we do prove a Resolution Theorem in the special case that G is centrally supplemented, that is G = Z(G)A(g, G) irrespective of the compactness of Z(G). Examples of metabelian connected pro-Lie groups which are not centrally supplemented were given in 9.42. The chapter contains a significant result, that illustrates quite well our motivation to invest so much energy into the investigation of compact and notably maximal compact subgroups. The core theorem is the following (12.81). Theorem. Let G be an arbitrary connected pro-Lie group. Then: (i) (ii) (iii) (iv)

G has at least one maximal compact subgroup C. Every maximal compact subgroup is connected. All maximal compact subgroups are conjugate under inner automorphisms. There is a set J and a homeomorphism ε : C × RI → G such that ε(C × {0}) = C.

None of these statements is obvious. Our proof gives many more details about Statement (iv). This last one, taken together with the Borel–Scheerer–Hofmann Splitting Theorem (see [102, Theorem 9.39], and see also Theorem 11.8 of this book) gives the following statement: Theorem. Every connected pro-Lie group is homeomorphic to a direct product of a compact connected semisimple group, a compact connected abelian group, and a space RJ for a set J .

Postscript

565

Clearly, such a group is locally compact iff J is finite. These theorems have been anticipated a few times in the course of the book, so for instance in the structure theorem for connected abelian pro-Lie groups 5.12 or the Structure Theorem for Simply Connected Pro-Lie Groups 8.14 (plus 8.15). An obvious but striking consequence is that all the information that algebraic topology (homotopy, homology, cohomology,) gives us on compact connected groups yields this precise information on connected pro-Lie groups, since according to the preceding theorems each maximal compact subgroup C of a connected pro-Lie group G is a homotopy deformation retract and thus is homotopically equivalent (that is, isomorphic in the homotopy category) to G. But the topological decomposition theorem also allows us to give a satisfying characterization of local compactness as follows (12.87): Corollary. For a connected pro-Lie group G, the following conditions are equivalent, where C is a maximal compact subgroup of G: (i) G is locally compact. (ii) The vector space codimension of L(C) in L(G) is finite. (iii) The factor group G/ MaxK(G) modulo the unique maximal compact normal subgroup is a Lie group. It is in the sense of these results that one might say with some justification that locally compact connected groups are nearly compact. With powerful structural results at our disposals we are able to prove some additional structure theorems that illustrate the power of the tools we now have. As a first sample we prove an Alternative Open Mapping Theorem (12.88) which now accompanies the Open Mapping Theorem 9.60: Theorem. Let f : G → H be a surjective morphism between pro-Lie groups and assume that (i) G/ ker f is a pro-Lie group, (ii) H is connected, (iii) L(f ) : L(G) → L(H ) is surjective. Then f is open. The proof of this Open Mapping Theorem uses Theorem 9.60. In each instance when we present an Open Mapping Theorem we recall the example of the identity morphism Rd → R and do a quick test which hypothesis fails to be satisfied; here it is hypothesis (iii). We apply the Alternative Open Mapping Theorem in the proof of a theorem which specializes to a theorem known in Lie group theory (see for instance [86, p. 189]) with a nontrivial proof there; in our case the proof is harder, considering the material entering it (12.87). Theorem. The center of a connected pro-Lie group is contained in some connected abelian subgroup.

Chapter 13

Iwasawa’s Local Splitting Theorem

We have seen many aspects in which the structure theory of connected pro-Lie groups reduces to either classical Lie theory, or the theory of compact groups, or both. We shall attempt yet another aspect of this kind of “reduction”. We shall say that a pro-Lie group G satisfies the Iwasawa Local Splitting Theorem if for every identity neighborhood U there is an N ∈ N (G) contained in U and a simply connected finite-dimensional Lie group L such that N ×L and G are locally isomorphic. Indeed Iwasawa proved in 1949 that every locally compact connected pro-Lie group has this property. We shall expose examples of connected pro-Lie groups that do not satisfy the Iwasawa Local Splitting Theorem and prove that a connected pro-Lie group G satisfies the Iwasawa Local Splitting Theorem if N (G)/Z(G)0 , the factor group of its nilradical modulo the identity component of its center is finite-dimensional. Since this factor group will be seen to be always simply connected pronilpotent, it must be finite-dimensional if G is locally compact. So our results imply Iwasawa’s original Local Splitting Theorem, but do apply to a wider class of pro-Lie groups, as every infinite product of noncompact Lie groups illustrates. Prerequisites. At this point the reader is probably aware that we shall unhesitatingly use whatever is needed from the information accumulated in the earlier portions of this book.

Locally Splitting Lie Group Quotients of Pro-Lie Groups We have made it our strategy to prepare a structure theoretical idea by first working as much as we can on the level of the pro-Lie algebras. In this chapter we proceed in exactly the same fashion. But first we shall prove a result that will motivate the one which we shall gain purely at the pro-Lie algebra level. In fact we start off with a result that might have fit into Chapter 11 where we discussed splitting. Let G be a connected pro-Lie group with Lie algebra g = L(G) and N a closed normal subgroup with Lie algebra n = L(N ). Assume that g is the semidirect sum def

n ⊕ h of the closed ideal n and a closed subalgebra h. Define H = (h). The  We set ϕ def = inclusion ih : h → g induces a morphism (ih ) : H → (g) = G. πG  (ih ) : H → G. Then ϕ is a morphism of topological groups. Now we can define a morphism α : H → Aut(N ) by α(h)(n) = ϕ(h)nϕ(h)−1 and see immediately that the automorphic action (h, n)  → α(h)(n) : H × N → N

Locally Splitting Lie Group Quotients of Pro-Lie Groups

567

is continuous. Therefore the semidirect product N α H , having the multiplication (n1 , h1 )(n2 , h2 ) = (n1 α(h1 )(n2 ), h1 h2 ) is well defined. The reader should be alerted to the fact that for a subalgebra h of a Lie algebra g, the inclusion ih may not induce an embedding (ih ) : (h) → (g) in  is SS 3 ∼ general even in the case of Lie groups: If G = G = SU(2), the unit quaternions under multiplication and with g = R · i + R · j + R · k the set of pure quaternions as Lie algebra and with the exponential function exp z = ez = 1 + z + 21 z2 + · · · , and if h = R · i ⊆ g, then (h) ∼ = R while im (ih ) = expSS 3 h = eR·i = SS 1 ⊆ C is a 3  is not injective. circle subgroup of SS and so the map (ih ) : (h) → (g) = G However, if h is an ideal, then (ih ) is an embedding by Corollary 6.9. We shall retain the notation we have just introduced in the following result: Proposition 13.1 (The Splitting and Sandwich Theorem). Let N be a closed normal subgroup of a pro-Lie group G and assume that the Lie algebra g of G is the semidirect sum of the Lie algebra n of N and some closed subalgebra of G. Then:  with Lie algebra g/n, a semidi(i) There is a simply connected pro-Lie group G/N  rect product N α G/N and two morphisms πN   μ α G/N  −−  −−−−→ G, G −−−−→ N α G/N  → G. whose composition μ  πN   is the universal morphism πG : G α G/N (ii) Both kernels ker πN   and ker μ are totally disconnected, and all of the α G/N morphisms πN   , μ, and πG induce isomorphisms. L(πN α H ), L(μ), and α G/N L(πG ) on the Lie algebra level.  → G by ϕ(h) = μ(1, h). The function (iii) Define ϕ : G/N h  → (ϕ(h)−1 , h) : ϕ −1 (N ) → ker μ,

 → G, ϕ : G/N

is an isomorphism of prodiscrete groups.  is a pro-Lie group. (iv) The semidirect product N α G/N Proof. (i) We note that h ∼ = g/n, and that therefore h may be identified with the Lie algebra L(G/N ) ∼ = L(G)/L(N ) by Corollary 4.21 (i). For the purposes of the  by H . By the remarks preceding the theorem we have proof we shall abbreviate G/N H = (h) and an automorphic action of H on N via the morphism α : H → Aut(N ) so  with (N α H ) that N α H is well defined. It remains to show that we may identify G in a natural way, to define μ, and to verify μ  πN α H = πG . Quite generally we recall that by Theorem 6.6 (vi), the functors  : proLieAlg → proSimpConLieGr and L : proSimpConLieGr → proLieAlg implement an equivalence of categories between the category of pro-Lie algebras and that of simply connected pro-Lie groups. By  = (n) with a closed normal the Strict Exactness Theorem 6.7, we may identify N  . For any h ∈ H , we shall subgroup of G = (g) in such a fashion that πN = πG |N

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13 Iwasawa’s Local Splitting Theorem

abbreviate (ih )(h) by  h. Now by Theorem 6.11 on the Preservation of Semidirect Products, there is a natural isomorphism  ι H = (n) ι (h) → G  = (g), m: N  ⊆ G,  h ∈ H , where the action of H on the normal m(x, h) = x(ih )(h), x ∈ N  of G  is by inner automorphisms so that ι(h)(x) =  subgroup N hx h−1 . Let us define μ : N α H → G by μ(n, h) = nϕ(h). Then we claim to have a commutative diagram of morphisms  ι H −−m  N −→ G ⏐ ⏐ ⏐ ⏐π πN ×idH  (∗)  G N α H

−−−→ μ

G.

 × H we have μ  (πN × idH )(x, h) = πN (x)ϕ(h) = Indeed, for (x, h) ∈ N   πG (x)πG (h) = πG (x h) = πG (m(x, h)), which establishes our claim. If we iden with g so that L(πG ) becomes the identity and therefore L(N ) becomes tify L(G) identified with n, applying L to the diagram (∗) we obtain the commuting diagram L(m)

 ι H ) −−−−→ L(N ⏐ ⏐ idn×h  L(N α H ) −−−−→ L(μ)

g ⏐= n⊕h ⏐id  g g

(∗∗)

 = (g) → of isomorphisms of pro-Lie algebras. It follows that (πN × id H )  m−1 : G N α H is πNα H . This completes the proof of (i). (ii) We have just observed that L(πN α H ), L(μ), and L(πG ) are isomorphisms. This implies that their kernels are zero. But since L preserves kernels, it follows from 3.30 = 4.23 that the kernels of πN α H , μ, and πG are totally disconnected. (iii) We have (n, h) ∈ ker μ iff n = ϕ(h)−1 ∈ N ∩ ϕ(H ) iff h ∈ ϕ −1 (N ) and n = ϕ(h)−1 . Thus β : ϕ −1 (N ) → ker μ, β(h) = (ϕ(h)−1 , h) is a bijective morphism having the inverse given by β −1 (n, h) = h and therefore being an isomorphism between totally disconnected pro-Lie groups and consequently prodiscrete groups. (iv) Let M ∈ N (G) and K ∈ N (H ). Then (M ∩ N ) × K is a normal subgroup of N α H , as is readily verified, (For instance, (1, h)(m, k)(1, h−1 ) = (ϕ(h)mϕ(h)−1 , hkh−1 ) ∈ (M ∩ N × K) if m ∈ M ∩ N and k ∈ K). But (N α H )/(M × K) ∼ = N/(N ∩ M)  H /K is a semidirect product of Lie groups and is therefore a Lie group. Recall that semidirect products of pro-Lie groups need not be pro-Lie groups as is exemplified by such locally compact groups as (Z/2Z)Z σ Z with Z acting automorphically on the power (Z/2Z)Z via shift operation. In this regard, conclusion 13.1 (iv) is special. What we have done in this proposition and its proof emulates Theorem 4.22 (iv) and its proof applied to the quotient morphism f : G → G/N ; we may and will identify the

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569

morphism σ : L(H ) = h → L(G) = g of weakly complete vector spaces satisfying L(f )  σ = idh with the inclusion morphism ih : h → g so that σ is in fact a morphism of pro-Lie algebras. h

ih =σ

exp G/N

  G/N

L(f )

/g

expG˜ (ih )

 /G 

CC CC ϕ CC πG/N πG CC C  !  G/N G

/h expH

f

 / G/N 

(†)

πG/N

f

 / G/N

where the top horizontal rows compose to the respective identity morphism and where we recall ϕ = πG  (ih ). In Theorem 4.22 (iv) however, we have assumed that G/N is a Lie group, and this is what we shall do shortly after the following corollary resumes the hypotheses of the preceding proposition and imposes additional ones. Recall that a semidirect sum splitting g = n⊕h implies that h ∼ = g/n which allowed  in view of the further natural isomorphism us to identify (h) and (g/n) = G/N g/n = L(G)/L(N ) ∼ = L(G/N ) of Corollary 4.21 (i). Corollary 13.2. Assume the hypotheses of Proposition 13.1 and the following hypotheses:  → G/N is surjective. (i) The universal morphism πG/N : G/N  ker μ is a pro-Lie group. (ii) The proto-Lie group (N α G/N)/  → G defined by μ(n, h) = nϕ(h) as in ProposiThen the morphism μ : N α G/N tion 13.1 is a quotient morphism with a kernel isomorphic to the central prodiscrete  subgroup ϕ −1 (N ) of G/N. Proof. From the diagram (†) we extract the commutative diagram id def   H = G/N ⏐ −−−−→ G/N ⏐ ⏐ ⏐πG/N ϕ  G −−−−→ G/N. f

Hypothesis (i) yields the surjectivity of πG/N , and thus that of f  ϕ. So let g ∈ G.  such that f (g) = f (ϕ(h)). Therefore gϕ(h)−1 ∈ ker f = Then there is an h ∈ G/N N , that is there is an n ∈ N such that gϕ(h)−1 = n; so g = nϕ(h) = μ(n, h). Hence μ : N α H → G is surjective. Now we invoke the Alternative Open Mapping Theorem 12.89. Indeed 12.89 (i) is satisfied by Assumption (ii) above, 12.89 (ii) is the connec and 12.89 (iii) is satisfied since L(μ) is an isomorphism by 13.1 (ii). tivity of G/N, Therefore μ is a quotient morphism and its kernel was identified in 13.1 (iii).

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13 Iwasawa’s Local Splitting Theorem

The situation is particularly clear if G/N is a Lie group. Recall that N (G) is the filter basis of all normal subgroups N such that G/N is a Lie group. Locally Splitting Lie Group Quotients Theorem 13.3. Assume, firstly, that G is a pro-Lie group and that N ∈ N (G). Assume, secondly, that the Lie algebra g of G is the semidirect product of the Lie algebra n of N  → G be the morphism defined and some closed subalgebra of g. Let μ : N α G/N by μ(n, h) = nϕ(h) as in Proposition 13.1. Then the morphism μ is an open morphism with a kernel isomorphic to a discrete  and implements a local isomorphism between N α G/N  central subgroup of G/N, and G. If g = n ⊕ h semidirectly, then the analytic subgroup A(h, G) of G having Lie  and has a countable intersection with N. algebra h agrees with ϕ(G/N), N/N0 is locally compact metric totally disconnected and N has an open almost connected subgroup N1 ∈ N (G) for which the same conclusions hold when N is replaced by N1 . Proof. Since G/N is a Lie group, its identity component (G/N )0 is open. There is therefore an open subgroup G of G containing N such that G/N = (G/N )0 . We notice  =G  /N. We observe that if the assertions of the theorem are proved for G in that G/N place of G then they hold for G since G is open in G. For the purposes of the proof we may therefore simplify notation by assuming that G/N is connected, which we shall do henceforth. Now we must verify the hypotheses of 13.2. Firstly, the assumptions of 13.1 are satisfied. Next Hypothesis 13.2 (i) is satisfied, since the universal morphism  → G/N πG/N : G/N of a Lie group is none other than the universal covering morphism which is surjective. Secondly, the kernel ker μ of μ is isomorphic to a totally disconnected central subgroup  by 13.1 (iii) and is therefore discrete. By 13.1 (iv), N α G/N  is a of the Lie group G/N  pro-Lie group. Then the quotient (N α G/N)/ ker μ is complete by Theorem 4.28 (ii) or (iii). Thus 13.2 (ii) is satisfied as well. Now 13.2 applies and shows that the morphism μ is a quotient morphism and that its kernel is a discrete subgroup; it therefore implements a local isomorphism. The subgroup H ({1} × H ) ker μ (N ∩ ϕ(H )) × H ∼ = = −1 −1 −1 ker μ {(ϕ(h) , h) : h ∈ ϕ (N )} ϕ (N ) is mapped bijectively onto the analytic subgroup A(h, G) of G with Lie algebra h. It  is finitely intersects N in a countable subgroup since the subgroup ker πG/N of G/N generated abelian and thus is countable.

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By Definition 9.43 and Theorem 9.44, the factor group G/N0 is finite-dimensional, hence locally compact metric and it is no loss of generality that N/N0 corresponds to  in 9.44. Therefore we may assume that N is almost connected if we wish. Corollary 13.4. If, in the circumstances of Theorem 13.3, the sum g = n ⊕ h is  and an open morphism direct with an ideal h, then there is a direct product N × G/N  → G with a discrete kernel implementing a local isomorphism. In μ : N × G/N particular G is locally isomorphic to the direct product of N and the Lie group G/N . Proof. We continue to write H and again assume without loss of generality that G/N is connected.. The closed connected subgroup ϕ(H ) is the minimal analytic subgroup with Lie algebra h and is closed. Now h is not only a subalgebra, but an ideal. This implies that ϕ(H ) is normal. Since N is normal, [N, ϕ(H )] ⊆ N ∩ ϕ(H ). Since ϕ(H ) is connected, [N, ϕ(H )] is connected and contains 1. On the other hand, N ∩ ϕ(H ) is a countable Hausdorff topological group and is therefore totally disconnected. It follows that [N, ϕ(H )] is singleton and thus N and ϕ(H ) commute elementwise. Thus α : H → Aut(N ), α(h)(n) = ϕ(h)nϕ(h)−1 = n is the constant morphism. Therefore N α H is a direct product. With these results we are now poised to shift the emphasis of our research to the Lie algebra.

The Lie Algebra Theory of the Local Splitting Definition 13.5. An ideal n in a pro-Lie algebra is called complemented, respectively, well-complemented if there a finite-dimensional subalgebra h such that g is the semidirect sum, respectively, direct sum, n ⊕ h algebraically and topologically. It is called supplemented if there is a finite-dimensional subalgebra h such that g = n + h. Every well-complemented ideal is complemented, and every complemented ideal is supplemented. Every supplemented ideal n is cofinite-dimensional, that is, dim g/n < ∞. Remember that the essential structural ingredient and invariant of a pro-Lie algebra g is the filter basis  (g) of all cofinite-dimensional closed ideals j, and recall that  (g) converges to zero. So supplemented ideals always are elements of  (g). Definition 13.6. A pro-Lie algebra g is said to be rich meaning rich in complemented ideals, if g has arbitrarily small complemented ideals; that is, for every zero neighborhood U of g there is a complemented ideal n contained in U . It is called very rich if g has arbitrarily small well complemented ideals. Finally g is said to be extremely rich if all sufficiently small ideals are well complemented. Notice that the set C (g) of complemented ideals of a pro-Lie algebra contains g and thus is not empty. Therefore, an alternative way of expressing the condition of richness is saying that

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13 Iwasawa’s Local Splitting Theorem

g is rich iff C (g) is cofinal in  (g). Every finite-dimensional Lie algebra is extremely rich by default. Proposition 13.7. (i) Any ideal of a reductive pro-Lie algebra is a direct Lie algebra summand algebraically and topologically. (ii) Every reductive pro-Lie algebra is extremely rich. Proof. (i) follows from Theorem 7.27(a). (ii) By definition of the product topology, given a zero neighborhood U , there is a finite  subset F ⊆ J and a family of zero neighborhoods Uj ⊆ sj for j ∈ F such that j ∈J Wj is a zero neighborhood contained in U if " Wj =

sj Uj

for j ∈ J \ F , for j ∈ F ,

provided the finitely many Uj , j ∈ F are selected small enough. Now  for this particular set F we construct n as in (i) above. Then we indeed have n ⊆ j ∈J Wj ⊆ U and n is well-complemented as we saw in (i). A bit more generally, we can formulate the following result: Proposition 13.8. An arbitrary product of finite-dimensional Lie algebras is very rich. Proof. By following the line of argument of the proof of Proposition 13.7, the proof follows straightforwardly. It is important that we take note of some examples which show that every pro-Lie algebra is not rich, let alone very rich. We shall use some multilinear algebra for weakly complete vector spaces which we have provided in Appendix 2, notably in Lemma A2.21, Lemma A2.22, and Corollary A2.23. Examples 13.9. (i) We write g = RN0 , where N0 = {0, 1, 2, . . . }; and for any subset J ⊆ N0 we identify RJ with the obvious vector subspace of g. The bracket operation of g is written [(x, r1 , r2 , . . . ), (y, s1 , s2 , . . . )] = (x · (0, 0, s1 , s2 , . . . ) − y · (0, 0, r1 , r2 , . . . )) = (0, 0, xs1 − yr1 , xs2 − yr2 , . . . ). Then g has a flag of ideals g[n] = R{n+1,n+2,... } , n = 1, 2, . . . , but none of them is supplemented, let alone complemented. Thus g is a pronilpotent centerfree algebra which is not rich. (ii) For n ∈ N, let Cn be the R-module defined on the underlying real vector space of C defined by the action (r, c) → r ·n c = 2π nirc : R × C → C. This is the Lie algebra action corresponding to the group action of the circle group R/Z given by def  (r +Z, c)  → e2π inr c : R/Z×C → C. We consider the product module V = n∈Z Cn with the morphism α : R → Der(V )given by α(r)(cn )n∈Z = (r·n cn )n∈Z . Now we form

The Lie Algebra Theory of the Local Splitting

573

def

the semidirect product g = V α R, that is, the product V × R with componentwise addition and the bracket [(c, r), (d, s)] = (α(r)(d) − α(s)(c), 0). Then g is a metabelian rich pro-Lie algebra that is not very rich. (iii) This example will present the pro-Heisenberg algebra h(V ) over a given weakly complete vector space V . We define h(V ) = V ×

2 >

V,

[(v, x), (w, y)] = (0, v ∧ w)),

with the componentwise topological vector space structure. Then dim V > 1 implies that h(V ) is a class 2 nilpotent pro-Lie algebra, and [h(V ), h(V )] = z(h(V )) = {0} × ; ;2 def V since the span of the v ∧ w is dense in 2 V . We record that g = h(V ) is its own nilradical n(g) and n(g)/z(g) ∼ = V. def Now assume that g = h(V ) is an ideal direct sum g1 ⊕ g2 . Assume g1 ⊆ z(g), then z(g) = [g1 + g2 , g1 + g2 ] = [g2 , g2 ] ⊆ g2 and so g1 ⊆ g1 ∩ g2 = {0}. Thus suppose that neither of the summands is zero. Then none of them is contained in z(g). Let (v1 , z1 ) ∈ g1 \ z(g) and (v2 , z2 ) ∈ g2 \ z(g). Then (0, 0) = [(v1 , z1 ), (v2 , z2 )] = (0, v1 ∧ v2 ). Thus v2 ∈ R · v1 by Corollary A2.23, that is, there is a nonzero r ∈ R such that v2 = r · v1 , and so v1 ∧ V = v2 ∧ V = {0}. Since g1 and g2 are ideals, we have {0} × (v1 ∧ V ) = [(v1 , z1 ), g] ⊆ g1 and {0} × (v2 ∧ V ) = [(v2 , z2 ), g] ⊆ g2 . Hence {(0, 0)} = {0} × (v1 ∧ V ) ⊆ g1 ∩ g2 , and this is a contradiction. Thus h(g) does not allow an ideal direct sum decomposition. In particular, if dim V = ∞, then h(V ) is a nilpotent class 2 pro-Lie algebra that is not very rich. On the other hand, if V = span{e1 , e2 }, then V is the semidirect product of the ideal z(h(V )) ⊕ R · (e1 .0) = R · e1 × R · (e1 ∧ e2 ) and the subalgebra R · (e2 , 0). The Campbell–Hausdorff multiplication on h(g) is given by (v, x) ∗ (w, y) = (v + w, x + y + 21 (v ∧ w)). The pro-Lie group (h(g), ∗) is called the pro-Heisenberg group H (V ) over the weakly complete vector space V . If dim V is infinite, H (V ) is an example of a nilpotent pro-Lie group of class 2 which does not satisfy Iwasawa’s Local Splitting Theorem. If dim V = 2, then H (V )N is a class 2 nilpotent pro-Lie group satisfying the Iwasawa Local Splitting Theorem. Exercise E13.1. Verify the details of Example 13.9 (i).  [Hint. For a finite subset F ⊆ Z identify VF = n∈F Cn in the obvious way with a subgroup of V . Show that (VZ\F × {0})  (VF × R) with the action of VF × R on VZ\F × {0} via R. Show that a direct product decomposition as required in a rich algebra is not possible.] Lemma 13.10. Let g be a pro-Lie algebra. (i) Assume that f is a finite-dimensional ideal f such that g/f is rich, respectively, very rich. Then g is rich, respectively, very rich. Indeed there are arbitrarily small

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13 Iwasawa’s Local Splitting Theorem

ideals i ∈  (g) for which there is a finite-dimensional subalgebra, respectively, ideal h containing f such that g = i ⊕ h. (ii) Assume that dim ncored (g) < ∞ then g is very rich. Proof. (i) Since dim f < ∞ and lim  (g) = 0 we find a j0 ∈  (g) such that j0 ∩f = {0}. Then j ∈  (g) and j ⊆ j0 implies j ∩ f = {0}. Since f is finite-dimensional, j0 + f is closed and since f is an ideal, j0 ⊕ f is an ideal direct sum and (j0 ⊕ f)/f ∈  (g/f). Since g/f is rich, there is an ideal i of g containing f such that i ⊆ j0 ⊕ f and that g/f is a semidirect sum i/f ⊕ h/f for a subalgebra h of g such that dim h/f < ∞, implying that dim h < ∞. So g = i+h and i∩h = f while, on the other hand, f ⊆ i ⊆ j0 ⊕f. Then the modular law implies i = (i∩j0 )⊕f. Therefore g = i+h = (i∩j0 )⊕f+h = (i∩j0 )+h and i∩j0 ∩h = j0 ∩f = {0}. Thus, since h is finite-dimensional, g is the semidirect sum (i ∩ j0 ) ⊕ h with i ∩ j0 ∈  (g). Since j0 may be taken arbitrarily small, this shows that g is rich. If g/f is very rich, h may be chosen to be an ideal, and so the sum (i ∩ j) ⊕ h is direct, showing that g is very rich. If g/f is extremely rich, the argument works for all sufficiently small i ∈  (g), and this shows that g is extremely rich. (ii) is an immediate consequence of (i) Proposition 13.11. Let g be a pro-Lie algebra, g = [g, g] its commutator algebra, and z(g) its center. Then the following statements are equivalent: dim g < ∞, dim g/z(g) < ∞.

(A) (B)

If these conditions are satisfied, then g is finite-dimensional, and therefore closed. Let z ⊆ z(g) be a cofinite-dimensional closed vector subspace and v be a finitedimensional vector subspace such that g = z ⊕ v. (i) f = g ⊕ v is a finite-dimensional ideal containing g and satisfying g = z + f. In particular, there is a central ideal a ∈  (g) such that g = a ⊕ f and a ⊆ z. (ii) g is very rich. Proof. (B) ⇒ (A): If we write g = z(g)⊕v with a finite-dimensional vector subspace v, then g = [g, g] = span{[z+v, z +v  ] : z, z ∈ z(g), v, v  ∈ v} = span{[v, v  ] : v, v  ∈ v} def

is finite-dimensional since v is finite-dimensional. (A) ⇒ (B): Assume that dim g < ∞. Denote by b : g × g → g the continuous bilinear map given by b(x, y) = [x, y]. Then it follows from Lemma A2.21 of Appendix 2 that there is a cofinite-dimensional closed vector subspaces E of g such that [E, g] = {0} and that therefore E ⊆ z(g). Thus z(g) is cofinite-dimensional as asserted. (i) We note that f contains g and recall that every vector subspace of a Lie algebra g containing g is an ideal. We find a closed vector subspace a of z such that z = a⊕(z∩f). Then g = a ⊕ f is a direct sum of pro-Lie algebras and a ⊆ z ⊆ z(g).

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(ii) This follows from (i) and Lemma 13.10. (It is also immediate from (ii) directly.) The implication (B) ⇒ (A) is always true while the implication (A) ⇒ (B) fails in general as is shown by the example g = R(N) × R with the bracket ∞ 3 6 n=1

∞ ∞  6 2  6  un , r , vn , s = 0, det(un , vn ) . n=1

n=1

Here {0} × R is g and z(g) at the same time, and the center therefore is far from being cofinite-dimensional. Exercise E13.2. Verify the following observation: Lemma. (i) Let g be a pro-Lie algebra and z an ideal contained in the center z(g) of g. If g/z is pronilpotent, then g is pronilpotent. (ii) The nilradical n(g/z(g)) of g/z(g) is n(g)/z(g). [Hint. (i) Let j ∈  (g). Then (g/j)/((z + j)/j) ∼ = g/(z + j) is a finite-dimensional quotient of g/z and is, therefore, a nilpotent Lie algebra. Also, (z + j)/j is contained in the center of g/j . Therefore g/j is nilpotent. Thus g is pronilpotent. (ii) Let m be that ideal of g containing z(g) for which m/z(g) = n(g/z(g)). Since this quotient is pronilpotent, by (i) above we know that m is pronilpotent. Hence m ⊆ n(g). Since n(g)/z(g) is pronilpotent, by Lemma 7.56 (ii) and Theorem 7.57, we have n(g) ⊆ m. Thus m = n(g).] Lemma 13.12. (i) Let p : h → k be a surjective morphism of pro-Lie algebras. Then p(ncored (h)) = ncored (k), and if ker p ⊆ z(h), then p(n(h)) = n(k). (ii) Let j be an ideal of a pro-Lie algebra g and assume that it contains the center. Then n(j)/z(j) is a quotient of an ideal of n(g)/z(g). In particular, if n(g)/z(g) is finite-dimensional, so is n(j)/z(j). Proof. (i) Let p : h → k be a surjective morphism of pro-Lie algebras. Then p(r(h)) = r(k) by Proposition 7.54. Thus p maps [h, r(h)] onto [k, r(k)]. The image of [h, r(h)] by p is closed according to Theorem A2.12 (b) in Appendix 2. Hence, in view of Theorem 7.67, we get p(ncored (h)) = p([h, r(h)] = [k, r(k)] = ncored (k). Moreover, p(n(h)) is a pronilpotent ideal and therefore is contained in n(k). Now assume that ker p is central; then p−1 (n(k))/z(h) ∼ = n(k) is pronilpotent and thus p −1 (n(k)) is pronilpotent by Exercise E13.2. (ii) According to (i), the quotient map j/z(g) → j/z(j) maps n(j)/z(g) onto n(j)/z(j). Since n(j) as a characteristic ideal of j is a pronilpotent ideal of g, we have n(j) ⊆ n(g) and thus n(j)/z(g) ⊆ n(g)/z(g). This proves the first assertion of the (ii), and the second is an immediate consequence. If we apply 13.12 (i) to the quotient morphism p : g → g/z(g),

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13 Iwasawa’s Local Splitting Theorem

we obtain (ncored (g) +z(g))/z(g) = ncored (g/z(g)), and so the full inverse image p−1 (ncored (g) /z(g)) agrees with ncored (g) + z(g). Definition 13.13. The pronilpotent pro-Lie algebra n(g)/z(g)) = n(g/z(g)) is called the nilcore of g. It is written nilcore(g). g  r(g)   n(g)   z(g) + n cored (g)  ncored  (g)  {0}

g  r(g)   n(g)   ncored (g)  +z(g)  z(g)   {0}

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

= nilcore(g)

We recall the following elementary examples showing that all containments are proper in general. In the Heisenberg algebra span{X, Y, Z}, [X, Y ] = Z, we have z(g) = ncored (g) = z(g) + ncored (g) = R · Z = g = r(g) = n(g); in the motion algebra span{X, Y, Z}, [X, Y ] = Z, [X, Z] = −Y , {0} = z(g) = ncored (g) = span{Y, Z} = z(g) + ncored (g) = n(g) = r(g) = g; in span{U, X, Y, Z}, [U, X] = Y , [U, Y ] = Z we have {0} = z(g) = R · Z = ncored (g) = z(g) + ncored (g) = span{X, Y, Z} = n(g) = g; in the direct sum of the motion algebra with R we have z(g) ⊆ ncored (g); in so(3) we have r(g) = g. In the direct sum of all of these, all of the containments in the tall Hasse diagram are proper. The following lemma deals with the case that ncored (g) ⊆ z(g). Lemma 13.14. Let g be a pro-Lie algebra satisfying ncored (g) ⊆ z(g). Then: (i) g = r(g) ⊕ s(g) is a direct sum algebraically and topologically of the radical and a unique Levi summand s(g).

The Lie Algebra Theory of the Local Splitting

577

(ii) [r(g), r(g)] = [g, ' r(g)] = ncored (g) & ⊆ z(g) and r(g) is nilpotent of class ≤ 2, that is r(g)[2] = r(g), [r(g), r(g)] = {0}. We have n(g) = r(g). (iii) [g, g] = [r(g), r(g)] ⊕ s(g), and this is a reductive pro-Lie algebra. If a is chosen so that z(g) = ncored (g) ⊕a, and r(g) = z(g) ⊕ v for a closed vector subspace v then ncored (g) ⊕v is a nilpotent ideal of class ≤ 2 and g = s(g)⊕a⊕ncored (g) ⊕v. (iv) If dim r(g) < ∞, then there is a finite-dimensional ideal f of g and a closed vector subspace a of z(g) such that g is the ideal direct sum s(g) ⊕ a ⊕ f, and g is very rich. If dim nilcore(g) < ∞ then dim r(g) < ∞. Proof. (i) By Corollary 7.75 and Theorem 7.77 (i) two Levi summands are conjugate under an inner automorphism of the form α = ead x for some x ∈ ncored (g). By our hypothesis x ∈ ncored (g) implies ad x = 0 and thus α = idg . Hence there is only one Levi summand. Thus, by Theorem 7.77 (ii) we conclude (i). (ii) From Theorem 7.67 we know that ncored (g) = [g, g] ∩ r(g) = [g, r(g)] and by hypothesis ncored (g) is central. Conclusion (i) and Lemma 7.26 imply g = [g, g] = r(g) + s. The factor algebra r(g)/z(g) is the center of g/z(g). Thus (ii) follows. (iii) is now straightforward from (ii). (iv) Since [r(g), s(g)] = {0} by (i), we have z(g) = z(r(g)). Therefore, if r(g) is finite-dimensional, then 13.11 shows that r(g) = a ⊕ f with a central ideal a and a finite-dimensional ideal f of r(g) and then also of g by (i). Then (i) also implies g = s(g) ⊕ a ⊕ f. Since s(g) ⊕ a is reductive and dim f < ∞, Proposition 13.7 (ii) and Lemma 13.10 imply that g is very rich. The nilcore nilcore(g) is r(g)/z(g); so if this algebra is finite-dimensional then r(g) is finite-dimensional by 11.13. Now we are ready for the structure theorem that is the Lie algebra nucleus of the local splitting theorems of this section. The Structure of Pro-Lie Algebras with Finite Dimensional Nilcore Theorem 13.15. For a pro-Lie algebra g, the following two conditions are equivalent: (i) The nilcore nilcore(g) = n(g)/z(g) of g is finite-dimensional. (ii) g is an ideal direct sum, algebraically and topologically, of a reductive pro-Lie algebra and a finite-dimensional Lie algebra. If these conditions are satisfied, then all sufficiently small cofinite-dimensional ideals of g are direct summands. If the maximal compactly embedded ideal m(g) is cofinite-dimensional in g then dim nilcore(g) < ∞. Proof. Propositions 13.7 (ii) and Lemma 13.10 show that Condition (ii) implies that g is very rich. If (ii) is satisfied, then g is a direct sum g1 ⊕ g2 with a reductive ideal g1 and a finite-dimensional ideal g2 . Then n(g) = n(g1 ) ⊕ n(g2 ) and z(g) = z(g1 ) ⊕ z(g2 ) and, accordingly, nilcore(g) ∼ = nilcore(g1 ) × nilcore(g2 ). But nilcore(g1 ) = {1} and dim g2 < ∞. Accordingly, dim nilcore(g) < ∞.

