Developments and Trends in Infinite-Dimensional Lie Theory (Progress in Mathematics, Volume 288)

Progress in Mathematics Volume 288 Series Editors Hyman Bass Joseph Oesterlé Alan Weinstein Karl-Hermann Neeb Arturo...

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Progress in Mathematics Volume 288

Series Editors Hyman Bass Joseph Oesterlé Alan Weinstein

Karl-Hermann Neeb Arturo Pianzola Editors

Developments and Trends in Infinite-Dimensional Lie Theory

Editors Karl-Hermann Neeb Department of Mathematics Friedrich-Alexander-Universität Erlangen-Nürnberg Bismarckstrasse 1 1/2 91054 Erlangen, Germany [email protected]

Arturo Pianzola Department of Mathematical Sciences University of Alberta Edmonton, Alberta T6G 2G1, Canada [email protected] and Instituto Argentino de Matemática Saavedra 15 1083 Buenos Aires, Argentina [email protected]

ISBN 978-0-8176-4740-7 e-ISBN 978-0-8176-4741-4 DOI 10.1007/978-0-8176-4741-4 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010938915 Mathematics Subject Classification (2010): 17Bxx, 20Gxx, 22Exx, 58Bxx, 58Dxx © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

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Preface

Since the advent of Kac–Moody algebras in the late 1960s, inﬁnite-dimensional Lie theory has become one of the core areas of modern mathematics. The global theory of inﬁnite-dimensional Lie groups developed through several stages, starting with the class of Banach–Lie groups in the 1960s. In the 1980s it became apparent that more general classes of groups, such as smooth loop groups, were needed to meet the requirements of various kinds of applications. The core of inﬁnite-dimensional Lie theory lies at the translation of the global group picture to the inﬁnitesimal Lie algebra picture, where the means are often based on geometric structures such as ﬁber bundles on (possibly inﬁnitedimensional) manifolds on which the groups act by transformations and on which the Lie algebra is realized by vector ﬁelds. The idea for the present book came up during an Oberwolfach meeting on inﬁnite-dimensional Lie theory. Its purpose is to describe some of the current developments and trends in inﬁnite-dimensional Lie theory. Many of these developments have been motivated by connections with other branches of mathematics. Accordingly, the book is divided into three main parts: (A) Inﬁnite-dimensional Lie (super-)algebras; (B) Geometry of inﬁnite-dimensional Lie (transformation) groups; (C) Representation theory of inﬁnite-dimensional Lie groups. The diﬀerent contributions clearly demonstrate the breadth of Lie theory in the inﬁnite-dimensional context. It is also our intention to display the connectedness among the algebraic, geometric, and analytic branches, which are diﬀerent faces of a general theory with many diﬀerent aspects. Acknowledgments. All contributions to this book have been refereed, with some of them going through several stages of rewriting until they reached their ﬁnal form. The editors are grateful to the referees for their eﬃcient help. We thank the Mathematische Forschungsinstitut Oberwolfach for hosting the conference where this book was conceived.

vi

Preface

Last but not least we express our appreciation to Birkh¨ auser Boston, in particular Ann Kostant and Tom Grasso as acting representatives for a remarkably pleasant cooperation.

Karl-Hermann Neeb Arturo Pianzola Banﬀ March 2008

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

Part A Inﬁnite-Dimensional Lie (Super-)Algebras Isotopy for Extended Aﬃne Lie Algebras and Lie Tori Bruce Allison and John Faulkner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

Remarks on the Isotriviality of Multiloop Algebras Philippe Gille and Arturo Pianzola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Extended Aﬃne Lie Algebras and Other Generalizations of Aﬃne Lie Algebras – A Survey Erhard Neher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Tensor Representations of Classical Locally Finite Lie Algebras Ivan Penkov and Konstantin Styrkas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Lie Algebras, Vertex Algebras, and Automorphic Forms Nils R. Scheithauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Kac–Moody Superalgebras and Integrability Vera Serganova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Part B Geometry of Inﬁnite-Dimensional Lie (Transformation) Groups Jordan Structures and Non-Associative Geometry Wolfgang Bertram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Direct Limits of Inﬁnite-Dimensional Lie Groups Helge Gl¨ ockner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

viii

Contents

Lie Groups of Bundle Automorphisms and Their Extensions Karl-Hermann Neeb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Gerbes and Lie Groups Christoph Schweigert and Konrad Waldorf . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Part C Representation Theory of Inﬁnite-Dimensional Lie Groups Functional Analytic Background for a Theory of InﬁniteDimensional Reductive Lie Groups Daniel Beltit¸a ˘ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 Heat Kernel Measures and Critical Limits Doug Pickrell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Coadjoint Orbits and the Beginnings of a Geometric Representation Theory Tudor S. Ratiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 Inﬁnite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces Joseph A. Wolf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Part A

Infinite-Dimensional Lie (Super-)Algebras

Isotopy for Extended Aﬃne Lie Algebras and Lie Tori Bruce Allison1∗ and John Faulkner2 1

Department of Mathematics and Statistics, University of Victoria, Victoria BC V8W 3P4 Canada, [email protected] 2 Department of Mathematics, University of Virginia, Charlottesville VA 22904-4137 USA, [email protected]

Summary. Centreless Lie tori have been used by E. Neher to construct all extended aﬃne Lie algebras (EALAs). In this article, we study isotopy for centreless Lie tori, and show that Neher’s construction provides a 1–1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to isomorphism. Also, centreless Lie tori can be coordinatized by unital algebras that are in general nonassociative, and, for many types of centreless Lie tori, there are classical deﬁnitions of isotopy for the coordinate algebras. We show for those types that an isotope of a Lie torus is coordinatized by an isotope of its coordinate algebra, thereby connecting the two notions of isotopy. In writing the article, we have not assumed prior knowledge of the theories of EALAs, Lie tori or isotopy. Key words: isotope, extended aﬃne Lie algebra, Lie torus, nonassociative algebra, Jordan algebra, alternative algebra. 2000 Mathematics Subject Classiﬁcations: Primary 17B65, 17A01. Secondary 17B60, 17B70, 17C99, 17D05.

1 Introduction If A is a unital associative algebra and u is invertible in A, one can deﬁne an algebra A(u) , called the u-isotope of A, which is equal to A as a vector space but has a new product x ·u y = xuy. This isotope A(u) is again unital and associative but with a shifted identity element u−1 . More generally there are deﬁnitions of isotope for several other classes of unital nonassociative algebras, notably Jordan algebras [Mc2, §I.3.2], alternative algebras [Mc1] and associative algebras with involution [Mc2, §I.3.4]. In each case, the isotope is obtained very roughly by shifting the identity element in the algebra, and two algebras are said to be isotopic if one is isomorphic to an isotope of the ∗

Bruce Allison gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada.

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_1, © Springer Basel AG 2011

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Bruce Allison and John Faulkner

other. In the associative case, the u-isotope A(u) is isomorphic to A under left multiplication by u, and therefore isotopy has not played a role in associative theory. That is not true in general though, and in particular isotopy plays an important role in Jordan theory (see for example [Mc2, §II.7]). In contrast, isotopes and isotopy have not been deﬁned for Lie algebras, for the evident reason that Lie algebras are not unital.2 In this article we study notions of isotope and isotopy, which were recently introduced in [ABFP2] for a class of graded Lie algebras called Lie tori. The point to emphasize here is that forming an isotope of a Lie torus does not change the multiplication at all, but rather it shifts the grading. We are primarily interested in the case in which the Lie torus is centreless, and there are two basic reasons why we are interested in isotopy in this case. First, centreless Lie tori arise naturally in the construction of families of extended aﬃne Lie algebras (EALAs), and we see in this article that isotopes and isotopy play a natural and fundamental role in the theory of both of these classes of Lie algebras. In fact, we show in section 6 using a construction of E. Neher, that there is a 1–1 correspondence between centreless Lie tori up to isotopy and families of EALAs up to bijective isomorphism. (This terminology is explained in section 6.) Second, any Lie torus is by deﬁnition graded by the group Q × Λ, where Q is the root lattice of a ﬁnite irreducible root system ∆ and Λ is (in this article) a ﬁnitely generated free abelian group. Depending on the type of ∆, centreless Lie tori are coordinatized by unital algebras that are in general nonassociative. For example, in types A1 , A2 , Ar (r ≥ 3) and Cr (r ≥ 4) the coordinate algebras are respectively Jordan algebras, alternative algebras, associative algebras and associative algebras with involution. It turns out that an isotope of a centreless Lie torus is coordinatized by an isotope of its coordinate algebra, thereby connecting our notion of isotopy for Lie tori with the classical notions for unital nonassociative algebras. This fact can then be used to obtain necessary and suﬃcient conditions for two centreless Lie tori of the same type to be isotopic (in terms of their coordinate algebras). We describe these results in detail for types A1 , A2 , Ar (r ≥ 3) and Cr (r ≥ 4) in sections 8, 9, 10, and 11, and we remark very brieﬂy on the other types in section 12. Section 12 also includes a brief discussion of an alternate approach, using multiloops, to constructing centreless Lie tori and studying isotopy. In writing this article, we have not assumed prior knowledge of the theories of EALAs, Lie tori or isotopy. In fact, we hope that this article will serve as a useful introduction to these theories and their interconnections. For this reason, we begin in sections 2, 3, 4 and 5 by recalling some of the important deﬁnitions and facts about EALAs, Lie tori, and isotopy for Lie tori. For the same reason, we have included a brief discussion of coordinatization in 2

There is a notion of isotopy that makes sense for nonunital algebras and hence Lie algebras [A], but it is quite diﬀerent from the one discussed in this article.

Isotopy for EALAs and Lie Tori

5

section 7; and in sections 8 and 9 we have included some arguments that are familiar to experts in Jordan and alternative algebra theory, but may be less familiar to others. Acknowledgements: We thank Stephen Berman, Erhard Neher, and Arturo Pianzola for stimulating conversations about the topics of this paper. We also thank Erhard Neher for his helpful comments about a preliminary version of this article. Assumptions and notation: Throughout this work we assume that F is a ﬁeld of characteristic 0. All algebras are assumed to be algebras over F, and all algebras (except Lie algebras) are assumed to be unital. We also assume that Λ is a ﬁnitely generated free abelian group. We denote the rank of Λ, which is an integer ≥ 0, by rank(Λ). If E is a Lie algebra and H is an ad-diagonalizable subalgebra of E, we let Eρ = { x ∈ E | [h, x] = ρ(h)x for h ∈ H } for ρ in the dual space H ∗ of H. Then, we deﬁne ∆(E, H) = { ρ ∈ H ∗ | Eρ = 0 } and we call elements of ∆(E, H) roots of E relative to H; so in particular, 0 is a root (if E = 0). Some terminology for graded algebras: If A = ⊕λ∈Λ Aλ is a Λ-graded algebra, the Λ-support of A is suppΛ (A) = { λ ∈ Λ | Aλ = 0 }. If Γ is a subgroup of Λ and A is a Γ -graded algebra or vector space, we regard A as Λ-graded by setting Aλ = 0 for λ ∈ Λ \ Γ . If A is a Λ-graded algebra and A is a Λ -graded algebra, we say that A and A are isograded-isomorphic, written A ig A , if there exists an algebra isomorphism η : A → A and a group isomorphism ηgr : Λ → Λ such that η(Aλ ) = Aηgr (λ) for λ ∈ Λ; in that case ηgr is uniquely determined by η if suppΛ (A) = Λ. (Here suppΛ (A) denotes the subgroup of Λ generated by suppΛ (A).) If A and A are Λ-graded algebras, we say that A and A are graded-isomorphic, written A Λ A , if there is an algebra isomorphism η : A → A that preserves the grading (that is ηgr = id). A Λ-graded algebra with involution is a pair (A, ι), where A is a Λ-graded algebra and ι is an involution (antiautomorphism of period 2) of A that is graded. There is an evident extension of the terms isograded-isomorphic and graded-isomorphic and the notations ig and Λ for graded algebras with involution. Finally, the group algebra F[Λ] = ⊕λ∈Λ Ftλ of Λ with tλ tλ = tλ+λ is naturally a Λ-graded algebra. If we choose a basis { λ1 , . . . , λn } for Λ, ±1 then F[Λ] is isograded-isomorphic to the algebra F[t±1 1 , . . . , tn ] of Laurent n polynomials with its natural Z -grading.

2 Extended aﬃne Lie algebras The deﬁnition of an EALA has evolved from [HT], where these algebras were introduced under the name elliptic irreducible quasi-simple Lie algebra, and from [BGK] and [AABGP], where many of their properties were developed. We will use the deﬁnition given in [N2], which has the important advantage that it makes sense over any ﬁeld of characteristic 0. Some of the facts about EALAs mentioned below were proved in the setting of [AABGP], but they can be veriﬁed in a similar fashion in our more general setting.