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13 Iwasawa’s Local Splitting Theorem

By Corollary 12.34 (iii), m(g) ∩ r(g) = z(g). Therefore, if g/m(g) is finitedimensional, then r(g)/z(g) = r(g)/(r(g) ∩ m(g)) ∼ = (r(g) + m(g))/m(g) is finitedimensional and thus certainly n(g)/z(g) is finite-dimensional, that is, (i) holds. It therefore remains to prove (i) ⇒ (ii): By Exercise E13.2 (ii), n(g/z(g)) = n(g)/z(g). So, by hypothesis, g/z(g) has a finite-dimensional nilradical. Since the nilradical contains the coreductive radical (see Theorem 7.66ff.), g/n(g) is reductive. So g/z(g) is very rich by Proposition 13.7 (ii). Thus Lemma 13.10 (i) applies with g/z(g) in place of g and with n(g)/z(g) in place of f and shows that there are two closed ideals j and h of g, both containing z(g) such that g = j + h,

j ∩ h = z(g),

n(g) ⊆ h,

and

dim h/z(h) < ∞.

(1)

We notice that [j, h] ⊆ j ∩ h = z(g). Thus the commutator function (x, y)  → [x, y] : j × h → z(g) is a Z-bilinear continuous morphism. Thus for each b ∈ h the morphism x  → [x, b] : j → z(g) vanishes on [j, j], and for each a ∈ j, the morphism y  → [a, y] : h → z(g) vanishes on [h, h]. Thus & ' & ' [j, j], h + j, [h, h] = {0}. (2) Next we discuss the ideal j: The nilradical n(j) of j is a characteristic ideal and is therefore a pronilpotent ideal of g. Hence it is contained in n(g) ⊆ h, and so it is contained in fact in j ∩ h = z(g). Since ncored (j) ⊆ n(j) this entails ncored (j) ⊆ z(j). Now Lemma 13.14 applies to j and shows that r(j) is nilpotent of class ≤ 2 and j = r(j) ⊕ s(j)

(3)

is a direct sum with a unique Levi summand s(j). Now s(j) ⊆ [j, j] commutes elementwise with h by (2) and with r(j) by (3). Hence it is an ideal of g = j + h while g is the semidirect sum of r(g) and any Levi summand s by the Levi–Mal’cev Theorem 7.52 (i). Now s(i) ⊆ s by the Levi–Mal’cev Theorem 7.77 (iv). Since s is semisimple, s is a direct sum s(i) ⊕ t with a semisimple direct factor t of s, and so g = s(j) ⊕ g1 where the ideal direct summand g1 is r(g) ⊕ t. If g1 has the structure asserted in Statement (ii) of the theorem, then (ii) holds for g. We may and will therefore assume that s(j) = {0} so that j is nilpotent of class ≤ 2, that is, j = n(j) and dim j/z(g) < ∞. Then (1) implies dim g/z(g) < ∞. By Proposition 13.11 applied to g where z(g) takes the place of z, there is a finitedimensional ideal f of g containing g = s + ncored (g) (where s is any Levi summand of g), and there is a central ideal a ⊆ z(g) such that g = a ⊕ f = (a ⊕ r(f)) ⊕ s.

(4)

This completes the proof of (ii). In Example 13.9 (iii) we saw that for every infinite-dimensional weakly complete vector space V , the class 2 nilpotent pro-Heisenberg algebra h(V ) fails to be very rich, and its nilcore nilcore(h(V )) is isomorphic to V . This example shows that Theorem 13.15 is likely to come very close to the best possible if one looks for a reasonably general sufficient condition for a pro-Lie algebra to be very rich.

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Splitting on the Group Level We now return from the pure algebra level to the pro-Lie group level. Recall that the nilradical N(G) of a connected pro-Lie group is the largest connected pronilpotent normal subgroup. We notice that if G is a pro-Lie group, then N (G0 )/Z(G0 )0 is a pro-Lie group by Theorem 4.28 (i). This pro-Lie group is called the nilcore of G. It is written nilcore(G). Proposition 13.16. Let G be a connected pro-Lie group. Then: (i) Its nilcore nilcore(G) is a simply connected pronilpotent pro-Lie group. (ii) Its Lie algebra L(nilcore(G)) is naturally isomorphic to the nilcore nilcore(g) of its Lie algebra g = L(G). (iii) The function exp : (nilcore(g), ∗) → nilcore(G) is a natural isomorphism of pro-Lie groups. Proof. By Theorem 9.50, the maximal compact subgroup of N (G) is contained in the center Z(G). From 11.26 it follows that nilcore(G) has no nontrivial compact subgroups. Then (i) and (iii) follow from Theorem 11.27. (ii) By Theorem 10.42 we have L(N (G)) = n(g), and by Proposition 9.23 we know L(Z(G)) = z(g). Now Corollary 4.21 (i) proves the claim. (iii) A combination of Theorem 8.5, Proposition 8.8 and Theorem 8.15 shows, that for a simply connected nilpotent pro-Lie Group N , the exponential function expN : (L(N), ∗) → N is an isomorphism of topological groups. The assertion is then a consequence of conclusion (ii) above. The structure of the nilcore of a connected pro-Lie group is therefore completely known when its Lie algebra is known. Trivially, Z(G)0 ⊆ N (G) ∩ Z(G). Conversely, N(G) ∩ Z(G) ⊆ Z(N (G)), and by Proposition 11.20 (i), the center of N (G) is connected. So Z(G)0 ⊆ N (G) ∩ Z(G) ⊆ Z(N )0 . But Example 13.9 (ii) shows that Z(G) = {0} and Z(N )0 = Z(N ) = {0} may occur. Since N(G)/Z(N (G)) is a quotient of nilcore(G), the finite-dimensionality of N(G)/Z(N(G)) is a weaker condition than that of nilcore(G). We are now ready for the Iwasawa Local Splitting Theorem. The Local Splitting Theorem Theorem 13.17. Let G be a pro-Lie group whose nilcore def

nilcore(G) = N(G0 )/Z(G0 )0 is finite-dimensional. Then every identity neighborhood contains a closed normal almost connected subgroup N such that G/N is a Lie group and that there is a morphism  → G such that the morphism μ : N × G/N  → G, μ(n, x) = nϕ(x) is an ϕ : G/N

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13 Iwasawa’s Local Splitting Theorem

 , and N × G/N open morphism having a discrete kernel. In particular, G, N × G/N are locally isomorphic. def

Proof. Let g = L(G) be the Lie algebra of G. By Proposition 13.16, nilcore(g) = g/n(g) is finite-dimensional. Then by 13.15, the Structure Theorem of Pro-Lie algebras with Finite Dimensional Nilcore, g is extremely rich (see Definition 13.6.) Thus all sufficiently small cofinite-dimensional ideals are well complemented. By Corollary 4.21 (ii), {L(N ) | N ∈ N (G)} converges to zero. Hence for all sufficiently small N ∈ N (G), the ideal L(N ) is well complemented. Now Theorem 13.3 of the Local Splitting of Lie Group Quotients and its Corollary 13.4 together prove the theorem. By the definition of analytic subgroups in Chapter 9 (see Definition 9.5, Proposition 9.6), im(ϕ) is an analytic subgroup whose Lie algebra f is the direct summand for which g = n ⊕ f. We note that (n, x) ∈ ker μ iff n = ϕ(x)−1 ∈ N ∩ im ϕ = A(f). That is, x  → (ϕ(x)−1 , x) : ϕ −1 (N ) → ker μ is an isomorphism. Let L be the underlying Lie group of the analytic group A(f). Then, the morphism (n, )  → n : N × L → G is a surjective morphism implementing a local isomorphism. Its kernel {(−1 , ) |  ∈ N ∩ A(f)} is algebraically isomorphic to a discrete and hence central normal subgroup of L and is therefore a finitely generated abelian group; it is algebraically isomorphic to N ∩ A(f). We retain the notation of the Local Splitting Theorem 13.17 and provide a bit of additional information on the local splitting. Proposition 13.18 (The Sandwich Theorem for Local Splitting). Let G be a connected def

pro-Lie group whose nilcore nilcore(G) = N (G)/Z(G)0 is finite-dimensional. We assume that A(f) = im ϕ is a closed analytic subgroup L. Then 13.17 applies, and the quotient map q : G → G/D, D = N ∩ L implements a covering morphism. Further, ν : N/D × L/D → G/D,

ν((nD, D)) = nD,

 → L/D be the quotient is an isomorphism. Let qN : N → N/D and πL/D : G/N morphisms, each with kernel isomorphic to D. Then the sandwich diagram  N ×⏐G/N ⏐ qN ×πG/D  N/D × L/D

μ

−−→

G ⏐ ⏐q  −−→ G/D ν

commutes. Proof. Given the information in Theorem 13.17, all of the assertions of the corollary amount to a straightforward verification. We should record that the Structure Theorem of Pro-Lie algebras with Finite Dimensional Nilcore 13.15 contains some information which is not yet reflected in the theorems formulated above.

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The Structure Theorem for Groups with Finite Dimensional Nilcore Theorem 13.19. Let G be a connected pro-Lie group. Then the following statements are equivalent: (i) The nilcore nilcore(G) is finite-dimensional. (ii) G is locally isomorphic to the direct product of a closed normal almost connected subgroup N with reductive identity component N0 and a connected Lie group L. Proof. (i) ⇒ (ii): By The Local Splitting Theorem 3.16, we find a closed normal subgroup N (which we may choose as small as we like) a connected Lie group L such that N × L and G are locally isomorphic and that g = n ⊕ f where n = L(N ) and f = L(L). By Theorem 13.15, if N and then n is small enough, n is an ideal of a reductive algebra and is, therefore, reductive itself. By Theorem 10.29 (i) this shows that N0 is reductive. (ii) ⇒ (i): If (ii) holds, then g = n ⊕ f with a reductive Lie algebra n = L(N ) and a finite-dimensional ideal f = L(L). Then nilcore(g) = nilcore(n) ⊕ nilcore(f) = nilcore(f ) is finite-dimensional since f is finite-dimensional. Information on the structure of reductive connected pro-Lie groups is to be found in Theorems 10.29, 10.32, 10.48, and 11.8. Our discussion now leads back to the domain of locally compact groups, where the original theorem of Iwasawa’s started. It was concerned with connected locally compact groups. The following consequence of our present theory is, therefore, more general. The Strong Iwasawa Local Splitting Theorem Corollary 13.20. Let G be a locally compact pro-Lie group. Then there is an open subgroup G of G and there are arbitrarily small compact normal subgroups N of G such that G/N is a Lie group and G and N × G/N are locally isomorphic. Proof. We recall from [145, p. 175], that a locally compact group is automatically a pro-Lie group if it is almost connected. So let G be an open almost connected subgroup of G which always exists because every locally compact totally disconnected group (such as for instance G/G0 ) has compact open subgroups. Let g = L(G) = L(G). By Corollary 12.88, g/m(g) the factor algebra modulo the largest compactly embedded ideal is finite-dimensional. Hence from the Structure of Pro-LieAlgebras with Finite Dimensional Nilcore 13.15 and from Proposition 13.16 (ii) we know that nilcore(g) = L(nilcore(G)) is finite-dimensional. So the Local Splitting Theorem 13.17 applies to G and proves the assertion. The hypotheses of the Local Splitting Theorem 13.17 allow us to draw additional structural conclusions which are important. We recall that in the preceding chapter

582

13 Iwasawa’s Local Splitting Theorem

we have discussed the special structure of centrally supplemented groups: see Definition 12.72ff. We recall that every connected pro-Lie group G has a unique characteristic largest compact normal abelian and hence central subgroup KZ(G), and that G/ KZ(G) has no nontrivial compact central subgroups (see Definition 9.47ff., notably Theorem 9.50). Theorem 13.21 (Theorem on Central Supplementation). Let G be a connected pro-Lie def

group whose nilcore nilcore(G) = N (G)/Z(G)0 is finite-dimensional. Then G = KZ(G)A(g, G). In particular, G is centrally supplemented. Proof. We factor KZ(G) and assume KZ(Z) = {1}; we must show G = A(g, G). Let g = j ⊕ h where j is a reductive ideal and h is a finite-dimensional ideal according to Theorem 13.15. Since {L(N )|N ∈ N (G)} converges to 0 in g by 4.21 (ii), there is an N ∈ N (G) such that L(N ) ⊆ j. Since j is a product of simple Lie algebras and copies of R by the Structure Theorem of Reductive Pro-Lie algebras 7.27, we have j = L(N ) ⊕ f with a finite-dimensional semisimple ideal f and so we have g = L(N ) ⊕ f ⊕ h. It is therefore no loss of generality if we assume henceforth that there is an N ∈ N (G) such that L(N) = j. Then G/N is a Lie group and thus G = NA(g, G)

(1)

by 4.22 (iii). The analytic subgroup A(g, G) is the product of the normal analytic subgroups A(j, G) and A(h, G) which commute elementwise. By 4.22 (i), N0 = A(j, G). Then A(h, G) ⊆ Z(N0 , G) and thus G = N0 Z(N0 , G) and Z(N0 ) ⊆ Z(G). Further, N0 is reductive by Theorem 10.48, and therefore satisfies N0 = KZ(N0 )A(j, N0 ) and is centrally supplemented in its own right. Since KZ(N0 ) ⊆ KZ(G) = {1}, we have N0 = A(j, G) ⊆ A(j, G)A(h, G) = A(g, G).

(2)

G = NA(h, G).

(3)

Then (1) implies Now L(G/N0 ) = L(G)/L(N0 ) = L(G)/L(N ) = g/j ∼ = h by Corollary 4.21 (i). Therefore G/N0 is finite-dimensional by Definition 9.43. Then G/N0 is locally compact metric by Theorem 9.44, and we may assume that N/N0 is compact totally discon = (h) → G be a morphism such that im ϕ = A(h, G). Then nected. Let ϕ : G/N def −1  and is therefore finitely generated. Z = ϕ N is a discrete central subgroup Z of G/N def Thus Theorem 5.32 (iv) applies to A = ϕ(Z) and shows that A ∼ = comp(A) × Zn for some nonnegative integer. However, A ⊆ Z(G) whence comp(A) ⊆ KZ(G) = {1}. Thus A is free discrete and therefore agrees with ϕ(Z) = N ∩ A(h, G). Then the map i : A(h, G)/A → G/N, i(aA) = aN is a bijective morphism by (3). Let ψ be the corestriction of ϕ to its image. Then the composition quot ψ i  −− G/N → A(h, G) −−−−→ A(h, G)/A −−→ G/N

Splitting on the Group Level

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is a surjective morphism between locally isomorphic Lie groups and is, therefore, open and implements itself a local isomorphism. This implies that ψ is open and def H = A(h, G) is a Lie group and G = NH , N ∩ H = A. Thus N/A ∼ = G/H is connected and is locally isomorphic to N and is, therefore a connected pro-Lie group. By (2) N0 = expN j and by Corollary 4.22 (iii), im expN/A  = N0 A. Therefore N0 A/N0 is a connected space which is completely regular as a Hausdorff topological group. Its cardinality is therefore 1 or at least 2ℵ0 (see for instance Exercise E12.13ff.). Since A is countable, it follows that N ∩ H = A ⊆ N0 . By Theorem 9.44, AN0 /N0 = ϕ(Z)N0 /N0 is dense in N/N0 and therefore N = N0 , that is N = A(j, G). Now (1) shows that G = A(g, G) which is what we had to show. From [102, p. 379, Theorem 8.20 (i)] it follows that every compact abelian group K contains a compact totally disconnected subgroup D such that K = Ka D (see also [102, p. 470, Theorem 9.41]). Therefore, in the circumstances of Theorem 13.21, we find a compact zero-dimensional central subgroup D of G such that G = DA(g, G) = DexpG g. We can therefore write Corollary 13.22. Let G be a pro-Lie group with a finite-dimensional nilcore. Then there is a compact totally disconnected central subgroup D of G and an open surjective morphism  → G, μ(d, x) = dπG (x) μ: D × G with a prodiscrete kernel.  → G whose image is Proof. We recall that we have a universal morphism πG : G exactly A(g, G) = im expG . The existence of the surjective morphism μ then follows from the preceding remarks in view of Theorem 13.21. The Open Mapping Theorem 9.60 finally shows that μ is open. Since every connected locally compact is a pro-Lie group byYamabe’s Theorem, and since the nilcore of a locally compact group is always finite-dimensional, Corollary 4.2 implies at once the following consequence: Corollary 13.23. Let G be a connected locally compact group. Then there is a compact totally disconnected central subgroup D of G and an open surjective morphism  → G, μ: D × G

μ(d, x) = dπG (x)

with a prodiscrete kernel.  while being a simply connected pro-Lie group, is in general We should recall that G, not locally compact. This is illustrated by any infinite-dimensional compact connected  is the additive group of the pro-Lie algebra L(G) ∼ abelian group G, for which G = RJ , card J = dim G. (See [102], Chapter 8, notably 8.20.) The structure of simply connected pro-Lie groups was rather fully exposed in Theorem 8.14 and Corollary 8.15:

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13 Iwasawa’s Local Splitting Theorem

 is the semidirect product R  S of its radical A simply connected pro-Lie group G I R, which is homeomorphic to R for a suitable set I , and a Levi complement S for which there is a family of simply connected simple Lie groups Sj , j ∈ J such that S ∼ = j ∈J Sj .

Some Comments on Connectedness We observe that, classically, the Iwasawa Local Splitting Theorem 13.20 required the hypothesis of connectivity. The presentation of the local splitting theory in the frame work of pro-Lie group theory, culminating in Theorem 13.17 does not require this hypothesis, as long as we are dealing with pro-Lie groups. This is plausible because the conclusion is a local one. The global version concludes the existence of an open  → G, and not a surjective one. We should recall that by the morphism N × G/N  of a pro-Lie group H (see 6.6ff.) it is always a definition of the universal group H connected, indeed simply connected group (see Theorem 8.15) regardless of whether H is connected or not. If the underlying space of H has a universal covering space, then the universal group agrees with the universal covering group (see Theorem 8.21). In particular, if H is a Lie group (such as in the case for G/N whenever N ∈ N (G)),  is the universal covering group of H0 . then H In the case of a locally compact group G, by the results of Yamabe, there is always an open subgroup G of G which is a pro-Lie group. Then the pro-Lie group result applies to G and produces arbitrarily small normal subgroups N of G such that G and N × G/N are locally isomorphic. But since G and G trivially are locally isomorphic, one can still say that a locally compact group, quite generally, is locally isomorphic to the product of an arbitrarily small compact subgroup (not necessarily normal!) and some Lie group. In [103] this was argued directly from the “connected” version of Iwasawa’s Local Splitting Theorem for Locally Compact Groups; but that required some technical arguments.

Postscript The litmus test for the entire Lie theory of pro-Lie groups is how the Iwasawa Local Splitting Theorem (1949, [120, p. 547, Theorem 11]) arises from the information provided in this book. Iwasawa Local Splitting Theorem for Locally Compact Groups. In every identity neighborhood of a connected locally compact group G there is a closed normal subgroup N such that G/N is a Lie group and G and N × G/N are locally isomorphic groups.

Postscript

585

Historical accuracy demands our noting that Iwasawa himself in 1949 required the hypothesis of G to be a pro-Lie group. Only the concerted combined effort of Gleason, Montgomery, Zippin, Yamabe resulted in showing in the end (1953) that local compactness and almost connectivity implied the property of being a pro-Lie group. (See [145, p. 175]). The significance of the Iwasawa Local Splitting Theorem is that like no other result it shows how the structure theory of locally compact connected groups reduces to Lie group theory and the structure theory of compact groups both of which are very well understood. In this chapter we have seen three different types of connected pro-Lie groups that fail to have local splitting: one being pronilpotent center free, one being metabelian centerfree, and one being nilpotent of class 2 with a large center equal to its closed commutator group (see Examples 13.9). All of these examples have one feature in common. If N(G) denotes the largest pronilpotent connected normal subgroup and Z(G) the center def

of the pro-Lie group G, then the so-called nilcore, written nilcore(G) = N (G)/Z(G)0 , is infinite-dimensional. The nilcore is canonically attached to any pro-Lie group, and it is a simply connected pronilpotent pro-Lie group and is therefore isomorphic to (n, ∗), where n denotes its pronilpotent pro-Lie algebra and ∗ the Campbell–Hausdorff multiplication (as discussed in Theorem 8.5). This means that it is homeomorphic to the underlying topological vector space of a weakly complete vector space ∼ = RJ for some set J . We can therefore call card J the dimension of the nilcore; this is a cardinal invariantly associated to G, called the nildimension, written ν(G). In all of the examples in 13.9, ν(G) is infinite, while, on the other hand, if G is locally compact, ν(G) is necessarily finite. The principal result of this chapter is our Local Splitting Theorem (13.17): Local Splitting Theorem for Pro-Lie Groups. If the nildimension ν(G) of a proLie group G is finite, then every identity neighborhood G contains a closed normal almost connected subgroup N such that G/N is a Lie group and the two groups G and N × G/N are locally isomorphic. In fact we show that, under the hypotheses of the Local Splitting Theorem, for all sufficiently small N, the identity component N0 is reductive. See Chapter 10 and 11 for the structure of reductive pro-Lie groups. This illustrates the fact that pronilpotency is an obstruction without which the structure theory of connected pro-Lie groups would be a clear-cut affair; it further illustrates that, once this obstruction is minimized, the structure theory of connected pro-Lie groups reduces to Lie group theory and the theory of reductive pro-Lie groups which we have seen to be tractable. In view of the fact, that for any locally compact group G, the nildimension ν(G) is finite, our Local Splitting Theorem for Pro-Lie Groups immediately implies the Iwasawa Local Splitting Theorem. Indeed the proof of this implication is now deceptively short (see Corollary 13.19). However, the proof of the Local Splitting Theorem is done with the aid of the Lie theory of pro-Lie groups. At the heart of the proof is

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13 Iwasawa’s Local Splitting Theorem

the following theorem on pro-Lie algebras g, for which one defines the nildimension ν(g) completely analogously as on the group level, and which is proved in the middle section of the chapter: Structure Theorem of Pro-Lie Algebras with Finite Nildimension. For a pro-Lie algebra g, the following two conditions are equivalent: (i) The nildimension ν(g) is finite. (ii) g is an ideal direct sum, algebraically and topologically, of a reductive pro-Lie algebra and a finite-dimensional pro-Lie algebra. If these conditions are satisfied, then all sufficiently small cofinite-dimensional ideals of g are direct summands. Moreover, there is a compact totally disconnected  → G. central subgroup D of G such that there is a quotient morphism μ : D × G If the maximal compactly embedded ideal m(g) is cofinite-dimensional in g then ν(g) < ∞. It does require a bit of work to translate this infinitesimal result to the group level. A portion of this work is done in the first section, and another portion, the smaller one, in the third section. In the process we resume a theme that we dealt with in Chapter 12 under the name of central supplementation (see Definition 12.72ff.). Indeed we show now that for a connected locally compact group G there is a compact totally disconnected central subgroup D of G and an open surjective morphism  → G, μ: D × G

μ(d, x) = dπG (x)

with a prodiscrete kernel. This is probably the next best surrogate, for locally compact connected groups, to the simply connected covering of connected Lie groups.

Chapter 14

Catalog of Examples

As we proceeded through the book we encountered many examples. As is usual, in each spot where an example occurs, it is meant to illustrate one or two particular points. Often such points are perfectly well illustrated by examples which are, in principle, well known, be it that they are drawn from the supply of finite-dimensional classical Lie groups or be it that they originate as compact groups (frequently indeed compact abelian groups). But we also saw a considerable collection of pro-Lie groups which neither are Lie groups, nor are compact nor even locally compact. It therefore serves a meaningful purpose for the user of this book to have a catalog of examples assembled in one place, and the place is here in this chapter. In many cases we are recording counterexamples here, counterexamples to conjectures that seem to be plausible as we proceed through the theory of pro-Lie groups. Prerequisites. This catalog requires material covered in this book and some items discussed in [102].

Classification of the Examples in the Catalog We coarsely group examples as follows. 1. 2. 3. 4. 5. 6. 7. 8. 9.

Abelian pro-Lie groups. Nilpotent and pronilpotent pro-Lie groups that are not abelian. Solvable and prosolvable pro-Lie groups that are not pronilpotent. Semisimple and reductive pro-Lie groups. Mixed pro-Lie groups. Examples concerning the definition of Lie and pro-Lie groups. Analytic subgroups of pro-Lie groups. Examples concerning simple connectivity. Example concerning g-module theory.

For each of the examples, as a rule, we record the coordinates of the example in this book and occasionally some extra references. Often we complement the description of the example with comments on the significance of the particular example in the fabric of the theory of pro-Lie groups. A word on the numbering system in this chapter: We have classified the examples into sections and subsections. Attention. Examples which contain particular pro-Lie groups which are not locally compact are marked by an asterisk ∗ .

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14 Catalog of Examples

Abelian Pro-Lie Groups Examples Illustrating Projective Limits The reader should be reminded that all locally compact abelian groups are pro-Lie groups. This fact provides us with a large supply of “traditional” pro-Lie groups. In particular, every almost connected abelian pro-Lie group is a product of a finite-dimensional vector group ∼ = Rn for some n ∈ N and a compact group. Thus the great variety of such groups arises from the variety of compact abelian groups. (See for instance Chapter 8 of [102].) Examples 14.1. Assume that we have a sequence ϕn : Gn+1 → Gn , n ∈ N of morphisms of topological groups: ϕ1

ϕ2

ϕ3

ϕ4

G1 ←− G2 ←− G3 ←− G4 ←− · · · We obtain a projective system of topological groups by defining, for natural numbers j ≤ k, the morphisms the morphisms fj k : Gk → Gj by " if j = k, idGj fj k = ϕj  ϕj +1  · · ·  ϕk−1 if j < k. Then G = limn∈N Gn is simply given by {(gn )n∈N | (∀n ∈ N) ϕn (gn+1 ) = gn }. Citation. 1.20. Comment. This is a simple but widespread and fairly representative example of the method to construct topological groups through a projective limit construction. Specific examples will follow. Examples 14.2. Let Z denote the discrete additive group of integers and Q the additive group of rational numbers. Choose a natural number p, set Q(pn ) = Q/pn Z and Z(p n ) = Z/p n Z for each n ∈ N, and give these groups the discrete topology. The best known case is when p is a prime number; but the construction does not depend on such a restriction. Define ψn : Z(pn+1 ) → Z(pn ) by ψn (z + pn+1 Z) = z + pn Z and ϕn : Q(pn+1 ) → Q(pn ) by ϕn (q + pn+1 Z) = q + pn Z. We have an infinite commutative diagram Z(p) ⏐ ⏐ incl Q(p)

ψ1

←−−

Z(p 2 ) ⏐ ⏐ incl

ψ2

←−−

Z(p 3 ) ⏐ ⏐ incl

ψ3

←−−

Z(p 4 ) ⏐ ⏐ incl

ψ4

←−− · · ·

←−− Q(p2 ) ←−− Q(p3 ) ←−− Q(p4 ) ←−− · · · ϕ1

ϕ2

ϕ3

ϕ4

The projective limit of the upper row in the category TopGr is called the group Zp of p-adic integers and the projective limit of the second row is called the group Qp of

Abelian Pro-Lie Groups

589

p-adic rationals . By the definition of the limits we see that Zp ⊆ Qp and thus there is an inclusion morphism making the completed diagram commutative Z(p) ←−− ⏐ ⏐ incl

ψ1

Z(p 2 ) ⏐ ⏐ incl

←−−

ψ2

Z(p 3 ) ⏐ ⏐ incl

←−−

ψ3

Z(p 4 ) ⏐ ⏐ incl

←−− · · ·

Zp ⏐ ⏐ incl

←−−

Q(p 2 )

←−−

Q(p 3 )

←−−

Q(p 4 )

←−− · · ·

Qp .

Q(p)

ϕ1

ϕ2

ϕ3

ψ4

ϕ4

Properties. The group Zp is compact and open in Qp . Thus Qp is locally compact. Since all Z(pn ) = Z/pn Z are rings and the maps ψn are morphisms of rings, Zp is a compact ring. The filter basis {pn Z : n ∈ N} is a basis for the zero neighborhoods of the topology of Qp , called the p-adic topology. A sequence (qn )n∈N ∈ QN gives an element (qn + p n Z)n∈N ∈ Qp = limn Q(pn ) if and only if for m ≤ n we have qm − qn ∈ pm Z, that is, if and only if cofinally (qn )n∈N agrees with a Cauchy sequence for the p-adic topology. Thus Qp is a completion of Q and can be shown to be a field (see for instance [199]). The additive group G of the field Qp of p-adic rationals (see Example 1.20 (A)) is a locally compact noncompact nondiscrete abelian group with KZ(G) = comp(G) = G. In particular, G has no largest compact subgroup. Citation. Exercise E1.13. Comment. The groups Zp and Qp play a crucial role in algebraic number theory. (See for instance [199].) Examples 14.3. For any additively written abelian group A and any natural number p we can define an endomorphism μp : A → A by μp (g) = p · g for g ∈ A. It is customary, however, to write p in place of μp . Now for each n ∈ N we can set Gn = A and define ϕn : Gn+1 → Gn to be μp . Accordingly, one defines Ap = limn Gn . Comment. Significant examples in the locally compact domain will follow at once. def

Examples 14.4 (Special Case). Now set Gn = R and Hn = T = R/Z and consider the quotient map qn : Gn → Hn . Since this morphism is equivalent to the exponential def

function t → e2π it : R → SS 1 = {z ∈ C : |z| = 1} (which induces an isomorphism T → SS 1 ) we shall call it exp. Then we have an infinite commuting diagram p

p

p

p

p

p

p

p

←− R ⏐ ←− R ⏐ ←− R ⏐ ←− · · · ⏐ ⏐ ⏐ exp exp exp ←− T ←− T ←− T ←− · · ·

R ⏐ ⏐ exp T

The projective limit of the upper row is R, identified with the subgroup of RN of all (p−n r)n∈N and the projective limit of the lower row is called the p-adic solenoid Tp . There is a morphism expTp : R → Tp induced by the morphism expTN : RN → TN , and we have the enlarged commutative diagram R ⏐ ⏐ exp T

p

p

p

p

←− R ⏐ ←− R ⏐ ←− R ⏐ ←− · · · ⏐ ⏐ ⏐ exp exp exp ←− T ←− T ←− T ←− · · · p

p

p

p

R ⏐ ⏐expT  p Tp .

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14 Catalog of Examples

Properties. The p-adic solenoid is compact, connected, 1-dimensional. Its identity arc component Ga is not closed. Citation. 1.20 A(ii) (See also [102, 1.28, 1.38], and Chapter 8 of [102] in general.) Example 14.5. Let H = Tp be the p-adic solenoid let G = L(H ) = R, and let f : G → H be the exponential function expH : L(H ) → H . Properties. Then L(f ) : L(G) = R → L(H ) is an isomorphism, but f is far from surjective. In fact, im f = Ha is the identity arc componentof H and as abelian groups, the arc component factor group π0 (H ) = H /Ha and Ext p1∞ · Z, Z are isomorphic groups of continuum cardinality. Citation. 1.20 (ii). (See also [102, Chapter 8].) Example 14.6. Let G = R × Zp for the group of p-adic integers Zp , which we assume to contain Z. Further let H = G/{(n, −n) | n ∈ Z} ∼ = Tp , and let f : G → H be the corresponding quotient morphism. Note that H is compact and connected. Properties. The identity component of G is G0 = R × {0}, and its image f (G0 ) = Ha under the surjective morphism f is not equal to H = H0 . Therefore, the relation H0 = f (G0 ) for a quotient morphism f such as in 4.22 (iii) cannot be improved. Citation. Paragraphs following the proof of 4.22. See also [102, Exercise E1.11 following Definition 1.30]. Example 14.7. Let G = Z and let N be the filter basis of all infinite cyclic subgroups. We may index N by N, where we partially order N by m|n (meaning that there is a k ∈ N such that n = km). Then GN = limn∈N Z/Zn is a compact group, into which Z is continuously and densely injected but not embedded.  Properties. N = {0}, but N does not converge to 0 in the discrete topology of Z. n is finite, but none of the members of N If Zm ⊇ Zn, then m|n and Zm/Zn ∼ = Z/Z m is compact. With Pontryagin Duality as expounded e.g. in [102, Chapter 7 and 8], it is easy to identify the group GN explicitly. The character group of Z/Zm may be identified with m1 · Z/Z and the dual of the quotient map Z/Zn → Z/Zm if m|n with the inclusion map m1 Z/Z → n1 Z/Z. Since GN = limn∈N Z/nZ, the dual is the direct  limit n∈N n1 · Z/Z = Q/Z = p prime Z(p∞ ), where Z(p∞ ) = p1∞ · Z/Z, p1 · Z = #m $  ∼ % p prime Zp . The dual of the p n · Z | m ∈ Z, n ∈ N . Hence GN = Q/Z = % inclusion morphism Q/Z → R/Z = T is a morphism Z → Q /Z and this is none other than the morphism γG : G → GN . Citation. 1.37. Comment. From Theorem 1.30 we know the significance of the convergence of the filter basis N . Proposition 1.38 provides further elucidation.