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We recall that an extended aﬃne Lie algebra (EALA) is a triple (E, ( | ), H), where E is a Lie algebra over F, ( | ) is a nondegenerate invariant symmetric bilinear form on E, and H is a ﬁnite dimensional nonzero self-centralizing ad-diagonalizable subalgebra of E, such that a list of axioms labeled as (EA3)– (EA6) are satisﬁed [N2]. For the purposes of this article we do not need the precise statement of these axioms, except to say that they are modeled after the properties of ﬁnite dimensional split simple Lie algebras and aﬃne KacMoody Lie algebras. If (E, ( | ), H) is an EALA, we also say that E is an EALA with respect to ( | ) and H, or simply that E is an EALA. Roots of E relative to H are simply called roots of E. If E is an EALA, we can as usual transfer the restriction of ( | ) to H to a nondegenerate form ( | ) on the dual space H ∗ . If R = ∆(E, H), V = spanF (R), and rad(V ) is the radical of the restriction of ( | ) to V , then the image of R in V / rad(V ) is a ﬁnite irreducible root system (including 0) whose type is called the type of E. (See the beginning of §3 below for our conventions about ﬁnite irreducible root systems.) A root ρ of an EALA E is called isotropic if (ρ, ρ) = 0, and otherwise called nonisotropic. According to one of the axioms for an EALA, the group generated by the isotropic roots is a free abelian group of ﬁnite rank, and its rank is called the nullity of E. Then the extended aﬃne Lie algebras of nullity 0 and 1 are precisely the ﬁnite dimensional split simple Lie algebras and the aﬃne Kac-Moody Lie algebras, respectively. (See [ABFP2, Remark 1.2.4] for nullity 0 and [ABGP] for nullity 1.) If (E, ( | ), H) is an EALA, then so is (E, a( | ), H) for a ∈ F× , and it is sometimes convenient to adjust the form in this way. For this reason, the following is a natural notion of isomorphism for EALAs. Definition 2.1. If (E, ( | ), H) and (E , ( | ) , H ) are EALAs, an isomorphism from (E, ( | ), H) onto (E , ( | ) , H ) is a Lie algebra isomorphism χ : E → E such that (χ(x)|χ(y)) = a(x|y) for some a ∈ F× and all x, y ∈ E, and χ(H) = H . If such a map exists we say that (E, ( | ), H) and (E , ( | ) , H ) are isomorphic. For short we often say that χ is an EALA isomorphism from E onto E and that E and E are isomorphic as EALAs. Definition 2.2. If E is an EALA, the core of E is the subalgebra Ec of E that is generated by the root spaces of E corresponding to nonisotropic roots, and the centreless core of E is Ecc := Ec /Z(Ec ).3

3 Lie tori With the construction of families of EALAs in mind, Yoshii gave a deﬁnition of a Lie torus in [Y6]. An equivalent deﬁnition, which we recall next, was given by Neher in [N1]. 3

We denote the centre of a Lie algebra L by Z(L). L is called centreless if Z(L) = 0.

Isotopy for EALAs and Lie Tori

7

It will be convenient for us to work with root systems that contain 0. So by a ﬁnite irreducible root system we will mean a ﬁnite subset ∆ of a ﬁnite dimensional vector space X over F such that 0 ∈ ∆ and ∆× := ∆ \ { 0 } is a ﬁnite irreducible root system in X in the usual sense (see [B2, Chap VI, §1, no. 1, Deﬁnition 1]). With this convention we will assume for the rest of this section that ∆ is a ﬁnite irreducible root system in a ﬁnite dimensional vector space X over F. Recall that ∆ is said to be reduced if 2α ∈ / ∆× for α ∈ ∆× . If ∆ is reduced then ∆ has type A ( ≥ 1), B ( ≥ 2), C ( ≥ 3), D ( ≥ 4), E6 , E7 , E8 , F4 , or G2 , whereas if ∆ is not reduced, ∆ has type BC ( ≥ 1) [B2, Chapter VI, §4 and 14]. We let ∆ind := { 0 } ∪ { α ∈ ∆× | 12 α ∈ / ∆ }, in which case ∆ind is an irreducible reduced root system in X, and ∆ind = ∆ if ∆ is reduced. Also we let Q = Q(∆) := spanZ (∆) be the root lattice of ∆. If L = (α,λ)∈Q×Λ Lλα is a Q × Λ-graded algebra, then L = λ∈Λ Lλ is Λ-graded and L = α∈Q Lα is Q-graded, with Lλ =

α∈Q

Lλα for λ ∈ Λ

and

Lα =

λ∈Λ

Lλα for α ∈ Q.

Definition 3.1. A Lie Λ-torus of type ∆ is a Q × Λ-graded Lie algebra L = λ (α,λ)∈Q×Λ Lα over F which satisﬁes: (LT1) Lα = 0 for α ∈ Q \ ∆. (LT2) (i) If 0 = α ∈ ∆ind , then L0α = 0. (ii) If 0 = α ∈ Q, λ ∈ Λ and Lλα = 0, then there exist elements eλα ∈ Lλα and fαλ ∈ L−λ −α such that Lλα = Feλα , and

λ L−λ −α = Ffα ,

[[eλα , fαλ ], xβ ] = β, α∨ xβ

(1)

for xβ ∈ Lβ , β ∈ Q. (LT3) L is generated as an algebra by the spaces Lα , α ∈ ∆× . (LT4) Λ is generated as a group by suppΛ (L). In that case the Q-grading of L is called the root grading of L, and the Λgrading of L is called the external grading of L. If ∆ has type X , we also say that L has type X . Remark 3.2. Suppose L is a Lie Λ-torus of type ∆. (i) It is shown in [ABFP2, Prop. 1.1.10] that suppQ (L) equals either ∆ or ∆ind .

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Bruce Allison and John Faulkner

(ii) It is sometimes convenient to assume the following additional axiom: (LT5) suppQ (L) = ∆. Note that by (i), (LT5) holds automatically if ∆ is reduced. Also, if (LT5) does not hold, then we can replace ∆ by ∆ind , in which case (LT5) holds. Thus, when convenient there is no loss of generality in assuming (LT5). In the study of Lie tori it is not convenient to ﬁx a particular realization of the root system ∆ or a particular identiﬁcation of the group Λ with Zn . For this reason, the following is a natural notion of isomorphism for Lie tori. Definition 3.3. If L is a Lie Λ-torus of type ∆ with Q = Q(∆) and L is a Lie Λ -torus of type ∆ with Q = Q(∆ ), L and L are said to be bi-isogradedisomorphic or bi-isomorphic for short, if there is an algebra isomorphism ϕ : L → L , a group isomorphism ϕr : Q → Q, and a group isomorphism ϕe : Λ → Λ such that ϕ (λ) ϕ(Lλα ) = L ϕer (α) for α ∈ Q and λ ∈ Λ. In that case we write L bi L and we call ϕ a bi-isograded-isomorphism, or a bi-isomorphism for short. Since suppQ (L) = Q and suppΛ (L) = Λ, the maps ϕr and ϕe are uniquely determined. Remark 3.4. Suppose that ϕ : L → L is a bi-isomorphism as in Deﬁnition 3.3. We did not assume that ϕr carries ∆ onto ∆ . However, this holds automatically, if L and L satisfy (LT5). In particular, if L and L satisfy (LT5) and ∆ = ∆, then ϕr ∈ Aut(∆). Definition 3.5. Suppose that L is a centreless Lie Λ-torus of type ∆. We set g = L0

and

h = L00 .

Then g is a ﬁnite dimensional simple Lie algebra and h is a splitting Cartan subalgebra of g [N1, ABFP2]. Moreover, h acts ad-diagonally on L. Furthermore, we can and do identify ∆ with a root system in the dual space h∗ in such a way that ∆(g, h) = ∆ind , ∆(L, h) = ∆ or ∆ind , and, for α ∈ Q, the root space of L relative to α is Lα [ABFP2, Prop. 1.2.2]. In particular, the type of the split simple Lie algebra g is the type of the root system ∆ind . We call (g, h) the grading pair for L.4

4

In [N1], the grading pair is deﬁned for not necessarily centreless Lie tori. For centreless Lie tori the two deﬁnitions are equivalent [ABFP2, Prop. 1.2.1].

Isotopy for EALAs and Lie Tori

9

Remark 3.6 ([BN, Prop. 3.13]). Suppose that L is a Λ-torus, and let Cent(L) be the centroid of L.5 Then, Cent(L) is Λ-graded, with Cent(L)λ = { c ∈ Cent(L) | c(Lµ ) ⊆ Lλ+γ for µ ∈ Λ }. Let Γ (L) := suppΛ (Cent(L)) be the support of Cent(L), in which case Γ (L) is a subgroup of Λ called the centroidal grading group of L. In that case, Cent(L) is graded-isomorphic to the group algebra F[Γ ] with its natural Γ -grading (and hence also with its Λ-grading). Example 3.7 (The untwisted centreless Lie torus). Let g be a ﬁnite dimensional split simple Lie algebra, let h be a splitting Cartan subalgebra of g, let g = ⊕α∈∆ gα be the root space decomposition of g relative to h, where ∆ = ∆(g, h). Set L = g ⊗ F[Λ] with Q × Λ-grading given by Lλα = gα ⊗ F[Λ]λ for α ∈ Q, λ ∈ Λ. Then L is a centreless Lie Λ-torus of type ∆ which we call the untwisted centreless Lie Λ-torus of type ∆.6 The grading pair for L is (g ⊗ 1, h ⊗ 1). The centroid of L consists of multiplications by elements of F[Λ], so Γ (L) = Λ. In later sections, we will need the following lemma about bi-isomorphisms of centerless Lie tori of the same type. Lemma 3.8. Let L be a Lie Λ-torus of type ∆ with grading pair (g, h), and let L be a Lie Λ -torus of type ∆ with grading pair (g , h ). Let Π = { α1 , . . . , α } be a base for the root system ∆, choose 0 = ei ∈ gαi , 0 = fi ∈ g−αi with [[ei , fi ], ei ] = 2ei , and choose 0 = ei ∈ gαi , 0 = fi ∈ g−αi with [[ei , fi ], ei ] = 2ei . Suppose there is a bi-isomorphism ϕ : L → L with ϕr in the Weyl group of ∆. (Here we are using the notation ϕr and ϕe of Deﬁnition 3.3.) Then, there exists a bi-isomorphism ϕ˜ : L → L such that ϕ˜e = ϕe , ϕ˜r = 1, ϕ(e ˜ i ) = ei and ϕ(f ˜ i ) = fi for all i. Proof. Since ϕr is in the Weyl group of ∆, there exists an automorphism η of g such that η(h) = h and η(gα ) = gϕr (α) for all α ∈ ∆ [B3, Theorem 2(ii), Ch. VIII, § 2, no. 2]. Moreover, η can be chosen in the form η = i exp(adg (xi )), where each xi is in a root space of g corresponding to a nonzero root. Thus each adL (xi ) is nilpotent and so we may extend η to η = i exp(adL (xi )). Then η : L → L is a bi-isomorphism with ηe = 1 and ηr = ϕr . So replacing ϕ by ϕη −1 , we can assume that ϕr = 1. Thus, ϕ(gαi ) = ϕ(g ∩ Lαi ) = g ∩ Lαi = gαi for all i. So ϕ(ei ) = ai ei and × × ϕ(fi ) = a−1 i fi for some ai ∈ F . Choose a group homomorphism ρ : Q → F such that ρ(αi ) = ai for all i, and deﬁne τ ∈ Aut(L) by τ (xα ) = ρ(α)xα for xα ∈ Lα , α ∈ Q. Then τ : L → L is a bi-isomorphism with τe = 1 and τr = 1. So ϕτ −1 is the required ϕ. ˜

5 6

Recall that the centroid of an algebra is the associative algebra of all endomorphisms of the algebra that commute with all left and right multiplications. The universal central extension of g⊗F[Λ] is called the toroidal Lie algebra [MRY], which is one of the origins of the term Lie torus.

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4 The construction of EALAs from Lie tori In [BGK], Berman, Gao, and Krylyuk gave a construction of a family of EALAs starting from a centreless Lie tori of type Ar , r ≥ 3. In [N2], Neher simpliﬁed this construction and extended it to all types. Since we will be working with this construction in some detail, we give a careful description of it in this section. Facts that we note without reference are either straightforward or can be found in [N1, N2]. We ﬁrst need to establish some notation and assumptions. Throughout the section, we assume that L is a centreless Lie Λ-torus of type ∆ and we let Q = Q(∆). Let Γ = Γ (L) be the centroidal grading group of L. We identify Cent(L) = F[Γ ] (see Remark 3.6) by ﬁxing a basis { tγ }γ∈Γ for Cent(L) satisfying tγ tδ = tγ+δ . Next let Hom(Λ, F) be the group of group homomorphisms from Λ into F. Then for θ ∈ Hom(Λ, F), let ∂θ ∈ Der(L) be the degree derivation deﬁned by ∂θ (xλ ) = θ(λ)xλ for xλ ∈ Lλ , λ ∈ Λ. Put D = { ∂θ | θ ∈ Hom(Λ, F) } and

CDer(L) = Cent(L)D.

Then CDer(L) is a subalgebra of the Lie algebra Der(L) with product given by [tγ1 ∂θ1 , tγ2 ∂θ2 ] = tγ1 +γ2 (θ1 (γ2 )∂θ2 − θ2 (γ1 )∂θ1 ), (2) and CDer(L) is Γ -graded with CDer(L)γ = Cent(L)γ D for γ ∈ Γ . Since L is a centreless Lie torus, there is a nondegenerate invariant Λgraded form ( | ) on L, and that form is unique up to multiplication by a nonzero scalar [Y6, Thm. 2.2 and 7.1]. We ﬁx a choice of ( | ) on L. Then, since L is perfect and ( | ) is invariant we have (c(x), y) = (x, c(y)) for x, y ∈ L, c ∈ Cent(L). Now let SCDer(L) be the subalgebra of CDer(L) consisting of the derivations in CDer(L) that are skew relative to the form ( | ). (This subalgebra does not depend on the choice of the form ( | ).) Then, SCDer(L) is a Γ -graded subalgebra of CDer(L) with SCDer(L)γ = tγ { ∂θ ∈ D | θ(γ) = 0 }

(3)

for γ ∈ Γ . SCDer(L) is called the algebra of skew-centroidal derivations of L. Now suppose that D = ⊕γ∈Γ Dγ is a Γ -graded subalgebra of SCDer(L). The graded-dual space of D is the subspace Dgr∗ = ⊕γ∈Γ (Dγ )∗ of D∗ , where

Isotopy for EALAs and Lie Tori

11

(Dγ )∗ is embedded in D∗ by letting its elements act trivially on Dδ for δ = γ. We let C = Dgr∗ , and we give the vector space C a Γ -grading by setting Cγ = (D−γ )∗ .