Abelian Pro-Lie Groups

591

The Arc Component of the Identity Example 14.8. In [102] it was shown that for any locally compact abelian group G, the identity arc component Ga agrees with the image exp L(G) of the exponential function (see [102, Theorem 8.30 (ii)]), and that for a compact abelian group G, the factor group  Z) (loc. cit. (iii)). G/Ga , as an abstract abelian group, is isomorphic to Ext(G, Properties. In the case of the p-adic solenoid Tp of Example 14.4, we have L(Tp ) ∼ = R, and expTp : L(Tp ) → Tp is injective. Thus (Tp )a is a copy of R endowed with a properly coarser topology. By Proposition 2.22 we have R ∼ = L(Tp ) = L((Tp )a ). In particular, (Tp )a is a topological group with a Lie algebra (due to Theorem 2.14), and its Lie algebra is R. The solenoid Tp has continuum many arc components. We say that an abelian group A is an S-group if every finite rank pure subgroup is free and splits. The group ZN is an ℵ1 -free group which is not a Whitehead group: see e.g. [102, Example A1.65]. Example 14.9. The group A = ZN has the following properties: (i) A is an S-group. (ii) A is not a Whitehead group. (iii) The subgroup Z(N) of A is a countable free pure subgroup which does not split. Citation. 4.4. Comment. In this area of the theory of abelian groups, ZN is a universal test example. For instance, Proposition 4.2 cannot be complemented by another equivalent condition which would say: Every countable pure subgroup splits. The significance of S-groups for the theory of pro-Lie groups is the fact that for a compact connected abelian group G the corestriction expG : L(G) → Ga of the exponential function is open (that is, is a quotient morphism) if and only if the character  of G is an S-group. (See Propositions 4.7, 4.10, 4.12, 4.13 and the summary group G in Theorem 4.14.) We can apply Example 14.9 and obtain def

Example∗ 14.10. The group G = RR with the product topology has a closed subgroup K which is free with countably many generators such that G/K is incomplete. Properties. G is an abelian connected arcwise connected and locally connected proLie group with an incomplete quotient. The weight w(G) is 2ℵ0 . We do show in Theorem 4.28 (i) that a quotient of a connected pro-Lie group modulo a connected closed subgroup is complete. (“Almost connected” suffices in both places.) The group K is a countable pro-Lie (in fact prodiscrete) group of weight w(K) = 2ℵ0 . In particular, it fails to be a Baire space, and so fails to be locally compact and it also fails to be Polish. There is a bijective morphism of pro-Lie groups Z(N) → K with discrete domain which shows that the Open Mapping Theorem fails for pro-Lie groups in general. We do show that a surjective morphism H → H1 between pro-Lie

592

14 Catalog of Examples

groups is open if H /H0 is compact (Theorem 9.60). Every compact subset of K is contained in a finite rank subgroup. If C is the completion of G/K then C is compact and the natural morphism f : G → C has the quotient property in the sense that every morphism ϕ : G → H into a complete group with kernel K factors in the form ϕ = F  f with a morphism F : C → H ; but F is not injective. Citation. 4.11, 5.2. Comment. This example shows that the formation of quotient groups of a pro-Lie group does pose delicate problems unknown in the theory of locally compact groups. Example 14.11. There is an abelian group B with a subgroup C ∼ = Z such that B/C ∼ = ZN and that every morphism B → Z annihilates C. The group B is an ℵ1 -free group which is not an S-group. Citation. 4.5. Comment. The example shows that the class of S-groups is properly smaller than that of ℵ1 -free groups and is not contained in the class of Whitehead groups. Every Whitehead group is an S-group (see [49, p. 226]). Thus the class of S-groups is properly bigger than the class of Whitehead groups and thus a fortiori properly bigger than the class of free groups. ℵ1 -free groups  S-groups   Whitehead  groups  free groups In [34] it is shown that the class of abelian topological groups which are isomorphic to a product C × RX × ZY where C is a compact abelian group and X and Y are countable sets is closed under the passage to closed subgroups and to quotient groups. This points out the fact that quotient groups of RX are complete if X is countable. In this sense the example of RR is minimal in the class of weakly complete topological vector spaces, if one momentarily accepts the Continuum Hypothesis. In [116] it is proved that a closed connected subgroup of RX is a closed vector subspace; consequently it is a direct summand algebraically and topologically (see Theorem A2.11 (i)); in particular the quotient is a weakly complete topological vector space. Example∗ 14.12. There is a connected proto-Lie group H whose Lie algebra is alge2 ℵ0 braically isomorphic to Rℵ0 , while L(H ) ∼ = R2 . Citation. 9.64. Comment. In Theorem 4.20 it is shown that for a proto-Lie group H which is a quotient of a pro-Lie group, the inclusion morphism γH : H → HN (H ) = limN ∈N (H ) H /N of

Abelian Pro-Lie Groups

593

H into its completion induces an isomorphism L(γH ) : L(H ) → L(HN (H ) ). Example 14.12 shows that this fails completely without the assumption that H is the quotient of a pro-Lie group. The result cited from Theorem 4.20 is used significantly in the proof of the Open Mapping Theorem for Pro-Lie Groups (9.60). The following example belongs to the circle of ideas of the Open Mapping Theorem and is based on the following general set-up. Let ω : H → G1 be a group homomorphism between connected pro-Lie groups such that the graph def

def

G = {(g, ω(g)) : g ∈ G} ⊆ G = H × G1 def

is dense. Then f = pr H |G : G → H is a bijective but nonopen morphism and def

extends to a surjective open morphism F = G → H of pro-Lie groups. Example∗ 14.13. Let H = RN with its product topology; then H is a pro-Lie group agreeing with L(H ) (up to a natural isomorphism). Let G1 = R and let ω : H → G1 be a discontinuous R-linear form. Such a linear map certainly exists, since the topological N is isomorphic to R(N) ∼ dual R = R(ℵ0 ) , while the algebraic dual (RN )∗ is isomorphic 2ℵ0

to R2 . Now the graph G of ω in H × G1 is a real hyperplane, being the image of RN × {0} under the real automorphism (x, y)  → (x, y + ω(x)) of RN × R. Since ω is discontinuous, G is dense, since hyperplanes in a topological vector space are either closed or dense. Citation. 9.65 (b). Properties. G is a proto-Lie group, H is a pro-Lie group, and the morphism f : G → H is a continuous bijective morphism which is not open. Comment. In the Open Mapping Theorem 9.60 one cannot relax the hypothesis that both domain and codomain of f are pro-Lie groups by allowing the domain to be merely a proto-Lie group. We also see from the examples that a dense connected subgroup need not be a proto-Lie group even if its exponential function is surjective. It is worth noting in passing, that the general set-up on which the preceding example is based, yields an interesting example in the world of finite-dimensional Lie groups: We may take H = G1 = R and find a bijective Q-linear map ω : R → R that is not a multiplication by a real number. This can even be done in such a fashion that the graph G of ω is a connected subgroup of R2 . (For the existence of such an f see [124].) Here G is a connected abelian topological group which is arcwise totally disconnected, causing L(G) to be zero; its completion is R2 , but it is certainly not a proto-Lie group because it has no small subgroups and is not a Lie group. We have L(G) ∼ = R2 , L(H ) ∼ = R.

594

14 Catalog of Examples

Abelian Pro-Lie Groups Recall here (see for instance [199]) the following basic types of locally compact nondiscrete fields: (a) The field R of real numbers. (b) The field Qp of p-adic rationals for some prime p. (c) The field GF(p)[[X]] of Laurent series in one variable with the exponent valuation over the field with p elements.   ∞ 1 Let Z(p∞ ) = n=1 p n · Z /Z denote the Prüfer group of all elements of p-power order in T = R/Z. We consider the groups Z of integers, Q of rationals, and the Prüfer group Z(p∞ ) with their discrete topologies. Example∗ 14.14. Let J be an arbitrary infinite set. The following examples are abelian pro-Lie groups. (i) All locally compact abelian groups. (ii) All products of locally compact abelian groups, specifically: (a) (b) (c) (d) (e)

the groups RJ ; the groups (Qp )J ; the groups QJ ; the groups ZJ ; the groups Z(p∞ )J .

Citation. 5.1. Properties. An infinite product of noncompact locally compact groups is not locally compact, so none of the groups in (ii)(a)–(e) is locally compact, but if J is countable, they are Polish (that is, completely metrizable and second countable). A countable product of discrete infinite countable sets in the product topology is homeomorphic to the space R \ Q of the irrational numbers in the topology induced by R (see [26, Chap. IX, §6, Exercise 7], or [52, p. 279, Exercise 4.3.G]). So QN , ZN , and Z(p∞ )N are abelian prodiscrete groups on the Polish space of irrational numbers. Comment. These elementary examples show that the category of abelian pro-Lie groups is considerably larger than that of locally compact groups. The groups in (ii)(a), (b) and (c) are divisible and torsion-free, and the groups in (ii)(e) are divisible and have a dense torsion group. We shall call a topological abelian group G semireflexive if   ηG : G → G,

ηG (g)(χ ) = χ (g)

 for g ∈ G, χ ∈ G,

is bijective, and we call G reflexive if ηG is an isomorphism of topological groups; in the latter case G is also said to have duality (see [102, Definitions 7.8]). Recall that Z(2) = Z/2Z. If I is any set and K is any group, then K (I ) is the subgroup of K I consisting of all functions f : I → K with finite support.

A Simple Construction

595

def

Example∗ 14.15. Let  denote the first uncountable ordinal and let J = {α : α < }. def Set G = Z(2)(J ) . Let M denote the filter basis of all subgroups Nα = Z(2)({ν:ν<α}) , α < . The filter generated by M in G is the filter of identity neighborhoods for a group topology on G making G into a nondiscrete prodiscrete group such that M is a basis of N (G). Properties. The group G is a nondiscrete prodiscrete (and thus, in particular, proLie) abelian torsion group of exponent 2 (that is, every element has order 2) and G semireflexive but not reflexive. Citation. See Banaszczyk [5, p. 159, Example 17.11]. Comment. We have discussed the duality theory of abelian pro-Lie groups in 5.33ff. The example shows right away, that the category of abelian pro-Lie groups is not self-dual under Pontryagin duality.

A Simple Construction We use a simple construction which is itself not restricted to abelian groups: Example 14.16. Let P and Q be topological groups and P1 and Q1 closed central subgroups of P and Q, respectively. Let f : P1 → Q1 be an injective morphism of topological groups and let  = {(p, f (p)) : p ∈ P1 }. Then  is a closed central subgroup of P × Q and the factor group G = P ×Q  has the following two normal subgroups: P × f (P1 ) P1 × Q and Q∗ = P∗ =   Properties. (i) G = P ∗ Q∗ , and P ∗ ∩ Q∗ = G/Q∗ = G/P ∗ = P∗ P ∗ ∩ Q∗ Q∗ P ∗ ∩ Q∗

∼ = ∼ =

P1 × f (P1 ) ∼ = f (P1 ),  P ×Q ∼ = P /P1 , P1 × Q P ×Q ∼ Q , = P × f (P1 ) f (P1 ) P × f (P1 ) ∼ P , = P1 × f (P1 ) P1 P1 × Q ∼ Q . = P1 × f (P1 ) f (P1 )

(ii) The functions p → (p, 1) :P1 → P ∗ ∩ Q∗ , p  → (p, 1) :P → P ∗ , q → (1, q) :Q → Q∗ are bijective morphisms of topological groups, and the last one is an isomorphism.

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14 Catalog of Examples

G G ∗ ∗ ∗ ∗ (iii) π : P ∗G ∩Q∗ → Q∗ × P ∗ , π(g(P ∩ Q )) = (gQ , gP ) is an isomorphism of topological groups. (iv) There is a strict exact sequence of abelian topological groups

0 → Q → G → P /P1 → 0, where Q → G is an embedding and G → G/G1 a quotient morphism. Citation. 11.10. Comment. We shall use this construction to expose later some rather bizarre reductive pro-Lie groups. First we illustrate the construction by some simple abelian examples. Example 14.17.√Take P = P1 = R, Q = Q1 = T2 for T = R/Z, and let f : R → T2 , f (r) = (r + Z, a · r + Z), for instance, a = 2. ∼ T2 , P ∗ = P ∗ ∩ Q∗ ∼ Properties. G = = f (R), Q∗ = G, G/P ∗ ∼ = T2 /f (R) and ∗ G/Q = {0}. Comment. This example shows that none of the bijective morphisms in the construction of Example 14.16 need be isomorphisms. Example 14.18. Take P = R, P1 = Z, Q = Q1 = Zp the additive group of padic integers (see Example 14.2 = 1.20 (A)(i)), and let f : Z → Zp be the canonical injection given by f (z) = (z + pn Z)n∈N ∈ limn Z/pn Z ⊆ n∈N Z/pn Z.

Properties. G is the p-adic solenoid Tp (see Example 14.4 = 1.20 (A)(ii)), P ∗ = P ∗ ∩ Q∗ = (R × f (Z))/ is the dense one-parameter subgroup of SSp and Q∗ ∼ = Q = Zp is a compact subgroup of Tp such that G/Q∗ ∼ = P /P1 = R/Z ∼ = T.

Comment. This is a more canonical example that is familiar to workers in topological groups; it is explicitly treated in [102, Exercise E1.11 following Definition 1.30], complete with picture: A detailed theory of finite-dimensional pro-Lie groups subsuming this example was given in this book in Chapter 9, 9.43, 9.44. Example 14.19. Let denote the power ZN but equipped with the discrete topology and not the product topology. As in Chapter 4 in Example 4.9, we consider the compact, connected, and strongly locally connected (see Definition 4.8) abelian character group  , call it A, and let Aa = expA L(A) be the arc component of the identity. The exponential function expA : L(A) → A has the kernel K(A) ∼ = Hom( , Z) while L(A) ∼ = Hom( , R) and the exponential function gives a corestriction expA : L(A) → Ga .

Properties. expA is a quotient morphism by Proposition 4.10. So it induces an isomorphism of topological groups L(A)/K(A) ∼ = Aa . The prodiscrete group K(A) is algebraically isomorphic to the free abelian group with countably many generators Z(N) (see Corollary 4.11; this goes back to the fact that Z is a “slender” abelian group (see [49, pp. 51ff., notably, p. 60, Corollary 2.4])). The group K(A) is an interesting example of a pro-Lie group itself as we pointed out in Proposition 5.2.

597

A Simple Construction

The inclusion function Z → R induces the inclusions function Hom( , Z) → Hom( , R) = L(A). We recalled from Chapter 4 that there is a strict exact sequence expA

0 → Hom( , Z) → L(A) −−−−→ Aa → 0, where the corestriction expA : L(A) → Aa of the exponential function to its image is a quotient morphism onto a proto-Lie group which is not a pro-Lie group; the completion of Aa is A. The inclusion Z(N) → induces the restriction morphism Hom( , Z) → Hom(Z(N) , Z) ∼ = ZN and since Z is slender ([49, p. 51ff.]) every morphism ZN → Z is uniquely determined def

by its action on Z(N) and thus we have an injection f : K(A) = Hom( , Z) → ZN . This injection is not an embedding, because if it were, K(A) would be a Polish countable group and would therefore be discrete by the Baire Category Theorem; but, on the contrary, it is not discrete (see Proposition 5.2). The image f (K(A)) is none other than (Z(N) )p ⊆ ZN , the countably generated free abelian group with the induced topology from the product. It is important to keep the three topologies on Z(N) apart: the discrete one, the one transported from K(A) via f , and the one induced from ZN , each  = (L(A)) and L(A). one properly coarser than the previous one. We may identify A  We now apply Example 14.16 with P = A, P1 = K(A), Q = Q1 = ZN , and with  × ZN is an abelian pro-Lie group and  = {(k, f (k)) : the injection f above. Then A k ∈ K(A)} is a closed totally disconnected subgroup and so we can form the factor group  × ZN def A G = .  The connected subgroup P ∗ =

 (K(A)) A×f 

 and is a bijective homomorphic image of A 



N

(K(A)) it is not closed but arcwise connected. Its closure P ∗ is A×f  = A×Z = G. Thus G, having a dense arcwise connected subgroup is a connected abelian proLie group. The automorphism ϕ of the group K(A) × ZN , ϕ(k, z) = (k, z + f (k)) def K(A)×ZN 

maps K(A) × {0} onto , and induces an isomorphism from H = K(A)×ZN ∼ N = Z . Also,

onto

K(A)×{0}

 × ZN?K(A) × ZN A ∼ ∼  G/H = = A/K(A) = Aa .   Hence G is a connected abelian pro-Lie group embedded into an exact sequence q

0 → ZN → G −−→ Aa → 0. Since L(H ) = {0} we have an isomorphism L(q) : L(G) → L(Aa ) = L(A) ∼ = RR . Citation. 11.11, E11.2.

598

14 Catalog of Examples

Comment. Let us summarize this example as follows, recalling that denotes the power ZN with the discrete topology: Let A =  , and let Aa be the arc component of 0 in A. There is a connected abelian proto-Lie group G with a Lie algebra g isomorphic to Hom(( , R), and there is a closed subgroup H of G isomorphic to such that G/H ∼ = Aa , that is, the quotient G/H is a connected arcwise and locally arcwise connected abelian proto-Lie group whose completion is the compact connected and strongly locally connected group A =  . The minimal analytic subgroup A(g) with Lie algebra g is a faithful homomorphic image of g, and G = A(g)H .  × ZN . The group G is an incomplete quotient of the pro-Lie group A The completion G of G satisfies L(G) = g and A(g, G) = A(g, G). There is a strict exact sequence 0 → ZN → G → A → 0.

Pronilpotent Pro-Lie Groups Pronilpotent Pro-Lie Groups Which Are Not Nilpotent Example∗ 14.20. Let n = RN and define ν : n → n by ν(r1 , r2 , . . . ) = (0, r1 , r2 , . . . ). Define g to be the semidirect sum n  R with [(X, r), (Y, s)] = (r · α(Y ) − s · α(X), 0). We write g = RN0 , where N0 = {0, 1, 2, . . . }; and for any subset J ⊆ N0 we identify RJ with the obvious subgroup of g. The bracket operation of g is now written [(x, r1 , r2 , . . . ), (y, s1 , s2 , . . . )] = (x · (0, 0, s1 , s2 , . . . ) − y · (0, 0, r1 , r2 , . . . )) = (0, 0, xs1 − yr1 , xs2 − yr2 , . . . ). Properties. (i) g[n] = R{n+1,n+2,... } , n = 1, 2, . . . . In particular, g is countably nilpotent and countably topologically nilpotent, that is, pronilpotent, but not nilpotent. (ii) z(g) = {0}, that is, g is center-free. (iii) g = g[1] is abelian, that is, g is metabelian. (iv) g is the smallest closed subalgebra containing (1, 0, 0, 0, . . . ) and (0, 1, 0, 0, . . . ). (v) None of the ideals g[n] is supplemented, let alone complemented. (See Definition 13.15.) Thus g is not rich. (See Definition 13.16.) def (vi) G = (g, ∗) is a simply connected pronilpotent pro-Lie group which is not nilpotent. Citation. 8.5, 8.6, 13.9.

Pronilpotent Pro-Lie Groups

599

Comment. Example 14.20 shows, in particular, that center-free pronilpotent (nonrich) pro-Lie algebras exist. It was shown in Theorem 7.57 that every pronilpotent pro-Lie group is countably nilpotent and in Example 7.63 we observed that a pro-Lie algebra g is pronilpotent iff its coadjoint module is a locally finite-dimensional module such that all finite-dimensional submodules are nilmodules.

Class 2 Nilpotent Pro-Lie Groups If G is a group such that G ⊆ Z(G), we say that G is nilpotent of class 2. ; Example 14.21. Let V = Rn and C = 2 V the second exterior power of V and let (v, w)  → v ∧ w : V × V → C the canonical symplectic map. Then dim C = n2 and C = span im ∧. Let Nn = V × C with the multiplication (v, c)(w, d) = v + w, c + d + 21 v ∧ w). Then Nn is a nilpotent group of class 2. Then the commutator comm((v, c), (w, d)) of two elements equals (0, v ∧ w). The commutator subgroup Nn of Nn is {0} × C and this equals the center Z(Nn ). Now let B be the Bohr compactification of C and η : C → B the Bohr compactification morphism. In the product Nn × B let D be the closed central subgroup {((0, −c), η(c)) : c ∈ C}. Set def

G = (Nn × C)/D. Properties. G is a locally compact nilpotent pro-Lie group of class 2 and f : Nn → G, f (v, c) = ((v, c), 0)D is an injective morphism of pro-Lie groups with dense image def

(Nn × {0})D/D = (Nn × f (C))/D. In particular, H = im f is a dense analytic subgroup. The commutator subgroups H  and G agree, and equal (({0} × C) × {0})D/D = {(({0}, c), f (c))D : c ∈ C}. The center is Z(G) = ({0} × B)D/D = def ˙ ∼ G = (G) = B. The nilcore nilcore(G) = N (G)/Z(G)0 is 2-dimensional. Citation. 9.31 (i). Comment. Since G is a locally compact connected group, Iwasawa’s Local Splitting Theorem 13.20 applies. The example illustrates that a proof of local splitting is not trivial even in such a special case. The example is also a good illustration of Theorem 13.21 on Central Supplementation. ; Example∗ 14.22. We let n ≥ 2 be a natural number and denote the group Rn × 2 Rn of Example 14.21 above by Nn . The set Sn = {x ∧ y : x, y ∈ Rn } is a variety in ;2 n n R ; it is in fact the image under the function (x, y)  → x ∧ y : R2n → R(2) (where ; n we identified Rn × Rn with R2n and 2 Rn with R(2) ). Set G = ∞ n=2 Nn . Set comm(G, G) = {xyx −1 y −1 : x, y ∈ G}. Properties. The commutator subgroup Nn of the class 2 nilpotent group Nn contains  an element an which is a product of no fewer than n−1 4 commutators. Also Nn = ;2 n Z(Nn ) = {0} × R .

600

14 Catalog of Examples

Now Z(G) =

∞

n=2 Z(Nn )



G =

∞ 

and comm(G, G) =

comm(G, G) = n

n=1

We identify

n

m=1 Nm

 n=2 Nn

Further

comm(Nm , Nm )n .

with its canonical image in G so that Nn =

n=1

Now G =

n=2 comm(Nn , Nn ).

n=1 m=2

∞ 6



∞ ∞  

∞

= Z(G) and

n ∞  

Nm ⊆ G.

n=1 m=1

5∞

 n=2 Nn

⊆ G . Since an ∈ / comm(Nn , Nn )k for def

p, the element a = k < n−1 4 , it is readily seen that for any given natural number  p ˙ = G is not contained in comm(G, G)p = ∞ (an )n=2,3,... ∈ G n=2 comm(Nn , Nn ) ; ˙ \ G . therefore a ∈ G ; The vector space nn = Rn × 2 Rn together with the bracket [(x, v), (y, w)] = (0, x ∧ y) may be identified with the Lie algebra of Nn such that expNn : nn → Nn is none other thanthe identity function. Accordingly, G is a pro-Lie group with Lie algebra g = ∞ n=2 nn and the identity function as exponential function. The commutator algebra g is not closed and agrees as a topological space with G . The ˙ = G . Lie algebra g˙ = g agrees as a space with G Thus the algebraic commutator subgroup G of the simply connected pronilpotent pro-Lie group G is not an analytic subgroup. Citation. 9.31 (ii). Comment. Example 14.22 is a pro-Lie group analog of the example in [102, Exercise E6.6, following Corollary 6.12], of a compact, totally disconnected class 2 nilpotent group in which the algebraic commutator subgroup is not closed. It shows that the algebraic commutator group may fail to be an analytic subgroup. For the following important example we recall that in Proposition A2.22 we defined for every weakly ; complete topological vector spaces V , its exterior square or second exterior power 2 V having the universal property that every symplectic bilinear map b : V × V → W into a weakly complete topological vector space factors with a ; continuous linear map b : 2 V → W in the form b(v1 , v2 ) = b (v1 ∧ v2 ). Example∗ 14.23. Let V be a weakly complete topological vector space. Define def

g = h(V ) = V ×

2 >

V,

[(v, x), (w, y)] = (0, v ∧ w)),

with the componentwise topological vector space structure. Properties. If ; V is nonsingleton, g is a class 2 nilpotent pro-Lie ; algebra, and [g, g] = z(g) = {0} × 2 V since the span of the v ∧ w is dense in 2 V . Also, g is its own nilradical n(g) and nilcore(g) = n(g)/z(g) ∼ = V . The pro-Lie algebra g does not allow an ideal direct sum decomposition.

Pronilpotent Pro-Lie Groups

601

In particular, if dim V = ∞, then g is a nilpotent class 2 pro-Lie algebra that is not very rich (see Definition 13.6). On the other hand, if V = span{e1 , e2 }, then V is the semidirect product of the ideal z(h(V )) ⊕ R · (e1 .0) = R · e1 × R · (e1 ∧ e2 ) and the subalgebra R · (e2 , 0). The Campbell–Hausdorff multiplication on g is given by (v, x) ∗ (w, y) = (v + w, x + y + 21 (v ∧ w)). The pro-Lie group (g, ∗) is called the pro-Heisenberg group H (V ) over the weakly complete topological vector space V . If dim V is infinite, H (V ) is an example of a nilpotent pro-Lie group of class 2 which does not satisfy Iwasawa’s Local Splitting Theorem. If dim V = 2, then H (V )N is a class 2 nilpotent pro-Lie group, which is not locally compact, but satisfies the Iwasawa Local Splitting Theorem. Citation. 13.9 (iii). Comment. Theorem 13.15 is likely to come very close to the best possible if one looks for a reasonably general sufficient condition for a pro-Lie algebra to be very rich. Example 14.24. Let g = R(N) × R with the bracket ∞ 3 6 n=1

∞ ∞  6 2  6  un , r , vn , s = 0, det(un , vn ) . n=1

n=1

Properties. g is a class 2 nilpotent algebra such that g = {0} × R = z(g) is onedimensional and thus the center is far from being cofinite-dimensional. Citation. 13.11ff. Comment. It was shown in Proposition 13.11 that for a pro-Lie algebra g, the following statements are equivalent: dim g < ∞, dim g/z(g) < ∞.

(A) (B)

If these conditions are satisfied, then g is finite-dimensional, and therefore closed and g is very rich. The implication (B) ⇒ (A) is always true while the implication (A) ⇒ (B) fails in general as is shown by our example here.

Compactly Embedded Subalgebras Example 14.25. Let g = h(R2 ) be the three-dimensional Heisenberg algebra. Properties. A noncentral one-dimensional subalgebra of g is a compact (hence procompact) Lie algebra which is not compactly embedded into g. In g, every 2-dimensional vector subspace containing the 1-dimensional center (= commutator subalgebra) is a maximal abelian subalgebra and is also an ideal.

602

14 Catalog of Examples

Citation. 12.10ff. Comment. This example shows that maximal abelian subalgebras and ideals are not unique. It also shows that the analytic subgroup belonging to a maximal abelian subalgebra may not be compact in a Lie group having the Heisenberg algebra as Lie algebra. It shows that maximal procompact subalgebras need not lead us to maximal compact subgroups. We need to consider the ambient pro-Lie algebra as well. Indeed this is what we have to do in the finite-dimensional situation, too, and this is where the concept of compactly embedded subalgebras comes in. The Heisenberg algebra is a good simple test for the viability of conjectures on procompact subalgebras on the one hand and compactly embedded ones on the other.

Prosolvable Pro-Lie Groups Metabelian Examples def

Example 14.26. Let L be any pro-Lie group. Then P = LZ is a pro-Lie group. Let α : Z → Aut(P ) be the representation given by α(n)((xm )m∈Z ) = (xm−n )m∈Z . Then Z acts automorphically on P via n · p = α(n)(p) in such a fashion that the only Z-invariant identity neighborhood of P is P itself. def

Let G = P α Z. Properties. G is a semidirect product of pro-Lie groups which is not a pro-Lie group, because LZ × {0} is the unique smallest normal open subgroup. If L is abelian, then G is metabelian, in particular, solvable. Citation. 4.29. Comment. The category of pro-Lie groups is not closed under the formation of semidirect products. Example 14.27. In Example 14.26, let L be a compact Lie group, for instance L = Z/2Z or L = R/Z. Properties. The classical example of a semidirect product LZ σ Z with the shift action of Z on the product is a locally compact group which is not a pro-Lie group. Citation. 3.37, 4.29 (i). Comment. The following question is in a sense a converse to that which is answered by the Closed Subgroup Theorem 1.34 and the Quotient Theorem 4.1: Let G be a topological group and N a closed normal subgroup of G such that both N and G/N are pro-Lie groups. Under what circumstances is G a pro-Lie group? Example 14.26 illustrates that this fails to be the case under rather simple circumstances.

Prosolvable Pro-Lie Groups

603

Example 14.28. Let D be a discrete group, and A a compact nontrivial group of automorphisms of D. Then the compact group AN acts automorphically on the discrete group V = D (N) via (an )n∈N · (dn )n∈N = (an dn )n∈N . Set G = V  AN . Citation. 4.29 (ii). Properties. G is not a pro-Lie group, but when A is abelian, G is metabelian and has a discrete normal subgroup N = V × {1}, that is, a normal Lie subgroup such that the factor group G/N ∼ = AN is compact and therefore a pro-Lie group. Example 14.29. Let G = C  R with multiplication (c, r)(d, s) = (c + e2π ir d, r + s). This group may be isomorphically represented as the group of all matrices ⎛ 2π ir ⎞ e c 0 (c, r)  → ⎝ 0 1 0 ⎠ , c ∈ C, r ∈ R. 0 0 er Properties. G is a three-dimensional metabelian Lie group in which (c, 2π ) is in the image of the exponential function iff c = 0. Thus G is not exponential. Citation. 8.9. Comment. The exponential function of simply connected pronilpotent pro-Lie groups is bijective by Theorem 8.13. This fails for prosolvable pro-Lie groups from 3-dimensional metabelian Lie groups on up. Example 14.30. Let α : R2 → Gl(2, C) be the representation given by , 2π ir 0 e . α(r, s) = 0 e2π is  be the real Lie group C2 α R2 with the multiplication (v, w)(v  , w ) = (v + Let G  is g = C2 × R2 with [(v, w), (v  , w )] = α(w)v  , w + w ). The Lie algebra of G   (δ(w)v , r2 )(c1 , c2 ) = 2π i · (r1 c1 , r2 c2 ). Now let h = C2 × √ − δ(w )v, 0) 2where δ(r1√  R(1, 2) and H = C × R · (1, 2).  is a 6-dimensional metabelian Lie group and H  is a 5-dimensional Properties. G  is D def = {(0, 0)} × Z2 , and accordingly, g is center-free. subgroup. The center of G 2 2 ∼  D/D. Then H is the analytic subgroup A(h) Set G = G/D = C  T and H = H of G with Lie algebra h. The center of G is trivial, and the commutator groups of G and H agree and equal (C2 × {(0, 0)})D/D ∼ = C2 . Citation. 9.29. Comment. If one considers the numerous examples of dense analytic subgroups arising in reductive Lie groups and pro-Lie groups one might be tempted to surmise that dense analytic subgroups have to be centrally supplemented (as defined in 12.72). The point of this example is that we can very well have a dense analytic subgroup of a centerless Lie group. In Theorem 12.70 it is shown how centrally supplemented dense analytic subgroups arise.

604

14 Catalog of Examples

Center-free Embedding We generalize this set-up in the framework of pro-Lie groups. = Example∗ 14.31. Let C be any (additively written) compact abelian group and C  gives a simple C-module Hom(C, R/Z) its character group. Each character χ ∈ C Vχ ∼ = R2 with the module operation , , - , -, x cos 2π χ (c) − sin 2π χ (c) x 2π χ (c)·I x , c· = =e y y sin 2π χ (c) cos 2π χ (c) y   where I = 01 −1 0 and where we have written sin 2π(r + Z) as sin 2π r and so on. We consider the commutative diagram L(χ )

L(C) −−−−→ ⏐ expC ⏐  C −−−−→ χ

R = L(T) ⏐ ⏐quot  T = R/Z

which may be expressed as χ (exp X) = (χ  X)(1) = L(χ )(X) + Z in formula terms. The derivative of t  → e2π χ(exp tX) v = e2π tL(X)v : R → Vχ at t = 0 is 2π L(X)v. Let us form the weakly complete module W = χ ∈C Vχ with componentwise operations c · (vχ )χ ∈G  = (χ (c)vχ )χ ∈C  and X∗(vχ )χ ∈C  = (2π L(χ )vχ )χ ∈C . We define G = W C to be the semidirect product with the multiplication (v, c)(w, d) = (v +c · w, c+d). Then g = W ×L(C) with the Lie bracket [(v, X), (w, Y )] = (X∗w−Y ∗v, 0) is its Lie algebra. Properties. G is a metabelian pro-Lie group such that g = span{Xv : X ∈ L(C), v ∈ ˙ = W ×{0}. The W } and g˙ = W ×{0}; likewise G = span{c·w : c ∈ C, w ∈ W } and G group (g) is the semidirect product W L(C) with the multiplication (v, X)(w, Y ) = (v + X∗w, X + Y ) and πG : (G) → G is given by πG (v, X) = (v, expC X). The arc component of the identity in G is Ga = W × Ca , and this subgroup is the minimal analytic subgroup of G with Lie algebra g. It is dense in G but not equal to G in general. Citation. 9.30. Comment. This construction allows us to embed any compact abelian group C into a center-free metabelian pro-Lie group. Within the category of Lie groups this is possible only for a compact abelian Lie group, and in general this is impossible to implement inside the category of locally compact groups. More examples follow at once. Let us consider the special case of a connected compact abelian group and use the complex manifestation of the module Vχ . Example∗ 14.32. Let K be any compact connected abelian group. Then each character  determines an irreducible representation πχ : K → C ∼ χ ∈K = R2 . The construction  of 9.41 provides us with a center-free metabelian pro-Lie group G = CK π K.

Prosolvable Pro-Lie Groups

605

 R), and the image exp k Properties. The Lie algebra k of K is isomorphic to Hom(K, of the exponential function is the dense proper analytic subgroup A(k, K), which is  exactly the arc component Ka of K and Ga = CK π |Ka Ka is the unique dense subgroup A(g, G) of G with Lie algebra g.  Z). Thus The abstract quotient group π0 (K) = K/K0 is isomorphic to Ext(K, whenever this group is nonzero, the analytic subgroup A(k, K) is proper and dense. It then follows that A(g, G) is a proper subgroup as well. All nontrivial Lie group homomorphic images G/N of G are of the form R2p × Tq with q > 0 and thus have an infinite Poincaré group. Citation. 9.30, 9.42. Comment. This entire class of examples yields center-free metabelian pro-Lie groups with a dense analytic subgroup which is proper in most cases. In particular, the dense analytic subgroup A(g, G) is not centrally supplemented in these cases. This illustrates the workings of the Central Supplementation Theorem 12.70. The following special case is representative. d of the discrete additive group of Example∗ 14.33. Let K be the character group Q  of K may be rational numbers. Then by Pontryagin Duality the character group K identified with Qd . Properties. Here for each N ∈ N (G) the factor group G/N is a circle group and so the Poincaré group P (G/N) is ∼ = Z while G/N0 is of the form R2p × K for some p ∈ N and thus has a nontrivial center. These examples show that not much improvement of Proposition 9.39 can be expected. On several occasions before we noted that the category of compact connected abelian groups K provides us with the prototype of pro-Lie groups with a unique minimal analytic subgroup A(k, K) having the same Lie algebra k as the group K itself, but being a proper dense subgroup, namely the arc component Ka = exp k of  Z) = {0} (see [102, Chapter 8, Theorem 8.30]). Since the identity whenever Ext(K, d is a good special case. In Example 9.42 we Ext(Q, Z) = {0} the group K = Q saw that the Center-Free Embedding Lemma 9.41 permits us to create for each such group K a center-free metabelian group G = V π K with a dense proper analytic group A(g, G). Example∗ 14.34. For n ∈ N, let Cn be the R-module defined on the underlying real def

vector space of C defined by the action (r, c)  → r ·n c = 2π nirc : R × C → C. This is the Lie algebra action corresponding to the group action of the circle group R/Z given by def  (r +Z, c)  → e2π inr c : R/Z×C → C. We consider the product module V = n∈Z Cn with the morphism α : R → Der(V )given by α(r)(cn )n∈Z = (r·n cn )n∈Z . Now we form def

the semidirect product g = V α R, that is, the product V × R with componentwise addition and the bracket [(c, r), (d, s)] = (α(r)(d) − α(s)(c), 0).

606

14 Catalog of Examples

Properties. g is a metabelian rich pro-Lie algebra that is not very rich. (See Definition 13.5.) Citation. 13.9. Recall from Definition 9.47 that the characteristic subgroup KZ(G) of a topological group is the union of all compact normal solvable subgroups and that for a connected pro-Lie group G, it is the unique largest compact central subgroup by Theorem 9.50. Using the Center-Free Embedding Lemma we can construct the following Example∗ 14.35. Let A be an abelian pro-Lie group and assume that comp(A) is compact. Then there is a connected metabelian pro-Lie group G such that A is the center of G. Citation. 9.51. Comment. By Theorem 9.50, a connected pro-Lie group G has maximal compact normal abelian subgroups. If C = comp(C) is a central subgroup of G, then C ⊆ KZ(G) and so C must be compact. Therefore, if A is a closed central abelian subgroup of a connected pro-Lie group, then comp(A) is necessarily compact. Example 14.35 shows that, conversely, every abelian pro-Lie group A with comp(A) compact arises as the center of some connected (even metabelian) pro-Lie group.

Preservation of Characteristic Subgroups under Homomorphisms def

def

Example 14.36. Let G = R and f : G → H = R/Z be the quotient morphism. Properties. G and H are connected abelian Lie groups such that KZ(G) = {0} and KZ(H ) = H , while f is a surjective morphism such that so f (KZ(G)) = KZ(H ). Comment. Note that the kernel ker f is neither compact nor connected. This calls for another example. def

Example 14.37. Let G = R2 ι SO(2) with ι(T )(v) = T (v); that is in G we multiply according to (v, S)(w, T ) = (v + T (w), ST ), that is, G is the group of rigid motions of the euclidean plane. Let H = SO(2) and let f : G → H be the projection onto the second factor. Properties. G is a solvable center-free and H an abelian connected Lie group such that KZ(G) = {(0, idR2 )} and KZ(H ) = H . Again f (KZ(G)) = KZ(H ), and ker f = R2 × {idR2 } ∼ = R2 is connected. Citation. 9.50, E9.4ff. Comment. In Exercise E9.4 following Theorem 9.50 it was shown that for any morphism f from a connected topological group G with KZ(G) compact onto a topological group we have f (KZ(G)) = KZ(H ) provided that ker f is compact. Thus while passing from 14.34 to 14.35 we succeeded in creating a connected kernel, it would not have been possible to create a connected and compact kernel.

Prosolvable Pro-Lie Groups

607

Example 14.38. Let G be the Lie group of all real 2 × 2 matrices , r s def e (s; r) = , r, s ∈ R. 0 1 Define f : G → R, f ((s; r)) = r. Properties. G is a connected metabelian Lie group and f is a quotient morphism. The nilradical N (G) of G is the set of all (s; 0), s ∈ R, that is, the kernel of f . The nilradical N(R) is R. Thus f (N (G)) = {0} = R = N (R). Citation. 10.42ff. Comment. This very simple example, involving no more than metabelian Lie groups shows, that the nilradical does not behave too well with respect to quotient morphisms. We have seen in Theorem 10.25 and in Theorem 10.43 that the radical and the coreductive radical behave better in this regard. Example 14.39. Let H = R2 × R denote the Heisenberg group defined by the multiplication (v, r) ∗ (w, s) = (v + w, r + s + 21 det(v, w)) with det(v, w) = v1 w2 − v2 w1 . Let G = H and H = H/Z(H), and let f : G → H be the quotient morphism. Properties. G is a class 2 nilpotent Lie group, and its commutator group (and coreductive radical) agrees with center of H . We also notice f (Ncored (G)) = {1}. Citation. 10.44. Comment. If G is a class 2 nilpotent pro-Lie group, then the closed commutator subgroup and the coreductive radical agree. The quotient morphism f : G → G/Ncored (G) maps the coreductive radical to the singleton group as it should by Theorem 10.43. Example 14.40. We let Zp denote the p-adic completion of Z and SSp the p-adic solenoid (Zp × R)/D, D = {(−n, n) : n ∈ Z}. Let z  → z∗ : Zp → SS p , be the embedding given by z∗ = (z, 0) + D. Then we get a quotient morphism q : SSp → SO(2) with kernel Z∗p , giving us an automorphic action of SSp on R2 defined by z · v = q(z)(v). So we set M = R2 q SSp , multiplication being (v1 , z1 ) + (v2 , z2 ) = (v1 +z1 ·v2 , z1 +z2 ). Thus M is a 3-dimensional locally compact connected metabelian group whose commutator group (and coreductive radical) is R2 × {0} and whose center is {0} × Z∗p . Now let G = M × H. Then Ncored (G) = [G, G] = (R2 × {0}) × ({0} × R) ∼ = R3 . The subgroup D = {((0, n∗ ), (0, −n)) : n ∈ Z} is discrete and central. We set H = G/D and let f : G → H denote the quotient morphism. Then H contains the cendef def tral subgroup S = (({0}×Z∗p )×({0}×R))/D ∼ = SSp . Let T = {((v, n∗ ), (0, n)) : v ∈ R2 , n ∈ Z}/D ∼ = R2 . The commutator subgroup of H is f ([G, G]) = f (Ncored (G)) whose closure is S + T = ((R2 × Z∗p ) × ({0} × R))/D ∼ = R2 × SSp .

608

14 Catalog of Examples

Properties. G is a 6-dimensional connected locally compact metabelian group and f : G → H is a covering morphism. We have f (Ncored (G)) = Ncored (H ) but f (Ncored (G)) = Ncored (H ). Citation. 10.44. Comment. In Theorem 10.43 it was shown that for a quotient morphism f : G → H of connected pro-Lie groups we have f (Ncored (G)) = Ncored (H ). Example 14.40 shows that the closure is indispensable.

Semisimple and Reductive Pro-Lie Groups Examples Based on Finite-Dimensional Lie Groups Example 14.41. The ground field will be the reals, Sl(2) = Sl(2, R) the group of all real 2 × 2 matrices of determinant 1, and sl(2) = sl(2, R) the Lie algebra of all real 2 × 2 matrices of trace 0. The Lie algebra of Sl(2) is sl(2).   def  = Sl(2), the universal coveringgroup Let H  of Sl(2) and let p : H → Sl(2) be 1 0 the universal covering morphism. Set E = 0 1 . The center Z(Sl(2)) is {E, −E} and def

C = p −1 Z(Sl(2)) is an infinite cyclic group with identity e and a generator c such that p(e) = E and p(c) = −E. The simply connected 4-dimensional reductive Lie  = R × H has center R × C. group G √ √ Let D be the subgroup generated by (1,√e) and ( 2, c), that is, D = {(m+n 2, cn ) : m, n ∈ Z}. The projection P = {(m + n 2, cn ) : m, n ∈ Z} into the first component  is dense in R. Now we define G = G/D. def  Set h = {0}×sl(2). Properties. The Lie algebra of G is g = R ×sl(2), and (g) = G. Also, πG : (g) → G agrees with p, the commutator algebra g is h, and (h) = . We note (g) = (g ). The unique analytic subgroup with Lie algebra {0} × H  h = g is  P ×H A(h) = (h)D/D = = G . D Then ih : (h) → G, ih (0, x) = (0, x)D induces a bijective morphism (h) → A(h), and the image is dense in G. Also G = A(h). The center Z(G) is (R × C)/D ∼ = R/Z = T, and

/C ∼ G/Z(G) ∼ = Sl(2)/{±E} = PSl(2, R). =H Further, Z(G) ∩ A(h) = Z(G) ∩ G = P × C/D = ((P × {e}) + D)/D = (P × {e})/((P × {e}) ∩ D) ∼ = P /Z. Let K and B be the subgroups of Sl(2) given by ., / ., r / cos t − sin t s e K= :t ∈R and B = : r, s ∈ . R sin t cos t 0 e−t

Semisimple and Reductive Pro-Lie Groups

609

 = p−1 ∼  = p−1 B. Then C ⊆ K.  The maps (x, y)  → xy : K → Set K = R and B      B → H and p|B : B → B are diffeomorphisms. If we identify H and R × B then we have G homeomorphic with R×R×B √ {(m + n 2, n, E) : m, n ∈ Z}

which is diffeomorphic to T2 × R2 .

The group T × PSl(2) is likewise homeomorphic to R×R×B Z × Z × {E}

which is diffeomorphic to T2 × R2 .

Citation. 9.28. Comment. The example of this group G is still of low dimension, namely, 4, but shows that the commutator group G which is the analytic subgroup whose Lie algebra is the unique Levi complement of the radical may be nonclosed and dense, intersecting the center in a dense subgroup. The group T × PSl(2, R) has the same center and the same factor group modulo the center; the maximal compact subgroups are isomorphic to T2 in both cases, and their topological structures are the same, but the two groups are radically different in their global algebraic structure. Example 14.42. Let SS 1 denote the complex unit circle group and SS 3 the group of quaternions of norm 1. We have SS 1 ⊆ SS 3 . For n ∈ Z we set " SS 3 if n ≥ 0, Gn = SS 1 if n < 0.  Define G = n∈Z Gn . Define ϕ : G → G by ϕ((gn )n∈N ) = (gn−1 )n∈N . Then ϕ is an injective endomorphism of a compact connected group. The radical is "  {1} if n ≥ 0, R(G) = Hn , Hn = SS 1 if n < 0, n∈Z and ϕ(R(G)) =

 n∈Z

" Kn ,

Kn =

{1} SS 1

if n ≥ 1, if n < 1.

Properties. R(G) is not left invariant by ϕ. Therefore the radical R(G) is not fully characteristic. Citation. 10.26 (a). Comment. The radical of a compact connected group may not be invariant under all endomorphisms.

610

14 Catalog of Examples

Example 14.43. Let S denote the simply connected covering group of Sl(2, R) and let ϕ : Z → S be one of the two possible isomorphisms from Z onto Z(S) and let ψ : Z → Zp be the injection of Z into its p-adic completion. Set def

D = {(0, ψ(m), −ϕ(m)) : m ∈ Z} ⊆ R × Zp × S, and def

 = {(n, ψ(m − n), −ϕ(m)) : m, n ∈ Z} ⊆ R × Zp × S. Define G = (R × Zp × S)/D, H = (R × Zp × S)/ and let f : G → H be the quotient morphism given by f (xD) = x. Properties. G is a four-dimensional locally compact connected reductive group whose center is isomorphic to R × Zp , while H is a quotient of G modulo a subgroup isomorphic to Z, and Z(H ) ∼ = SSp where SSp is the p-adic solenoid. Thus R(G) = Z(G)0 ∼ = R and R(H ) = Z(H ) ∼ = SSp . In this case f (R(G)) is dense in, but not equal to, R(H ). Citation. 10.26 (b). Comment. Surjective morphisms of connected reductive locally compact groups need not map the radical onto the radical. Thus the second part of Theorem 10.25 cannot be improved.  → G has to be surjective, perhaps even a quotient One might surmise that πG : G morphism. However, this fails to be true even for three-dimensional locally compact groups as the following example shows:  be the universal covering group of Sl(2, R). (See for instance Example 14.44. Let G  Let ϕ : Z → Z(G)  be one of the two possible [113] for details on the structure of G.)  ∼  isomorphisms onto the center G. Let βZ : Z → ζ (Z) = q prime Zq be the universal totally disconnected compactification of Z where Zq is the additive group of q-adic integers. (See for instance 14.2 above.) The subgroup  = {(ϕ(n), β(n)−1 ) : n ∈ Z} def

 × ζ (Z) is discrete central and n  → (ϕ(n), β(n)−1 ) : Z →  is an isomorphism of G of topological groups. We set  × ζ (Z))/. G = (G def

Then G and G × ζ (G) are locally isomorphic. Properties. On the Lie algebra level,  × L(ζ (Z)) ∼  × ζ (Z)) ∼ L(G = L(G) = sl(2, R), and (in view of Corollary 4.21 (i))  × ζ (Z))/L() ∼ L(G) ∼ = sl(2, R) = L(G

Semisimple and Reductive Pro-Lie Groups

611

 × ζ (Z) → G induces an isomorphism L(q). The and the quotient morphism q : G morphism →G g  → (g, 0) : G may be identified with πG . Since ζ (Z) = β(Z), the morphism πG is not surjective, and thus A(sl(2, R)) = G. Citation. 10.30, 10.32. Comment. The example can be summarized as follows: (i) G is semisimple and L(G) = sl(2, R).  = A(L(G), G) = (G  × β(Z))/ is a bijective image of G,  is dense, but (ii) πG (G) is not equal to G. In particular, πG is not a quotient map. Assume that G = A(s) for a semisimple closed subalgebra of L(G). Then R(G) = Z(G)0 by Theorem 10.32. One might surmise that G = R(G)A(s) as is the case for Lie groups, whence G = R(G)A(s)N for all N ∈ N (G). However, Example 14.44 shows that this is not even the case when G is semisimple (that is, R(G) = {0}) and g is simple. Thus Theorem 10.32 (v) illustrates that there is a significant difference between Z(G) and Z(G)0 = R(G) for a reductive pro-Lie group G. Example 14.45. Let A be the abelian Lie group R × Z × Z and define the subgroup D ⊆ A by √ D = {(n1 + n2 2, n1 , n2 ) : (n1 , n2 ) ∈ Z2 }. Then D is a discrete subgroup of A such that D ∩ (R × {0} × {0}) = {(0, 0, 0)} = D ∩ ({0} × Z × Z). The group G is the direct sum, algebraically and topologically, of A0√= R × Z × Z and D. Thus A/D ∼ = R. The subgroup ({0} × Z × Z) + D = (Z + 2Z) × Z × Z is dense in A  = R ×  × . Now let  be the universal covering group of Sl(2, R) and set G one of the two isomorphisms from Z onto the center of  and Let ϕ : Z → Z() be √  : (n1 , n2 ) ∈ Z2 }. Then  is a discrete define  = {(n1 + n2 2, ϕ(n1 ), ϕ(n2 )) ∈ G  The  and is contained in the center R × Z() × Z() ∼ subgroup of G = A of G. √ def  subgroup T = ({0} ×  × ) equals (Z + 2Z) ×  ×  and thus is dense in G.  Now we set G = G/. Properties. G is a 7-dimensional reductive connected Lie group in which the analytic def Levi subgroup S = T / is dense where L(S) ∼ = sl(2, R)2 . The radical R(G) is ∼ (R ×{1}×{1})/ = (R ×Z()×Z())/ = A/D ∼ = R. Note that Z(G) = R(G). In particular, G has no nontrivial compact normal subgroup. However, the analytic subgroup A(g, G) with Lie algebra g is G itself. Citation. 10.33. Comment. One might surmise that, for some reason or another, the radical of the closure of a semisimple analytic subgroup of a Lie group would have to be compact. This is false as Example 14.45 shows.

612

14 Catalog of Examples

For a topological group G we let α(G) be the Bohr compactification of G. Example 14.46. Let J and K be sets. A product of J copies of the group constructed in Example 14.45 realizes a connected reductive group with dense minimal analytic Levi subgroup whose radical is isomorphic to RJ . Next consider a product of K copies of the simply connected covering group  of Sl(2, R) in order to have a simply connected semisimple pro-Lie group with center ∼ = ZK . Then use the construction of K K K K Example 14.45 with α(Z ) in place √ of R2 and 2the injective morphism η : Z → α(Z ) in place of (n1 , n2 )  → n1 + n2 2 : Z → R . Properties. There is a pro-Lie group G and a minimal analytic Levi subgroup A(s, G) = expG s which is dense in G0 such that the radical R(G) = R(G0 ) is isomorphic to RJ × α(ZK )0 . Citation. 10.34.  R) the simply connected covering group of Sl(2, R) Example 14.47. Let S = Sl(2, def

and z one of the two generators of its cyclic center Z(S). Let  = {(−n, zn ) ∈ R × S : n ∈ Z} be a cyclic subgroup of the center R × Z(S) of R × S. We let X ∈ sl(2, R) 0 π and identify L(S) with sl(2, R) so that π be −π Sl(2,R) : S → Sl(2, R) induces the 0 identity on the Lie algebra level. Then exp SX ∈ {z, z−1 }, say expS X = z. Now we def

let G = (R × S)/ and set C = {(−r, expS r · X) : r ∈ R} ⊆ G. Properties. G is a 4-dimensional reductive Lie group, and C ∼ = R/Z and C is a maximal compact subgroup of G. The radical R(G) = (R × Z(S))/ is isomorphic to R; the unique Levi complement (Z × S)/ is isomorphic to S, and R(G) ∩ S = (Z × Z(S))/ ∼ = Z while R(G) ∩ S ∩ C = {1}. We have L(G) = R × sl(2, R) and L(C) = R · (−1, X) is a subalgebra which is not an ideal. In particular, the maximal compact subgroups of G fail to meet the center and the Levi complement nontrivially. Citation. 12.75.

A Bizarre Reductive Proto-Lie Group We shall now use Example 14.16 and other ingredients we have prepared in the course of this book to construct a somewhat bizarre reductive connected proto-Lie group. We shall first discuss the construction and then summarize the salient features in Example 14.44 below. The idea is to embed Example 14.16 into the center of a reductive group. def

Let S = Sl(2, R) be the group of 2 × 2-matrices of determinant 1 and s = sl(2, R) the Lie algebra of all 2 × 2-matrices with trace 0. Let  S = (s) be the universal covering of S and πS :  S → S the covering morphism. We identify the Lie algebra L( S) with s via the natural isomorphism ηs : s → L((s)) so that πS  expS˜ = expS . We consider the injective morphisms , 0 1 , ι : R → s, ι(r) = π r −1 0

Semisimple and Reductive Pro-Lie Groups

613

 j : Z → R and notice that exp S ι  j : Z → S is one of the two possible injective morphisms with image Z( S). We write z = ι(Z) ⊆ s. N N  The power (s ) = S is a simply connected (Polish) semisimple pro-Lie group. S N with componentwise expoThe power sN may be considered as the Lie algebra of  N S N and thus the natural nential function. We have an injection (expS˜ ι  j ) : ZN →  injection ϕ : K(A) = Hom( , Z) (compare Example 14.19) yields an injection f = (expS˜  ι  j )N  ϕ : K(A) →  SN, which is not an embedding but has the uncountable center Z( S)N as range.  P1 = K(A), Q =  We now apply Examples 14.16 and 14.19 with P = A, SN, ×  S)N , and with the injection f above. Then A S N is a reductive simply Q1 = Z( connected pro-Lie group and  = {(k, f (k)) : k ∈ K(A)} is a closed central totally disconnected subgroup and so we can form the factor group def

G =

×  A SN . 

By Theorem 4.1, this factor group is a connected proto-Lie group. The central con (K(A))  and it is is a bijective homomorphic image of A nected subgroup P ∗ = A×f  not closed but is arcwise connected. It is the minimal analytic subgroup A(L(A), G)  (K(A))   S)N = A×Z( = Z(G), and this is a with Lie algebra L(A). Its closure P ∗ is A×f   copy of the group constructed in Example 14.19. Hence Z(G) is a connected abelian pro-Lie group embedded into an exact sequence 0 → ZN → Z(G) → Aa → 0. SN ˙ = K(A)× is isomorphic to Q =  SN. The closed commutator subgroup Q∗ = G  (K(A)) ∼ The intersection P ∗ ∩ Q∗ is (K(A)×f = f (K(A)) ∼ = (Z(N) )p . We compute the  factor groups

 ×  SN ∼ A G∼ A ∼ = = = Ga N ˙ K(A) G K(A) ×  S

and

×   A SN G ∼ SN ∼ . = =  × f (K(A)) P∗ f (K(A)) A

˙ is an incomplete abelian proto-Lie group whose completion is the compact Thus G/G connected strongly locally connected abelian group A. In particular, this shows that G cannot be a pro-Lie group, because if it were, then by Theorem 4.28 (i) the quotient ˙ would have to be a pro-Lie group. The factor group  G/G S N /f (K(A)) contains the factor group Z( S)N /f (K(A)) ∼ = ZN /f (K(A)) ∼ = ZN /(Z(N) )p . This last group is not Hausdorff, and thus not unexpectedly, G/P ∗ is not Hausdorff; its Hausdorffisation is G/Z(G) ∼ = PSl(2, R)N . Recall that by a reductive proto-Lie group we understand a proto-Lie group whose Lie algebra is a reductive pro-Lie algebra. We now summarize the salient features of this discussion as follows:

614

14 Catalog of Examples

Example∗ 14.48. There exists a connected reductive proto-Lie group G with the following properties: ˙ is isomorphic to  (i) The closed commutator group G S N where  S is the universal covering group of Sl(2, R). ˙ is an arcwise connected and arcwise locally connected (ii) The factor group G/G abelian proto-Lie group Aa whose completion is the connected, strongly locally  N. connected compact abelian group A = Z (iii) There is an analytic subgroup Z of G such that L(Z) = L(A) = Hom(ZN , R) ∼ = RR which is dense in the center Z(G) which is a connected abelian pro-Lie group, and the exponential function expZ : L(Z) → Z is bijective. There is an exact sequence 0 → ZN → Z(G) → Aa → 0. ˙ and Z ∩ G ˙ is isomorphic to the proto-discrete countably generated free (iv) G = Z G ( N ) ˙ group (Z )p whose completion is ZN . Notably, G = Z(G)G. ˙ The non-Hausdorff group G/(Z ∩ G) is isomorphic to the direct product ˙ ˙ Aa × (G/D) where D is a countable dense subgroup of Z(G). N   (v) G is a factor group of the reductive pro-Lie group A× S modulo a closed central subgroup isomorphic to ZN . In particular, L(G) ∼ = (R×sl(2))N . = L(A)×sl(2)N ∼ def

def

(vi) The completion G = GN (G) is a reductive Polish pro-Lie group with g = L(G) = L(G) such that there is a strict exact sequence ˙ 1→G= S N → G → A → 1.

˙ ˙ ∼ We have G = G S N and R(G) = A(z(g), G) = A(z(g), G) = Z(G) = =  ˙ Z(G). Thus R(G)G = R(G)A(˙g, G) = G = G. Citation. 11.12. Comment. The Borel–Scheerer–Hofmann Splitting Theorem saying that a compact ˙ connected group G is the semidirect product of the commutator subgroup G = G  , was generalized to certain reductive and some closed abelian subgroup A ∼ G/G = connected pro-Lie groups in the Splitting Theorem for Reductive Pro-Lie Groups 11.8. ˙ Here, in Example 14.48, the closed commutator group G is far from splitting semidirectly in G. Thus splitting the commutator subgroup in reductive pro-Lie groups does require special assumptions. Also, the reductive pro-Lie group G is an example of a connected pro-Lie group H with Lie algebra h = r(h) + s for a Levi summand s such that H = R(H )A(s, H ). (For more information on central supplementation, see 12.62ff.) In the absence of simple connectivity, we have a variety of examples of connected reductive groups that show what a great variety of groups with curious properties can

Mixed Groups

615

be constructed if there are simple atoms (that is, simple factors in the Lie algebra) which are of the unbounded type; a very representative example is the case that in the Lie algebra is a product of Lie algebras isomorphic to sl(2, R), that is s = sl(2, R)J . In this case, Z((sl(2, R))) ∼ = Z and the remarkable properties of the abelian groups J cause the structure theory of H = A(s; G) to be so complex. As we Z((s)) ∼ Z = said in Chapter 10, serious complications in the global structure theory arise whenever elements of infinite order are present in the center of the simply connected pro-Lie group (s) generated by a semisimple Lie algebra. On the other hand, if all Z((sj )) are finite, we obtain the remarkable Splitting Theorem 11.8.

Mixed Groups Example 14.49. (a) We let g be the semidirect product of euclidean 3-space R3 and the rotation algebra so(3): def

g = R3 ⊕ so(3),

[(v, X), (w, Y )] = (X(w) − Y (v), [X, Y ]).

Equivalently, we might consider ⎧ ⎛ ⎞⎫ x ⎬ ⎨, X v g= : X ∈ so(3), v = ⎝y ⎠ . ⎩ 0 0 ⎭ z The group belonging to this Lie algebra is ⎧ ⎛ ⎞⎫ x ⎬ ⎨, R v G= : R ∈ SO(3), v = ⎝y ⎠ . ⎩ 0 1 ⎭ z ' & Consider S = (v, 0) and T = (0, Y ). Then [S, T ] = (−Y (v), 0) and S, [S, T ] = 0 (0, 0). Accordingly, ead S T = T + [S, T ] = (−Y (v), Y ). Now take v = 0 , 1  0 −1 0  Y0 = 1 0 0 . Set k = {0} × so(3) and t = {0} × R · Y0 ⊆ k. Then Y0 (v) = 0 and 0 0 0

thus ead S t = t. Further, we set k1 = ead S k = {(−Y (v), Y ) : Y ∈ so(3)}. def

(b)  Let g be sl(2, C) considered as a 6-dimensional real Lie algebra. Then t = 0 Ri · 01 −1 is a maximal compactly embedded subalgebra, and its centralizer z(t, g)  0   0  is C · 01 −1 . For each t ∈ R, set θ (t) = ad 0t −t . Then eθ (t) su(2) is a maximal compactly embedded subalgebra of g containing t for each t, providing many different maximal compactly embedded subalgebras of g containing t.

616

14 Catalog of Examples

Properties. Both examples of 6-dimensional real Lie algebras contain 2-dimensional maximal compactly embedded subalgebras k and k1 of g such that t = k ∩ k1 is a 1-dimensional maximal compactly embedded abelian subalgebra of g. Citation. 12.28. Example 14.50. In the Heisenberg algebra span{X, Y, Z}, [X, Y ] = Z, we have z(g) = ncored (g) = z(g) + ncored (g) = R · Z = g = r(g) = n(g); in the motion algebra span{X, Y, Z}, [X, Y ] = Z, [X, Z] = −Y , {0} = z(g) = ncored (g) = span{Y, Z} = z(g) + ncored (g) = n(g) = r(g) = g; in span{U, X, Y, Z}, [U, X] = Y , [U, Y ] = Z we have {0} = z(g) = R · Z = ncored (g) = z(g) + ncored (g) = span{X, Y, Z} = n(g) = g; in the direct sum of the motion algebra with R we have z(g) ⊆ ncored (g); in so(3) we have r(g) = g. In the sum of all of these, all of the containments in the tall Hasse diagram are proper. g  r(g)   n(g)   z(g) + n cored (g)  ncored  (g)  {0}

g  r(g)   n(g)   ncored (g)  +z(g)  z(g)   {0}

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

= nilcore(g)

Examples Concerning the Definition of Lie and Pro-Lie Groups In this section we discuss examples that illustrate the very definition of Lie and pro-Lie groups and one’s intuition when a topological group “is amenable to Lie theory,” that is when it has a Lie algebra that somehow significantly determines the structure of the group. We begin the listing by pointing out examples that somehow should have Lie algebras, but in the end badly fail to possess one.

Examples Concerning the Definition of Lie and Pro-Lie Groups

617

Example 14.51. Let U be the group of unitary operators on a separable infinitedimensional Hilbert space H. With respect to the operator norm topology, U is a Lie group, and its Lie algebra is (up to a natural identification) the Lie algebra of all skewadjoint bounded operators with the bracket [X, Y ] = XY − Y X and the operator norm topology. As a connected Lie group, U is algebraically generated by expU L(U ). Let G be U endowed with the strong operator topology, that is, the topology of pointwise convergence. Then G is a topological group in which L(G) may be identified with the set of all not necessarily bounded, but closed skewadjoint operators X (for which the domains of definition of X and X∗ agree). This is the content of Stone’s Theorem (see e.g. [69, p. 32, Theorem, 4.7]). Then L(U ) ⊆ L(G) and thus G is algebraically generated by expG L(G). However, L(G) is not a vector space because X + Y in general fails to be a closed operator with dense domain of definition. Properties. G is an arcwise connected topological group generated algebraically by the union expG L(G) of its one-parameter subgroups, but which does not have a Lie algebra. Citation. 2.17ff. Comment. The unitary groups in the strong operator topology thus provide examples of topological groups which have an ample supply of one-parameter subgroups but fail to have a Lie algebra. However, if the closure X + Y is a closed skewadjoint operator, then the Trotter Product Formula (4) of Chapter 2 does hold; see e.g. [69, p. 53, Theorem 8.12]. Example 14.52. Consider the additive Lie group G of the separable Banach space L1 ([0, 1], λ), λ = Lebesgue measure on [0, 1] and the subgroup H of all equivalence classes [f ] ∈ L1 ([0, 1], λ) of L1 -functions f taking their values almost everywhere in Z ⊆ R. Properties. H is a complete arcwise connected (indeed contractible) group such that L(H ) = {0}, that is, the group H has no nondegenerate one-parameter subgroups and thus has a Lie algebra by default. There is a Lie group G such that H ⊆ G. Citation. 2.17ff. See also [102, Exercise E7.17 following Example 7.39]. Comment. G is in the category LieAlgGenGr of topological groups having a generating Lie algebra, while H is not in LieAlgGenGr. Since G and G/H are in LieAlgGenGr the subgroup H , as a kernel of the quotient morphism G → G/H is a limit in the category theoretical sense. Thus the category LieAlgGenGr is not closed under the passage to all limits and hence fails to be complete. Example 14.53. Let X be an arcwise connected compact pointed space and F (X) the def

free compact abelian group on X (see [102, p. 407ff.]). The subgroup G = X of F (X) is free as an abelian group (see loc. cit. p. 410, Proposition 8.52) and thus as a topological group satisfies L(G) = {0} while being arcwise connected. Properties. G is an arcwise connected topological group containing a given arcwise connected compact space X with base point x0 such that G is algebraically free over

618

14 Catalog of Examples

X \ {x0 }. But G has no nondegenerate one-parameter subgroups and thus has a Lie algebra by default. Citation. 2.21ff. Comment. Similar properties are exhibited by the free (nonabelian) compact group. There is an abundance of arcwise connected topological groups (even abelian ones) which have a Lie algebra by default, due to the fact that they have no nontrivial one parameter subgroups. This is the turning point when we actually note genuine pro-Lie groups. Example∗ 14.54. If G belongs to any class of topological groups in the following list, then G is a pro-Lie group. (i) All finite-dimensional Lie groups. (ii) Products of any family of finite-dimensional Lie groups. (iii) Compact groups.  (∗) Groups of the form j ∈J Lj × K where the Lj are members of a family of finite-dimensional Lie groups, and K is a compact group. (∗∗) Groups of the form RJ for an arbitrary set J . (iv) Locally compact abelian groups. (v) Almost connected locally compact groups. Properties. As a pro-Lie group every member of any of these classes has a generating Lie algebra which is a pro-Lie algebra. Citation. 3.4. Comment. The results of Chapter 3 show that a topological group is a pro-Lie group if and only if it is isomorphic to a closed subgroup of a member of class (ii). So if we add to the list (iiia) closed subgroups of products of finite-dimensional Lie groups, then the catalog is made exhaustive – by brute force. However the list distinguishes some obvious prototypes of pro-Lie groups, like the ones in (∗) and (∗∗) which simply arise as special cases or products of groups listed before. The examples in (∗∗) are simply the additive groups of our weakly complete topological vector spaces which pervade the entire theory of connected pro-Lie groups: firstly as the underlying topological vector spaces of the relevant topological Lie algebras secondly, as topological prototypes occurring in such theorems as that which states that a connected pro-Lie group is homeomorphic to a pro-Lie group of the form RJ ×K for some set J and some maximal compact subgroup K (see Chapter 12).

Examples Concerning the Definition of Lie and Pro-Lie Groups

619

Examples of type (iv) or (v) are less obvious but nevertheless quite important, because they allow the Lie and structure theory of pro-Lie groups to be applied to the more conventional classes of locally compact groups. Most counterexamples of a topological nature originate from class (iii) and indeed from commutative compact groups as we shall see in the next section. def

Example 14.55. The group G = Sl(2, Qp ) is a p-adic Lie group. The only normal subgroup is the 2-element subgroup {±1}. Properties. G is a totally disconnected locally compact group which is not a proLie group (in the sense of this book) because it has no nontrivial normal subgroups except {±1} Citation. 3.38. Comment. A compact totally disconnected group is profinite, hence prodiscrete and so, in particular, a pro-Lie group. This conclusion is elementary (see for instance [102, Theorem 1.34]). Of course, all compact groups are pro-Lie groups, but this is proved on what most people would consider a higher level ([102, Corollary 2.34]). Most proofs proceed via Haar measure, and they are still the most natural ones. In Example 14.55 we see that locally compact totally disconnected groups are not pro-Lie groups in general. Example 14.56. (i) The additive group of any dense vector subspace of a weakly complete topological vector space is a proto-Lie group. (ii) In particular, for any set X, the restricted product (in this case the direct sum) R(X) is a proto-Lie group when it is equipped with the topology induced from the product RX . (iii) The additive group of any hyperplane in a weakly complete topological vector space (closed or not) is a proto-Lie group. (iv) The  restricted product of any given family {Lj | j ∈ J } of Lie groups in the product j ∈J Lj is a proto-Lie group. Properties. Except in the case of finite dimensions the examples arising in this fashion fail to be complete and thus cannot be pro-Lie groups. But they are proto-Lie groups. Citation. 3.26ff., E3.6. Comment. The example of the group of rationals in its natural topology shows that a dense subgroup of a Lie group need not be a proto-Lie group. The concept of a local isomorphism is tricky even for the simplest of all Lie groups. Example 14.57. Let T = R/Z be the circle group. Then G = T is a Lie group: Set g = R which is a Dynkin algebra with [X, Y ]&= 0 and ' ' hence  X ∗ Y = X + Y for & all X, Y ∈ R, and take B = − 21 , 21 and U = − 21 , 21 + Z /Z; define the function e : B → U by e(X) = X + Z.

620

14 Catalog of Examples

Properties. B is a Campbell–Hausdorff neighborhood of g and U is an open symmetric identity neighborhood of G such that e is a homeomorphism and satisfies e(X ∗ Y ) = X + Y + Z = (X + Z) + (Y + Z) = e(X) + e(Y ). Now let g = h = 13 + Z ∈ U ; then e−1 (g) = e−1 (h) = 13 and since gh = 23 + Z =   − 13 + Z, we have e−1 (g) ∗ e−1 (h) = 13 + 13 = 23 = − 13 = e−1 − 13 + Z = e−1 (gh). Citation. 2.3. Comment. This example shows that we have to be careful with the definition of a Lie group if we wish to base it on the Campbell–Hausdorff mechanism, as we do. General agreement is easily reached as to when two topological groups are locally isomorphic. However, local groups and local isomorphisms are more delicate concepts than meets the eye at first glance, as we see by looking at Example 14.57. For the following recall first that we say that two subgroups M and N of a group are said to be disjoint if M ∩ N = {1}. Example 14.58 (H. Glöckner). There is a complete topological abelian group G with two disjoint closed subgroups Nk , k = 1, 2 such that both quotient groups G/Nk are Lie groups, while G itself fails to be a Lie group. So N1 , N2 ∈ N (G) but N1 ∩N2 ∈ / N (G). Citation. 3.40. Comment. The example has some interesting additional aspects for which we refer the reader to 3.40.