(4)

With this grading, C is a Γ -graded D-module by means of the contragradient action given by (d · f )(e) = −f ([d, e]), for d, e ∈ D, f ∈ C. We also regard C as a L-module with trivial action. Now deﬁne σD : L × L → C by σD (x, y)(d) = (dx|y). Then σD is a Γ -graded 2-cocycle for L with values in the trivial L-module C.7 Finally, let ev : Λ → C0 denote the evaluation map deﬁned by (ev(λ))(∂θ ) = θ(λ) for λ ∈ Λ and ∂θ ∈ D0 ⊆ SCDer(L)0 = D. With this background we now present the construction. Construction 4.1 ([N2]). Let L be a centreless Lie Λ-torus of type ∆. In order to construct an EALA from L we need two additional ingredients. We assume that (4.1a) D is a Γ -graded subalgebra of SCDer(L) such that the map ev : Λ → C0 is injective, where Γ = Γ (L) and C = Dgr∗ . (4.1b) τ : D × D → C is a Γ -graded invariant 2-cocycle of D with values in the D-module C and τ (D0 , D) = 0. Note that the assumption in (4.1b) that τ is invariant means that τ (d1 , d2 )(d3 ) = τ (d2 , d3 )(d1 ) for di ∈ D. Let

E = E(L, D, τ ) = D ⊕ L ⊕ C,

where C = D . We identify D, L and C naturally as subspaces of E. Deﬁne a product [ , ]E on E by gr∗ 8

7

8

In this section and in §6 we use the terminology (cochain, cocycle, coboundary and extension) from the cohomology of Lie algebras. See for example [B1, Chapter I, Exercise 12 for §3]. We have changed the order used in [N2] of the components of E. This is convenient in the proof of Corollary 6.3 below.

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[d1 + l1 + f1 , d2 + l2 + f2 ]E = [d1 , d2 ] + ([l1 , l2 ] + d1 (l2 ) − d2 (l1 )) + (d1 · f2 − d2 · f1 + σD (l1 , l2 ) + τ (d1 , d2 )) for di ∈ D, li ∈ L, fi ∈ C. Then, E is a Λ-graded algebra with the direct sum grading. Next we extend the bilinear form ( | ) on L to a graded bilinear form ( | ) on E by deﬁning: (d1 + l1 + f1 | d2 + l2 + f2 ) = (l1 | l2 ) + f1 (d2 ) + f2 (d1 ). Finally let H = D0 ⊕ h ⊕ C0

(5)

in E, where (g, h) = (L0 , L00 ) is the grading pair for L. Thus we have constructed a triple ((E(L, D, τ ), ( | ), H), which we often denote simply by E(L, D, τ ). Neher has announced the following fundamental result on this construction: Theorem 4.2 ([N2, Thm. 6]). (i) If L is a centreless Lie Λ-torus, D satisﬁes (4.1a) and τ satisﬁes (4.1b), then E(L, D, τ ) is an extended aﬃne Lie algebra of nullity rank(Λ). (ii) If E is an EALA, then E is isomorphic as an EALA to E(L, D, τ ) for some L, D and τ as in (i). Remark 4.3. (i) It is easy to check that the EALA E(L, D, τ ) in part (i) of the theorem does not depend, up to EALA isomorphism, on the choice of the form ( | ) on L. (ii) Neher actually states more than we have stated in part (ii) of the theorem. Indeed, given an EALA E, then Ecc , with a suitable grading, is a Lie torus satisfying (LT5) and E is isomorphic as an EALA to E(Ecc , D, τ ) for some D and τ as in (i) [N2, Thm. 6(ii)]. (iii) In particular, we can always choose L in part (ii) of the theorem satisfying (LT5). In that case, the type of E and L are the same. (iv) The Lie torus L in part (ii) of the theorem is not uniquely determined up to bi-isomorphism. However, it is uniquely determined up to isotopy, a fact that has motivated our work on this topic. We discuss this in detail in §6 below. Remark 4.4. Suppose L is a centreless Lie torus, Γ = Γ (L), and D is a Γ graded subalgebra of SCDer(L). Let M := D ⊕ L and identify D and L naturally as subspaces of M . Then, M is a Λ-graded Lie algebra with product [ , ] given by [d1 + l1 , d + l2 ] = [d1 , d2 ] + ([l1 , l2 ] + d1 (l2 ) − d2 (l1 )) , and C is an M -module via (d + l) · f = d · f . The cocycles σD and τ can be extended to cocycles for the Lie algebra M with values in the M -module C

Isotopy for EALAs and Lie Tori

13

by letting σD (D, M ) = σD (M, D) = 0 and τ (L, M ) = τ (M, L) = 0. If D and τ satisfy (4.1a) and (4.1b), then as a Lie algebra E(L, D, τ ) is equal to the extension Ext(M, C, σD + τ ) of M by C using the cocycle σD + τ . If L is a centreless Lie torus, we let P(L) = { (D, τ ) | D satisﬁes (4.1a) and τ satisﬁes (4.1b) }. We note P(L) is nonempty since in particular (SCDer(L), 0) ∈ P(L). With this notation Theorem 4.2(i) tells us that Construction 4.1 builds a family { E(L, D, τ ) }(D,τ )∈P(L) of EALAs from L. Also, Theorem 4.2(ii) tells us that any EALA E occurs (up to isomorphism of EALAs) in the family { E(L, D, τ ) }(D,τ )∈P(L) constructed from some centreless Lie torus L.

5 Isotopy for Lie tori In this section, we recall the deﬁnition of isotopy from [ABFP2]. Throughout the section, we assume that L is a Lie Λ-torus of type ∆ and we let Q = Q(∆). If α ∈ Q, we let Λα = Λα (L) := { λ ∈ Λ | Lλα = 0 }. We call Λα the Λ-support of L at α. Note that 0 ∈ Λα if 0 = α ∈ ∆ind , by (LT2)(ii). Definition 5.1. Suppose that s ∈ Hom(Q, Λ), where Hom(Q, Λ) is the group of group homomorphisms from Q into Λ. We deﬁne a new Q × Λ-graded Lie algebra L(s) as follows. As a Lie algebra, L(s) = L. The grading on L(s) is given by (L(s) )λα = Lλ+s(α) (6) α for α ∈ Q, λ ∈ Λ. The following necessary and suﬃcient conditions for L(s) to be a Lie torus are obtained in [ABFP2, Prop. 2.2.3]. Proposition 5.2. Let s ∈ Hom(Q, Λ), and let Π be a base for the root system ∆. The following statements are equivalent: (a) L(s) is a Lie torus. (b) s(α) ∈ Λα for all 0 = α ∈ ∆ind . (c) s(α) ∈ Λα for all α ∈ Π. Definition 5.3 (Isotopes of Lie tori). Suppose that s ∈ Hom(Q, Λ). If s satisﬁes the equivalent conditions in Proposition 5.2, we say that s is admissible for L. In that case, we call the Lie torus L(s) the s-isotope of L.

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Remark 5.4. Suppose that Π = { α1 , . . . , αr } is a base for ∆. By Proposition 5.2, to specify an admissible s ∈ Hom(Q, Λ) for L, and hence an isotope L(s) of L, one can arbitrarily choose λi ∈ Λαi for 1 ≤ i ≤ r, and then deﬁne s ∈ Hom(Q, Λ) with s(αi ) = λi for 1 ≤ i ≤ r. Definition 5.5. Suppose that L is a Lie Λ-torus of type ∆ and L is a Lie Λ -torus of type ∆ . We say that L is isotopic to L , written L ∼ L , if some isotope L(s) of L is bi-isomorphic to L . Using the facts that L(0) = L, (L(s) )(t) = L(s+t) and (L(s) )(−s) = L as Q × Λ-graded algebras for s, t ∈ Hom(Q, Λ), it is shown in [ABFP2, §2] that isotopy is an equivalence relation on the class of Lie tori.

6 Isotopy in the theory of EALAs As promised at the end of §4, we now describe the role that isotopy plays in the structure theory of EALAs. Our ﬁrst theorem on this topic was actually the starting point of our investigation of isotopy. Theorem 6.1. Suppose L is a centreless Lie Λ-torus of type ∆ and L is a centreless Lie Λ -torus of type ∆ . If some member of the family { E(L, D, τ ) }(D,τ )∈P(L) is isomorphic as an EALA to some member of the family { E(L , D , τ ) }(D ,τ )∈P(L ) , then L is isotopic to L . Proof. Let (g, h) be the grading pair of L. Since ∆(L, h) = ∆ or ∆ind (see Deﬁnition 3.5), we can if necessary replace ∆ by ∆ind so that ∆(L, h) = ∆. Suppose now that E = E(L, D, τ ), where (D, τ ) ∈ P(L). We use the notation Q, C, ev, H from Construction 4.1. Recall that H = D0 ⊕ h ⊕ C, so we can identify H ∗ = (D0 )∗ ⊕ h∗ ⊕ (C0 )∗ = C0 ⊕ h∗ ⊕ D0 . We use ev to identify Λ with a subgroup of C0 ⊆ H ∗ . Let R be the set of roots of E and let R× be the set of nonisotropic roots in R. Then by [N2, §5], we have R× = ∪α∈∆× (Λα + α) ⊆ R ⊆ Λ ⊕ Q,

(7)

where Λα = Λα (L). It follows from this fact and (LT2)(i) that R = Λ ⊕ Q.

(8)

Eλ+α = Lλα

(9)

Now by [N2, §5], we have

Isotopy for EALAs and Lie Tori

15

for λ ∈ Λ, 0 = α ∈ Q. It is easy to show using this equation, (7) and (LT3) that the core of E (see Deﬁnition 2.1(ii)) is given by Ec = L ⊕ C.

(10)

(In fact this is implicit in [N2].) Hence, since L is centreless, we have Z(Ec ) = C.

(11)

Next, let E = E(L , D , τ ), where (D , τ ) ∈ P(L ), and we use all of the above notation (with primes added) for L and E . Suppose that χ : E → E is an EALA isomorphism. So χ(H) = H , and χ preserves the forms up multiplication by a nonzero scalar. It follows that χ(Ec ) = Ec . So, by (10) and (11) (and the corresponding primed equations), we have χ(L ⊕ C) = L ⊕ C and χ(C) = C . Thus, there exists a unique Lie algebra isomorphism ϕ : L → L with χ(l) ≡ ϕ(l)

(mod C )

(12)

for l ∈ L. It then follows that χ−1 (l ) ≡ ϕ−1 (l ) (mod C) for l ∈ L . Now, by (12), we have ϕ(h) ⊆ (H ⊕ C ) ∩ L = h , and similarly ϕ−1 (h ) ⊆ h. So ϕ(h) = h . (13) Hence, if h ∈ h, χ(h) − ϕ(h) ∈ H ∩ C = C0 , so χ(h) ≡ ϕ(h)

(mod C0 ).

(14)

Also, χ(C0 ) ⊆ H ∩ C = C0 and similarly χ−1 (C0 ) ⊆ C0 . So χ(C0 ) = C0 .

(15)

But then, by (13), (14) and (15), χ(h ⊕ C0 ) ⊆ h ⊕ C0 ; so using the corresponding property for χ−1 , we have χ(h ⊕ C0 ) = h ⊕ C0 . ∗

Next deﬁne χ ˆ : H ∗ → H by (χ(ρ))(χ(t)) ˆ = ρ(t)

(16)

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Bruce Allison and John Faulkner

for ρ ∈ H ∗ , t ∈ H. (χ ˆ is the inverse dual of χ|H : H → H .) Similarly, deﬁne ∗ ∗ ϕˆ : h → h by (ϕ(α))(ϕ(h)) ˆ = α(h) for α ∈ h∗ , h ∈ h. Then, using R = ∆(E, H) and ∆ = ∆(L, h) (and the corresponding primed equations), we see that χ(R) ˆ = R

and ϕ(∆) ˆ = ∆ .

Thus, since R = Λ ⊕ Q and ∆ = Q, we have χ(Λ ˆ ⊕ Q) = Λ ⊕ Q , ϕ(Q) ˆ = Q .

(17) (18)

Let α ∈ Q, and write, using (17), χ(α) ˆ = λ + α , where λ ∈ Λ , α ∈ Q . Then, (λ + α )(χ(h)) = α(h) for h ∈ h (in fact for h ∈ H). Thus, by (14), (λ + α )(ϕ(h)) = α(h), so α (ϕ(h)) = α(h) for h ∈ h. Hence, α = ϕ(α). ˆ Thus, since ϕˆ is invertible, χ(α) ˆ = s (ϕ(α)) ˆ + ϕ(α) ˆ (19) for α ∈ Q and some s ∈ Hom(Q , Λ ). Next let λ ∈ Λ, and as above write χ(λ) ˆ = λ + α , where λ ∈ Λ , α ∈ Q . Then, (λ +α )(χ(h+C0 )) = λ(h+C0 ) = 0. Hence, by (16), (λ +α )(h +C0 ) = −1 (Λ ) ⊆ Λ. So 0, so α = 0. Thus, χ(Λ) ˆ ⊆ Λ and similarly χ ˆ−1 (Λ ) = χ χ(Λ) ˆ = Λ .

(20)

Using (18) and (20), we let ϕr = ϕ| ˆ Q : Q → Q and ϕe = χ| ˆ Λ : Λ → Λ . Then, using (9) and (19), we have, for 0 = α ∈ Q, λ ∈ Λ, χ(Lλα ) = χ(Eλ+α ) = Eχ(λ+α) = Eχ(λ)+s (ϕ(α))+ ˆ ϕ(α) ˆ ˆ ˆ χ(λ)+s ˆ (ϕ(α)) ˆ

= (L )ϕ(α) ˆ So by (12), we have

ϕ (λ)

= ((L )(s ) )ϕer (α) .

ϕ (λ)

ϕ(Lλα ) = ((L )(s ) )ϕer (α) for 0 = α ∈ Q, λ ∈ Λ. But then, by (LT3), this equation holds for all α ∈ Q, λ ∈ Λ. Therefore, since L is a Lie Λ-torus of type ∆, so is (L )(s ) . Hence s is admissible for L , and ϕ is a bi-isomorphism of L onto (L )(s ) .

We next consider the relationship between the family of EALAs constructed from a Lie torus L and the family of EALAs constructed from an isotope of L. For this purpose, we assume now that L is a centreless Lie Λtorus of type ∆, Q = Q(∆), s ∈ Hom(Q, Λ) is admissible for L, and L(s) is the s-isotope of L. We ﬁx a nondegenerate Λ-graded invariant symmetric bilinear form ( | ) on L and we use that same form on L(s) .

Isotopy for EALAs and Lie Tori

17

(s)

Let ∂θ be the degree derivation of L(s) determined by θ ∈ Hom(Λ, F). Deﬁne the Λ-graded Cent(L)-module isomorphism Ψ : CDer(L) → CDer(L(s) )

(s)

by Ψ (c∂θ ) = c∂θ .

Using (2) we see that Ψ is also a Lie algebra isomorphism. Let hθ ∈ h be given by α(hθ ) = θ(s(α)) for α ∈ Q, and deﬁne the Λ-graded Cent(L)-module monomorphism Ω : CDer(L) → L by λ+s(α)

for c ∈ Cent(L). For l ∈ (L(s) )λα = Lα (s) ∂θ (l) + [hθ , l]. Thus, (s)

∂θ

Ω(c∂θ ) = chθ , we have ∂θ (l) = θ(λ + s(α))l =

= ∂θ − adL (hθ ) and Ψ (d) = d − adL (Ω(d))

for d ∈ CDer(L). Since the derivations in adL (L) are skew, we see that Ψ maps SCDer(L) into SCDer(L(s) ). We also note using (2) that Ω([tγ1 ∂θ1 , tγ2 ∂θ2 ]) = tγ1 tγ2 (θ1 (γ2 )hθ2 − θ2 (γ1 )hθ1 ) = tγ1 ∂θ1 (tγ2 hθ2 ) − tγ2 ∂θ2 (tγ1 hθ1 ) = tγ1 ∂θ1 (Ω(tγ2 ∂θ2 )) − tγ2 ∂θ2 (Ω(tγ1 ∂θ1 )), for γi ∈ Γ and θi ∈ Hom(Λ, F), so Ω is a derivation of CDer(L) into L. If (D, τ ) ∈ P(L), let D (s) = Ψ (D), (21) in which case D(s) is a Γ -graded subalgebra of SCDer(L). Also let ψ : D → D(s)

and ω : D → L

be the restrictions of Ψ and Ω to D. Then ψ is a Λ-graded Lie algebra isomorphism, ω is a Λ-graded vector space monomorphism, and ψ(d) = d − adL (ω(d)) for d ∈ D. Finally let

(22)

C (s) = (D(s) )gr∗ ,

which is Γ -graded as in (3), and deﬁne a Λ-graded cochain τ (s) : D(s) × D(s) → C (s) for di ∈ D.

by τ (s) (ψd1 , ψd2 )(ψd3 ) = τ (d1 , d2 )(d3 )

(23)

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Theorem 6.2. Suppose that L is a centreless Lie Λ-torus, s ∈ Hom(Q, Λ) is admissible for L, (D, τ ) ∈ P(L), and (D(s) , τ (s) ) is deﬁned by (21) and (23). Then (D(s) , τ (s) ) ∈ P(L(s) ). Moreover, E = E(L, D, τ ) and E (s) = E(L(s) , D(s) , τ (s) ) are isomorphic as EALAs. To be more precise, deﬁne χ : E → E (s) by χ(x) = ψ(d) + (ω(d) + l) + η(x) for x = d + l + c ∈ E, d ∈ D, l ∈ L, f ∈ C, where η : E → C (s) is given by 1 η(x)(ψ(d )) = f (d ) − (l + ω(d) | ω(d )) 2 for d ∈ D. Then χ is an isometry and a Λ-graded Lie algebra isomorphism (s) (s) that maps H to H (s) = D0 ⊕ h ⊕ C0 . ˆ )(ψ(d)) = f (d) for d ∈ D, f ∈ C. Clearly, Proof. Deﬁne ψˆ : C → C (s) by ψ(f ˆ ψ is a Γ -graded vector space isomorphism. Let ev(s) be the evaluation map (s) (s) for L(s) . Then, for λ ∈ Λ and ∂θ ∈ D0 , we have (s) ˆ ˆ (ψ(ev(λ)))(∂ θ ) = (ψ(ev(λ)))(ψ(∂θ )) = (ev(λ))(∂θ ) = θ(λ),

so ψˆ |C0 ◦ ev = ev(s) . Hence, ev(s) is injective; i.e., D(s) satisﬁes (4.1a). It will be convenient to view E as Ext(M, C, σD + τ ), the extension of M = D ⊕ L by C using the cocycle σD + τ as in Remark 4.4. Let M (s) = D(s) ⊕ L and set ξ(d + l) = ψ(d) + ω(d) + l for d ∈ D, l ∈ L. We claim ξ : M → M (s) is a Lie algebra isomorphism.

(24)

Clearly, ξ is bijective since ψ is bijective. Since ψ : D → D(s) is an isomorphism and L is an ideal in M (s) , the map d + l → ψ(d) + L = ξ(d + l) + L is a homomorphism M → M (s) /L. Thus, for m1 , m2 ∈ M , z := ξ([m1 , m2 ]) − [ξ(m1 ), ξ(m2 )] ∈ L. On the other hand, ζ : M → Der(L) given by ζ(m) = adM (m) |L and similarly ζ (s) : M (s) → Der(L) are homomorphisms. Moreover, ζ (s) (ξ(d + l)) = ψ(d) + adL (ω(d)) + adL (l) = d + adL (l) = ζ(d + l), so ζ (s) ◦ ξ = ζ. Thus, adL (z) = ζ (s) (z) = ζ([m1 , m2 ]) − [ζ(m1 ), ζ(m2 )] = 0. Since L is centreless, z = 0 and ξ is an isomorphism.

Isotopy for EALAs and Lie Tori

19

We also note that for d, d ∈ D, l ∈ L, f ∈ C, ˆ ))(ψ(d )) = −ψ(f ˆ )([ψ(d), ψ(d )]) ˆ ))(ψ(d )) = (ψ(d) · ψ(f (ξ(d + l) · ψ(f ˆ = −f ([d, d ]) = (d · f )(d ) = ψ((d + l) · f )(ψ(d ), so ˆ ) = ψ(m ˆ · f) ξ(m) · ψ(f

(25)

for m ∈ M , f ∈ C. Recall that the group Z 2 (M, C) of 2-cocycles on M with values in C consists of all alternating bilinear maps µ : M × M → C such that (µ([mi , mj ], mk ) − mi · µ(mj , mk )) = 0 (i,j,k)

for m1 , m2 , m3 ∈ M where (i, j, k) means (i, j, k) is a cyclic permutation of (1, 2, 3). We can use ξ and ψˆ to transfer µ to µ ˜ : M (s) × M (s) → C (s) with ˆ µ ˜ (ξ(m1 ), ξ(m2 )) = ψ(µ(m 1 , m2 )). Using (24) and (25), we see that ρ(µ) = µ ˜ deﬁnes a group isomorphism ρ : Z 2 (M, C) → Z 2 (M (s) , C (s) ). Furthermore, the map ξ ⊕ ψˆ : Ext(M, C, µ) → Ext(M (s) , C (s) , ρ(µ)) is a Lie algebra isomorphism. ˆ (M, L)) = 0 and Let τ˜ = ρ(τ ). Then τ˜(M (s) , L) = τ˜(ξ(M ), ξ(L)) = ψ(τ thus also τ˜(L, M (s) ) = 0. Moreover, for d1 , d2 , d3 ∈ D, we have ˆ (d1 , d2 ))(ψ(d3 )) = τ (d1 , d2 )(d3 ), τ˜(ψ(d1 ), ψ(d2 ))(ψ(d3 )) = ψ(τ so τ˜ is the extension of τ (s) to M (s) and τ (s) is a cocycle. The invariance of τ (s) (s) is immediate from the invariance of τ , while the equation τ (s) (D0 , D(s) ) = (s) 0 is immediate from the deﬁnition of τ (s) and ψ(D0 ) = D0 . This shows (s) (s) (s) (D , τ ) ∈ P(L ). It remains to prove the statement about χ. To do this it will be helpful for us to write an element x = d + l + f ∈ E, where d ∈ D, l ∈ L, f ∈ C, as a column vector ⎡ ⎤ d x = ⎣ l ⎦. f Similarly we express elements of E (s) as column vectors. This notation allows us to write linear maps (for example from E to E (s) ) as matrices. Deﬁne ω # : L → C by ω # (l)(d) = (l | ω(d)). We see that for x = d + l + c ∈ E, d ∈ D, l ∈ L, f ∈ C and for d ∈ D, we have 1 ˆ − ω # (l + 1 ω(d)))(ψ(d )), η(x)(ψ(d )) = (f − ω # (l + ω(d)))(d ) = ψ(f 2 2

20

Bruce Allison and John Faulkner

ˆ ) − (ψˆ ◦ ω # )(l) − 1 (ψˆ ◦ ω # ◦ ω)(d). Hence, in matrix form so η(x) = ψ(f 2 ⎡

ψ ⎣ ω χ= 1 ˆ − 2 ψ ◦ ω# ◦ ω Let

0 Id −ψˆ ◦ ω #

⎤ 0 0 ⎦. ψˆ

⎤ 0 0 0 ν = ⎣ω 0 0⎦, 0 −ω # 0 ⎡

so ν : E → E. The deﬁnition of ω # shows that ν is a skew transformation of E. Thus, exp(ν) is an isometry of E. We can write ⎡ ⎤⎡ ⎤ ⎡ ⎤ ψ 0 0 ψ 0 0 Id 0 0 ω Id 0 ⎦ = ⎣ 0 Id 0 ⎦ exp(ν). χ = ⎣ 0 Id 0 ⎦ ⎣ 1 # ˆ − 2 ω ◦ ω −ω # Id 0 0 ψ 0 0 ψˆ ˆ Thus, χ : E → The ﬁrst factor is an isometry E → E (s) by the deﬁnition of ψ. (s) E is an isometry. ˆ ω # , η and χ. Also, ω(D0 ) ⊆ h, so Since ψ and ω are Λ-graded, so are ψ, (s) χ(H) ⊆ ψ(D0 ) ⊕ (ω(D0 ) + h) ⊕ η(E0 ) ⊆ H (s) . But since D0 = ψ(D0 ) and (s) ˆ 0 ), we have dim(H (s) ) = dim(H), so χ(H) = H (s) . C0 = ψ(C It remains to show that χ is a homomorphism of Lie algebras. Let ˆ (m1 , m2 )) σ = ρ−1 (σD(s) ) ∈ Z 2 (M, C). Then, σD(s) (ξ(m1 ), ξ(m2 )) = ψ(σ for m1 , m2 ∈ M , so σ (m1 , m2 )(d) = σD(s) (ξ(m1 ), ξ(m2 ))(ψ(d)) for d ∈ D. We will show that σ = σD +δ(κ) where κ : M → C is the 1-cocycle κ(d + l) =

1 # ω (ω(d)) + ω # (l) 2

and δ(κ) is the coboundary of κ; i.e., δ(κ)(m1 , m2 ) = m2 · κ(m1 ) − m1 · κ(m2 ) − κ([m1 , m2 ]). Indeed, using linearity and skew symmetry, it suﬃces to check this for the pairs (l1 , l2 ), (l1 , d2 ), and (d1 , d2 ). For this recall that L · C = 0, [ω(D), ω(D)] ⊆ [Cent(L)h, Cent(L)h] = 0, and that ω is a derivation of D into L. We have σ (l1 , l2 )(d) = σD(s) (l1 , l2 )(ψ(d)) = (ψ(d)(l1 ) | l2 ) = ((d − adL (ω(d))(l1 ) | l2 ) = (d(l1 ) − [ω(d), l1 ] | l2 ) = (d(l1 ) | l2 ) − (ω(d) | [l1 , l2 ]) = (σD (l1 , l2 ) − ω # ([l1 , l2 ]))(d),