Analytic Subgroups of Pro-Lie Groups As is explained in [102, Definition 8.27ff.], a compact connected abelian group, typdef  is not arcwise connected, and S is the only closed, ically, like the solenoid S = Q hence maximal analytic subgroup with Lie algebra s ∼ = R. The arc component Sa is a bijective image of s and is equal to expS s; it is the smallest analytic subgroup expS s. Example 14.59. Let G be the metrizable compact connected abelian group S N and X an arbitrary subset of N. We form HX = SaX × S N\X considered in the obvious way as a subgroup of G = S N . Properties. Then HX is an analytic subgroup with Lie algebra g = sN ∼ = RN . Citation. 9.8 (iv). Comment. G has 2ℵ0 different analytic subgroups with the same Lie algebra as the whole group. If we adhere to our definition of an analytic subgroup of a pro-Lie group we have to accept that many analytic subgroups may have the same Lie algebra. It is only the minimal and, wherever they exist, then maximal analytic subgroups with a given Lie algebra that are uniquely determined by their Lie algebra.  N. Example 14.60. Let G = Z

Analytic Subgroups of Pro-Lie Groups

621

Properties. G is a compact connected abelian group and thus is, in particular, a proLie group. The corestriction of the exponential function exp : g → Ga is a quotient morphism. Thus the proto-Lie group topology τproto-Lie on the analytic group Ga = expG g with Lie algebra g is the subgroup topology, and it is not a pro-Lie topology because it is not a complete topology. Citation. 4.9, 4.10. Comment. The concept of an analytic subgroup of a pro-Lie group is vastly more complicated than that of an analytic subgroup of a finite-dimensional Lie group – which is complicated enough. An analytic subgroup of a finite-dimensional Lie group always has a unique Lie group topology that is frequently finer than the subspace topology. An analytic subgroup of a pro-Lie group always has a canonical proto-Lie group topology that is often finer than the subspace topology, but the present example shows that this topology may very well not be a pro-Lie group topology – even if it happens to be the subgroup topology. Let G be the additive subgroup of a weakly complete topological vector space. Then a subgroup A of G is analytic if and only if it is the additive group of a closed vector subspace. The following example illustrates this. Example 14.61. Let G = RN , and H = R(N) , the group of all real sequences of finite support. Properties. H is an arcwise connected, indeed contractible, subgroup of the pro-Lie group G, and H = exp L(H ), but it is not an analytic subgroup. Citation. 9.8ff. Comment. In this context we record the Theorem of Yamabe and Gotô saying that a subgroup of a finite-dimensional Lie group is analytic if and only if it is arcwise connected (see e.g. [17, Chap. 3, §7, Exercise 4], or [70]). The example shows that this is no longer the case for pro-Lie groups. In Proposition 9.14 we saw that for a pro-Lie group G and an analytic subgroup H , an automorphism α of G satisfying L(α)(h) = h also satisfies α(H ) = H , provided at least one of the following conditions is satisfied: (a) H is the minimal analytic subgroup A(h) with Lie algebra h. (b) H is a closed analytic subgroup with Lie algebra h. (c) Hproto-Lie is complete, that is, is a pro-Lie group Hpro-Lie , and α|A(h) is an automorphism with respect to the topology τ (Hpro-Lie ) of Hpro-Lie on A(h). The following example shows that we cannot do much better. 2 , and H = (Q )a × Q  ⊆ G. Let α be the automorphism Example 14.62. Let G = Q of G which switches the two factors, that is, α(x1 , x2 ) = (x2 , x1 ). ∼ R2 and H is an analytic subgroup. Also,  2 = Properties. L(H ) = (L(Q)) , and thus is a pro-Lie group. Further L(α) : g → g Hproto-Lie is isomorphic to R × Q leaves L(H ) invariant. But it does not leave H invariant.

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14 Catalog of Examples

Citation. 9.14ff. Remember, the additive group of p-adic rationals (see Example 1.20 (A)(i)) is a nondiscrete locally compact but noncompact abelian group which is a union of an ascending chain of compact (open) subgroups; thus there is no maximal compact subgroup in such a pro-Lie group. Ofcourse, there are simple discrete abelian examples:  ∞ 1 The groups Z(p∞ ) = n=1 pn Z /Z and n=1 Z/mn Z (for a family of positive integers mn ) are countably infinite torsion groups which are the union of ascending towers of finite groups. It is therefore not a priori clear whether, for instance, connected pro-Lie groups have maximal compact subgroups at all.

Examples Concerning Simple Connectivity Examples 14.63. Any product of simply connected finite-dimensional Lie groups is simply connected in any sense and is also a prosimply connected pro-Lie group. Citation. [102, Theorem 9.29]. Examples 14.64. Let G be a compact connected abelian group. Properties. G is a pro-Lie group. Every Lie group quotient G/N , N ∈ N (G) is a torus. Thus G is never prosimply connected. Citation. [102, Corollary 2.43, Theorem 8.62, Theorem 9.29].  Z). Comment. For a locally compact abelian group G we have π1 (G) ∼ = Hom(G, ∼   Thus if G is the 1-dimensional solenoid Q, then G = Q and π1 (G) = {0}, that is, G is loopwise simply connected. However, no compact connected abelian group is simply connected.

Example Concerning g-Module Theory Example 14.65. Let s = so(3) and ej ∈ s, j ∈ Z(3) = Z/3Z, a basis of s such that ' [ej , e&j +1 ] = ej +2 for j =N 0, 1, 2 mod 3. Then [s, e0 ] contains e1 and e2 and s, [s, e0 ] = s. Now set g = s and x = (xn )n∈N with xn = e0 for all n. Properties. In the adjoint module gad we have g · x = [s, e0 ]N and g · (g · x) = ' &N s, [s, e0 ] = sN = g. Thus the adjoint module is singly generated and yields a profinite-dimensional g-module whose underlying vector space does not yield a locally finite-dimensional g-module since {x} is not contained in a finite-dimensional submodule. Citation. 7.8ff. Comment. Let g be a finitely generated topological Lie algebra on an infinite-dimensional vector space. Then the underlying vector space |g| with the adjoint module

Postscript

623

structure cannot be locally finite-dimensional, since a finite set of generators of g is not contained in a finite-dimensional subalgebra, let alone an ideal, and thus it is not contained in a finite-dimensional submodule.

Postscript It may strike the reader that the section on commutative examples is the biggest, but this reflects the fact that the abelian portion of the theory supplies the most significant counterexamples. We remind the reader of the fact that by the end of Chapter 12 we know that topologically a connected pro-Lie group is a product of a weakly complete topological vector space and a compact connected group, and by the end of Chapter 13 we know that, excepting an obstruction which is a simply connected pronilpotent group, a connected pro-Lie group is covered by a direct product of a compact group and a Lie group. From the theory of compact groups it is known (see [102]) that all complications in the area of general topology originate in the abelian portions of the group. This statement remains true for pro-Lie groups. On the other hand, connected reductive pro-Lie groups, if they are not simply connected, are more complicated than compact connected groups by orders of magnitude, as is seen in Chapters 10 and 11 and as is also reflected in the examples of the subsection on semisimple and reductive pro-Lie groups. Some of the most genuine nonlocally compact pro-Lie groups are pronilpotent as the section on pronilpotent example shows. Fortunately, we have a good structure theory for these modulo the theory of pronilpotent pro-Lie algebras which is as impenetrable as the theory of finite-dimensional nilpotent Lie algebras, and a good deal much more so.

Appendix 1

The Campbell–Hausdorff Formalism

Definition A1.1. (i) A Lie algebra is a real vector space L together with a bilinear multiplication [·, ·] : L × L → L satisfying for all x, y, z ∈ L and

'

[x, x] = 0,

(∗)

& ' & ' & x, [y, z] + y, [z, x] + z, [x, y] = 0.

(∗∗)

The equation (∗∗) is called the Jacobi identity. The multiplication in the Lie algebra is called the Lie bracket. (ii) A Dynkin algebra is a completely normable real topological vector space L and a Lie algebra such that the Lie bracket [·, ·] : L × L → L is continuous. The concept of Lie algebra can be defined over any field in place of the field of real numbers. A Lie algebra is said to be abelian if the Lie bracket vanishes identically. Proposition A1.2. Let A be a real Banach algebra. Then the underlying topological vector space together with the Lie bracket (x, y)  → [x, y] = xy − yx : A × A → A is a Dynkin algebra. For the notion of a Banach algebra, see any textbook on analysis or functional analysis; readers having [102] at hand find a reference there in the paragraph preceding Proposition 1.4. Exercise EA1.1. Prove Proposition A1.2. Let L be a Lie algebra. For each x ∈ L, the linear map ad x : L → L is defined by (ad x)(y) = [x, y]; the validity of the Jacobi identity (2) is equivalent to saying that the following statement holds: (∀x, y, z)

(ad x)[y, z] = ((ad x)y)z + y((ad x)z).

(1)

Definition A1.3. A linear map D : E → E of a vector space E with a bilinear multiplication (x, y)  → xy is called a derivation, if (∀x, y ∈ E)

D(xy) = (Dx)y + x(Dy)

(2)

holds. The set of all derivations on E is denoted by Der(E). In a Lie algebra, the derivation ad x (see (1) above) is called an inner derivation. Exercise EA1.2. Prove that Der(E) is a Lie algebra with respect to the Lie bracket [D1 , D2 ] = D1 D2 − D2 D1 .

Appendix 1 The Campbell–Hausdorff Formalism

625

The concept of exponential function is of central importance for everything we are doing. Initially, the proper setting for it is a Banach algebra with identity. Proposition A1.4. Let A be a real Banach algebra with identity, and let A−1 denote the topological group of all elements of A. Then the exponential function 5 invertible 1 n is an analytic function defined by an absolutely exp : A → A−1 , exp x = ∞ · x n=0 n! convergent power series. Let U1 denote the open unit ball around 0; then x  → 5 (−1)n n − ∞ n=1 n · x is an analytic function such that the analytic function log : 1 + U1 → A,

log z = −

∞ 6 (−1)n n=1

n

· (z − 1)n

satisfies exp log z = z for z − 1 < 1 and log(exp x) = x whenever  exp x − 1 < 1. Proof. Exercise EA1.3. Exercise EA1.3. Prove Proposition A1.4. [Hint. If nothing else helps, check [102, Definition 5.1ff.]] Proposition A1.5. Let A be a real Banach algebra with identity. Then there is an open ball B around 0 on which the function (x, y)  → x ∗ y = log(exp x exp y) : B → A is defined by an infinite 5 series of homogeneous polynomial functions Hn : A × A → A of degree n as x ∗ y = ∞ n=1 Hn (x, y) for x, y ∈ B. (i) The coefficients of the summands of Hn are rational numbers which do not depend on the structure of A. (ii) For n = 1, 2 one has H1 (x, y) = x + y, H2 (x, y) = 21 [x, y], so that we can write 1 (∀x, y ∈ B) x ∗ y = x + y + [x, y] + H3 (x, y) + H4 (x, y) + · · · . 2 (iii) For all x, y ∈ B, the following functional equation of the exponential function is satisfied: exp(x ∗ y) = (exp x)(exp y). (iv) For all x, y ∈ A the elements n1 ·x and n1 ·y are in B and the following formulae hold: , 1 1 ·x∗ ·y , (3) x + y = lim n · n→∞ n n , 1 1 [x, y] = lim n2 · comm∗ · x, · y , (4) n→∞ n n where comm∗ (a, b) = a ∗ b ∗ −a ∗ −b for sufficiently small a and b.

626

Appendix 1 The Campbell–Hausdorff Formalism

Proof. Exercise EA1.4. Exercise EA1.4. Prove Proposition A1.5. [Hint. See [102], notably Propositions 5.5, 5.10, and 5.11.] The following fact is remarkable (see for instance [17, Chap. II, §3, no 2, Corollaire]): Proposition A1.6. In any associative algebra, the polynomials Hn (x, y) are linear combinations of polynomials of the form (ad x)m1 (ad y)n1 (ad x)m2 (ad y)n2 . . . (ad x)mp (ad y)np (x), or (ad x)m1 (ad y)n1 (ad x)m2 (ad y)n2 . . . (ad x)mp (y). By A1.6, the polynomials Hn (x, y) are defined in any Lie algebra (over a ground field of characteristic 0). This allows us to formulate the following theorem. The Fundamental Theorem on Dynkin Algebras Theorem A1.7. On any Dynkin algebra L there is a convex symmetric open identity neighborhood B and a function ∗ : B × B → L such that the following conclusions hold: (i) The infinite series x +y + 21 [x, y]+H3 (x, y)+· · · is summable for each x, y ∈ B and its sum is x ∗ y. (ii) If x, y, z ∈ L satisfy x ∗ y, y ∗ z ∈ B then x ∗ (y ∗ z) = (x ∗ y) ∗ y. (iii) The following formulae define addition and Lie brackets L in terms of ∗: , 1 1 x + y = lim n · ·x∗ ·y , (5) n→∞ n n , 1 1 [x, y] = lim n2 · comm∗ · x, · y , (6) n→∞ n n where comm∗ (a, b) = a ∗ b ∗ −a ∗ −b for sufficiently small a and b. Proof. See for instance [17, Chap. II, §7, no 2] and [17, Chap. III, §4, no 3, Proposition 4]. The Definition of Campbell–Hausdorff Multiplication Definition A1.8. A partial multiplication ∗ on a Dynkin algebra as in the Fundamental Theorem A1.7 is called a Campbell–Hausdorff multiplication, and an open zero neighborhood B on which it is defined is called a Campbell–Hausdorff neighborhood. The formulae (5) and (6) (respectively (3) and (4)) are called the Trotter formulae. As an immediate consequence of A1.6 we observe that In an abelian Dynkin algebra L, we have x ∗ y = x + y for all x, y ∈ L.

Appendix 1 The Campbell–Hausdorff Formalism

627

In [102, Theorem 5.21 (ii)] this was proved for Dynkin algebras which are Banach algebras with the Lie bracket of associative algebras without resorting to A1.6. Corollary A1.9. Let f : L1 → L2 be a morphism of Dynkin algebras, that is, a continuous linear map preserving Lie brackets. If B2 is a Campbell–Hausdorff neighborhood in L2 , then there is a Campbell–Hausdorff neighborhood B1 ⊆ L1 such that f (B1 ) ⊆ B2 . Whenever Bj ⊆ Lj , j = 1, 2 are Campbell–Hausdorff neighborhoods such that f (B1 ) ⊆ B2 , then for all x, y ∈ B1 the following equation holds f (x ∗ y) = f (x) ∗ f (y).

(7)

Proof. Since f is a morphism of Lie algebras, by A1.8, we have f (Hn (x, y)) = Hn (f (x), f5 (y)) for all x, y ∈ L1 and all n = 1, 2, . . . . If x, y ∈ B1 , then the infi∞ converges, and as f is we have f (x ∗ y) = nite n=1 Hn (x, y)  5series 5∞ 5continuous, ∞ ∞ f n=1 Hn (x, y) = n=1 f (Hn (x, y)) = n=1 Hn (f (x), f (y)) = f (x) = f (y). The Definition of Partial Groups and Local Groups Definition A1.10. (i) A partial group is a set P with a subset D ⊆ P × P , an element (1, 1) ∈ D, and functions x → x −1 : P → P and (x, y)  → xy : D → P such that the following statements hold: (a) (∀x ∈ P ) (1, x), (x, 1) ∈ D and 1x = x1 = x, (b) (∀x ∈ P ) (x, x −1 ), (x −1 , x) ∈ D and xx −1 = x −1 x = 1, (c) (∀x, y, z ∈ P ) (x, y), (xy, z) ∈ D iff (y, z), (x, yz) ∈ D, and each of these conditions implies (xy)z = x(yz). The set D is called the domain of definition of the multiplication (x, y)  → xy, and x  → x −1 is called the inversion of the partial group (P , D). Two partial groups (Pj , Dj ), j = 1, 2, are said to be isomorphic if there is a bijection f : P1 → P2 such that (f × f )(D1 ) = D2 satisfying f (xy) = f (x)f (y) for (x, y) ∈ D1 and f −1 (uv) = f −1 (u)f −1 (u) for (u, v) ∈ D2 . (ii) A local group is a partial group (P , D) which is also a Hausdorff topological space such that D is open in P ×P and that multiplication and inversion are continuous. Two local groups are said to be isomorphic if they are isomorphic as partial groups and the bijection implementing this isomorphism is a homeomorphism. Every group gives rise to a partial group with D = G × G and every topological group yields a local group in this fashion. If G is a topological group, then every symmetric open identity neighborhood U becomes a local group with domain of definition def

DU = {(x, y) ∈ U × U | xy ∈ U } and the multiplication and inversion induced from those of G. We call (U, DU ) the local group associated with G and U . Equipped with all of this terminology we can formulate a corollary of TheoremA1.7.

628

Appendix 1 The Campbell–Hausdorff Formalism

Corollary A1.11. Let L be a Dynkin algebra and B a Campbell–Hausdorff neighdef borhood with a Campbell–Hausdorff multiplication ∗ : B × B → L. Define DU = {(x, y) ∈ B × B | x ∗ y ∈ B}. Then B is a local group with domain of definition DU for the multiplication (x, y) → x ∗ y : D → B and the inversion x  → −x : B → B. We shall say that (B, DB ) is the local group associated with L and B. In [102] most of the facts concerning Dynkin algebras and the Campbell–Hausdorff multiplication were established for Dynkin algebras arising from Banach algebras with identity with the exception of A1.6 which was only established for n = 1 and n = 2; ostensibly this sufficed for the development of a viable theory of linear Lie groups. The Theorem of Ado 7.102 allows us to say that every finite-dimensional Lie algebra is a subalgebra of a Lie algebra covered by the theory in [102].

Appendix 2

Weakly Complete Topological Vector Spaces Character Groups of Topological Vector Spaces For topological vector spaces the study of vector space duals turned out to be eminently fruitful. We want to make the connection between character theory and vector space duality. A first step is the following: Proposition A2.1. (i) Assume that E1 and E2 are R-vector spaces such that the underlying additive groups are topological groups and that all functions r  → r ·v : R → Ej , v ∈ Ej , j = 1, 2 are continuous. Then every morphism f : E1 → E2 of abelian topological groups is linear. (ii) Let E be a real topological vector space and E  = HomR (E, R) the space of all continuous linear forms E → R endowed with the compact open topology. Then E  = Hom(E, R) (in the sense of topological Hom-groups), and if q : R → T is the  is an quotient morphism, then Hom(E, q) : E  = Hom(E, R) → Hom(E, T) = E isomorphism of topological vector spaces. n Proof. (i) Let f : E1 → E2 be additive. If m ∈ N, n ∈ Z, then m·f ( m ·v) = f (n·v) = n m n · f (v), whence f ( m · v) = n · f (v). Thus f is Q-linear, i.e. r · f (v) = f (r · v) for r ∈ Q. By the continuity of all r → r · v and the continuity of f we get the desired R-linearity. (ii) Each continuous linear form E → R is trivially a member of Hom(E, R). Conversely, every member f of Hom(E, R) is R-linear by (i). It follows that E  =  is a morphism of topological groups as is Hom(E, R). Now Hom(E, q) : E  → E readily checked. The additive topological group of E as that of a real topological vector space is simply connected (see for instance [102, Definition A2.6, Proposition A2.9, Proposition A2.10 (i)]). Hence every character χ : E → T has a unique lifting χ : E → R such that Hom(E, q)( χ) = q  χ = χ  → E  is an inverse of (see e.g. [102, Appendix 2, A2.32]). Thus χ  → χ : E Hom(E, q). It remains to be verified that it is continuous. We set D = {z ∈ K : |z| ≤ 1}. Let C be a compact subset of E and U = ]−ε, ε[ ⊆ R with 0 < ε ≤ 41 . Now D · C is compact connected and contains C. Consider χ ∈ VE (D · C, q(U )). Then χ (D · C) is a connected subset of q −1 (q(U )) containing 0. The component of 0 in q −1 (q(U )) = U + Z is U . Hence χ (D ·C) ⊆ U . Thus χ  ∈ VE (D ·C, U ) ⊆ VR (C, U ),  → E  . This completes the proof. proving the continuity of χ  → χ : E

Let us recall some basic introductory facts on topological vector spaces in an exercise. In this Appendix we shall denote the real or complex ground field, alternatively,

630

Appendix 2 Weakly Complete Topological Vector Spaces

by K. A subset U in a K-vector space is called balanced if D · U = U . It is called absorbing if (∀v ∈ E)(∃r > 0)(∀t ∈ K)

(|t| > r) ⇒ (v ∈ t · U ).

A balanced set is absorbing if every vector is contained in a multiple of the set. Recall that a topological vector space over K (or over any topological field K, for that matter) is an abelian topological group E with a continuous scalar multiplication (t, v)  → t · v : K × E → E. Exercise EA2.1. Show that the filter of zero neighborhoods U in a topological vector space satisfies the following:  (0) U = {0}. (i) (∀U ∈ U)(∃V ∈ U) V − V ⊆ U . (ii) (∀U ∈ U)(∃V ∈ U) D · V ⊆ U . (iii) Every U ∈ U is absorbing. Conversely show that, if a filter U satisfies (i), (ii), (iii), then the set O of all subsets U of E such that for v ∈ U there is a W ∈ U with v + W ⊆ U is a vector space topology whose filter of identity neighborhoods in U. If it also satisfies (0), then O is a Hausdorff topology. (We shall always assume that.) A subset P of a topological group is called precompact, if for every open subset U there is a finite subset F such that P ⊆ F U . We call an abelian topological group G locally precompact if there is a precompact identity neighborhood. For topological vector spaces over the reals (or indeed locally compact fields), the finite-dimensional ones form a topologically distinguished class. Proposition A2.2. (i) On a one-dimensional K-vector space E, outside the indiscrete topology {Ø, E} there is only one vector space topology. For each 0 = v ∈ E the map r  → r · v : K → E is an isomorphism of topological vector spaces. (ii) A locally compact subgroup H of a Hausdorff topological group G is a closed subset. (iii) A finite-dimensional K-vector space admits one and only one vector space topology. If E is a K-vector space with dim E = n and E → Kn is an isomorphism then it is an isomorphism of topological vector spaces where Kn has the product topology. (iv) A locally precompact topological vector space over K is finite-dimensional. Proof. (i) Observe that {0} is a vector subspace. Since dim E = 1 there are only two possibilities: {0} = E, yielding the indiscrete topology, or {0} = {0} which we assume. Then the topology on E is Hausdorff. Let 0 = e ∈ K, set f : K → E, f (t) = t · e. Let def

V = f −1 (U(E)) be the inverse image of the filter of zero neighborhoods of E. Since f is linear, V has a basis of balanced and absorbing sets. In K a set V is balanced iff it is of the form V = {r ∈ K : |r| < ε} or of the form V = {r ∈ K : |r| ≤ ε}. The

Character Groups of Topological Vector Spaces

631

only one among these which is not absorbing is {0}. Note that V intersects in {0} and conclude that V is the neighborhood filter of 0 in K. (ii) It is no loss of generality to assume G = H ; show H = G. Let K be a compact identity neighborhood of H and let U be an open identity neighborhood of G with U ∩ H ⊆ K. Consider g ∈ G; we must show g ∈ H . Since H is dense in G there is an h ∈ H ∩ U −1 g, say h = u−1 g. Since U is open, U ∩ H is dense in U . Thus U ∩ K is dense in U , i.e. U ⊆ K = K (as K is compact in a Hausdorff space). Thus u ∈ U ⊆ K ⊆ H , whence g = uh ∈ H H ⊆ H . (iii) Let e1 , . . . , en be a basis of E and set f : Kn → E, f (x1 , . . . , xn ) = x1 · e1 + · · · + xn · en . Prove by induction on n that f is an isomorphism of topological vector spaces. In (i) one dealt with n = 1. Consider vector subspaces def

def

N = K · e1 ⊕ · · · ⊕ K · en and H = K · en+1 . Show that (n, h)  → n + h : N × H → E is an isomorphism of topological vector spaces. This map is a continuous algebraic isomorphism. We must show that its inverse is continuous. By the induction hypothesis N ∼ = Kn , H ∼ = K, and so both vector spaces are locally compact hence closed by (ii). Then E/N is Hausdorff, and hence isomorphic to K by (i). Thus h  → h + N : H → E/N is an isomorphism of topological vector spaces by (i). Hence the projection of pH : E → H , pH (x1 · e1 + · · · + xn+1 · en+1 ) = xn+1 · en+1 , factoring through the quotient E → E/N and the isomorphism E/N → H is continuous. Hence the projection pN = id −pH onto N is continuous. So (n, h)  → n + h has a continuous inverse. (iv) Let U be a balanced zero neighborhood of E such that U +U is precompact. The sets U +u, u ∈ U form an open cover of U +U by translates of an identity neighborhood. By precompactness there is a finite subset F ⊆ U such that U + U ⊆ U + F . Let E1 = spanK F . Then U + U ⊆ U + E1 . Since the vector space E1 is finite-dimensional and therefore locally compact by (iii), it is closed by (ii). Hence E/E1 is Hausdorff. Set V = (U + E1 )/E1 , then V is a balanced 0-neighborhood of E/E1 satisfying V + V ⊆ V = −V . Then V is a vector space. However, V is precompact since U is precompact. Claim: V is a singleton. Suppose it is not a singleton; then by (i) the K-vector space contains a vector subspace isomorphic to K. Hence K would have to be precompact. But then the subspace N ⊆ K would have to be precompact, but it is not because it cannot be covered by a finite number of translates of a disc of radius 21 . Thus V is singleton and E = E1 . For a vector space E over K we shall denote the set of all finite-dimensional vector subspaces by Fin(E). For a topological vector space E over K we denote the set of cofinite-dimensional closed vector subspaces (i.e. closed vector subspaces M with dim E/M < ∞) by Cofin(E). A topological vector space E over K is called locally convex if every zero neighborhood contains a convex one. Now let E be any real vector space and let B(E) denote the set of all balanced, absorbing and convex subsets of E. Let us observe that there are plenty of those, in fact enough to allow only 0 in their intersection. Let F be a basis

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of E and ρ : F → ]0, ∞[ any function. Then the set 6  U (F ; ρ) = re · e : |re | < ρ(e) e∈F

is balanced, absorbing and convex. We call it a box neighborhood with respect to F . The box neighborhoods with respect to a single basis already intersect in 0. Thus the filter of all supersets of sets from B(E) satisfies (0), (i), (ii), and (iii) of E7.12. If we set O(E) = {W ⊆ E | (∀w ∈ W )(∃U ∈ B(E)) w + U ⊆ W }, then O(E) is a locally convex vector space topology. From its definition it is immediate that it contains every other locally convex vector space topology. It is clearly an algebraic invariant in so far as it depends only on the vector space structure of E. A convex subset U of E belongs to O(E) if and only if for every u ∈ U and every x ∈ E the set {r ∈ R | u + r · x ∈ U } is an open interval of R containing 0. It follows that a convex subset U of E belongs to O(E) if and only if for each finite-dimensional vector subspace F and each v ∈ E the intersection F ∩ (U − v) is open in F (in the unique vector space topology of F ). Let us record some of the basic properties of O(E). We shall see that O(E) is, in essence, a purely algebraic invariant of the real vector space E. Proposition A2.3. Let E be an arbitrary vector space over K. (i) If E1 and E2 are vector spaces, T : E1 → E2 is a linear map, and E2 is a locally convex topological vector space, then T is continuous for the topology O(E1 ). In particular, every algebraic linear form E → K is O(E)-continuous; i.e. the algebraic dual E ∗ = HomK (E, K) is the underlying vector space of the topological  (which is considered to carry the compact open topology). dual E  = Hom(E, K) ∼ =E (ii) Every vector subspace of E is O(E)-closed and is a direct summand algebraically and topologically. Moreover, the topology induced on each vector subspace is its finest locally convex topology. (iii) Let F be a linearly independent subset of E. Then there is a zero neighborhood U ∈ O(E) such that {v + U | v ∈ F } is a disjoint open cover of F . In particular, any linearly independent subset of (E, O(E)) is discrete. (iv) If C is an O(E)-precompact subset, then spanK (C) is finite-dimensional. Proof. (i) If U is any balanced and convex zero neighborhood of E2 then T −1 (U ) is balanced, absorbing and convex and thus belongs to B(E1 ). This shows the continuity of T with respect to O(E1 ). The remainder of (i) then follows at once. (ii) Let E1 be an arbitrary vector subspace of E and let E2 be a vector space complement; i.e. E = E1 ⊕E2 . The function x  → (pr 1 (x), pr 2 (x)) : E → E1 ×E2 is a vector space isomorphism and then is continuous by (i). The function α : E1 ×E2 → E, α(x, y) = x + y, is its inverse. Since it is the restriction of the continuous addition (x, y)  → x + y : E × E → E to the subspace E1 × E2 it is continuous. Hence α is an isomorphism of topological vector spaces. This proves assertion (ii).

Character Groups of Topological Vector Spaces

633

(iii) Let F be a linearly independent subset. Since by the Axiom of Choice, F can be supplemented to a basis {ej | j ∈ J } we may just as well assume that F is this basis. Let σ : F → ]0, ∞[ be the function with the constant value 21 . We claim that for different elements e ∈ F we have   (e + U (F ; σ )) ∩ f ∈F, f =e f + U (F ; σ ) = Ø. Indeed if ω : E → K is the linear functional defined by ω(f ) = 0 for e = f ∈ F and by ω(e) = 1, then    ' 1 ' ω  f ∈F, f =e f + U (F ; σ ) ⊆ 0, 2 and |ω(e + U (F ; σ )| ⊆

&1

2, 1

'

.

This proves (iii). (iv) Let K be a precompact subset of (E, O(E)). We want to show that dim spanK K < ∞. It is no loss of generality to assume that E = spanK K. Then we can select a basis B = {ej | j ∈ J } of elements ej ∈ K. Then B is precompact. By (iv) we find an open zero neighborhood U such that ej + U is a disjoint cover of B by translates of U . Since B is precompact this implies that J is finite. Hence dim E = card J < ∞. Our main interest will be with vector spaces dual to those we just discussed. Their topology was determined by the finite-dimensional vector subspaces. Dually we may consider vector space topologies which are determined by the cofinite-dimensional closed vector subspaces. Proposition A2.4. Let E be a topological vector space. Then for M, N ∈ Cofin(E) with N ⊆ M, there is a canonical quotient map qMN : E/N → E/M. Since Cofin(E) is a filter basis, there is an inverse system and, in the category of topological vector spaces, there is a projective limit ECofin(E) = limM∈Cofin(E) E/M, the vector subspace of all  E/M (vM + M)M∈Cofin(E) ∈ M∈Cofin(E)

such that N ⊆ M implies vN − vM ∈ M. The function γE : E → ECofin(E) , γE (v) = (v + M) M∈Cofin(E) is a morphism of topological vector spaces which is injective if and only if Cofin(E) = {0}. Proof. This is straightforward. The Definition of Weakly Complete Topological Vector Spaces Definition A2.5. The topology O(E) is called the finest locally convex vector space topology on E. A topological vector space E is called weakly complete if γE : E → ECofin(E) is an isomorphism of topological vector spaces. The topology O  (E) on E making γE : E → ECofin(E) a topological embedding is called the weak topology on E.

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The weak topology on a K-vector space is the smallest topology making all linear functionals f : E → K continuous. All finite-dimensional vector spaces are weakly complete. On a weakly complete topological vector space, the continuous functionals separate points. Our definition in terms of the projective limit establishes completeness in the weak topology as the name of a “weakly complete topological vector space” suggests. Recall from A2.3 (i) that the algebraic dual E ∗ of a real vector space E is at the same time the topological dual E  , consisting of all continuous linear functionals on E when E is endowed with the finest locally convex topology. On the basis of bare linear algebra one always has the weak ∗-topology on E ∗ , that is, the topology of pointwise convergence induced by the natural inclusion E ∗ → RE . The first item in the following lemma will show that this topology agrees with the topology of uniform convergence on compact sets which is the topology we consider in order to have the isomorphism  according to A2.1. E ∼ =E Lemma A2.6. Let E be a real vector space endowed with its finest locally convex vector space topology.  is the weak∗ -topology, i.e. the (i) Then the compact open topology on E  ∼ = E topology of pointwise convergence (ii) Every continuous linear functional  : E  → R is of the form ω  → ω(x) :  E → R for a unique x ∈ E. (iii) F  → F ⊥ : Fin(E) → Cofin(E  ) is an order reversing bijection. (iv) E  is weakly complete. Proof. (i) The compact open topology on E  is generated by the basic zero-neighborhoods VE (C, U ) for a compact subset of E and some zero neighborhood U of R and it is always equal to or finer than the weak∗ -topology. It is no loss of generality to assume that C is convex balanced, since C is contained in a finite-dimensional subspace by A2.3 (iv), where the closed convex circled hull C ∗ of a compact set C is compact and then VE (C ∗ , U ) ⊆ VE (C, U ). Assume C is convex balanced now. There is an ε > 0 such that ] − ε, ε[ ⊆ U , and then , 1 VE C, ] − 1, 1[ = VE (C, ] − ε, ε[) ⊆ VE (C, U ). ε Thus we may assume that the filter of zero-neighborhoods of E  for the compact open topology is generated by basic zero neighborhoods of the form VE (C, ] − 1, 1[) as C ranges through the compact convex and balanced subsets of E. Now span C is def

finite-dimensional; then there is a basis e$1 , . . . , en of span C such that C ⊆ K = # 5n r · e | |r | ≤ 1, j = 1, . . . , n , where K is the convex balanced hull of j j j j =1 {e1 , . . . , rn }. Then VE ({e1 , . . . , en }, ] − 1, 1[) = VE (K, ] − 1, 1[) ⊆ VE (C, ] − 1, 1[). Hence the topology of E  and the weak∗ -topology agree.

Character Groups of Topological Vector Spaces

635

(ii) Let  : E  → E be a continuous linear functional. By (i), its continuity implies the existence of a finite set F ⊆ E of vectors in E such that (VE (F, ] − 1, 1[) ⊆] − 1, 1[. Since F ⊥ ⊆ E  is contained in VE (F, ] − 1, 1[) we have (F ⊥ ) ⊆] − 1, 1[, and since (F ⊥ ) is a vector space, we conclude F ⊥ ⊆ ker(). Therefore we have a linear functional  : E  /F ⊥ → R such that  =   q for the quotient map q : E  → E  /F ⊥ . If x∈ E then x ⊥ = {ω ∈ E  : ω(x) = 0} is a closed hyperplane in E  , and thus F ⊥ = x∈F x ⊥ is a finite intersection of hyperplanes. Thus F ⊥ ∈ Cofin(E  ) and thus E  /F ⊥ is a finite-dimensional vector space. Its dual may be identified with span F in the sense that every linear functional of E  /F ⊥ is of the form ω + F ⊥  → ω(x) for an x ∈ span F . We apply this to  and find some vector x ∈ span F such that  (ω) = ω(x). Then (ω) = ω(x) for all ω ∈ E  , and this is what we had to show. (iii) If F ∈ Fin(E), then F ⊥ ∈ Cofin(E  ) as we saw in the proof of (i). Conversely let M ∈ Cofin(E  ). Then M is the intersection of finitely many closed hyperplanes. Each one of these is the kernel of a continuous functional ; we saw in (ii) that each one of these is of the form ω  → ω(x). Hence M is the annihilator M = F ⊥ of some F ∈ Fin(E). Hence F  → F ⊥ : Fin(E) → Cofin(E  ) is a containment reversing bijection. (iv) The function κE,F : E ∗ /F ⊥ → F ∗ given by κE,F (ω + F ⊥ ) = ω|F for ω ∈ E ∗ is a bijection by an elementary annihilator mechanism of linear algebra. (The map is clearly seen to have zero kernel and to be surjective, since by the Axiom of Choice E = F ⊕ F1 with a vector space complement so that every functional of F extends such has F1 in its kernel.) In order for (ωF + F ⊥ ))F ∈Fin(E) ∈  that the∗ extension ⊥ ∗ ⊥ F ∈Fin(E) E /F to be in lim F ∈Fin(E) E /F it is necessary and sufficient that F1 ⊆ F2 in Fin(E) implies ωF2 |F1 = ωF1 . Thus in view of (i), (ii) above there is an isomorphism ϕ:

lim

M∈Cofin(E ∗ )

E ∗ /M →

lim

F ∈Fin(E)

F ∗,

ϕ((ωF + F ⊥ )F ∈Fin(E) ) = (ωF |F )F ∈Fin(E) .

Now ψ : E ∗ → limF ∈Fin(E) F ∗ , ψ(ω) = (ω|F )F ∈Fin(E) is a morphism which is injective, because E = F ∈Fin(E) F , whence ω|F = 0 for all F implies ω = 0. Moreover, if (ωF )F ∈Fin(E) ∈ limF ∈Fin(E) F ∗ , then F1 ⊆ F2 in Fin(E) implies ωF2 |F1 = ωF1 . Thus we may define unambiguously ω ∈ E ∗ by setting ω(x) = ωR·x (x) and obtain ψ(ω) = (ωF )F ∈Fin(E) . Then ψ is surjective and so is an isomorphism of vector spaces. We observe ψ = ϕ ◦ γE and thus γE ∗ : E ∗ → limM∈Cofin(E ∗ ) E ∗ /M is an isomorphism of vector spaces. Let Fin1 (E) be the set of one-dimensional subspaces  , j (ω) = (ω|F )F ∈Fin (E) is an embedding of of E. Then j : E ∗ → F ∈Fin1 (E) F 1 ∗ topological vector spaces if E is given the weak ∗-topology. Consider the projection π:

 F ∈Fin(E)

→ F

 F ∈Fin1 (E)

, F

π((ωF )ωF ∈Fin(E) ) = (ωF )F ∈Fin1 (E) .

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Appendix 2 Weakly Complete Topological Vector Spaces

The diagram

 F limF ∈Fin(E) ⏐ ⏐ incl  ∗ F ∈Fin1 (E) F

ψ −1

−−−−→ −−−−→ π



E⏐∗ ⏐j  F ∈Fin1 (E) F

∗

is commutative. Since j is an embedding and j ◦ ψ −1 = π ◦ incl is continuous, it follows that ψ −1 is continuous and so that ψ is an isomorphism of topological vector spaces. Hence γE ∗ is an isomorphism of topological vector spaces. This shows that E ∗ is a weakly complete topological vector space in the weak ∗-topology, and thus E  is weakly complete in the compact open topology by (i). Let E be a locally convex topological vector space over K and E  its topological dual. If ηE : E → E  , ηE (x)(ω) = ω(x), denotes the evaluation morphism, then for  def each subset H ⊂ E we set H ◦ = {ω ∈ E  : |ω(H )| ⊆ [0, 1]} = h∈H ηE (h)−1 B1 with B1 = {r ∈ K : |r| ≤ 1} and call this set the polar of H in E  . Similarly for a def

subset  ⊆ E  we define the polar of  in E to be ◦ = {x ∈ E : |(x)| ⊆ [0, 1]} =  −1 ω∈ ω (B1 ). Again as in the case of annihilators of subsets of abelian topological groups one must specify where the polars are taken. Polars are always closed. For the following recall that E is called semireflexive if the morphism ηE : E → E  , ηE (x)(ω) = ω(x) is bijective. It is called reflexive if ηE is an isomorphism of topological groups. Lemma A2.7 (The Bipolar Lemma). Let E be a locally convex vector space and U be a convex balanced subset of E. Let  be a convex balanced subset of E  . Then (i) U  = U , and (ii) if E is semireflexive,  = . Proof. (i) The taking of polars is containment reversing. Hence U ⊆ U  and since polars are closed we have U ⊆ U  . In order to prove the converse containment let x ∈ U  . This means that |U ◦ (x)| ⊆ [0, 1]. We claim that this implies x ∈ U . Suppose it does not. Then by the Theorem of Hahn and Banach (see e.g. [15]), there is a real linear functional ρ : E → R with ρ(x) > 1 and ρ(U ) ⊆ [−1, 1]. If K = R, then this says that ρ ∈ U ◦ , contradicting |U ◦ (x)| ⊆ [0, 1]. If K = C, then ω(y) = ρ(x) − iρ(i · x) defines a complex linear functional such that |ω(x)| ≥ Re |(ω(x))| = ρ(x) > 1, while, on the other hand, for 0 = u ∈ U we define z with |z| = 1 by zω(u) = |ω(u)|. Then zu ∈ U and |ω(u)| = zω(u)ω(zu) = Re ω(zu) = ρ(zu) = |ρ(zu)| ≤ 1. Thus ω ∈ U ◦ . This proves the claim. (ii) If E is semireflexive, then E may be identified with the vector space of all continuous linear functionals of E  . Since E  is locally convex, the proof of part (i) applies here and proves the assertion. We shall stay in the category AbTopGr of all abelian topological groups, and we shall denote the full subcategory of all reflexive abelian topological groups by RefTopGr,

Character Groups of Topological Vector Spaces

637

 η (x)(χ ) = χ(x), is  that is all abelian topological groups G such that ηG : G → G, G an isomorphism of abelian topological groups. In the following theorem we shall see that for each real vector space E endowed with its finest locally convex topology, the additive group of E belongs to RefTopGr, and that for each weakly complete topological vector space V , the additive group of V belongs to RefTopGr. Duality of Real Vector Spaces Theorem A2.8. Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then:   is an isomorphism of topological vector (i) E is reflexive; that is, ηE : E → E spaces.   is an isomorphism of topological vector (ii) V is reflexive; that is, ηV : V → V spaces. (iii) The contravariant functor  · : RefTopGr → RefTopGr exchanges the full subcategory of real vector spaces (given the finest locally convex topology) and the full subcategory of weakly complete topological vector spaces.  are isomorphic by 7.5 (iii). Proof. (i) The topological dual E  and the character group E Since the linear functionals separate the points, ηE is injective. By A2.6 (ii), ηE is   may be identified. surjective. As vector spaces, therefore, E, E  , and E  ∼  is compact if and only if it is closed in E  and for each compact A subset K in E = E subset C of E and each ε > 0 there is an M ∈ B(E) such that |K(C ∩ M)| ⊆ [0, ε[ (see for instance [102, Proposition 7.6]). In view of A2.3 (iv) we may express this as follows: (∀F ∈ Fin(E), ε > 0)(∃M ∈ B(E))

K(F ∩ M) ⊆ [0, ε].

(∗)

We shall now show that B(E) has a basis of basic zero neighborhoods VE  (K, Bε ) = {x ∈ E : |K(x)| ⊆ [0, ε]} for the compact open topology on E when E is identified   For a proof of the claim let U ∈ B(E). Set K def = {ω ∈ E  : |ω(U )| ⊆ with E  ∼ = E. ◦ [0, 1]} = U . We claim that K is compact. Clearly, K is closed in E  since E  has the topology of pointwise convergence by (i) above. Let F ∈ Fin(E) and ε > 0. Then |K(F ∩ ε · U )| ⊆ ε · [0, 1] by the definition of K. Hence (∗) is satisfied and K is compact as asserted. Now we observe that VE  (K, B1 ) = U  . By the Bipolar Lemma A2.7 we have VE  (K, B1 ) = U . Since the filter basis B(E) has a basis of closed sets, we have shown that it has a basis of sets VE  (K, B1 ). Thus the compact open topology on E is finer than or equal to the given topology of E which is the finest locally convex topology. But since the sets VE  (K, B1 ) are convex, the two topologies agree and thus E is the dual of E  and E is reflexive. (ii) Set E = V  . Then each nonzero ω ∈ E has a closed hyperplane as kernel, and each closed hyperplane is the kernel of such an ω. It follows that M  → M ⊥ :

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Appendix 2 Weakly Complete Topological Vector Spaces

Cofin(V ) → Fin(E) is an order reversing bijection. Now let  ∈ E  and consider F ∈ Fin(E). Set M = F ⊥ . Now F may be canonically identified with the dual of V /M so that ω, v + M = ω(v) for ω ∈ F = M ⊥ , v ∈ V (see the Annihilator Mechanism [102, Lemma 7.17 (i)]). But F and V /M are finite-dimensional vector spaces which are reflexive. Hence there is a unique element M ∈ V /M such that ω, |F  = ω, M  for all ω ∈ F . Moreover, if F1 ⊆ F2 then M2 = F2⊥ ⊆ F1⊥ = M1 and since map qM1 M2 : V /M2 → V /M1 maps M2 to M1 . (|F2 )|F1 = |F1 the quotient  Hence (M )M∈Cofin(V ) ∈ M∈Cofin(V ) V /M is contained in limM∈Cofin(V ) V /M. By hypothesis, V is weakly complete; hence there is an element v ∈ V such that v + M = def

M . Let ω ∈ E. Then F = K · ω ∈ Fin(E). Thus, letting M = F ⊥ we get (ω) = ω, |F  = ω, M  = ω, v + M = ω(v). Hence  = ηV (v). This suffices to show that V is semireflexive. Now we investigate the compact open topology of E and show that it agrees with the finest locally convex topology. Let U be in the set B(E) of all balanced, absorbing, def

and convex subsets of E. Set U ◦ = {x ∈ V : |U (x)| ⊆ [0, 1]} and consider 0 = ω ∈ E. Set M = ω−1 (0). Since U is absorbing, Bε · ω ⊆ U for some def

ε > 0, Bε #= {r ∈ K : |r| ≤ ε}. $Thus U ◦ ⊆ (Bε · ω)◦ = {x ∈ V : |(Bε · ω)(x)| ⊆ [0, 1]} = x ∈ V : |ω(x)| ≤ 1ε = ω−1 B1/ε . Since V = limM∈Cofin(V ) V /M, the sets ω−1 (Br ), ω ∈ E and r > 0 are subbasic zero neighborhoods of V (meaning that the collection of all finite intersections of these form a basis of the filter of zero neighborhoods), and the topology of V is the smallest making all ω ∈ E continuous. E\{0} is an embedding and U ◦ is mapped onto a Thus x  → (ω(x)) ω∈E\{0} : V → K closed subset of ω∈E\{0} Br(ω) for a family of positive numbers r(ω). Hence U ◦ is compact in V . Now U  = {ω ∈ E : ω(U ◦ ) ⊆ B1 } is a zero neighborhood for the compact open topology. Since V is semireflexive, by the Bipolar Lemma A2.7 (ii), U  = U . Thus the closed sets which are members of B(E) are zero neighborhoods of the compact open topology. This says that the compact open topology of E and the finest locally convex topology of E agree. Now E  carries the topology of pointwise convergence by A2.6 (i); by the characterisation of the topology of V just derived we know that ηV : V → V  = E  is an algebraic and topological embedding. Since we have seen ηV to be bijective, it is an isomorphism of topological vector spaces. (iii) is a consequence of (i) and (ii). Since we know, using the Axiom of Choice, that a real vector space has a basis and thus is a direct sum of copies of R, duality instantly gives us the following corollary Weakly Complete Topological Vector Spaces Explicitly Corollary A2.9. For a topological vector space V the following conditions are equivalent: (i) V is a weakly complete topological vector space. (ii) There is a set J such that V is isomorphic as a topological vector space to RJ .

Duality: Structure Theory of Weakly Complete Topological Vector Spaces

639

Proof. The dual of a sum, that is a coproduct of vector spaces in the category of real vector spaces is a product in the category of topological vector spaces. The cardinal card J is called the topological dimension of W . (See [103].) Thus the topological dimension of a weakly complete topological vector space is the linear dimension of its dual. In this spirit, the weakly complete topological vector spaces are generalisations of the familiar euclidean vector spaces Rn , and we provide enough evidence in this book that they are the correct generalisation. As an exercise on the duality of a vector spaces we present a lemma which we use in Chapter 3 where we amply use limits of projective systems of Lie groups and Lie algebras. Corollary A2.10. Let g = limk∈J gk be a projective limit of a projective system {γj k : gk → gj | j ≤ k, (j, k) ∈ J × J } of finite-dimensional real vector spaces in the category of weakly complete topological vector spaces. Let γj : g → gj denote the limit maps. Then for each j ∈ J there is an index kj ≥ j such that γj kj (gk ) ⊆ γj (g). Proof. Since gj is finite-dimensional γj (g) is a closed vector subspace of gj . By the Duality Theorem A2.8 (iii), the corollary is equivalent to the following assertion: (∗) Let E = colimk∈J Ek be the direct limit of a direct system {ηj k : Ej → Ek | j ≤ k, (j, k) ∈ J × J } of real vector spaces. Fix an index j ∈ J . Then there is an index kj ≥ j such that ηj kj vanishes on ker ηj . Now E is the directed union of the images ηk (Ek ). If x ∈ Ej is such that ηj k (x) = 0 for all k, then ηj (x) = 0. Thus for each x ∈ ker ηj there is a kx ≥ j such that ηj kx (x) = 0. Since dim ker ηj ≤ dim Ej is finite, ker ηj is finitely generated. Statement (∗) follows.

Duality at Work for the Structure Theory of Weakly Complete Topological Vector Spaces Duality: Vector Subspaces Theorem A2.11. Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then (i) Every closed vector subspace V1 of V is algebraically and topologically a direct summand; that is there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces.

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Appendix 2 Weakly Complete Topological Vector Spaces

(ii) For every closed vector subspace H of E, the relation H ⊥⊥ = H ∼ = (E  /H ⊥ )  ⊥ ⊥  holds and E /H is isomorphic to H . The map F  → F is an antiisomorphism of the complete lattice of vector subspaces of E onto the lattice of closed vector subspaces of E  . E   = E/H  H   ∼ /H ⊥ = V  {0}

 = E/H



{0}    H⊥   V.

Proof. (i) Let E = V  . Then V  is a vector space with its finest locally convex topology by (iii). Let E1 = V1⊥ . Then there is a vector subspace E2 such that E = E1 ⊕ E2 algebraically and topologically by A2.3 (ii). Set V2 = E2⊥ and conclude V = V2 ⊕ V1 2 and V2 ∼ 1 . This proves (ii). with V1 ∼ =E =E (ii) We consider a closed vector subspace H of E and note that E is reflexive by (i) above and that the linear functionals of E/H , all of which are continuous by A2.3 (i), separate the points. Then the Annihilator Mechanism applies (see for instance [102, Lemma 7.17 (iii)]) and shows H = H ⊥⊥ . Since E is reflexive and the continuous functionals, and thus the characters of E/H , separate points, [102, 7.17 (v)] shows that (E  /H ⊥ ) and H are isomorphic topological vector spaces. By A2.3 (ii) we have E = H ⊕ K, and by A2.8 (ii) and its proof we obtain E  = K ⊥ ⊕ H ⊥ with a  of E  . Then closed and hence weakly complete topological vector subspace K ⊥ ∼ =H  ⊥ ⊥ E /H ∼ = K is a weakly complete topological vector space and thus is reflexive  (see for instance [102, Lemma 7.17 (vi)]). In by (ii). This implies that E  /H ⊥ ∼ =H particular, E  /H ⊥ is reflexive by (ii). The last assertion of (iii) is now a consequence of this. Duality: Morphisms Theorem A2.12. (a) Let f : V → W be a morphism of weakly complete topological vector spaces. Assume that f has a dense image. Then f : V → W splits; that is, there is a morphism σ : W → V such that f  σ = idW . (b) Let f : V → W be a morphism of weakly complete topological vector spaces. Then f (V ) is a closed vector subspace of W , and the natural bijection V / ker f → f (V ) is an isomorphism of topological vector spaces. V ⏐ ⏐ q V / ker f

f

−−−→

W  ⏐j ⏐ −−−→ f (V )  f

(c) (The Second Isomorphism Theorem) If V and W are closed vector subspaces of a weakly complete topological vector space U , then V + W is closed, and the function

Duality: Structure Theory of Weakly Complete Topological Vector Spaces

641

f : V /(V ∩ W ) → (V + W )/W , f (v + (V ∩ W )) = v + W is an isomorphism of topological vector spaces. Proof. (a) Let f : V → W be a morphism of weakly complete topological vector spaces with dense image. Then the morphism f is an epimorphism in the category of →V  is a monomortopological vector spaces since it has a zero cokernel. Then f: W phism of real vector spaces with their finest locally convex topology by A2.8 (iii). Thus fhas a zero kernel and so fis injective. Now the linear algebra of vector spaces  → W  such provides, with the use of the Axiom of Choice, a linear function τ : V that τ  f = idW . By A2.3 (i), the map τ is continuous and is thus a morphism of     . By topological vector spaces. From (iii) we know that f  τ is the identity map of W (ii) we get a morphism σ : W → V such that   σ = τ and that f  σ = idW . (b) In the category of weakly complete topological vector spaces, we have a canonical decomposition f V −−−→ W ⏐  ⏐ ⏐j q ⏐ V / ker f

−−−→ f (V )  f

f  (v

+ ker f ) = f (v). After replacing f by f  where q(v) = v + ker f , j (w) = w, we may assume without loss of generality that f is injective and has a dense image. Then f is both a monic and an epic in the category of weakly complete topological vector spaces. By (a) it is also a retraction, and a monic retraction is an isomorphism. In particular, it is surjective and thus (b) is proved. (c) We define a morphism of weakly complete topological vector spaces F : V /(V ∩ W ) → U/W

by F (v + (V ∩ W )) = v + W.

By Part (a), the vector subspace (V +W )/V = F (V ) is closed in U , and thus V +W , the full inverse image under the quotient morphism U → U/W is closed in U . Moreover, the corestriction of the injective morphism F to the image. Namely, f : V /(V ∩W ) → (V + W )/W , is an isomorphism of weakly complete topological vector spaces. Filter Bases of Closed Linear Subspaces Theorem A2.13. Let W be a weakly complete topological vector space and F a filter basis of closed linear subspaces. (a) Assume that F is a closed vector subspace of W . Then   F+ F = F + H. (∗) H ∈F

If, in addition, dim E/F < ∞, then   F+ F = (F + H ). H ∈F

(∗∗)

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Appendix 2 Weakly Complete Topological Vector Spaces

(b) The following conditions are equivalent: (i)  lim F = 0. (ii) F = {0}. (c) Assume that F ⊆ N (W ). Then the following conditions are equivalent: (i) lim F = 0. (ii) F is a basis of N (W ). Proof. (a) By duality of weakly complete topological vector spaces according to Theorems A2.8 (iii) and A2.11 (ii) by passing to annihilators, the assertion is equivalent to the following assertion in a vector space E with a vector subspace S and a directed (ascending) set D of vector subspaces   S ∩ U. S∩ D= U ∈D

This relation holds for elementary set-theoretic reasons. Now assume that dim W/F < ∞. All vector subspaces of a finite-dimensional (Hausdorff topological) vector space are closed. Hence every vector subspace of W containing F is the full inverse image under the quotient morphism W → W/F of a vector subspace of W/F and thus is closed. Thus (∗∗) is equivalent to (∗) in the present situation. (b) Let U denote the filter of zero neighborhoods of W . Then by definition, (i) is equivalent to (i) The filter F  generated by F contains U. This in turn is equivalent to (i) (∀U ∈ U)(∃V ∈ F ) V ⊆ U .   Now (i) ⇒ (ii) is clear since F ⊆ U = {0}. We prove (ii) ⇒ (i) . Let U be a zero neighborhood. Since W is weakly complete the filter basis V of cofinite-dimensional vector spaces V converges to 0. Hence we may assume that there is a cofinite-dimensional vector space V contained in U , and we may even assume that V is the only vector space containing V and being contained  in U . By (a) we have V  ∈F (V + V  ) = V + F = V + {0} = V . Now {(V + V  )/V : V  ∈ F } is a filter basis of vector subspaces of the finite-dimensional vector space W/V intersecting in {0}, and thus there is a V  ∈ F such that (V + V  )/V is zero, that is, V  ⊆ V ⊆ U . This proves (i) . (c) Trivially, (ii) implies (i) because lim N (W ) = 0. For a proof of (i) implies (ii), let V ∈ N (W ). Then W/V is finite-dimensional, and by the continuity of the quotient map W → W/V the filter basis {B + V /V : B ∈ B} of vector subspaces of W/V converges to zero. Since W/V has no small subgroups, there is a B ∈ B such that (B + V )/V = V /V , that is B + V = V or, equivalently, B ⊆ V .

Topological Properties of Weakly Complete Topological Vector Spaces

643

In lattice theoretic terminology conclusion (a) reads: The lattice of closed vector subspaces of V is meet continuous. For the following, recall that an affine subspace A of a vector space W is a subset of the form A = g + V for some vector subspace V . The affine subspace is linear iff g ∈ V. Filter Bases of Affine Subspaces Theorem A2.14. Let W be a weakly complete topological vector space and F a filter  basis of closed affine subspaces. Then F = Ø. Proof. (a) We write the affine members of F in the form gj + Vj with closed vector subspaces Vj and elements gj , j ∈ J . We claim that the set {Vj : j ∈ J } is a filter basis. Indeed let i, j ∈ J , then there is a k ∈ J such that gk + Vk ⊆ (gi + Vi ) ∩ (gj + Vj ), since F is a filter basis. Therefore gi + Vi = gk + Vi and gj + Vj = gk + Vj . Now gk +Vk ⊆ (gk +Vi )∩(gk +Vj ), and hence Vk ⊆ Vi ∩Vj . Let V = j ∈J Vj . Then W/V is a weakly complete topological vector space and F /V = {(gj +V )+Vj /V : j ∈ J } is a filter basis of closed affine subsets. It clearly suffices to show that F /V has a nonempty intersection. Thus we assume from here on that V = {0}, that is the filter def

basis V = {Vj : j ∈ J } has the intersection {0}. But then lim V = 0 in W by (b) above. This implies that F is a Cauchy filter: Let U be an identity neighborhood; then there is a j ∈ J such that Vj ⊆ U . Then (gj + Vj ) − (gj + Vj ) = Vj ⊆ U . Since W is a complete topological vector space, every Cauchy filter basis converges. Let g = lim F . Since all gj + Vj are closed, we have g ∈ gj + Vj for all j ∈ J and this completes the proof of the lemma. In terms of a terminology that has been used for situations resembling the one we have in the previous theorem, this result can be expressed in the following form: Weakly complete topological vector spaces are linearly compact.

Topological Properties of Weakly Complete Topological Vector Spaces A topological group G is topologically compactly generated if there is a compact subset C ⊆ G such that G = C. It is compactly generated if there is a compact subset C ⊆ G such that G = C. We begin by observing that the idea of a weakly complete topological vector space to be a topologically compactly generated pro-Lie group is not very restrictive. Remark A2.15. Any weakly complete topological vector group is topologically compactly generated.

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Appendix 2 Weakly Complete Topological Vector Spaces

Proof. For the purposes of the proof we may and will assume that W = RJ for some set J . For any subset I of J we identify RI naturally with a subgroup of RJ . The def  may and will be identified with R(J ) , the set of all f : J5→ R with finite dual E = W support, in such a fashion that f ∈ E and g ∈ W gives us f, g = j ∈J f (j )g(j ). Let K = {δj ∈ RJ : j ∈ J } ∪ {0}. Let V be a cofinite-dimensional vector subspace def  of W . Then V ⊥ is a finite-dimensional vector subspace of the dual E = W . Let Fin(J )  denote the set of finite subsets of J . Since E = I ∈Fin(J ) R(I ) and since V ⊥ is finite-dimensional, there is an I ∈ Fin(J ) such that V ⊥ ⊆ R(I ) and thus V ⊆ (R(I ) )⊥ = R J \I . Hence K \ V = {δi : i ∈ I } is finite. Therefore K is compact. On the other hand, W = R(J ) = [0, 1] · K and [0, 1] · K is a compact subset of R(J ) . Hence W is topologically compactly generated. Since δj ∈ ZJ ⊆ RJ , the assertion on ZJ follows analogously, as ZJ = Z(J ) = K. In Chapter 5 we specified the following concepts: A topological space is called a Polish space if it is completely metrizable and second countable. It is said to be σ -compact, if it is a countable union of compact subspaces. It is said to be separable if it has a dense countable subset. Countable products of Polish spaces are Polish. For instance, RN is Polish. Remark A2.16 (= 5.22). (i) Every almost connected locally compact group is compactly generated. (ii) Every compactly generated topological group is σ -compact. (iii) A topological group whose underlying space is a Baire space and which is σ -compact is a locally compact topological group. (iv) A σ -compact Polish group is locally compact. (v) A compactly generated Baire group is locally compact. Proof. See proof of 5.22. Proposition A2.17. For a weakly complete topological vector space W , the following statements are equivalent: (A) (B) (C) (D)

W W W W

is σ -compact. is locally compact. is finite-dimensional. is compactly generated.

Proof. The equivalence of (B) and (C) was shown in A2.2 (iii) and (iv). Locally compact connected groups are compactly generated by A2.16 = 5.22 (i) and so (B) implies (D); and (D) implies (A) by A2.16 = 5.22 (ii). In order to prove that (A) implies (C), let W be a weakly complete σ -compact topological vector space. Its dual is a vector space E and W is finite-dimensional iff E is finite-dimensional. Suppose that E is infinite-dimensional. Selecting from a basis an infinite countable subset we get a vector subspace F with a countable basis. Then

Topological Properties of Weakly Complete Topological Vector Spaces

645

W/F ⊥ is isomorphic to the dual of F ∼ = R(N) and therefore W/F ⊥ is a homomorphic image of W which is isomorphic to RN and therefore is a Polish topological vector space. Since it is also σ -compact as a homomorphic image of a σ -compact group, it is locally compact by A2.16 = 5.22. But then it is finite-dimensional, a contradiction. Proposition A2.18. For a weakly complete topological vector space W , the following statements are equivalent: (i) (ii) (iii) (iv) (v) (vi)

W W W W W W

∼ = RJ with card J ≤ ℵ0 . is locally compact or is isomorphic to RN . is finite-dimensional or is isomorphic to RN . is second countable. is first countable. is Polish.

Proof. By the remarks preceding the proposition, for each cardinal ℵ, there is, up to isomorphy of topological vector spaces and of topological groups one and only one weakly complete topological vector space of topological dimension ℵ, namely, Rℵ . Conditions (i), (ii), (iii) are ostensibly all equivalent to saying that ℵ is countable. The weight w(W ), that is the smallest cardinal representing the cardinality of a basis for the topology of W ∼ = Rℵ is ℵ0 if ℵ is countable, and is ℵ if ℵ is uncountable (see e.g. [102, Exercise EA4.3 following Proposition A2.8]), so (iv) is likewise equivalent to (ii), and implies (v). If the weakly complete topological vector space W is first countable, then the filter basis  (W ) of cofinite-dimensional closed vector subspaces has and thus W ∼ = limV ∈ (W ) W/V is a closed vector subspace of  a countable basis, N ∼ W/V R and thus (vi) implies (iv). If (iv) is satisfied then the complete = V ∈ (W ) topological vector space W is metrizable (see e.g. [102, Theorem A4.16]) and thus (vi) follows; trivially (vi) implies (iv). Proposition A2.19. For a weakly complete topological vector space W , the following statements are equivalent: (a) W is separable. (b) W contains a dense vector subspace of countable linear dimensions over R. (c) W is isomorphic as a topological vector space to RJ with card J ≤ 2ℵ0 . These conditions are implied by the equivalent statements of Proposition A2.17. Proof. A second countable space is always separable: It suffices to pick a point in every set in a countable basis for the topology: this yields a countable dense set. What remains therefore is to see the equivalence of (a), (b), and (c). We may safely assume that W is infinite-dimensional, since the finite-dimensional case is clear. (a) ⇒ (b): Let C be a countable dense subset of RJ . Then the real linear span of C is dense vector subspace of RJ whose linear dimension is countable. (b) ⇒ (c): Assume that ι : R(N) → RJ is a linear map between vector spaces such that im(ι) = RJ . We give R(N) the finest locally convex topology. The vector space

646

Appendix 2 Weakly Complete Topological Vector Spaces

dual of R(N) may be identified with RN , and that of RJ with R(J ) . The morphism ι is both an epic (and a monic) in the category of (Hausdorff) topological vector spaces. Its adjoint morphism ι : R(J ) → RN is a monic (and epic) and is therefore an injection (with dense image). Thus card(J ) ≤ dimR RN = 2ℵ0 . (c) ⇒ (a): Let W = RJ with card(J ) = 2ℵ0 . We shall show that W is separable; since RI with card(I ) ≤ card(J ) is a homomorphic image of RJ , this will entail the implication. The topological vector space dual of RJ may be identified with R(J ) and then there is a linear bijection β : R(J ) → RN . If we give R(J ) the finest locally convex topology and RN the product topology, then β is an epic (and a monic) in the category of topological vector spaces and thus its adjoint β  : R(N) → RJ has a dense image (and is injective). Even in the finest locally convex topology, Q(N) is dense in R(N) , and Q(N) is countable. Hence RJ is separable as asserted.

Summability in Weakly Complete Topological Vector Spaces Lemma A2.20. Let V be a weakly complete topological  vector space and F a filter basis consisting of closed vector subspaces satisfying F = {0}. Let J be an infinite index set and let {aj : j ∈ J } be a family of elements of V such that for5 each finite / E. Then j ∈J aj is subset E ⊆ J we find an F ∈ F such that aj ∈ F whenever j ∈ summable in V . Proof. Let Fin(J ) denote the directed set of finite subsets of J . Since V is a complete topological vector 5 space it suffices to show that the net (sE )E∈Fin(J ) of all finite partial sums sE = j ∈E aj is a Cauchy net. That is, we have to show that for each zero neighborhood U of V there is an EU ∈ Fin(J ) such that for all E, E  ⊇ EU we have sE  − sE ∈ U . Let a zero neighborhood U of V be given. By Theorem A2.13 (b) we have lim F = 0. Hence there is an F ∈ F such that F ⊆ U . By hypothesis on the family of the aj we find a finite set EU ∈ Fin(J ) such that j ∈ J \ EU implies aj ∈ F .    Now let E, E  ∈ Fin(J 5) and E, E ⊇5EU . Then (E \ E) ∩ EU = Ø = (E \ E ) ∩ EU and thus sE  − sE = j ∈E  \E aj + j ∈E\E  (−aj ) ∈ F ⊆ U and this is what we had to show.

Continuous Symplectic Maps on Weakly Complete Topological Vector Spaces Lemma A2.21. Let U , V and W be weakly complete topological vector spaces and b : U × V → W a continuous bilinear map. Assume dim W < ∞. Then there are cofinite-dimensional vector subspaces E of U and F of V such that b vanishes on E × V and U × F .

Continuous Symplectic Maps on Weakly Complete Topological Vector Spaces

647

Proof. Let W ∗ be a zero neighborhood of W not containing any vector subspaces other than {0}. We have b({0} × V ) = {0}. Hence for each y ∈ V there are open zero neighborhoods Uy and Vy in U and V , respectively, such that b(Uy ×(y+Vy )) ⊆ W ∗ for all y ∈ V . Now by the definition of the topology of weakly complete topological vector spaces there are cofinite-dimensional vector subspaces Ey and Fy contained in Uy and ∗ Vy , respectively, whence b(Ey × (y + Fy )) ⊆ W and therefore b(Ey × (y + Fy )) = b(Ey × {y}) + x∈Ey b({x} × Fy ) = {0} because U does not contain any nonsingleton linear subspaces. Since this set is stable under scalar multiplication we have in fact (∀y ∈ V ) b(Ey × (R · y + Fy )) = {0}.

(∗)

Vy . We find vectors y2 , . . . , yn in V such Let y5 1 = 0; then Fy1 is cofinite-dimensional in that nj=1 (R · yj + Fyj ) = V , and set E = nj=1 Eyj . As an intersection of finitely many cofinite-dimensional vector subspaces, E is cofinite-dimensional. Therefore (∗) implies b(E × V ) =

n 6

b(E × (R · yj + Fyj )) ⊆

j =1

n 6

b(Eyj × (R · yj + Fyj )) = {0}.

j =1

Exchanging the roles of U and V yields the second part of the assertion. If E and F are vector spaces, then a function s : E × E → F is called symplectic if it is bilinear and satisfies s(x, y) = −s(y, x) for all x, y ∈ E. Proposition A2.22. For each weakly complete ; topological vector space V there is a weakly complete topological vector space 2 V and a continuous symplectic map (v, w)  → v ∧ w : V × V →

2 >

V

such that for every continuous symplectic map b : V × V → W of weakly ; complete topological vector spaces there is a unique continuous linear map b : 2 V → W such that b(v, w) = b (v ∧ w). ; The linear span of {v ∧ w : v, w ∈ V } is dense in 2 V . Proof. For  a finite-dimensional V with ; dimension n, the construction is classical yielding an n2 -dimensional vector space 2 V with the required properties. For an arbitrary weakly complete topological vector space let  (V ) denote the filter basis of closed cofinite-dimensional vector subspaces; then V ∼ = V (V ) = limj∈ (V ) V /j. For j ⊇ k in ;  (V ), let pj,k : V /k → V /j be the quotient morphism. Since 2 • is a functor of the category of finite-dimensional vector spaces to itself, 2 >

pjk :

2 >

V /k →

2 >



V /j | k ⊆ j in  (V )

is a projective system. We define 2 >

V = lim

j∈ (V )

2 >

V /j.

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Appendix 2 Weakly Complete Topological Vector Spaces

Since the inclusion functor from topological vector spaces to topological spaces preserves limits, the limit property for limj∈ (V ) V /j in the category of topological spaces ; gives us a fill in map ∧ : V × V → 2 V filling in correctly in the infinite diagram ;2 ∧ −−−−→ V V ×V ⏐ ⏐ ;2 ⏐ ⏐ pj ×pj   pj ;2 V /j × V /j −−−−→ V /j. ∧V /j

We must verify the universal property: Let b : V ×V → W be a continuous symplectic map. Pick any j ∈  (W ). From Lemma A2.21 we deduce the existence of a k in  (V ) such that the symplectic map qj

b

V × V −−→ W −−→ Wj factors through V /k × V /k via V ×V ⏐ ⏐ pk ×pk 

b

−−−−→ fjk

;2 W ⏐ ⏐;2  qj ;2 W/j.