Isotopy for EALAs and Lie Tori

21

while δ(κ)(l1 , l2 ) = l1 · κ(l2 ) − l2 · κ(l1 ) − κ([l1 , l2 ]) = −κ([l1 , l2 ]) = −ω # ([l1 , l2 ])). Also, σ (d1 , d2 )(d) = σD(s) (ψ(d1 ) + ω(d1 ), ψ(d2 ) + ω(d2 ))(ψ(d)) = (ψ(d)(ω(d1 )) | ω(d2 )) = ((d − adL (ω(d))(ω(d1 )) | ω(d2 )) = (d(ω(d1 )) | ω(d2 )) and 2(δ(κ)(d1 , d2 ))(d) = (d1 · ω # (ω(d2 )) − d2 · ω # (ω(d1 )) − ω # (ω([d1 , d2 ])))(d) = −(ω(d2 ) | ω([d1 , d])) + (ω(d1 ) | ω([d2 , d])) − (ω([d1 , d2 ])) | ω(d)) = −(ω(d2 ) | d1 (ω(d)) − d(ω(d1 ))) + (ω(d1 ) | d2 (ω(d)) − d(ω(d2 ))) − (d1 (ω(d2 )) − d2 (ω(d1 )) | ω(d)) = (ω(d2 ) | d(ω(d1 ))) − (ω(d1 ) | d(ω(d2 ))) = 2(d(ω(d1 )) | ω(d2 )). Finally, σ (d1 , l2 )(d) = σD(s) (ψ(d1 ) + ω(d1 ), l2 )(d) = (d · ω(d1 ), l2 ) and 1 (δ(κ)(d1 , l2 ))(d) = (d1 · ω # (l2 ) − l2 · ω # (ω(d1 )) − ω # ([d1 , l2 ]))(d) 2 = −(l2 | ω([d1 , d])) + (l2 | d1 · ω(d)) = (l2 | d · ω(d1 )). So σ = σD + δ(κ) as claimed. Since σ + τ = σD + τ + δ(κ), the map m + f → m + f − κ(m) for m ∈ M , f ∈ C is a Lie algebra isomorphism of E = E(L, D, τ ) with Ext(M, C, σ + τ ). We can write this isomorphism as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Id 0 0 D D ⎣ 0 Id 0 ⎦ : ⎣ L ⎦ → ⎣ L ⎦ . C C − 21 ω # ◦ ω −ω # Id Since σD(s) + τ (s) = ρ(σ + τ ), we also have that ⎡ ⎤ ψ 0 0 ξ ⊕ ψˆ = ⎣ ω Id 0 ⎦ 0 0 ψˆ

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Bruce Allison and John Faulkner

is a Lie algebra isomorphism of Ext(M, C, σ + τ ) with E (s) . Thus, ⎡ ⎤⎡ ⎤ ψ 0 0 Id 0 0 0 Id 0 ⎦ χ = ⎣ ω Id 0 ⎦ ⎣ − 1 ω # ◦ ω −ω # Id 0 0 ψˆ 2

is a Lie algebra isomorphism from E to E (s) .

Corollary 6.3. Let L be a centreless Lie Λ-torus of type ∆, and let L be a centreless Lie Λ -torus of type ∆ . Suppose that L is isotopic to L . Then, there is a bijection (D, τ ) → (D , τ ) from P(L) onto P(L ) such that E(L, D, τ ) is isomorphic as an EALA to E(L , D , τ ) for all (D, τ ) ∈ P(L). Proof. If L and L are bi-isomorphic, the result is clear. So we can assume that L = L(s) is an isotope of L. Deﬁne T : P(L) → P(L ) by T (D, τ ) = (D(s) , τ (s) ) using the notation of Theorem 6.2. Similarly, we have a map T : P(L ) → P(L (−s) ) = P(L), and it is clear that T and T are inverses of one another. So T is a bijection.

Less precisely (but more succinctly), Corollary 6.3 says that if the centerless Lie tori L and L are isotopic, then the families { E(L, D, τ ) }(D,τ )∈P(L) and { E(L , D , τ ) }(D ,τ )∈P(L ) of EALAs are bijectively isomorphic. Using this language, Theorem 4.2, Theorem 6.1, and Corollary 6.3 together tell us that Construction 4.1 provides a one-to-one correspondence between centreless Lie tori up to isotopy and families of EALAs up to bijective isomorphism. Remark 6.4. Suppose that L is a centreless Lie torus. To construct all of the members of the family { E(L, D, τ ) }(D,τ )∈P(L) of EALAs corresponding to L, one must do the following: (i) Calculate the centroid Cent(L) and identify it explicitly with F[Γ ] (in which case SCDer(L) is completely understood from (3)); (ii) Determine the Γ -graded subalgebras D of SCDer(L) which satisfy (4.1a); (iii) For each D in (ii), determine all 2-cocycles of D satisfying (4.1b). We note that (i) can be carried out for each type ∆ using the coordinatization theorems (see §7 below) and the results and methods in [BN, §5]. However, ﬁnding a general approach to (ii) and (iii) seems to be much more diﬃcult. (See [BiN, p.3] for some remarks on problem (iii) and see [BGK, Remark 3.71(b)] for an example of a nontrivial choice of τ .) We do however note that the tasks (ii) and (iii) are independent of the type ∆; in fact they depend only on the rank of Γ .

7 Coordinatization of Lie tori In the classical study of ﬁnite dimensional split simple Lie algebras, one can construct the algebras using special linear, orthogonal and symplectic matrix

Isotopy for EALAs and Lie Tori

23

constructions along with some exceptional constructions starting from Jordan algebras and alternative algebras. (See for example [S, §III.8, IV.3 and IV.4].) Similar coordinatization theorems have been proved for centreless Lie tori. Very roughly speaking these results show that a centerless Lie torus of a given type can be constructed as a “matrix algebra” over a “coordinate torus”, which for some types is nonassociative. References for each type are: A1 : [Y1]; A2 : [BGKN, Y4]; Al (l ≥ 3), Dl (l ≥ 4), E6 , E7 and E8 : [BGK, Y2]; B2 = C2 : [AG, BY]; Bl (l ≥ 3): [AG, Y5, AB]; Cl (l ≥ 3): [AG, BY]; F4 and G2 : [AG]; BC1 [AFY]; BC2 : [F1]; and ﬁnally BCl (l ≥ 3): [AB]. We note that these theorems were in some cases proved before the notion of Lie torus had been introduced and were instead formulated in the language of EALAs. Also, in some cases the theorems were proved over the complex ﬁeld. Nevertheless the translation to the language of Lie tori and the extension to arbitrary base ﬁelds of characteristic 0 are not diﬃcult. In the following sections, we recall the coordinatization theorems for types A1 , A2 , Al (l ≥ 3) and Cl (l ≥ 4), which use respectively Jordan tori, alternative tori, associative tori and associative tori with involution as coordinates. First we recall the deﬁnitions of these types of tori. Recall that a Jordan algebra is an algebra with commutative product satisfying a2 (ab) = a(a2 b), and an alternative algebra is an algebra with product satisfying a(ab) = a2 b and (ba)a = ba2 (see for example [S] or [Mc2]). λ Definition 7.1. Suppose that A = λ∈Λ A is a Λ-graded Jordan algebra, alternative algebra or associative algebra. A is said to be a Jordan, alternative, or associative Λ-torus, respectively if: (i) For λ ∈ suppΛ (A), Aλ is spanned by a single invertible element of A.9 (ii) Λ is generated as a group by suppΛ (A). Note of course that any associative torus is an alternative torus. An associative Λ-torus with involution is a pair (A, ι), where A is an associative Λ-torus and ι is a graded involution of A. Remark 7.2. Suppose that A is a Jordan or alternative Λ-torus, and let S = suppΛ (A). Then, it is clear that −S = S. Moveover, if A is alternative, it is easy to see that S is closed under addition, so S = Λ. However, this is not true in the Jordan case, although S is closed under the operation (λ, µ) → λ + 2µ. Example 7.3. Let q = (qij ) be an n × n-matrix over F with qii = 1 and −1 ±1 qij = qji for all i, j. Let Fq = Fq [x±1 1 , . . . , xn ] be the associative algebra over −1 F presented by the generators xi , xi , i = 1, . . . , n, subject to the relations xi x−1 = x−1 i i xi = 1 and xj xi = qij xi xj , 9

1 ≤ i, j ≤ n.

We are using the standard deﬁnitions of invertibility for Jordan algebras [Mc2, p.210], alternative algebras [S, p.38], and associative algebras.

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Then, Fq has a natural Zn -grading with (Fq )(l1 ,...,ln ) = Fxl11 . . . xlnn . The Zn graded algebra Fq is an associative Zn -torus called the quantum torus determined by q.10 Conversely, it is easy to see that any associative torus is isograded-isomorphic to a quantum torus. Recently, rational quantum tori, that is quantum tori Fq with each qij a root of unity, have been classiﬁed up to isograded-isomorphism by K.-H. Neeb in [Ne, Thm. III.4] as tensor products of rational quantum Z2 -tori and an algebra of Laurent polynomials. (In [H, Thm. 4.8], J.T. Hartwig gave a tensor product decomposition of rational quantum tori but he did not give a condition for isomorphism.) In the following sections, we will give some further examples of the tori deﬁned in Deﬁnition 7.1.

8 Type A1 In this section we assume that ∆ = { α, 0, −α } is the root system of type A1 , and Q = Q(∆) = Zα. To construct a centreless Lie Λ-torus of type ∆, we use the Tits–Kantor– Koecher construction. (See [Mc2, p.13] for the facts mentioned below about this construction.) We begin with a Jordan Λ-torus A and we deﬁne operators Vx,y ∈ End(A) for x, y ∈ A by Vx,y z = {x, y, z} := 2((xy)z + (zy)x − (zx)y). These operators satisfy the identity [Vx,y , Vz,w ] = V{x,y,z},w − Vz,{y,x,w} , so their span VA,A is a Lie algebra under the commutator product. Moreover, the map ∗ deﬁned by Vx,y → −Vy,x is a well-deﬁned automorphism of VA,A . We now set L(A) = TKK(A) := A1 ⊕ VA,A ⊕ A−1 , where, for i = ±1, Ai is a copy of the vector space A under the linear bijection x → xi . Then L(A) is a Lie algebra, called the Tits–Kantor–Koecher Lie algebra of A, under the product deﬁned by [T, x1 ] = (T x)1 , [T, y−1 ] = (T ∗ y)−1 , [xi , yi ] = 0 (i = ±1), [x1 , y−1 ] = Vx,y , [T, T ] = T T − T T, for x, y ∈ A, T, T ∈ VA,A . Finally, L(A) is a Q × Λ-graded algebra with VAµ ,Aν , L(A)λ−α = (Aλ )−1 , L(A)λα = (Aλ )1 , L(A)λ0 = µ+ν=λ

for λ ∈ Λ and L(A)λβ = 0 for λ ∈ Λ, β ∈ Q\∆. Yoshii has shown that L(A) is a centreless Lie Λ-torus of type ∆, and he proved the following coordinatization theorem. 10

If n = 0, we interpret q = ∅ and Fq = F.

Isotopy for EALAs and Lie Tori

25

Theorem 8.1 ([Y1, Thm. 1]). Any centreless Lie Λ-torus of type ∆ is graded-isomorphic to the Lie Λ-torus L(A) = TKK(A) constructed from a Jordan Λ-torus A. Yoshii went on to describe ﬁve families of Jordan Zn -tori and then he showed that every Jordan Λ-torus with rank(Λ) = n is isograded-isomorphic to a torus in one of the families [Y1, Thm. 2]. The simplest of these families + n consists of the tori F+ q , where Fq denotes the Z -graded algebra Fq with 1 multiplication x · y = 2 (xy + yx). Definition 8.2 (Isotopes of Jordan tori). Let A be Jordan Λ-torus and let u be a nonzero homogeneous element of A. So u ∈ A−ρ , where ρ ∈ suppλ (A). Let A(u) be the algebra with underlying vector space A and product ·u deﬁned by 1 x ·u y = {x, u, y}. 2 Then, since u is invertible in A, A(u) is a Jordan algebra and the identity element of A(u) is 1(u) = u−1 [Mc2, p.86]. So 1(u) ∈ Aρ . We endow A(u) with a Λ-grading by setting (A(u) )λ = Aλ+ρ for λ ∈ Λ. It is easily checked that A(u) is a Jordan Λ-torus which we call the u-isotope of the Jordan torus A. It is also easily checked that up to gradedisomorphism A(u) does not depend on the choice of nonzero u in A−ρ . Suppose that u, v are nonzero homogeneous elements of a Jordan Λ-torus. It is well known that A(1) = A,

(A(u) )(v) = A(Uu v)

−2

and (A(u) )(u

)

= A,

(26)

as algebras [Mc2, Proposition 7.2.1], where Uu v = 12 {u, v, u}. It is easy to check that these are also equalities of Λ-graded Jordan algebras. Definition 8.3. Suppose that A is a Jordan Λ-torus and A is a Jordan Λ torus. We say that A is isotopic to A if some isotope of A is isogradedisomorphic to A . In that case, we write A ∼ A . It follows from (26) that ∼ is an equivalence relation on the class of Jordan tori. The following lemma is our key to understanding the connection between isotopy for Lie tori and isotopy for Jordan tori. In fact its proof (see (27) and (28) below) shows that the deﬁnition of isotope for Jordan tori is determined without foreknowledge from the deﬁnition of isotope for Lie tori. We will see the same phenomenon for alternative tori in §9 and for associative tori with involution in §11. Lemma 8.4 (Key lemma for type A1 ). Suppose that A is a Jordan Λ torus, A is an Jordan Λ-torus, and s ∈ Hom(Q, Λ) is admissible for L(A). Suppose that ϕ : L(A ) → L(A)(s) is a bi-isomorphism with ϕr = 1. Then,

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−s(α) ∈ suppΛ (A) and for any nonzero u in A−s(α) there is an isogradedisomorphism η : A → A(u) such that ηgr = ϕe . Proof. Note ﬁrst that since s is admissible, we have s(α) ∈ Λα (L(A)) = suppΛ (A), so −s(α) ∈ suppΛ (A). Thus we can choose nonzero u in A−s(α) as suggested. Let 1 denote the identity in A . Then −s(α)

ϕ(1−1 ) ∈ ϕ(L(A )0−α ) = ((L(A)(s) )0−α = L(A)−α

= (A−s(α) )−1 .