V /k × V /k −−−−→ ; Then by the universal property of 2 V /k for finite-dimensional vector spaces we find ; a linear map fj k : 2 V /k → W/j such that fjk = fjk  ∧V /k . By the definition of ;2 V , we have a commutative diagram ;2 ∧ −−−−→ V V ×V ⏐ ⏐ ⏐ ⏐ ;2 pk ×pk   pk ∧V /k ;2 V /k × V /k −−−−→ V /k. Now we obtain a natural commutative diagram V ×V ⏐ ⏐ b W ⏐

qj ⏐ 

W/j

;2 V ⏐ ⏐ ;2  pk ;2 V /k ⏐ ⏐ id ;2 ←−− −− V /k ∧

−−−−→

fjk

for each j ∈  (W ). Since W ∼ = W (W ) = limj∈ (W ) W/j we get a unique morphism ; b : 2 V → W such that b = b  ∧. This proves the main assertion. ; It remains to show that the span of the v ∧ w is dense in 2 V as v and w range ; through V . But for each j ∈  (V ), the span of the v ∧ w is mapped by pj : 2 V → ;2 V /j onto the span of the vj ∧wj , vj , wj ∈ Vj . This span, however is V /j. Therefore, ; span{v ∧ w : v, w ∈ V } + j = 2 V for all j, and that proves the assertion.

Miscellaneous Observations Involving Weakly Complete Topological Vector Spaces

649

For S ⊆ V we set def

Ann(S) = {w ∈ V : (∀s ∈ S)s ∧ w = 0}, ; where (v, w)  → v ∧ w : V × V → 2 V is the universal symplectic map of Proposition A2.22. Corollary A2.23. Let W be a vector subspace of V . Then ⎧ ⎪ if dim W = 0, ⎨V Ann(W ) = W if dim W = 1, ⎪ ⎩ {0} if dim W ≥ 2. Proof. We begin by reviewing the assertion in the finite-dimensional case.5 Let v = 0 in n V and let v = e1 , . . . , e5 n be a basis of V . Assume v∧x = 0. We write x = m=1 rm ·em n and find 0 = e ∧ x = m=1 rm · e1 ∧ em . Because of the linear independence of the e1 ∧ em , m = 2, . . . , n, we get r2 = · · · = rn = 0, that is, x = r1 · v. If v1 and v2 are linearly independent and 0 = v1 ∧ x = v2 ∧ x, then x = r1 · v1 = s2 · v2 by the preceding and this implies x = 0. This establishes the claim in finite dimensions. Now let V be a weakly complete topological vector space of arbitrary dimension and let 0 = v;∈ V and assume that v ∧ x = 0 for some x ∈ V . Let j ∈  (V ). By the definition of 2 V and ∧ we have (v + j) ∧V /j (x + j) = (v ∧ x) + j = 0. It follows from the preceding discussion of the finite-dimensional case that x ∈ R · v + j as soon as v ∈ / j, as is the case for all sufficiently small j. Thus x ∈ j∈ (V ) (R · v + j) = R · v, since R · v is closed in V . As in the finite-dimensional case, this is enough to secure the assertion in the general case.

Miscellaneous Observations Involving Weakly Complete Topological Vector Spaces We have seen in Chapter 5 that weakly complete vector groups, that is, the additive groups of weakly complete topological vector spaces, play a significant role in the structure theory of abelian pro-Lie groups. For the sake of completeness, we sample some of the insights we have had. Lemma A2.24 (= 5.10). Let G be an abelian pro-Lie group, and assume that comp(G) is compact. Then the following conclusions hold. (i) There is a weakly complete vector group W and a compact abelian group C which is a product of circle groups such that G may be considered as a closed subgroup of W × C such that G ∩ ({0} × C) = comp(G). (ii) G/ comp(G) is embedded as a closed subgroup into the weakly complete vector group W .

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Appendix 2 Weakly Complete Topological Vector Spaces

Remark A2.25 (= 5.24). For a discrete abelian group, the following statements are equivalent: (i) (ii) (iii) (iv)

G is finitely generated free. G is isomorphic to a closed additive subgroup of Rn for some n = 0, 1, 2, . . . . G is isomorphic to a closed additive subgroup of RJ for some set J . G is isomorphic to a closed additive subgroup of a weakly complete topological vector space.

This result provides an alternative proof of the fact that the prodiscrete free abelian group F of infinite rank cannot be discrete. Example A2.26. There is a closed nondiscrete free abelian subgroup of countably infinitely many generators in RR . Proof. See Proposition 5.2. We remark in passing that an abelian Lie group, being isomorphic to a group Rp × Tq × D for some discrete group (see for instance [102], Exercise E5.18 following Proposition 5.54, or Corollary 7.58), is isomorphic to a subgroup of a connected abelian Lie group if and only if it is algebraically generated by a compact subset. In [34] it is shown that the class of abelian topological groups which are isomorphic to a product C × RX × ZY where C is a compact group and X and Y are countable sets is closed under the passage to closed subgroups and to quotient groups. This points out the fact that quotient groups of RX are complete if X is countable. In this sense the example of RR is minimal in the class of weakly complete topological vector spaces, if one momentarily accepts the Continuum Hypothesis. In [116] it is proved that a closed connected subgroup of RX is a closed vector subspace; consequently it is a direct summand algebraically and topologically by A2.11 (i). In particular, the quotient is a weakly complete topological vector space. For more information we refer the reader to Chapter 5 of this book and for additional information concerning the duality theory of closed subgroups of weakly complete topological vector spaces we refer the reader to [102, Propositions 7.31 through 7.35].

Appendix 3

Various Pieces of Information on Semisimple Lie Algebras

In this appendix we record some results on semisimple or reductive finite-dimensional Lie Algebras which we need in the body of the book. The first portion deals with the question whether in a semisimple Lie algebra every element is a bracket, the second with the question whether a maximal compactly embedded abelian subalgebra of a reductive Lie algebra is contained in only one compactly embedded subalgebra. Prerequisites. In this Appendix, in the first part, we shall make use of Kostant’s Convexity Theorem (see TheoremA3.1), and of basic facts of the theory of finite-dimensional real Lie algebras and their complexification for which we refer to standard sources such as Bourbaki [16]–[20]. We shall also refer to a result of Gordon Brown [33]. In the second part we shall refer to standard source books on semisimple Lie algebras.

Brackets in Semisimple Lie Algebras If A and B are subsets of a finite-dimensional real Lie algebra g, then [A, B] traditionally denotes the linear span of all brackets [x, y], x ∈ A, y ∈ B. We therefore write brack(A, B) = {[x, y] | x ∈ A, y ∈ B}. def

The set brack[g] = brack(g, g) is trivially invariant under scalar multiplication. The commutator algebra g = [g, g] is the sum of a minimal number m(g) of summands [g, g] = brack[g] + · · · + brack[g] . )* + (

(∗)

m(g)

From Example 14.21–14.22 we recall that the minimal number of commutators required to represent a member of the commutator subalgebra of a class 2 nilpotent finite-dimensional real Lie algebras is unbounded;    in fact, we constructed a nilpotent Lie algebra nn such that the commutator subgroup of (nn , ∗) of class 2 of dimension n + n2 = n+1 2 contains an element which is not the product of fewer than n−1 4 commutators, and that implies that the commutator algebra of nn contains an element that is not the sum of fewer than n−1 4 brackets. In other words, the number m(nn ) defined in (∗) above ≤ m(n satisfies n−1 n ). The point is that already in the class of nilpotent Lie algebras g 4 of the lowest possible class of nilpotency to make them nonabelian, the number m(g) is unbounded.

652

Appendix 3 Various Pieces of Information on Semisimple Lie Algebras

 This permitted us to argue that the product ∞ n=2 nn is a class 2 nilpotent proLie algebra with a nonclosed commutator subalgebra. From the structure theory of reductive pro-Lie algebras in Chapter 7 (see Theorem 7.27) we know that a reductive  pro-Lie algebra is a product of the form RI × j ∈J sj for suitable index sets I and J and a family sj of simple real Lie algebras. In Theorem 7.27 we showed that the (algebraic) commutator subalgebra of a reductive pro-Lie algebra is closed. In the proof we use a result which we present in this appendix (Corollary A3.5): For a real reductive Lie algebra g we have m(g) ≤ 2. This follows from a consequence of Gordon Brown’s Theorem of 1963 [33] which we shall quote, that every element in a complex semisimple Lie algebra is a bracket. What we shall prove in this appendix is m(g) ≤ 4 which is a rather direct consequence of the main result of this appendix which is more in the spirit of this book: In a semisimple compact real Lie algebra g every element is a bracket.

Brackets in Compact Lie Algebras The Lie algebra so(3) of all 3 × 3 skew symmetric matrices is isomorphic to the euclidian vector space R3 equipped with the vector product. In this vector space every nonzero element u is the vector product u = v × w of two orthogonal vectors v ⊥ w both of which are orthogonal to u. It follows that every element of so(3) is a bracket. The Lie algebra so(3) is an example of a compact Lie algebra. Recall that a real Lie algebra is called compact if it supports an invariant scalar product, equivalently, that it is isomorphic to the Lie algebra of a compact Lie group. (See [102, Proposition 6.2ff., notably, Theorem 6.6].) The example raises the following question: For which real Lie algebras g it is true that g = brack[g], that is, that every element is a bracket? We shall see that this is the case for instance for all semisimple compact Lie algebras. This is an “infinitesimal” analog of the theorem that in a compact connected semisimple Lie group every element is a commutator (Gotô’s Theorem, see for instance [102, Corollary 6.56]). Before we can show that every element of a compact real Lie algebra is a bracket we have to cite a result known as Kostant’s Convexity Theorem (see for instance [84, p. 257, Theorem III.8.4]). On a semisimple compact Lie algebra g we may and will take the invariant inner product on g given by (X|Y ) = − tr ad X ad Y , the negative of the Cartan–Killing form. In the following theorem we shall consider a compact Lie algebra g and let G be the compact group of its inner automorphisms ead g = ead g . We fix an arbitrary Cartan subalgebra t of g, and take the orthogonal projection p : g → t with respect to an invariant scalar product of g. Moreover, we let W be the Weyl group of g with respect to t. Assume that for x ∈ t, the G-orbit of x in g is denoted G · x and the W -orbit of x in t is denoted W · x. The convex hull of a set S in a real vector spaces is denoted conv(S). If S is compact then conv(S) is precompact and thus compact if the vector space is complete,

Maximal Compactly Embedded Subalgebras

653

for instance, if it is finite-dimensional. This applies, in particular, to the convex hull of the compact orbit G · x in g, and so conv(G · x) is compact. Theorem A3.1 (Kostant’s Convexity Theorem). If x is an arbitrary element of the Cartan subalgebra t of a compact Lie algebra g, then p(G · x) is the convex hull of the finite subset W · x in t. This is not a trivial result. It is very useful in many respects, and it will serve us well, too. The following is based on a communication of Karl-Hermann Neeb to the authors. Theorem A3.2. Every element of a semisimple compact Lie algebra is a bracket. Proof. The set of fixed points of the linear action of G on g is the center of g; since the Lie algebra g is semisimple, its center is {0}. Thus 0 is in the closed convex hull conv(G · x) of G · x by [102, Theorem 3.36]. Thus 0 = p(0) ∈ p(conv(G · x)) = conv(p(G · x)). Theorem A3.1 applies and yields conv(p(G · x)) = conv(W · x). Therefore 0 ∈ conv(W · x) = p(G · x) again by Theorem A3.1. Thus there is an inner automorphism γ ∈ G such that γ (x) ∈ ker p. Now assume that x0 is a regular element of g contained in t, that is, dim ker ad x0 is minimal. Then t = ker ad x0 and if t+ abbreviates ker p, the orthogonal complement of t in g, then ad x0 |t+ : t+ → t+ is bijective. Therefore ker p = t+ = [x0 , g]. We conclude that x ∈ γ −1 [x0 , g] = [γ −1 (x0 ), g] ⊆ brack[g]. Corollary A3.3. Let g be any semisimple real Lie algebra. Then every element is the sum of at most four brackets, that is g = brack[g] + brack[g] + brack[g] + brack[g]. Proof. Let g be a semisimple real Lie algebra. Then its complexification gC has a compact real form, that is, there is a real compact subalgebra k such that gC = k + i · k. From Theorem A3.2 we have k = brack(k, k) ⊆ brack(gC , gC ). Thus gC = brack[gC ] + brack[gC ]. So for any x ∈ g there are elements b1 , b2 ∈ k such that g = b1 + i · b2 . By Theorem A3.2 there are elements xj , yj , uj , vj ∈ g, j = 1, 2, such that xj + i · yj , uj + i · vj ∈ k and bj = [xj + i · yj , uj + i · vj ], j = 1, 2. Thus b1 = [x1 , u1 ] − [y1 , v1 ] + i · z1 with z1 ∈ g and i · b2 = −[y2 , u2 ] − [x2 , v2 ] + i · z2 with z2 ∈ g. Therefore x = [x1 , u1 ] − [y1 , v1 ] − [x2 , v2 ] − [y2 , u2 ]. Actually, the result in this corollary is not the best possible representing the status of our knowledge. From the result of Brown [33] we have the following consequence: Proposition A3.4. Every element in a semisimple complex Lie algebra is a bracket. By an even simpler calculation than that in the proof of Corollary A3.3 we then obtain Corollary A3.5. Let g be any semisimple real Lie algebra. Then every element of g is the sum of at most two brackets, that is g = brack[g] + brack[g]. Proof. Let x ∈ g ∈ gC . Then there are elements a, b, u, v ∈ g such that x = [a + ib, u + iv] by Proposition A3.4. Thus x = [a, u] − [b, v]. Corollary A3.5 is the result we used in Chapter 7 in the proof of Lemma 7.26.

654

Appendix 3 Various Pieces of Information on Semisimple Lie Algebras

Maximal Compactly Embedded Subalgebras Proposition A3.6. Let g be a finite-dimensional real reductive Lie algebra and k and k1 two maximal compactly embedded subalgebras containing one and the same maximal compactly embedded Cartan subalgebra t. Then k = k1 . Proof. There is a direct reference for this result: See [84, p. 217, Proposition II.5.6, notably Section (i)]. Another pertinent reference is [157, p. 260ff]. It is clear from Theorem 12.15 that every compactly embedded abelian subalgebra t is contained in a maximal compactly embedded subalgebra k. If t is a Cartan subalgebra, then k is unique by the preceding Proposition A3.6. Write G for the group Inn(g) of inner automorphisms of a finite-dimensional Lie algebra g. Definition A3.7. For a subalgebra k of G we set N(k, G) = {ϕ ∈ G : ϕ(k) = k} and call this group the normalizer of k in G. By Theorem 12.27, G operates transitively on the set of maximal compactly embedded subalgebras m, and the normalizer N(m, G) is the isotropy (or stability) group of this action at the point m. It also acts on the set of all maximal compactly embedded abelian subgroups a. Assume a ⊆ m and set A = ead a , and M = ead m . then N(a, M)/A is the Weyl group of m with respect to a. By Corollary 12.31, G acts transitively on the set  of pairs (a, m) where a is a maximal compactly embedded abelian subalgebra contained in the maximal compactly embedded subalgebra m. Thus N(a, G) acts transitively on the set  of all m containing a. Therefore Lemma A3.8. There is a continuous bijection β : N(a, G)/(N (a, G) ∩ N (m, G)) → .



It follows that  is singleton iff N(a, G) ⊆ N (m, G). If this is the case, then N(a, G)0 ⊆ N (m, G)0 which is the case iff n(a, g) ⊆ n(m, g).

(∗)

Since M is its own normalizer (see [78, p. 208, Theorem 3.3, and p. 252, Theorem 1.1]), we have n(m, g) = m. On the other hand n(a, g) = z(a, g) as follows from the decomposition of g into isotypic decomposition of the action of A on g. (See e.g. [102], Definition 4.21ff.) Therefore (∗) is equivalent to z(a, g) ⊆ m. However, z(a, g) contains all Cartan subalgebras containing a. Therefore (∗) implies that m contains a Cartan subalgebra c containing a. Since a is maximal abelian, we conclude a = c. Thus we have seen Proposition A3.9. If a is a maximal compactly embedded abelian subalgebra of a reductive Lie algebra g, then the following conditions are equivalent: (i) a is a Cartan subalgebra of g. (ii) a is contained in a unique maximal compactly embedded subalgebra of g.

Postscript

655

Postscript We started the Appendix with the most elementary example, namely, so(3). Here everyone knows from elementary linear algebra that every element is a commutator. The example of sl(2, R) is a bit less elementary,  0  but indeed, in sl(2,  element  R) every  is a bracket as well. For a proof set h = 01 −1 and p + = 00 01 , p − = 01 00 . Then def

def

h = [p+ , p− ] and the element k = p + + p − satisfies [h, k] = u = p + − p − . Also [h, p± ] = ±2 ·p ± . Since every element of sl(2, R) is conjugate to either h, or u, or p + (see for instance [84, p. 108, Proposition II.3.6]), every element of sl(2, R) is a bracket. So whenever an element of a Lie algebra is contained in a subalgebra isomorphic to either so(3) or sl(2, R), then it is a bracket. The fact that every element of sl(n, K) for a field K of characteristic zero is a commutator was shown by Shoda in 1937 [182]. G. Brown proves in 1963 [33] that each element is a commutator in any semisimple Lie algebra over any field of characteristic zero that is splittable (déployable in the sense of Bourbaki [19, p. 75]; in fact his result reaches further as far as the field is concerned [33]. Theorem A3.2 originates from correspondence between K. H. Hofmann and K.-H. Neeb [111]. We present it here because it is of independent interest and does not fall into any of the categories covered by Brown’s Theorem. It seems to be unknown whether in an arbitrary real semisimple Lie algebra every element is a bracket. The second portion of this Appendix provides some information which illuminates the role of a maximal compactly embedded abelian subalgebra a of a reductive Lie algebra and those maximal compactly embedded subalgebras m containing it. Only if a is in fact a Cartan subalgebra is it true that m is uniquely determined.

Bibliography

[1] Adámek, J., H. Herrlich, and G. E. Strecker, Abstract and concrete categories, Wiley, New York 1990. [2] Adams, J. F., Lectures on Lie groups, W. A. Benjamin, New York 1969. [3] Arkhangel’ski˘ı, A. V., and L. S. Pontryagin, eds., General topology I, Encyclopaedia Math. Sci. 17, Springer Verlag, Berlin 1990. 83 [4] Armacost, D. L., The structure of locally compact abelian groups, M. Dekker, New York 1981. [5] Banaszczyk, W., Additive subgroups of topological vector spaces, Lecture Notes in Math. 1466, Springer-Verlag, Berlin 1991. 238, 595 [6] Berestovskii, V., and C. Plaut, Covering group theory for topological groups, Topology Appl. 114 (2001), 141–186. 268 [7] Berestovskii, V., and C. Plaut, Covering group theory for locally compact groups, Topology Appl. 114 (2001), 187–199. 268 [8] Berestovskii, V., and C. Plaut, Covering group theory for compact groups, J. Pure Appl. Algebra 161 (2001), 255–267. 268 [9] Birkhoff, G., Lie groups simply isomorphic with no linear group, Bull. Amer. Math. Soc. 42 (1936), 883–888. 9 [10] Borel, A., Topology of Lie groups and characteristic classes, Bull. Amer. Math. Soc. 61 (1955), 397–432. [11] Borel, A., Seminar on transformation groups, Princeton University Press, Princeton, NJ, 1960. [12] Boseck, H., G. Czichowski, and K. Rudolph, Analysis on topological groups – general Lie theory, Teubner, Leipzig 1981. [13] Bourbaki, N., Algèbre, Chap. 8, Modules et anneaux semi-simples, Hermann, Paris 1958. [14] Bourbaki, N., Commutative algebra, Addison Wesley, Reading, Mass., 1972. [15] Bourbaki, N., Espaces vectoriels topologiques, Masson, Paris 1981. 636 [16] Bourbaki, N., Groupes et algèbres de Lie, Chap. 1, Hermann, Paris 1960. 269, 284, 291, 293, 303, 304, 305, 309, 312, 331, 496, 651 [17] Bourbaki, N., Groupes et algèbres de Lie, Chap. 2 et 3, Hermann, Paris 1972. 261, 287, 336, 337, 365, 375, 377, 378, 621, 626 [18] Bourbaki, N., Groupes et algèbres de Lie, Chap. 4, 5 et 6, Hermann, Paris 1968. 108 [19] Bourbaki, N., Groupes et algèbres de Lie, Chap. 7 et 8, Hermann, Paris 1975. 50, 322, 325, 327, 506, 655 [20] Bourbaki, N., Groupes et algèbres de Lie, Chap. 9, Masson, Paris 1982. 651 [21] Bourbaki, N., Intégration, Chap. 1, 2, 3, et 4, Hermann, Paris 1965. [22] Bourbaki, N., Intégration, Chap. 7 et 8, Hermann, Paris 1963.

658

Bibliography

[23] Bourbaki, N., Intégration, Chap. 9, Hermann, Paris 1969. 4 [24] Bourbaki, N., Théories spectrales, Chap. 1 et 2, Hermann, Paris 1967. [25] Bourbaki, N., Topologie générale, Chap. 1–4, nouvelle édition, Hermann, Paris 1971. 6, 63, 83, 86, 96, 106 [26] Bourbaki, N., Topologie générale, Chap. 5–10, nouvelle édition, Hermann, Paris 1974. 85, 203, 213, 233, 594 [27] Bourbaki, N., Variétés différentielles et analytiques, fascicule de résultats, Hermann, Paris 1971. [28] Bredon, G. E., Introduction to compact transformation groups, Academic Press, NewYork 1972. [29] Bredon, G. E., Topology and geometry, Springer-Verlag, New York 1993. [30] Bröcker, T., and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag, New York 1985. [31] Brooks, M. S., S. A. Morris, and S. A. Saxon Generating varieties of topological groups, Proc. Edinburgh Math. Soc. (2) 18 (1973), 191–197. [32] Browder, W., Torsion in H -spaces, Ann. of Math. (2) 74 (1961), 24–51. [33] Brown, G., On commutators in a simple Lie algebra, Proc. Amer. Math. Soc. 14 (1963), 763–767. 651, 652, 653, 655 [34] Brown, R., P. J. Higgins, and S. A. Morris, Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties, Math. Proc. Cambridge Philos. Soc. 78 (1975), 19–32. 182, 592, 650 [35] Chasco, M. J., and E. Martín-Peinador, Binz-Butzmann duality versus Pontryagin duality, Arch. Math. (Basel) 63 (1994), 264–270. [36] Chevalley, C., Theory of Lie groups, Princeton University Press, Princeton, NJ, 1946. [37] Chevalley, C., Théorie de groupes de Lie, groupes algébriques, théorèmes généraux sur les algèbres de Lie, Hermann, Paris 1968. [38] Clifford, A. H., and G. B. Preston, The algebraic theory of semigroups, vol. I, Amer. Math. Soc., Providence, RI, 1961. [39] Comfort, W. W., A survey of cardinal invariants, General Topology Appl. 1 (1971), 163–199. [40] Comfort, W. W., Topological groups, in Handbook of set-theoretic topology, North Holland, Amsterdam 1984, 1143–1263. [41] Comfort, W. W., K. H. Hofmann, and D. Remus, Topological groups and semigroups, in Recent progress in general topology (Prague, 1991), North-Holland, Amsterdam 1992, 57–144. 173 [42] Comfort, W. W., S. A. Morris, D. Robbie, S. Svetlichny and M. Tkaˇcenko, Suitable sets for topological groups, Topology Appl. 86 (1998), 25–46. [43] Dikranjan, D. N., I. R. Prodanov, and L. N. Stoyanov, Topological groups, Marcel Dekker, New York 1990. [44] Dixmier, J., Quelques propriétés des groupes abéliens localement compacts, Bull. Sci. Math. (2) 81 (1957), 38–48.

Bibliography

659

[45] Dixmier, J., Les C ∗ -algèbres et leurs représentations, 2e éd., Gauthier-Villars, Paris 1969. [46] Djokovi´c, D. Ž., An elementary proof of the Baker-Campbell-Hausdorff-Dynkin formula, Math. Z. 143 (1975), 209–211. [47] Djokovi´c, D. Ž., and K. H. Hofmann, The surjectivity question for the exponential function of real Lie groups: a status report, J. Lie Theory 7 (1997), 171–199. 342 [48] Dold, A., Lectures on algebraic topology, Springer-Verlag, Berlin 1980. [49] Eklof, P. C., and A. H. Mekler, Almost free modules, North-Holland, Amsterdam 1990. 175, 176, 179, 214, 470, 592, 596, 597 [50] Engelking, R. D., General topology, Polish Scientific Publishers, Warsaw 1977. [51] Engelking, R. D., Dimension theory, North-Holland, Amsterdam 1978. [52] Engelking, R. D. General topology, revised and completed edition, Heldermann Verlag, Berlin 1989. 210, 594 [53] Engelking, R. D., Topology: a geometric approach, Heldermann Verlag, Berlin 1992. [54] Freudenthal, H., Topologische Gruppen mit genügend vielen fastperiodischen Funktionen, Ann. of Math. (2) 37 (1936), 57–77. [55] Freudenthal, H., and H. de Vries, Linear Lie groups, Academic Press, New York 1969. [56] Freyd, P., Abelian categories: an introduction to the theory of functors, Harper & Row, New York 1964. [57] Fuchs, L., Abelian groups, Publishing House of the Hungarian Academy of Sciences, Budapest 1958. [58] Fuchs, L., Infinite Abelian groups, vol, I, Academic Press, New York 1970. 175 [59] Fuchs, L., Infinite Abelian groups, vol. II, Academic Press, New York 1973. [60] Fulton, W., Algebraic topology, Springer-Verlag, New York 1995. [61] George Michael, A. A., On inverse limits of finite dimensional Lie groups, J. Lie Theory 16 (2006), 221–224. 164 [62] Gilmore, R., Lie groups, Lie algebras, and some of their applications, John Wiley, New York 1974. [63] Gleason, A., Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193–212. 1 [64] Glöckner, H., Real and p-adic Lie algebra functors on the category of topological groups, Pacific J. Math. 203 (2002), 321–368. 4, 6 [65] Glöckner, H., Locally compact groups built up from p-adic Lie groups, for p in a given set of primes, J. Group Theory 9 (2006), 427–454. 162 [66] Glöckner, H., and K.-H. Neeb, Infinite-dimensional Lie groups, I, II, Springer-Verlag, in preparation. [67] Glushkov, V. M., Lie algebras of locally bicompact groups, Uspekhi Mat. Nauk 12 (1957), 137–142. [68] Glushkov, V. M., On the structure of connected locally bicompact groups, Mat. Sb. 48 (90) (1959), 75–92; Amer. Math. Soc. Transl. (2) 27 (1963) 159–177. [69] Goldstein, J., Semigroups of linear operators and applications, The Clarendon Press, New York 1985. 119, 617

660

Bibliography

[70] Gôto, M., On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc. 20 (1969), 157–162. 365, 621 [71] Graev, M. I., Free topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 279–324; Amer. Math. Soc. Transl. 35 (1951) and reprinted as Amer. Math. Soc. Transl. (1) 8 (1962), 305–364. 159, 188 [72] Griffith, P. A., Infinite abelian group theory, The University of Chicago Press, Chicago 1970. [73] Grosser, S., and M. Moskowitz, On central topological groups, Trans. Amer. Math. Soc. 127 (1967), 317–340. [74] Grosser, S., and M. Moskowitz, Compactness conditions in topological groups, J. reine u. angew. Math. 246 (1971), 1–40. [75] Grothendieck, A., Sur les espaces (F ) et (DF ), Summa Brasil. Math. 3 (1954), 57–123. 204 [76] Halmos, P. R., Measure theory, D. Van Nostrand, New York 1950. 4 [77] Heil, J., Lokal kompakte Gruppen und schwach vollständigeVektorräume, Diploma thesis, Technische Hochschule Darmstadt, Darmstadt 1997. [78] Helgason, S., Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York 1978. 454, 455, 466, 508, 654 [79] Hewitt, E., and K. A. Ross, Abstract harmonic analysis, vol. I, Springer-Verlag, Berlin 1963. 85, 198, 408, 410 [80] Hewitt, E., and K. A. Ross, Abstract harmonic analysis, vol. II, Springer-Verlag, New York 1970. [81] Heyer, H., Dualität lokal kompakter Gruppen, Lecture Notes in Math. 150, SpringerVerlag, Berlin 1970. [82] Higgins, P. J., Introduction to topological groups, Cambridge University Press, London 1974. [83] Hilbert, D., Mathematical problems, Bull. Amer. Math. Soc. 37 (2000), 407–436; reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479, originally published as “Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Congress zu Paris 1900”, Nachr. Ges. Wiss. Göttingen 1900, 253–297. [84] Hilgert, J, K. H. Hofmann, and J. D. Lawson, Lie groups, convex cones, and semigroups, The Clarendon Press, Oxford 1989. 340, 454, 456, 652, 654, 655 [85] Hilgert, J. and K.-H. Neeb, Lie-Gruppen und Lie-Algebren, Vieweg, Braunschweig 1991. 195, 261 [86] Hochschild, G., The structure of Lie groups, Holden-Day, Inc., San Francisco 1965. 195, 354, 565 [87] Hofmann, K. H., Introduction to Compact Groups I, II, Lecture Notes, Tulane University, New Orleans, 1966–67 and 1968–69. [88] Hofmann, K. H., Categories with convergence, exponential functors and the cohomology of compact abelian groups, Math. Z. 104 (1968), 106–140. [89] Hofmann, K. H., A general invariant metrization theorem for compact spaces, Fund. Math. 68 (1970), 281–296.

Bibliography

661

[90] Hofmann, K. H., Die Formel von Campbell, Hausdorff und Dynkin und die Definition Liescher Gruppen, in Theory of sets and topology, Deutscher Verlag der Wissenschaften, Berlin 1972, 417–424. [91] Hofmann, K. H., Théorie directe des groupes de Lie I–IV, Séminaire Dubreil (Algèbre) 27e année, 1973–74, no. 1, no. 2., no. 3, no. 4, Sécrétariat mathématique, Paris 1975. [92] Hofmann, K. H., Analytic groups without analysis, in Convegno sui gruppi topologici e gruppi di Lie (Rome, 1974), Sympos. Math. 16, Academic Press, London, 1975, 357–374. [93] Hofmann, K. H., Sur la décomposition semidirecte des groupes compacts connexes, in Convegno sui gruppi topologici e gruppi di Lie (Rome, 1974), Sympos. Math. 16, Academic Press, London, 1975, 471–476. [94] Hofmann, K. H., An essay on free compact groups, in Categorical aspects of topology and analysis (Ottawa, Ont., 1980), Lecture Notes in Math. 915, Springer-Verlag, Berlin 1982, 171–197. 160 [95] Hofmann, K. H., and S. A. Morris, Free compact groups I: Free compact abelian groups, Topology Appl. 23 (1986), 41–64; Errata, ibidem 28 (1988), 101–102. 160 [96] Hofmann, K. H., and S. A. Morris, Free compact groups II: The center, Topology Appl. 28 (1988), 215–231. 160 [97] Hofmann, K. H., and S. A. Morris, Free compact groups III: Free semisimple compact groups, in Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), World Sci. Publ., Singapore 1989, 208–219. 160 [98] Hofmann, K. H., Compact groups: lectures and labs, Lecture Notes, Technische Hochschule Darmstadt, Darmstadt 1989. [99] Hofmann, K. H., and S. A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1990), 181– 194. [100] Hofmann, K. H., and S. A. Morris, Free compact groups IV: Splitting the component and the structure of the commutator group, J. Pure Appl. Algebra 70 (1991), 89–96. 160 [101] Hofmann, K. H., and S. A. Morris, Free compact groups V: Remarks on projectivity, in Category theory at work (Bremen, 1990), Heldermann Verlag, Berlin 1991, 177–198. 160 [102] Hofmann, K. H., and S. A. Morris, The structure of compact groups, Walter de Gruyter, Berlin 1998; 2nd revised and augmented ed. 2006. 1, 4, 9, 13, 18, 25, 30, 31, 32, 42, 45, 52, 59, 63, 64, 73, 76, 77, 80, 85, 86, 87, 96, 100, 101, 103, 105, 107, 108, 109, 110, 113, 117, 118, 121, 122, 125, 126, 127, 128, 129, 132, 133, 137, 138, 142, 144, 146, 147, 151, 159, 160, 165, 168, 169, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 186, 187, 192, 193, 194, 195, 196, 203, 206, 208, 212, 214, 216, 217, 218, 219, 220, 223, 225, 227, 231, 232, 234, 237, 238, 239, 240, 241, 242, 243, 245, 246, 247, 249, 250, 252, 253, 255, 256, 257, 258, 259, 262, 267, 270, 271, 272, 276, 282, 335, 344, 345, 350, 351, 352, 353, 354, 355, 360, 363, 364, 377, 382, 390, 395, 396, 398, 400, 403, 405, 419, 436, 437, 438, 451, 453, 454, 455, 461, 469, 474, 476, 477, 479, 480, 491, 494, 497, 502, 503, 508, 515, 517, 518, 519, 520, 522, 523, 526, 533, 534, 535, 540, 541, 544, 546, 548, 553, 554, 564, 583, 587, 588, 590, 591, 594, 596, 600, 605, 617, 619, 620, 622, 623, 624, 625, 626, 627, 628, 629, 637, 638, 640, 645, 650, 652, 653, 654 [103] Hofmann, K. H., and S. A. Morris, Transitive actions of compact groups and topological dimension, J. Algebra 234 (2000), 454–479. 4, 235, 246, 396, 554, 584, 639

662

Bibliography

[104] Hofmann, K. H., and S. A. Morris, Projective limits of finite-dimensional Lie groups, Proc. London Math. Soc. (3) 87 (2003), 647–676. 4, 11 [105] Hofmann, K. H., and S. A. Morris, The structure of abelian pro-Lie groups, Math. Z. 248 (2004), 867–891. [106] Hofmann, K. H., S. A. Morris, and D. Poguntke, The exponential function of locally connected compact abelian groups, Forum Math. 16 (2004), 1–16. 4, 9, 181, 213, 214 [107] Hofmann, K. H., S. A. Morris, and M. Stroppel Locally compact groups, residual Lie groups, and varieties generated by Lie groups, Topology Appl. 71 (1996), 63–91. 6, 106, 149 [108] Hofmann, K. H., and P. S. Mostert, Splitting in topological groups, Mem. Amer. Math. Soc. 43 (1963). 10, 43, 491 [109] Hofmann, K. H., and P. S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio 1966. [110] Hofmann, K. H., and P. S. Mostert, Cohomology theories for compact abelian groups, VEB Deutscher Verlag der Wissenschaften, Berlin 1973. [111] Hofmann, K. H., and K.-H. Neeb, Letters on Lie algebras in which every element is a bracket, unpublished correspondence 2002. 655 [112] Hofmann, K. H., and K.-H. Neeb, Pro-Lie groups as infinite dimensional Lie groups, submitted (2006), 42 pp. [113] Hofmann, K. H., and W. A. F. Ruppert, Lie groups and subsemigroups with surjective exponential function, Mem. Amer. Math. Soc. 130 (1997). 340, 342, 438, 610 [114] Hofmann, K. H., T. S. Wu, and J. S.Yang, Equidimensional immersions of locally compact groups, Math. Proc. Cambridge Philos. Soc. 105 (1989), 253–261. [115] Humphreys, J. E., Introduction to Lie algebras and representation theory, Springer-Verlag, New York 1972. [116] Hunt, D. C., S. A. Morris, and A. J. van der Poorten, Closed subgroups of products of reals, Bull. London Math. Soc. 7 (1975), 124–128. 182, 592, 650 [117] Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin 1967. [118] Husain, T., Introduction to topological groups, W. B. Saunders, Philadelphia 1966. [119] Itzkowitz, G., and Ta Sun Wu, Iwasawa-type decomposition in compact groups, in Papers on general topology and applications (Amsterdam, 1994), Ann. New York Acad. Sci. 788, New York Acad. Sci., New York 1996, 138–146. [120] Iwasawa, K., On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. 3, 61, 403, 584 [121] Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Publ. 37, Amer. Math. Soc., Providence, RI, 1956; revised edition 1964. [122] Jacobson, N., Lie algebras, Interscience Publishers, New York 1962. 291 [123] Jech, T., Set theory, Academic Press, New York 1978. [124] Jones, F. Burton, Connected and disconnected plane sets and the functional equation f (x + y) = f (x) + f (y), Bull. Amer. Math. Soc. 48 (1942), 115–120. 415, 593

Bibliography

663

[125] Kampen, D. R. van, Locally bicompact abelian groups and their character groups, Ann. of Math. (2) 36 (1935), 448–463. 18 [126] Kaplan, S., Extensions of the Pontryagin duality I: Infinite products, Duke Math. J. 15 (1948), 649–658. [127] Kaplan, S., Extensions of the Pontryagin duality II: Direct and inverse sequences, Duke Math. J. 17 (1950), 419–435. [128] Kaplansky, I., Infinite abelian groups, revised ed., The University of Michigan Press, Ann Arbor, Mich., 1969. [129] Kaplansky, I., Lie algebras and locally compact groups, The University of Chicago Press, Chicago 1971. 2, 165, 167, 553 [130] Katz, E. and S. A. Morris, Free abelian topological groups on countable CW-complexes, Bull. Amer. Math. Soc. 41 (1990), 451–456. 188 [131] Kelley, J. L., General topology, D. Van Nostrand, Princeton, NJ, 1955. 85, 197 [132] Köthe, G, Die Quotientenräume eines linearen vollkommenen Raumes, Math. Z. 51 (1947), 17–35. 204 [133] Lang, S., Differentiable and Riemannian manifolds, Springer-Verlag, New York 1995. [134] Lashof, R. K., Lie algebras of locally compact groups, Pacific J. Math. 7 (1957), 1145–1162. 4 [135] Lee, D. H., Supplements for the identity component in locally compact groups, Math. Z. 104 (1968), 28–49. [136] Leptin, H., Zur Dualitätstheorie projektiver Limites abelscher Gruppen, Abh. Math. Sem. Univ. Hamburg 19 (1955), 264–268. [137] Leptin, H., and L. Robertson, Every locally compact MAP group is unimodular, Proc. Amer. Math. Soc. 19 (1968), 1079–1082. [138] Luminet, D., and A. Valette, Faithful uniformly continuous representations of Lie groups, J. London Math. Soc. (2) 49 (1994), 100–108. [139] Mackenzie, K., Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, Cambridge 1987. [140] MacLane, S., Categories for the working mathematician, Springer-Verlag, New York 1971. [141] Marin, E. I., Strengthening of a group topology of an abelian group up to a complete group, Mat. Issled. 105 (1988), 105–119. 162 [142] Martín-Peinador, E., A reflexive admissible topological group must be locally compact, Proc. Amer. Math. Soc. 123 (1995), 3563–3566 [143] Mimura, M., Homotopy theory of Lie groups, in Handbook of algebraic topology, NorthHolland, Amsterdam 1995, 951–991. [144] Montgomery, D., and L. Zippin, Small subgroups of finite dimensional groups, Ann. of Math. (2) 56 (1952), 213–241. 1 [145] Montgomery, D., and L. Zippin, Topological transformation groups, Interscience Publishers, New York 1955. 1, 165, 167, 493, 553, 556, 581, 585

664

Bibliography

ˇ [146] Morris, S. A., Free compact abelian groups, Mat. Casopis Sloven. Akad. Vied. 22 (1972), 141–147. [147] Morris, S.A., Quotient groups of topological groups with no small subgroups, Proc. Amer. Math. Soc. 31 (1972), 625–626. [148] Morris, S. A., Pontryagin duality and the structure of locally compact abelian groups, Cambridge University Press, Cambridge 1977. [149] Morris, S. A., Varieties of topological groups: a survey, Colloq. Math. 46 (1982), 147–165. [150] Morris, S. A., Free abelian topological groups, in Categorical topology (Toledo, Ohio 1983), Heldermann Verlag, Berlin 1984, 375–391. 188 [151] Morris, S. A. and H. B. Thompson, Metrizability of subgroups of free topological groups, Bull. Austral. Math. Soc. 33 (1986), 103–112. 188 [152] Moskowitz, M. and R. Sacksteder The exponential map and differential equations, J. Lie Theory 13 (2003), 291–306. 351 [153] Mostert, P. S., Sections in principal fiber spaces, Duke Math. J. 23 (1956), 57–71. [154] Nagami, K. R., Cross sections in locally compact groups, J. Math. Soc. Japan 15 (1963), 301–303. [155] Nagami, K. R., Dimension theory, Academic Press, New York 1970. [156] Nagata, J., Modern dimension theory, North-Holland, Amsterdam 1965. [157] Neeb, K.-H., Holomorphy and convexity in Lie theory, Walter de Gruyter, Berlin 1999. 654 [158] Neeb, K.-H., Towards a Lie theory of locally convex groups, Japan. J. Math. 1 (2006), 291–468. [159] Onishchik, A. L., ed., Lie groups and Lie algebras I, Encyclopaedia Math. Sci. 20, Springer-Verlag, Berlin 1993. [160] Onishchik, A. L., and E. B. Vinberg, eds., Lie groups and Lie algebras II, Encyclopaedia Math. Sci. 21, Springer-Verlag, Berlin 2000. [161] Onishchik, A. L., and E. B. Vinberg, eds., Lie groups and Lie algebras III, Encyclopaedia Math. Sci. 41, Springer-Verlag, Berlin 1994. [162] Onishchik, A. L., and E. B. Vinberg, Lie groups and algebraic groups, Springer-Verlag, Berlin 1990. [163] Palais, R. S., The classification of G-spaces, Mem. Amer. Math. Soc. 36 (1960). [164] Palmer, T. W., Classes of nonabelian, noncompact, locally compact groups, Rocky Mountain J. Math. 8 (1978), 683–741. [165] Palmer, T. W., Banach algebras and the general theory of ∗-algebras, vol. I, Encyclopedia Math. Appl. 49, Cambridge University Press, Cambridge 1994. [166] Palmer, T. W., Banach algebras and the general theory of ∗-algebras, vol. II, Encyclopedia Math. Appl. 79, Cambridge University Press, Cambridge 2001. [167] Pasynkov, B., The coincidence of various definitions of dimensionality for locally bicompact groups, Dokl. Akad. Nauk SSSR 132 (1960), 1035–1037 (Russian); translated as Soviet Math. Dokl. 1 (1960), 720–722.

Bibliography

665

[168] Pier, J.-D., L’appartition de la théorie des groups topologiques, Cahiers Sém. Hist. Math. 9, Univ. Paris VI, Paris 1988, 1–21. [169] Pontryagin, L. S., Topological groups, Gordon and Breach, New York 1966. 18 [170] Price, J., Lie groups and compact groups, Cambridge University Press, Cambridge 1977. [171] Ribes, L., and P. Zalesskii, Profinite groups, Springer-Verlag, Berlin 2000. [172] Rickert, N. E., Arcs in locally compact groups, Math. Ann. 172 (1967), 222–228. [173] Robertson, L., and T. W. Wilcox, Splitting in map groups, Proc. Amer. Math. Soc. 33 (1972), 613–618. [174] Roeder, D. W., Functorial characterizations of Pontryagin duality, Trans. Amer. Math. Soc. 154 (1971), 151–175. [175] Roeder, D. W., Category theory applied to Pontryagin duality, Pacific J. Math. 52 (1974), 519–527. [176] Roelcke, W., and S. Dierolf, Uniform structures on topological groups and their quotients, McGraw-Hill, New York 1981. 9, 63, 96, 156, 173, 203, 204, 211, 408 [177] Sagle, A. A., and R. E. Walde, Introduction to Lie groups and Lie algebras, Academic Press, New York 1973. [178] Samelson, H., Topology of Lie groups, Bull. Amer. Math. Soc. 58 (1952), 2–37. [179] Samelson, H., Notes on Lie algebras, Van Nostrand Reinhold, New York 1969. [180] Scheerer, H., Sur la topologie des groupes compacts connexes, C. R. Acad. Sci. Paris Sér. A Math. 274 (1972), 1441–1443. [181] Scheffer, W., Maps between topological groups that are homotopic to homomorphisms, Proc. Amer. Math. Soc. 33 (1972), 562–567. [182] Shoda, K., Einige Sätze über Matrizen, Japan J. Math. 13 (1937), 361–365. 655 [183] Simon, B., Representations of finite and compact groups, Amer. Math. Soc., Providence, RI, 1996. [184] Sipacheva, O. V., Zero-dimensionality and completeness in free topological groups I, II, Serdica 15 (1989), 119–140; 141–154. 173, 210 [185] Spanier, E., Algebraic topology, McGraw-Hill, New York 1966. [186] Specht, W., Gruppentheorie, Springer-Verlag, Berlin 1956. [187] Stroppel, M., Locally compact groups with many automorphisms, J. Group Theory 4 (2001), 427–455. [188] Stroppel, M., Locally compact groups, EMS Publishing House, Zürich 2006. [189] Tits, J., Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen, Lecture Notes in Math. 40, Springer-Verlag, Berlin 1967. [190] Tits, J., Liesche Gruppen und Algebren, Springer-Verlag, Berlin 1983. [191] Toda, H. A note on compact semi-simple Lie group, Japan. J. Math. 2 (1976), 355–360. [192] tom Dieck, T., Transformation groups, Walter de Gruyter, Berlin 1987. [193] Varadarajan, V. S., Lie groups, Lie algebras, and their representations, Prentice-Hall, Englewood Cliffs, NJ, 1974.

666

Bibliography

[194] Varopolous, N. Th., A theorem on the continuity of homomorphisms of locally compact groups, Proc. Cambridge Philos. Soc. 60 (1964), 449–463. [195] Varopolous, N. Th., Studies in harmonic analysis, Proc. Cambridge Philos. Soc. 60 (1964), 465–516. [196] Wagon, S., The Banach-Tarski paradox, Cambridge University Press, Cambridge 1985. [197] Warner, G., Harmonic analysis on semi-simple Lie groups I, Springer-Verlag, Berlin 1972. 454, 455, 466 [198] Weil, A., L’intégration dans les groupes topologiques et ses applications, Hermann, Paris 1940. 4 [199] Weil, A., Basic number theory, Springer-Verlag, Berlin 1967. 212, 589, 594 [200] Wendel, J. G., Haar measure and the semigroup of measures on a compact group, Proc. Amer. Math. Soc. 5 (1954), 923–929. [201] Wigner, D., Inverse limits and the completeness of quotient groups, Michigan Math. J. 22 (1975), 165–170. 204 [202] Wilson, J., Profinite groups, The Clarendon Press, Oxford 1998. [203] Wüstner, M., Lie groups with surjective exponential function, Shaker Verlag, Aachen 2001. 342 [204] Wüstner, M., A short survey on the surjectivity of exponential Lie groups, in Nuclear groups and Lie groups (Madrid, 1999), Heldermann Verlag, Lemgo 2001, 239–249. 342 [205] Wüstner, M., A connected Lie group equals the square of the exponential function, J. Lie Theory 13 (2003), 307–309. 351 [206] Yamabe, H., On the conjecture of Iwasawa and Gleason, Ann. of Math. (2) 58 (1953), 48–54. 2, 56, 165, 167, 493, 531, 556 [207] Yamabe, H., Generalization of a theorem of Gleason, Ann. of Math. (2) 58 (1953), 351–365. 2, 56, 165, 167, 493, 531, 556 [208] Young, G. S., The introduction of local connectivity by change of topology, Amer. J. Math. 68 (1946), 479–494.

List of Symbols

A, 115 [A, B], 420 AbTopGr, 119 Ad(g), 131 ad x, 624 α(G), 104 A(h), 366 A(h, G), 366 A(J ) , 150 Ann(S), 649 0BA : A → B, 169 B 1 (K, n), 488 C(N), 463, 488 Cofin, 631 Cofin, 634 comm(g, h), 115 comp(G), 17, 217 CompGr, 72 comp(L)(G), 217 ConnGr, 72 Der(E), 624 D(G), 405 Eeff , 280 E(G), 121 E(G), 121 E1 ⊥ E2 , 494 Ess , 280 expG , 110 E0 , 280 Fin, 631, 634 f (ϕ), 307 G , 420 Ga , 121  13 G, gad , 20

[g, g], 282 [g, g], 282 GN , 91 G0 , 2, 3, 115, 121 g0 (h), 49, 320 H 1 (K, n), 488 H (V ), 573, 601 h(V ), 573  (L), 138 Inn(g), 352, 359 j ∈J

Ej , 277

KZ(G), 399 LCGr, 72

Lcoad , 272 L(G), 110  272 L, LieAlg, 123 LieAlgGr, 119 LieGr, 119 LieProGr, 135 lim F = x, 88 limj ∈J Gj , 78 L , 272 MaxK(G), 530 M(G), 195 M ∼ N, 195 N, 73 , 91 N Ncored (G), 38, 447 n ⊕δ h, 264 N (G), 38, 447 n(g), 23, 303 N (G), 148, 164 N(H, G), 31, 373

668

List of Symbols

n(h, g), 31, 373 nilcore(G), 579 nilcore(g), 576 N π H , 15 ν(G), 61, 585 ν(g), 586 νMN : G/N → G/M, 91 νN : GN → G/N , 91 P (G), 32, 255 PN , 91 proLieAlg, 138 proLieAlgGr, 140 proLieComGr, 140 proLieGr, 2 proSimpConLieGr, 255 protoLie, 205 1 p ∞ · Z/Z, 101, 590 Q, 79, 588 Q(G), 57, 534 qN : G → G/N , 91 Qp , 589 R, 73 ran f , 69 RefTopGr, 272, 273, 637 R(G), 38, 432

r(g), 23, 291 Scal, 119 S ⊥ , 272

T (G), 15 Top, 119 TopGr, 2, 71, 119 Top∗ , 119 Tp , 80, 589 , 270 V V  , 270 V 0 (h), 316 x · ω, 271 Z, 72, 79, 588 Zp , 588 Z(p ∞ ), 100, 590 z(g), 282 ζ (G), 104 Z(G), 197 Z(H, G), 30, 371 Z 1 (K, n), 488 1 ZN ∩H (H, N ), 463 Z2 (G), 453

Index

abelian Lie algebra, 624 abelian sequence of ideals, 285 abelian sequence of normal subgroups, 420 absorbing, 630, 631 addition on L(G), 115 adjoint module, 20, 270 morphism, 131 representation, 131, 270 Adjunction Theorem for L, 127 Ado’s Theorem for Pro-Lie Algebras, 332 affine subspace, 226, 643 Alexander Subbasis Theorem, 197 algebra Banach, 109 pro-Lie, 138 algebraic dual, 632, 634 algebraically locally compact L-module, 45, 494, 561 almost connected, 2, 7, 101, 195, 218, 478 metrizable, 203 prosolvable, 486 Alternative Open Mapping Theorem for Pro-Lie Groups, 556 analytic morphism of pro-Lie groups, 360 semisimple group of bounded type, 455 semisimple group of unbounded type, 455 subgroup, 29, 360, 416 annihilator, 272, 281, 636, 638, 640 Annihilator Mechanism, 638, 640 antiisomorphism, 273, 275, 640 approximately nilpotent, 308 arbitrarily small co-Lie subgroups,

148, 160 associated representation, 331 atom, 454 of bounded type, 455 of unbounded type, 455 automorphic action, 222 automorphism, 462, 464 inner, 356 special, 24 Axiom of Choice, 633 back adjunction, 12, 127, 257 Baire Category Theorem, 85 Baire space, 85, 497 balanced, 630, 631, 636 Banach algebra, 109 bicommutant, 372 Bipolar Lemma, 636, 637 Bohr compactification, 104 Bohr compactification morphism, 104 bonding maps, 78 bounded type, 41, 455 bracket multiplication on L(G), 115 Campbell–Baker–Hausdorff–Dynkin series, 336 Campbell–Hausdorff multiplication, 626 Campbell–Hausdorff neighborhood, 107, 626 canonical decomposition, 132 Cartan pair, 322 Cartan subalgebra, 49, 322 category, 633 complete, 69 discrete, 65 has equalizers, 67 has finite limits, 67 has intersections of coherent retracts, 68 has limits, 67 has products, 67

670

Index

has projective limits, 67 has pullbacks, 67 of compact groups, 72 of connected groups, 72 of diagrams, 64 of locally compact groups, 72 of topological groups, 119 of topological vector spaces, 633 small, 64 Cauchy filter basis, 94, 111 Cauchy net, 95 center, 282 Center-free Embedding Lemma, 394 centralizer, 30, 371 centrally supplemented, 58, 541 Characterisation Theorem for Local Compactness, 555 for Potentially Compact Connected Pro-Lie Groups, 522 for Strong Local Connectivity, 181 of Reductive Pro-Lie Groups, 452 of Semisimple and Reductive Connected Pro-Lie Groups, 435 of Simply Connected Pronilpotent Pro-Lie Groups, 484 of Simply Connected Prosolvable Pro-Lie Groups, 487 closed commutator algebra, 282 closed subcategory, 69 Closed Subgroup Theorem, 8 Closed Subgroup Theorem for Pro-Lie Groups, 158 closeness relation, 195 Closure Theorem for Semisimple Analytic Subgroups, 439 co-Lie subgroup, 148 coadjoint module, 21, 272 coboundary, 488 cocycle, 71, 462–464, 488 coequalizer, 276 cofactor, 463 cofinal, 81 cofinal function, 74, 81

cofinal functor, 74 cofinality, 74 Cofinality Lemma, 75, 83, 86 Cofinality Lemma Revisited, 81 coherent coretractions, 68 cohomology, 488 cokernel, 169 colimit, 276 colimit cone, 276 commutant, 30, 371 commutator, 115 subalgebra, 282 subgroup, 420 Commutator Series Theorem for Pro-Lie Groups, 429 Commutator Subgroup Theorem for Dense Analytic Subgroups, 382 compact, 652 compact L-module, 45, 494, 561 Compact Generation Theorem for Abelian Pro-Lie Groups, 236 compact Lie algebra, 652 compact open topology, 270, 629, 632, 634, 636–638 compact-free, 217 compactly embedded, 47, 54, 500, 521, 561 generated, 218, 643 simple, 56, 483, 530 topologically generated, 218 compactly simple, 402 complemented, 571 complete, 2, 95 lattice, 273, 275, 640 complete category, 69 Complete Groups and Projective Limits Theorem, 96 completeness of locally compact groups, 95 Completeness Theorem for Pro-Lie Groups, 158 for Projective Limits of Lie Groups, 136 for Topological Groups with pro-Lie

Index

Algebras, 141 for Topological Groups with Pro-Lie Component, 142 of the Category of Groups with Lie algebras, 124 completion, 104 concomitant Dynkin algebra, 109 cone, 64 conilsubmodule, 302 Conjugacy Theorem of Cartan Subalgebras, 329 Connectivity Theorem for Maximal Compact Subgroups, 547 constant functor, 64 constant morphism, 169 Continuity Theorem for Functors, 77 continuous G-module, 497 continuous L-module, 269 continuous action, 222 continuous cross section, 340 continuous functor, 77 contravariant functor, 273, 637 convergence of a filter basis, 88 convex, 631, 636 coproduct, 276 coquotient, 170 coreductive radical, 38, 283, 303, 447 coretraction, 67 Correspondence Theorem for Subalgebras and Subgroups, 375 cotorus subgroup, 242 counit, 12 countably nilpotent, 36, 287, 421 countably solvable, 22, 35, 285, 420 countably topologically nilpotent, 36, 287, 423 countably topologically solvable, 22, 36, 286, 422 crossed homomorphism, 71 defining morphism, 360 derivation, 308, 624

671

descending nil-sequence, 296 diagram, 64 dimension, 554 direct product, 464 direct sum, 277, 313 directed set, 66 discrete category, 65 disjoint, 162, 620 divisible group, 404 domain of definition, 627 dual algebraic, 632, 634 topological, 270, 632, 634 dual module, 272 Duality Theorem for L-Modules, 274 for Real Vector Spaces, 272, 637 Dynkin algebra, 107, 624 Dynkin Algebra Fundamental Theorem, 626 effective submodule, 280 elementwise compact, 217 embedding, 94 embedding morphism, 94 Embedding Preservation Theorem in Lie’s Third Theorem, 263 epic-monic factorisation, 169 equalizer, 65, 66, 112 Equivalence Theorem for Nilpotency, 297 for Nilpotency of Connected Pro-Lie Groups, 444 for Prosimple and Simple Connectedness, 344 for Solvability, 295 for Solvability of Connected Pro-Lie Groups, 427 of the Definitions of Pro-Lie Group, 161 equivariant, 341 Existence Theorem for Cartan Subalgebras, 50 for Manifold Factors, 555

672

Index

for Maximal Compactly Embedded Subalgebras, 48 of Cartan Subalgebras, 325 Explicit Structure of Weakly Complete Topological Vector Spaces, 638 exponential, 340 strictly, 340 exponential function, 110, 625 of a group, 110 Exponential Generation Theorem for Simply Connected Pro-Lie Groups, 350 exponentially generated, 360 subgroup, 29 extended radical, 57, 534 extremely rich, 571 faithful, 331 faithful module, 331 fibered product, 65 filter basis, 89 convergence of, 89 Filter Basis Theorem for Closed Affine Subspaces, 643 for Closed Linear Subspaces, 641 filter of identity neighborhoods, 630 finest locally convex vector space topology, 273, 633, 637, 639 finite intersection property, 197 finite limit, 66 finite-dimensional, 396 finite-dimensional pro-Lie group, 32 finite-dimensional topological vector space, 630 First Axiom of Countability, 114 First Fundamental Lemma, 152 fixed point-free, 394 Fréchet space, 497 front adjunction, 12, 127 full closed subcategory, 102 full subcategory, 2, 69, 119, 637 full subcategory of C-objects, 119 fully characteristic, 405, 523 function

equivariant, 341 exponential, 110 functor, 273, 637 cofinal, 74 constant, 64 continuous, 77 contravariant, 273, 637 left adjoint, 126 procontinuous, 77 Fundamental Lemma First, 152 Second, 154 Third, 156 Fundamental Theorem on Projective Limits, 88 generating Lie algebra, 115, 121 G-module, 497 group compactly simple, 56 having duality, 238, 594 Lie, 107 of p-adic integers, 588 of p-adic rationals, 589 prodiscrete, 15, 149 protodiscrete, 149 with pro-Lie algebra, 140 Group Theoretical Characterisation Theorem of Procompact Pro-Lie Algebras, 517 Hahn–Banach Theorem, 636 Hausdorff series, 336 Hilbert L-module, 45, 494 idealizer, 31, 373 idempotent, 264 identity arc component, 121 identity component, 115, 121 inductive, 278 injective, 227 injective limit, 276 injective object, 227 injectivity, 464

Index

inner automorphism, 50, 131, 352, 356, 359 inner automorphism group, 357, 359 inner derivation, 356, 624 intersection, 66 inverse limit, 78 inversion, 627 involutive, 294 vector subspace, 294 isomorphy of local groups, 627 isomorphy of partial groups, 627 Jacobi identity, 624 jointly continuous topological G-module, 46 jointly topological G-module, 497 k-space, 83 kernel, 71, 169, 331 kernel of a module, 331 Largest Compact Normal Subgroup of a Pro-Lie Group, 530 Largest Compactly Embedded Connected Normal Subgroup of a Pro-Lie Group, 527 Largest Compactly Embedded Ideal of a Pro-Lie Algebra, 515 lattice, 273, 275, 640 left adjoint functor, 126 Levi summand, 293 Levi–Mal’cev Theorem for Compact Connected Groups, 518 for Pro-Lie Algebras: Conjugacy, 310 for pro-Lie Algebras: Existence, 294 for Reductive Pro-Lie Algebras, 284 Lie algebra, 624 abelian, 624 compact, 652 generating, 121 of a topological group, 115 stable, 407 Lie bracket, 624

673

Lie group, 107 Lie projective component, 140 Lie subalgebra, 109 Lie’s Third Fundamental Theorem for Pro-Lie Algebras, 252 Lie’s Third Theorem, Functorial Version, 253 limit, 64, 66 inverse, 78 projective, 65 strict projective, 86 strong projective, 86 limit cone, 64 limit diagram, 64 Limit Existence Theorem, 70 limit maps, 78 Limit Theorem via Equalizers and Products, 69 linear, 643 linear functional, 634, 637 linear Lie group, 1, 109 linear morphism, 629 linearly compact, 643 L-module, 19, 269 local group, 627 associated with G and U , 627 associated with L and B, 628 local one parameter subgroup, 110 locally arcwise simply connected, 344 locally compact subgroup, 630 locally convex, 631 locally finite-dimensional L-module, 45, 273 locally precompact, 630 loopwise simply connected, 250 Main Corollary on Maximal Compact Connected Subgroups, 525 Main Theorem on Maximal Compactly Embedded Subgroups: Existence and Conjugacy, 525 manifold factor, 554 manifold rank, 60, 555 map

674

Index

bonding, 78 limit, 78 perfect, 83 proper, 83, 86 symplectic, 647 Maximal Compactly Embedded Subalgebras: Conjugacy, 511 Maximal Compactly Embedded Subalgebras: Existence, 504 maximal connected abelian subgroup, 467 maximally compactly embedded subalgebra, 454 maximal zero submodule, 280 metabelian, 328, 339, 598 metrizability of topological groups, 86 minimal analytic subgroup, 368, 416 module, 19, 269, 497 morphism defined by h ⊆ g, 366 Morphism Duality Theorem for Weakly Complete Topological Vector Spaces, 640 morphism of G-modules, 497 morphism of L-modules, 20, 269 multiplication, 627 natural isomorphism, 64 natural transformation, 64 neighborhood filter, 88 niladapted, 331 niladapted module, 331 nilcore, 61, 576, 579, 585 nildimension, 585 nilmodule, 301 nilpotent, 36, 287, 421 nilpotent module, 301 nilpotent of class 2, 380, 599 nilradical, 23, 38, 61, 303, 447 nilsequence of ideals, 285 nilsequence of normal subgroups, 421 no compact central subgroups, 402 no small subgroups, 118 normalizer, 31, 321, 373, 462, 654 null morphism, 169

null object, 169 ω-nilmodule, 301 ω-nilpotent module, 301 one parameter subgroup, 110 one-dimensional topological vector space, 630 Open Mapping Theorem, 42, 85, 408 orthogonal subspaces, 494 orthogonal sum, 494 outer automorphism group, 357 outer derivation algebra, 356 p-adic integers, 79, 588 p-adic rationals, 79, 589 p-adic solenoid, 80, 589 p-adic topology, 589 partial group, 627 partial one parameter subgroup, 110 partially ordered set, 66 passing to intersections of coherent retracts, 69 perfect, 83, 282, 285, 420 Poincaré group, 32, 259, 352, 400 compactified, 400 point of adherence, 451 pointed category, 169 polar, 636 Polish space, 233, 644 potentially compact, 54, 521 Prüfer group, 405 pre-Hilbert L-module, 45, 493 pre-Hilbert module, 561 precompact, 630–633 Preservation Theorem for Semidirect Products, 265 primary components, 405 procompact, 45, 494 procompact L-module, 561 procompact pro-Lie algebra, 47, 500 procontinuous functor, 77 procountable, 209 prodiscrete, 15, 149 product, 65

Index

Product Theorem via Projective Limits and Finite Products, 67 profinite, 2 profinite-dimensional, 316 profinite-dimensional Lie algebra, 138 profinite-dimensional L-module, 20, 45, 273 pro-Heisenberg algebra, 573 pro-Heisenberg group, 573, 601 projection, 65 projective (abelian group), 242 projective limit, 65, 634 projective limit associated with (G, N ), 91 projective limit in C, 66 projective limit of Lie groups, 135, 161 projective system, 66, 82 projective system D, 6 projective system associated with (G, N ), 91 pro-Lie algebra, viii, 6, 138 procompact, 47, 500 reductive, 20, 277 semisimple, 20, 282 pro-Lie group, vii, 2, 149, 164 finite-dimensional, 32 prosimply connected, 11, 249 simply connected, 13 pro-Lie group: Definition A, 160 pro-Lie group: Definition B, 161 pro-Lie group: Definition C, 161 Pro-Lie Group Theorem, 157 Pro-Lie Groups with Pronilpotent Lie Algebra, 337 pronilpotent, 37, 289, 424 pro-open, 208 Pro-Open Mapping Theorem for Pro-Lie Groups, 209 proper, vii, 83, 86 prosimply connected pro-Lie group, 11, 249 prosolvable, 22, 37, 289, 424 proto-Lie group, 149, 407

675

protodiscrete, 149 pullback, 65, 66 Pullback Theorem via Products and Equalizers, 66 pure, 540 pushout, 276 quotient morphism, 169 Quotient Preservation Theorem in Lie’s Third Theorem, 261 Quotient Theorem for Pro-Lie Groups, 170 Quotient Theorem for Pro-Lie Groups Revisited, 202 radical, 23, 38, 291, 432 coreductive, 38, 283, 303, 447 extended, 57, 534 solvable, 23, 38, 291 range, 69 reductive, 38, 277, 282, 434 reductive module, 20, 277 reductive pro-Lie algebra, 20, 277 reductive proto-Lie group, 472, 613 Reflection Theorem to Simply Connected Pro-Lie Groups, 255 reflexive, 238, 272, 594, 636 topological vector space, 273, 637 relative injective for E , 227 relatively compact element, 217 relatively compact one parameter subgroup, 217 relatively open, 208 Resolution Theorem for Connected Pro-Lie Groups, 538 Resolution Theorem of Abelian Pro-Lie Groups, 244 restricted product, 150 retract, 67, 340 retraction, 67, 340 Retraction Theorem for Full Closed Subcategories, 102 Retraction Theorem for Subcategories of TopGr, 103 rich, 571

676

Index

right adjoint, 76 S-group, 175, 591 Sandwich Theorem, 40, 435 Sandwich Theorem for Abelian Pro-Lie Groups, 245 scalar morphism, 114 scalar multiplication, 111, 112 second center, 453 Second Fundamental Lemma, 154 Second Isomorphism Theorem for Pro-Lie Groups, 413 semidirect cofactor, 42, 71, 488 semidirect factor, 42, 71 semidirect product, 71, 462, 463 semidirect sum, 264, 313 semireflexive, 238, 594, 636 semisimple, 20, 38, 277, 282, 434 semisimple Lie algebra of bounded type, 455 semisimple Lie algebra of unbounded type, 455 semisimple module, 20, 277 semisimple pro-Lie algebra, 20 semisimple radical, 280 separable, 233, 644 separable space, 233, 644 separation of points, 634, 637 σ -compact, 84, 233, 644 simple module, 20, 277 simply connected, 11, 249, 343, 344, 355, 629 simply connected pro-Lie groups category of, 13, 256, 343, 344 small category, 64 small subgroups, 118 solvable, 22, 35, 285, 420 solvable radical, 23, 291 solvable radical of the group G, 38, 432 special automorphism, 309, 359 splits, 273, 275, 640 Splitting a Maximal Compact Subgroup topologically, 551 Splitting Theorem

for Reductive Lie Groups, 466 for Simply Connected Prosolvable Groups with Compact Quotients: Conjugacy, 489 for Simply Connected Prosolvable Groups with Compact Quotients: Existence, 476 for Vector Group with Compact Quotients: Conjugacy, 489 for Vector Group with Compact Quotients: Existence, 474 splitting theorem, 42, 490 Strict Exactness Theorem for , 259 Strict Exactness Theorem for L, 188 strict morphism, 188 strict projective limit, 86 strict projective system, 86 strictly exact, 188 strictly exponential, 340 strong operator topology, 132, 270, 498 strong projective limit, 86 strong projective system, 86 strongly locally connected, 177 Structure Theorem for Algebraically Locally Compact Modules, 495 for Almost Connected Pro-Lie Groups, 485 for Groups with Finite Dimensional Nilcore, 581 for Procompact Lie Algebras, 501 for Procompact Modules, 497 for Prosimply Connected Pro-Lie Groups, 343 for Prosolvable Pro-Lie Groups, 478 of Pro-Lie Algebras with Finite Dimensional Nilcore, 577 of Reductive L-Modules, 281 of Semisimple Compact Connected Groups, 518 of Semisimple Locally Finite Dimensional Modules, 277 of Semisimple Pro-Lie Algebras, 283

Index

on Simply Connected Pro-Lie Groups with prosolvable Lie algebra, 342 subbasis for the set of closed sets, 196 subgroup algebraically generated, 115 submodule, 273 Subspace Theorem for Weakly Complete Topological Vector Spaces, 639 Supplement for the Minimal Analytic Subgroup Generated by the Full Lie Algebra, 535 supplemented, 571 Sylow groups, 405 symplectic map, 647 tangent bundle of a pro-Lie group, 15 terminating, 285, 420 terminating nil-sequence of ideals, 296 Theorem Closed Subgroup for Projective Limits, 96 on Closed Connected Normal Subgroups of Simply Connected Pro-Lie Groups, 349 on Completeness of TopGr, 72 on the Compact Group Associated with a Procompact Module, 498 on the Descending Central Series of Pro-Lie Groups, 446 Weak Pro-Open Mapping for Pro-Lie Groups, 171 Third Fundamental Lemma, 156 topological G-module, 497 L-module, 20, 270 dimension, 235, 554 dual, 270, 632, 634 group with a Lie algebra, 115 Lie algebra, 123 topological group having enough compact sets, 239 having no small subgroups, 118

677

Topological Product Decomposition Theorem for Connected Pro-Lie Groups, 553 topologically compactly generated, 643 nilpotent, 36, 287, 423 perfect, 286, 422 solvable, 22, 36, 286, 422 topological vector space, 629, 630 finite-dimensional, 630 locally convex, 631 one-dimensional, 630 reflexive, 273 topology compact open, 636 generated by a subbasis, 197 of pointwise convergence, 132, 270, 634, 637, 638 of uniform convergence, 634 of uniform convergence on compact subsets, 270 toral homomorphic image, 242 transfinite nilmodule, 301 transfinitely nilpotent, 36, 287, 421 nilpotent module, 301 solvable, 22, 35, 285, 420 topologically nilpotent, 36, 287, 423 topologically solvable, 22, 36, 286, 422 Trotter formulae, 626 unbounded type, 455 uniqueness of limits, 65 unit, 12 universal covering morphism, 259 universal group, 259 universal morphism, 259, 352 vector group complement, 222 Vector Group Splitting Theorem for Abelian Pro-Lie Groups, 230 for Connected Abelian Pro-Lie Groups, 221 vector space manifold factor, 554

678

Index

very rich, 571 weak ∗-topology, 634 weak topology, 633, 634 weakly complete, 83, 138, 273, 633, 634, 637, 639 topological vector space, 246, 633 vector group, 212, 220, 246 vector space, 16

vector subgroup, 246 Weil’s Lemma for Pro-Lie Groups, 216 well-complemented, 571 Z-topology, 197 zero-dimensional Bohr compactification, 104 zero-dimensional Bohr compactification morphism, 104, 400