So replacing u by a scalar multiple we can assume that ϕ(1−1 ) = u−1 . Now deﬁne a linear bijection η : A → A by (ηx)1 = ϕ(x1 ) for x ∈ A . Then, for x, y ∈ A , we have (η(xy))1 = ϕ((xy)1 ) = =

1 ϕ({x, 1 , y}) = 12 ϕ([[x1 , 1−1 ], y1 ]) 2

1 [[(ηx)1 , u−1 ], (ηy)1 ] = ( 12 {ηx, u, ηy})1 = (η(x) ·u η(y))1 , 2

(27)

and, for λ ∈ Λ , λ

λ

ϕe (λ) (η(A ))1 = ϕ((A )1 ) = ϕ(L(A )λα ) = (L(s) )α

= (Aϕe (λ)+s(α) )1 = ((A(u) )ϕe (λ) )1 .

(28)

Proposition 8.5. Let A be a Jordan Λ-torus, and suppose that s ∈ Hom(Q, Λ) is admissible for L(A). Then, −s(α) ∈ suppΛ (A) and for any nonzero u in A−s(α) , we have L(A)(s) Q×Λ L(A(u) ). Proof. By Theorem 8.1, we have L(A)(s) Q×Λ L(A ) for some Jordan Λtorus A . So we have a bi-isomorphism ϕ : L(A ) → L(A)(s) with ϕr = 1 and ϕe = 1. Lemma 8.4 now tells us that A Λ A(u) , so L(A ) Q×Λ L(A(u) ). Hence L(A)(s) Q×Λ L(A(u) ).

Our main theorem in this section, gives necessary and suﬃcient conditions for two centreless Lie tori L(A) and L(A ) to be isotopic. To do this, we need to ﬁrst give necessary and suﬃcient conditions for these Lie tori to be biisomorphic. Theorem 8.6. Suppose that A is a Jordan Λ-torus and A is a Jordan Λ torus. Let L(A) = TKK(A) and L(A ) = TKK(A ). Then (i) L(A) bi L(A ) ⇐⇒ A ig A . (ii) L(A) ∼ L(A ) ⇐⇒ A ∼ A .

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27

Proof. Let L = L(A) and L = L(A ). (i): The proof of “⇐” is clear, so we prove “⇒”. Suppose ϕ : L → L is a bi-isomorphism. Then ϕr = ±1, so ϕr is in the Weyl group of ∆. Therefore, by Lemma 3.8, we can assume that ϕr = 1. Thus, by Lemma 8.4 with s = 0 and u = 1, we have A ig A. (ii): “⇒” By assumption, L bi L(s) for some s ∈ Hom(Q, Λ) that is admissible for L. So, choosing u as in Proposition 8.5, we have L bi L(A(u) ). By (i), A ig A(u) . “⇐” By (i) we can assume that A = A(u) , where u is a nonzero homogeneous element of A. Then u ∈ A−ρ , where ρ ∈ suppΛ L = Λα (L). So s : Q → Λ deﬁned by s(α) = ρ is admissible for L, and hence L bi L(s) by Proposition 8.5.

We will next give an example of two centreless Lie tori that are isotopic but not bi-isomorphic. For this we need a little preparation. Remark 8.7. Suppose that A is a Jordan Λ-torus, and let Cent(A) be the centroid of A. It is clear that Cent(A) is Λ-graded and that the support Γ = Γ (A) of Cent(A) is a subgroup of Λ [Y1, §3]. Also, if S = suppΛ (A), it is clear that S is a union of cosets of Γ in Λ, and we let S/Γ = { λ+Γ | λ ∈ S } ⊆ Λ/Γ . If S/Γ is ﬁnite, we denote the sum of the elements of S/Γ in Λ/Γ by Σ(S/Γ ). Lemma 8.8. Suppose A is a Jordan Λ-torus with S/Γ ﬁnite and Σ(S/Γ ) = 0, where Γ = Γ (A) and S = suppΛ (A). If ρ ∈ S with |S/Γ | ρ ∈ / Γ and 0 = u ∈ A−ρ , then A(u) ig A. Proof. Let Γ (u) = Γ (A(u) ) and S (u) = suppΛ (A(u) ). It is easily checked that Cent(A(u) ) = Cent(A), so Γ (u) = Γ . Also, S (u) = S − ρ, so Σ(S (u) /Γ ) = Σ(S/Γ ) − |S/Γ | (ρ + Γ ) = − |S/Γ | (ρ + Γ ) = 0. But then Σ(S/Γ ) = 0 and Σ(S (u) /Γ ) = 0 imply our conclusion.

Our example uses Jordan tori of Cliﬀord type [Y1, Example 5.2] as coordinate algebras. Example 8.9. Let Λ = Z3 and let R = F[2Λ] be the group algebra of 2Λ with its natural grading by 2Λ (and hence by Λ). Let { λ1 , λ2 , λ3 } be the 4 standard basis of Λ and let λ4 = λ1 + λ2 + λ3 . Let V = i=1 Rxi be the free R-module with base { x1 , x2 , x3 , x4 }, and give V the Λ-grading such that deg(tµ xi ) = µ + λi for µ ∈ 2Λ and 1 ≤ i ≤ 4. Deﬁne a Λ-graded R-bilinear form f : V × V → R by f (xi , xj ) = δi,j t2λi for 1 ≤ i, j ≤ 4. Finally, let A = R ⊕ V with the direct sum Λ-grading, and deﬁne a product in A by (r + v)(r + v ) = rr + f (v, v ) + rv + r v for r, r ∈ R, v, v ∈ V . Then A is a Jordan torus. (The algebra A is called a spin factor in [Mc2, p.74].) Further, using the notation of Lemma 8.8, Γ = 2Λ and S = (2Λ) ∪ (∪4i=1 (2Λ + λi )), so S/Γ is ﬁnite and Σ(S/Γ ) = 0. Also |S/Γ | ρ = 5ρ ∈ / Γ for ρ ∈ S \ Γ . So for any nonzero homogeneous element u in A \ R, we have, by Lemma 8.8, that A ig A(u) . Therefore, for any such u, we see by Theorem 8.6 that L(A) bi L(A(u) ) but L(A) ∼ L(A(u) ).

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9 Type A2 Suppose in this section that ∆ = { 0 } ∪ { εi − εj | 1 ≤ i = j ≤ 3 } is the root system of type A2 (where ε1 , ε2 , ε3 is a basis for a space containing ∆), and Q = Q(∆). We let α1 = ε1 − ε2 and α2 = ε2 − ε3 , in which case { α1 , α2 } is a base for the root system ∆. Let A be an alternative Λ-torus. We construct a centreless Lie Λ-torus of type ∆ from A using J. Faulkner’s 3 × 3-projective elementary construction. (See [F1, Appendix, (A6)] for the properties mentioned below about this construction. Note however that we are using the notation eij (x) in place of the notation vij (x) that is used in [F1].) Recall that the 3×3 projective elementary Lie algebra is the Lie algebra L(A) = pe3 (A) := L0 ⊕ ( 1≤i=j≤3 eij (A)), (29) where L0 and eij (A) are subspaces of L satisfying: (a) (b) (c) (d) (e) (f) (g)

For i = j, there is a linear bijection x → eij (x) from A to eij (A), [eij (x), ejk (y)] = eik (xy) for x, y ∈ A and distinct i, j, k, [eij (x), eij (y)] = 0 for x, y ∈ A, i = j, [L0 , L0 ] ⊆ L0 , [L0 , eij (A)] ⊆ eij (A) for i = j, L0 = 1≤i<j≤3 [eij (A), eji (A)], { x ∈ L0 : [x, eij (A)] = 0 for i = j } = 0.

(30) These properties characterize the Lie algebra L(A) in the following sense: If L (A) = L0 ⊕ ( 1≤i=j≤3 eij (A)) is another Lie algebra satisfying the conditions in (30), then there is a unique isomorphism from L(A) → L (A) such that eij (x) → eij (x) for i = j, x ∈ A. Finally L(A) is a Q × Λ-graded algebra with L(A)λεi −εj = eij (Aλ ), L(A)λ0 = µ+ν=λ 1≤k
A diﬀerent description of L(A) was used in [BGKN] and [Y4], but it is not diﬃcult to check that the two descriptions give Lie tori that are Q × Λ-graded-isomorphic.

Isotopy for EALAs and Lie Tori

29

An important example of an alternative torus is the octonion torus. A simple way to introduce this torus uses a presentation [Y7]. Example 9.2. Let C(3) be the alternative algebra that is presented by the −1 generators x±1 = x−1 i , 1 ≤ i ≤ 3, subject to the relations xi xi i xi = 1 for all i, xi xj = −xj xi for i = j,

and (x1 x2 )x3 = −x1 (x2 x3 ).

Although we will not need this, one can show more concretely that each element of C(3) can be written uniquely as a linear combination of monomials xi = (xi11 xi22 )xi33 , i = (i1 , i2 , i3 ) ∈ Z3 , with multiplication given by xi xj = (−1)κ(i,j) xi+j , where κ(i, j) = i3 j1 + i2 j1 + i3 j2 + i1 j2 j3 + i2 j1 j3 + i3 j1 j2 . ±1 n If n ≥ 3, the graded algebra C(3) ⊗ F[x±1 4 , . . . , xn ] is an alternative Z torus, where x1 , . . . , xn are assigned the degrees in the standard ordered ba±1 sis for Zn (identifying C(3) and F[x±1 4 , . . . , xn ] as subalgebras of the tensor ±1 n product). The Z -graded algebra C(3) ⊗ F[x4 , . . . , x±1 n ] is called the octonion Zn -torus. We have the following description of alternative tori: Theorem 9.3. ([BGKN, Thm. 1.25], [Y4, Cor. 5.13]) If A is an alternative Λ-torus, then either A is associative (and hence isograded-isomorphic to a quantum torus), or n ≥ 3 and A is isograded-isomorphic to the octonion Zn torus. Remark 9.4. Suppose A is a Λ-torus. Recall that by Remark 7.2, supp(A) = Λ. So Λα (L(A)) = Λ for all nonzero α ∈ ∆, and therefore any s ∈ Hom(Q, Λ) is admissible for L(A). Definition 9.5 (Isotopes of alternative tori). Let A be an alternative Λtorus and let u1 and u2 be nonzero homogeneous elements of A. Thus, ui ∈ A−ρi , where ρi ∈ Λ for i = 1, 2. Let A(u1 ,u2 ) be the algebra with underlying vector space A and with product ·u1 ,u2 deﬁned by: x ·u1 ,u2 y = (xu1 )(u2 y) for x, y ∈ A. Then, since u1 , u2 are invertible, A(u1 ,u2 ) is an alternative algebra with identity 1(u1 ,u2 ) = (u1 u2 )−1 [Mc1, Thm. 1 and Prop. 2]. We deﬁne a Λgrading on A(u1 ,u2 ) by setting (A(u1 ,u2 ) )λ = Aλ+ρ1 +ρ2 for λ ∈ Λ. Then A(u1 ,u2 ) is an alternative Λ-torus which we call the (u1 , u2 )isotope of A . It is easily checked that this torus does not depend up to gradedisomorphism on the choice of nonzero ui in A−ρi .

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Lemma 9.6 (Key lemma for type A2 ). Suppose that A is an alternative Λ -torus, A is an alternative Λ-torus, and s ∈ Hom(Q, Λ). Suppose that ϕ : L(A ) → L(A)(s) is a bi-isomorphism with ϕr = 1. Choose 0 = ui ∈ As(αi ) for i = 1, 2. Then there is an isograded-isomorphism η : A → A(u1 ,u2 ) such that ηgr = ϕe . Proof. Write L(A ) = L0 ⊕ ( 1≤i=j≤3 eij (A )) as in (29). Now ϕ(eij (Aλ )) = ϕ (λ)+s(ε −ε )

ϕ (λ)

i j = eij (Aϕe (λ)+s(εi −εj ) ) ϕ(L(A )λεi −εj ) = (L(A)(s) )εie−εj = L(A)εie−εj for λ ∈ Λ , i = j. Thus, for i = j, we have a linear bijection ϕij : A → A with

ϕ(eij (x)) = eij (ϕij (x))

(31)

for x ∈ A , in which case λ

ϕij (A ) = Aϕe (λ)+s(εi −εj )

(32)

for λ ∈ Λ . Using (31) and the relation (30)(b) in L(A ) and L(A), we see that ϕij (xy) = ϕik (x)ϕkj (y)

(33)

for x, y ∈ A and i, j, k =. Now let uij = ϕij (1 ) for i = j, where 1 is the identity in A . Then, taking y = 1 and separately x = 1 in (33), we have ϕij (x) = ϕik (x)ukj and ϕij (y) = uik ϕkj (y) for x, y ∈ A and i, j, k =. So ϕij (xy) = ϕik (x)ϕkj (y) = (ϕij (x)ujk )(uki ϕij (y)) = ϕij (x) ·ujk ,uki ϕij (y) for x, y ∈ A and i, j, k =. Also, by (32), u12 ∈ As(α1 ) and u23 ∈ As(α2 ) . So replacing u1 and u2 by scalar multiples, we may assume that u1 = u12 and u2 = u23 . Thus, ϕ31 is an algebra isomorphism of A onto A(u1 ,u2 ) . Moreover, for λ ∈ Λ , ϕ31 ((A )λ ) = Aϕe (λ)+s(ε3 −ε1 ) = Aϕe (λ)−s(α1 )−s(α2 ) = (A(u1 ,u2 ) )ϕe (λ) . So we have our conclusion with η = ϕ31 .

Proposition 9.7. Suppose A is an alternative Λ-torus and s ∈ Hom(Q, Λ). Choose 0 = ui ∈ As(αi ) for i = 1, 2. Then, L(A)(s) Q×Λ L(A(u1 ,u2 ) ). Proof. This follows from Lemma 9.6 as in §8. (See the proof of Proposition 8.4.)

We could introduce isotopy for alternative tori in the obvious fashion (as in Deﬁnition 8.3 in the Jordan case). However, in view of the following result, this will not be necessary. Proposition 9.8. Suppose that A is an alternative Λ-torus and u1 , u2 are nonzero homogeneous elements of A. Then A(u1 ,u2 ) ig A.

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Proof. Let A = A(u1 ,u2 ) . Suppose ﬁrst that A is associative. Then, one checks that the map x → (u1 u2 )−1 x is a graded algebra isomorphism of A onto A . So we can assume that A is not associative. Thus, A is also not associative [Mc1, p.256–257]. Therefore, by Theorem 9.3, A and A are both isogradedisomorphic to the octonion Zn -torus. So A ig A .

Remark 9.9. With the assumptions of Proposition 9.8 one can actually show that A(u1 ,u2 ) Λ A, but isograded isomorphism is all that we need. If A is an alternative Λ-torus, then the opposite torus of A is the alternative Λ-torus Aop , where Aop = A as graded vector spaces and the product on Aop is given by x ·op y = yx. Lemma 9.10. Suppose that A is an alternative Λ-torus. Then, there exists a unique bi-isomorphism from L(A) onto L(Aop ) such that eji (x) eij (x) → −˜ ˜ 0 ⊕ ( for x ∈ A and i = j, where L(Aop ) = L ˜ij (Aop )) as in (29). 1≤i=j≤3 e ˜ 0. eij (x) = −˜ eji (x) for x ∈ A, i = j, and put L0 = L Proof. In L(Aop ), put op Then, L(A ) = L0 ⊕ ( 1≤i=j≤3 eij (Aop )), and for x, y ∈ A and i, j, k =, we have [eij (x), ejk (y)] = [˜ eji (x), e˜kj (y)] = −[˜ ekj (y), e˜ji (x)] = −˜ eki (y ·op x) = eik (xy). The other properties in (30) are clear, so we have a unique Lie algebra isomorphism ϕ : L(A) → L(Aop ) such that ϕ(eij (x)) = eij (x) = −˜ eji (x) for x ∈ A and i = j. It is clear that ϕ is a bi-isomorphism with ϕr = −1 and ϕe = 1. The main result of this section is: Theorem 9.11. Suppose that A is an alternative Λ-torus and A is an alternative Λ -torus. Let L(A) = pe3 (A) and L(A ) = pe3 (A ). Then, the following are equivalent:12 (a) A is isograded-isomorphic to A or (A )op . (b) L(A) bi L(A ). (c) L(A) ∼ L(A ). Proof. “(a)⇒(b)” Since L(A ) bi L(A ) by Lemma 9.10, we can assume that A ig A . (b) is then clear. “(b)⇒(c)” is trivial. “(c)⇒(a)” By assumption, L(A ) bi L(A)(s) for some s ∈ Hom(Q, Λ) that is admissible for L(A). Choose 0 = ui ∈ As(αi ) for i = 1, 2. Then, by Propositions 9.7 and 9.8, we have L(A)(s) bi L(A(u1 ,u2 ) ) bi L(A). So there op

12

See [GN, Lemma 1.4] for a result related to Theorems 9.11 and 10.6 in the context of Jordan pairs covered by grids.

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is a bi-isomorphism ϕ : L(A ) → L(A). Now the automorphism group of ∆ is generated by −1 and the Weyl group of ∆. Hence, post-multiplying ϕ by the bi-isomorphism in Lemma 9.10 if necessary, we can assume that ϕr is in the Weyl group. Thus, by Lemma 3.8, we can assume that ϕr = 1. So, by Lemma 9.6 (with s = 0 and u1 = u2 = 1), we have A ig A.

Remark 9.12. If A is an alternative torus, it is not in general true that A is isograded-isomorphic to Aop . However, one can show that this is true if A is the octonion torus (since the octonion torus has an involution) and if A is a rational quantum torus (using the classiﬁcation in [Ne]).

10 Type Ar , r ≥ 3 Suppose in this section that r ≥ 3, ∆ = { 0 } ∪ { εi − εj | 1 ≤ i = j ≤ r + 1 } is the root system of type Ar (where ε1 , . . . , εr is a basis for a space containing ∆), and Q = Q(∆). We let αi = εi − εi+1 for 1 ≤ i ≤ r, in which case { α1 , . . . , αr } is a base for the root system ∆. Let A be an associative Λ-torus, and denote by [A, A] the Lie algebra spanned by commutators in A. Let L(A) = slr+1 (A) := { X ∈ Matr+1 (A) | tr(X) ∈ [A, A] }. Then L(A) is a Lie algebra under the commutator product called the (r + 1) × (r + 1)-special linear Lie algebra over A. For 1 ≤ i, j ≤ r + 1, we let eij be the (i, j)-matrix unit in Matr+1 (A) with 1 in the (i, j)-position and zeros elsewhere, and we let eij (x) = xeij for x ∈ A. Then L(A) is a Q × Λ-graded algebra with r+1 λ L(A)λεi −εj = eij (Aλ ), L(A)λ0 = { r+1 i=1 eii (xi ) | xi ∈ A , i=1 xi ∈ [A, A] }, for i = j, λ ∈ Λ, and L(A)λα = 0 for λ ∈ Λ, α ∈ Q \ ∆. It is easy to check that the matrices eij (x), x ∈ A, i = j, generate the Lie algebra L(A) and also, using the fact that the centre of A intersects trivially with [A, A] [BGK, Prop. 2.44], that L(A) is centreless.13 It then follows easily that L(A) is a centreless Lie Λ-torus of type ∆. Moreover, Berman, Gao, and Krylyuk (over C) and Yoshii (in general) proved the following coordinatization theorem: Theorem 10.1. ([BGK, Thm. 2.65], [Y2, §4]) Any centreless Lie Λ-torus of type ∆ is graded-isomorphic to the Lie Λ-torus L(A) = slr+1 (A) constructed from an associative Λ-torus A. 13

In view of these properties, it is natural to also use the notation per+1 (A) for L(A), which is consistent with the notation in rank 2 (see Remark 10.2).

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Remark 10.2. Exactly as above one can deﬁne centreless Lie tori sl2 (A) and sl3 (A) of type A1 and A2 respectively. Then it is not diﬃcult to show that sl2 (A) Q×Λ TKK(A+ ) and sl3 (A) Q×Λ pe3 (A). However, as we have seen, these constructions do not provide all examples of centreless Lie tori of type A1 or A2 . To understand isotopes of Lie tori of type ∆, we could proceed as in rank 1 and 2 to relate them to isotopes of the coordinate tori. However, we can take a simpler approach in view of the next proposition.14 Proposition 10.3. Suppose A is an associative Λ-torus and s ∈ Hom(Q, Λ). (Note that as in rank 2 any s ∈ Hom(Q, Λ) is admissible for L(A).) Then, L(A)(s) Q×Λ L(A). r+1 Proof. Let Q = i=1 Zεi . Then Q = Zε1 ⊕ Q, so we can extend s to a group homomorphism s : Q → Λ by setting s(ε1 ) = 0. Choose 0 = di ∈ As(εi ) for 1 ≤ i ≤ r + 1. Let d = diag(d1 , . . . , dr+1 ) and deﬁne ϕ : L(A) → L(A) by ϕ(X) = dXd−1 for X ∈ L(A). (Here we are using the fact that tr(X1 X2 ) ≡ tr(X2 X1 ) (mod [A, A]) for X1 , X2 ∈ Matr+1 (A), so ϕ maps L(A) into L(A).) Then, ϕ is a Lie algebra isomorphism. Also λ+s(εi )−s(εj ) ϕ(L(A)λεi −εj ) = ϕ(eij (Lλ )) = eij (di Aλ d−1 ) j ) = eij (A λ+s(ε −εj )

= L(A)εi −εj i

= (L(A)(s) )λεi −εj

for λ ∈ Λ; so, by (LT3), ϕ(L(A)λα ) = (L(A)(s) )λα for α ∈ Q, λ ∈ Λ.

The “key lemma” that we need is simpler than in rank 1 and 2 because of Proposition 10.3. Lemma 10.4 (Key lemma for type Ar , r ≥ 3). Suppose that A is an associative Λ -torus and A is an associative Λ-torus. Suppose that ϕ : L(A ) → L(A) is a bi-isomorphism with ϕr = 1. Then, there is an isogradedisomorphism η : A → A such that ηgr = ϕe . Proof. By Lemma 3.8, we can assume that ϕ(eij (1 )) = eij (1) for i − j = ±1, and hence for all i = j. Also, we have the relation [eij (x), ejk (y)] = eik (xy) in L(A) for x, y ∈ A and i, j, k =. We can thus argue as in the proof of Lemma 9.6 (with s = 0 and u1 = u2 = 1).

14

This proposition could also be proved at the level of coordinates using the fact that any isotope of an associative algebra A is isomorphic to A.

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The following is easily checked: Lemma 10.5. Suppose that A is an associative Λ-torus. Then, the map T → −T t is a bi-isomorphism from L(A) onto L(Aop ). Theorem 10.6. Suppose that A is an associative Λ-torus and A is an associative Λ -torus. Let L(A) = slr+1 (A) and L(A ) = slr+1 (A ). Then, the following are equivalent: (a) A is isograded-isomorphic to A or (A )op . (b) L(A) bi L(A ). (c) L(A) ∼ L(A ). Proof. If (a) holds then, by Lemma 10.5, we can assume that A ig A , so (b) follows. The implication “(b)⇒(c)” is trivial. Finally, suppose that (c) holds. Thus, L(A)(s) bi L(A ) for some s ∈ Hom(Q, Λ), so L(A) bi L(A ) by Proposition 10.3. We can now obtain (a) using the argument in the proof of Theorem 9.11 (and using Lemmas 10.4 and 10.5).

11 Type Cr , r ≥ 4 Suppose in this section that r ≥ 4, ∆ = { 0 } ∪ { εi − εj | 1 ≤ i = j ≤ r }{ ±(εi + εj ) | 1 ≤ i ≤ j ≤ r } is the root system of type Cr (where ε1 , . . . , εr is a basis for the space spanned by ∆), and Q = Q(∆). We let αi = εi − εi+1 , 1 ≤ i ≤ r − 1, and αr = 2εr , in which case { α1 , . . . , αr } is a base for the root system ∆. Let (A, ι) be an associative Λ-torus with involution. For convenience, write ¯ = (¯ x ¯ = ι(x) for x ∈ A and X xij ) for X = (xij ) ∈ Matr (A). We write A± = A±, ι := { x ∈ A | x ¯ = ±x }, in which case A = A+ ⊕ A− . Let

X Y L(A, ι) = ssp2r (A, ι) := { ¯ t | X, Y, Z ∈ Matr (A), Z −X Y¯ t = Y, Z¯ t = Z, tr(X) ∈ A+ + [A, A] }. Then L(A, ι) is a Lie algebra under the commutator product called the (2r) × (2r)-special symplectic Lie algebra over A. To deﬁne a grading on L(A, ι), let ¯ej+r,i+r if 1 ≤ i = j ≤ r, and

ij (x) = xei,j − x ¯ej,i+r , nij (x) = xei+r,j + x ¯ej+r,i if 1 ≤ i, j ≤ r, mij (x) = xei,j+r + x where eij is the (i, j)-matrix unit. With this notation, L(A, ι) is a Q×Λ-graded algebra with L(A, ι)λεi −εj = ij (Aλ )

for 1 ≤ i = j ≤ r,

L(A, ι)λεi +εj = mij (Aλ )

for 1 ≤ i, j ≤ r,

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35

L(A, ι)λ−(εi +εj ) = nij (Aλ ) for 1 ≤ i, j ≤ r, r r xi )) | xi ∈ Aλ , L(A, ι)λ0 = { i=1 (ei,i (xi ) − ei+r,i+r (¯ i=1 xi ∈ A+ + [A, A] } for λ ∈ Λ, and L(A, ι)λα = 0 for λ ∈ Λ, α ∈ Q \ ∆. Then L(A, ι) is a centreless Lie Λ-torus of type ∆. Moreover, Allison and Gao (over C) and Benkart and Yoshii (in general) proved: Theorem 11.1. ([AG, Prop. 4.87], [BY, Theorem 5.9]) Any centreless Lie Λtorus of type ∆ is graded-isomorphic to the Lie Λ-torus L(A, ι) = ssp2r (A, ι) constructed from an associative Λ-torus with involution (A, ι).15 Remark 11.2. Exactly as above one can deﬁne centreless Lie tori ssp4 (A, ι) and ssp6 (A, ι). However, these constructions do not provide all examples of Lie tori of type C2 = B2 or C3 [AG, Thm. 4.87]. Example 11.3. Let q = (qij ) ∈ Matn (F) with qii = 1 and qij = qji = ±1 for ±1 all i, j, and let Fq = Fq [x±1 1 , . . . , xn ] be the quantum torus corresponding to q as in Example 7.3. Suppose e = (e1 , . . . , en ) ∈ Fn with ei = ±1. Then, there is a unique involution ιe on Fq such that ιe (xi ) = ei xi for 1 ≤ i ≤ n; and (Fq , ιe ) is an associative Zn -torus with involution.16 It is not diﬃcult to show that any associative torus with involution (A, ι) is isograded-isomorphic to some (Fq , ιe ). More speciﬁcally, if we choose any basis { λ1 , . . . , λn } for Λ and choose 0 = yi ∈ Aλi for each i, then there is an isograded-isomorphism of (A, ι) onto (Fq , ιe ) so that yi → xi for each i, where q and e are determined by the equalities yj yi = qij yi yj and ι(yi ) = ei yi [AG, p.163–164]. Yoshii in [Y3, Thm. 2.7] classiﬁed the associative tori with involution (Fq , ιe ) up to isograded-isomorphism using successive changes of variables that transform the pair (q, e) into a canonical form. Alternatively, as we now see, one can classify associative tori with involution using quadratic forms following an approach discussed in [AFY, Remark 5.20]. ¯ Let Λ¯ = Λ/2Λ regarded as a vector space over Z2 = Z/2Z, and let λ → λ ¯ Recall that a map κ : Λ¯ → Z2 is called be the canonical map of Λ onto Λ. ¯ µ a quadratic form on Λ¯ if the map κp : Λ¯ × Λ¯ → Z2 deﬁned by κp (λ, ¯) = ¯+µ ¯ + κ(¯ κ(λ ¯) + κ(λ) µ) is biadditive. In that case, κp is called the polarization ¯ = a2 κ(λ) ¯ for a ∈ Z2 and that κp is of κ. (The usual assumptions that κ(aλ) Z2 -bilinear are automatic.) If Λ has basis { λ1 , . . . , λn }, then the quadratic forms on Λ¯ are precisely the maps κ : Λ¯ → Z2 of the form n n ¯ li λi ) = li b i + κ( i=1 15 16

i=1

li lj aij ,

(34)

1≤i≤j≤n

L(A, ι) was described diﬀerently in [AG] and also in [BY]. It is not diﬃcult to check that the three descriptions give Lie tori that are Q × Λ-graded-isomorphic. If n = 0, we interpret (Fq , ιe ) as (F, id).

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where (bi ) ∈ Zn2 and (aij ) is an alternating matrix in Matn (Z2 ) (that is aij = aji and aii = 0). Proposition 11.4. (i) If (A, ι) is an associative Λ-torus with involution, then there is a unique quadratic form κ : Λ¯ → Z2 , which we call the mod-2 quadratic form for (A, ι), such that ¯

ι(xλ ) = (−1)κ(λ) xλ

(35)

for xλ ∈ Aλ , λ ∈ Λ. In that case, we also have ¯

xλ xµ = (−1)κp (λ,¯µ) xµ xλ

(36)

for xλ ∈ Aλ , xµ ∈ Aµ , λ, µ ∈ Λ. (ii) Any quadratic form κ on Λ¯ is the mod-2 quadratic form for some associative Λ-torus with involution. In fact, if Λ = Zn with standard basis { λ1 , . . . , λn }, κ is given by (34) and we deﬁne e and q by ei = (−1)bi and qij = (−1)aij , then κ is the mod-2 quadratic form for (Fq , ιe ). (iii) If (A, ι) is an associative Λ-torus with involution and (A , ι ) is an associative Λ -torus with involution, then (A, ι) ig (A , ι ) if and only the corresponding mod-2 quadratic forms κ and κ are isometric, which means ¯ = κ(λ) ¯ that there is an Z2 -linear isomorphism τ¯ : Λ¯ → Λ¯ such that κ (¯ τ (λ)) ¯ ¯ for λ ∈ Λ. Proof. We may assume that Λ = Zn with standard basis { λ1 , . . . , λn }, and in (iii) that Λ = Zn . (i) The uniqueness of κ is clear. For the existence we may assume that (A, ι) = (Fq , ιe ) as in Example 11.3. Choose (bi ) ∈ Zn2 and alternating (aij ) ∈ Matn (Z2 ) such that ei = (−1)bi and qij = (−1)aij . Then an easy computation shows that (35) holds with κ given by (34). Also, if xλ ∈ Aλ , xµ ∈ Aµ , ¯ ¯ λ, µ ∈ Λ, then xλ xµ = ι(ι(xµ )ι(xλ )) = (−1)κ(λ+¯µ)+κ(λ)+κ(¯µ) xµ xλ , so (36) holds. (ii) It is enough to prove the second statement which follows from the proof of (i). (iii) We may assume that (A, ι) = (Fq , ιe ) as in Example 11.3. For “⇒”, let η : (A, ι) → (A , ι ) be an isograded-isomorphism and set τ := ηgr ∈ τ (λ) ¯ for λ ∈ Λ. Thus, GL(Λ). Then, η(Aλ ) = A and so κ (τ (λ)) = κ(λ) ¯ the map τ¯ ∈ GL(Λ) induced by τ is an isometry of κ onto κ . For “⇐”, ¯ be an isometry of κ onto κ . Then τ¯ is induced by some let τ¯ ∈ GL(Λ) τ ∈ GL(Λ). (This follows from the fact that any element of GLn (Z2 ) is the λ product of elementary matrices.) Let λi = τ (λi ) and 0 = xi ∈ A i for each ¯ ¯ i. Then, xj xi = (−1)κp (λi ,λj ) xi xj = (−1)κp (λi ,λj ) xi xj = qij xi xj and ι (xi ) =

¯

(−1)κ (λi ) xi = (−1)κ(λi ) xi = ei xi , so (A , ι ) ig (Fq , ιe ) (see Example 11.3).

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37

Finite dimensional quadratic forms over Z2 have been classiﬁed up to isometry [D, Chap. I, §16], and hence Proposition 11.4 gives a corresponding classiﬁcation of associative Λ-tori with involution. Remark 11.5. Suppose that (A, ι) is an associative Λ-torus with involution with A+ = A+, ι , and set Λ+ = Λ+ (A, ι) := { λ ∈ Λ | Aλ ⊆ A+ }. If 1 ≤ i = j ≤ r, then ij (A), mij (A), and nij (A) are subspaces of L(A, ι) and the maps x → ij (x), x → mij (x) and x → nij (x) are injections from A onto

ij (A), mij (A), and nij (A) respectively. On the other hand if 1 ≤ i ≤ r, we have mii (x) = mii (¯ x) and nii (x) = nii (¯ x) in L(A, ι) for x ∈ A, and the maps h → mii (h) and h → nii (h) are injections of A+ onto mii (A+ ) = mii (A) and nii (A+ ) = nii (A), respectively. Thus, if α ∈ ∆× , we have Λ, if α is short; Λα (L(A, ι)) = Λ+ , if α is long. Hence, if s ∈ Hom(Q, Λ), then s is admissible for L(A, ι) if and only if s(αr ) ∈ Λ+ . If (A, ι) is an associative Λ-torus with involution, elements of A+, ι are called hermitian elements of (A, ι). To form an isotope of (A, ι) along the lines of previous cases, we could a deﬁne new product (x, y) → xhy on A, where h is a nonzero homogeneous hermitian element of (A, ι), shift the grading, but keep the same involution. Following [Mc2, §I.3.4], an equivalent but simpler approach is to keep the product and grading and modify the involution. Definition 11.6 (Isotopes of involutions). Suppose that (A, ι) is an associative Λ-torus with involution. Let h be a nonzero homogeneous hermitian element of (A, ι), and deﬁne ι(h) : A → A by ι(h) (x) = h¯ xh−1 for x ∈ A. Then, (A, ι(h) ) is an associative Λ-torus with involution. (The multiplication and grading on A are unchanged.) We call ι(h) the h-isotope of ι. Lemma 11.7 (Key lemma for type Cr ). Suppose (A , ι ) is an associative Λ -torus with involution, (A, ι) is an associative Λ-torus with involution, and s ∈ Hom(Q, Λ) is admissible for L(A, ι). Suppose that ϕ : L(A , ι ) → L(A, ι)(s) is a bi-isomorphism with ϕr = 1. Choose 0 = h ∈ As(αr ) . Then there is an isograded-isomorphism η : (A , ι ) → (A, ι(h) ) such that ηgr = ϕe ∈ GL(Λ). Proof. In the argument i, j, k will always denote integers with 1 ≤ i, j, k ≤ r. As in the proof of Lemma 9.6, we see that ϕ( ij (Aλ )) = ij (Aϕe (λ)+s(εi −εj ) ) and ϕ(mij (Aλ )) = mij (Aϕe (λ)+s(εi +εj ) ) for λ ∈ Λ and i = j. So we may deﬁne

38

Bruce Allison and John Faulkner

linear bijections ϕij : A → A and ψij : A → A for i = jsuch that ϕ( ij (x)) = ij (ϕij (x)),

ϕ(mij (x)) = mij (ψij (x))

for x ∈ A , and λ

ϕij (A ) = Aϕe (λ)+s(εi −εj ) ,

λ

ψij (A ) = Aϕe (λ)+s(εi +εj )

for λ ∈ Λ. Next, direct multiplication in L(A, ι) gives the following identities: [ ik (x), kj (y)] = ij (xy)

and [ ij (x), [ ji (y), mij (z)]] = mij (x(yz + yz))

for x, y, z ∈ A, i, j, k =. We have similar equations in L(A , ι ), and applying ϕ to these yields ϕij (xy) = ϕik (x)ϕkj (y), ψij (x(yz + yz)) = ϕij (x)(ϕji (y)ψij (z) + ϕji (y)ψij (z) )

(37) (38)

for x, y, z ∈ A , i, j, k =. We let uij = ϕij (1 ) ∈ As(εi −εj ) and vij = ψij (1 ) ∈ As(εi +εj ) for i = j. Setting x = y = 1 in (37), we get uij = uik ukj for i, j, k =. Therefore, −1 −1 −1 uij uji = u−1 ki uki uij uji = uki ukj uji = uki uki = 1, so uij = uji for i = j. As in the proof of Lemma 9.6, it follows from (37) that we have ϕij (xy) = (ϕij (x)ujk )(uki ϕij (y)) for x, y ∈ A , i, j, k =. So ϕij (xy) = ϕij (x)uji ϕij (y) for i = j. Thus, we see that χij : A → A, deﬁned by χij (x) = uji ϕij (x),

(39)

is an algebra isomorphism of A onto A for i = j. Also, χij (Aλ ) = uji ϕji (Aλ ) = uji Aϕe (λ)+s(εi −εi ) ⊆ Aϕe (λ) for λ ∈ Λ, so χij is isograded with (χij )gr = ϕe . If we put x = y = 1 in (38), we get ψij (z + z¯) = uij (uji ψij (z)+uji ψij (z) ). Since uij uji = 1, this yields z ) = uij ψij (z) u¯ji ψij (¯

(40)

for z ∈ A , i = j. Also, if we take y = z = 1 in (38), we get ψij (2x) = ϕij (x)(uji vij + uji vij )

(41)

for x ∈ A , i = j. Next ur1 v1,r ∈ As(εr −ε1 ) As(εr +ε1 ) ⊆ As(2εr ) . So replacing h by a scalar multiple, we can assume that h = ur1 v1,r . Since s is admissible for L(A, ι), ¯ = h, so (41) tells us that we have h ψ1r (x) = ϕ1r (x)h

(42)

Isotopy for EALAs and Lie Tori

39

for x ∈ A . Finally, using (39), (40) and (42), we have x) = ur1 ϕ1r (¯ x) = ur1 ψ1r (¯ x)h−1 = ur1 u1r ψ1r (x) u ¯r1 h−1 χ1r (¯ = ψ1r (x) u ¯r1 h−1 = hϕ1r (x) u ¯r1 h−1 = hχ1r (x) h−1 for x ∈ A . Thus, χ1r is an isograded isomorphism of (A , ι ) onto (A, ι(h) ). As in types A1 and A2 , the following proposition now follows from Theorem 11.1 and Lemma 11.7. Proposition 11.8. Suppose that (A, ι) is associative Λ-torus with involution and s ∈ Hom(Q, Λ) is admissible for L(A, ι). Choose 0 = h ∈ As(αr ) . Then, we have L(A, ι)(s) Q×Λ L(A, ι(h) ). We could follow the pattern in type A1 and deﬁne two associative tori with involution (A, ι) and (A , ι ) to be isotopic if (A, ι(h) ) is isograded-isomorphic to (A , ι ) for some nonzero homogeneous hermitian element h of (A, ι). However, we don’t need to do this because of the following: Proposition 11.9. Suppose that (A, ι) is an associative Λ-torus with involution and h is a nonzero homogeneous hermitian element of (A, ι). Then (A, ι(h) ) ig (A, ι). Proof. Let κ and κ(h) be the mod-2 quadratic forms for (A, ι) and (A, ι(h) ) respectively. Now h ∈ Aµ , where µ ∈ Λ, and since h is hermitian we have κ(¯ µ) = 0. Also, if x ∈ Aλ , λ ∈ Λ, we have ι(h) (x) = hι(x)h−1 = ¯ ¯ ¯ κ(λ) ¯ = κ(λ) ¯ + κp (λ, ¯ µ ¯ ∈ Λ. ¯ (−1) hxh−1 = (−1)κ(λ)+κp (λ,¯µ) x, so κ(h) (λ) ¯ ) for λ Hence, since κ(¯ µ) = 0, we have ¯ = κ(λ) ¯ + κp (λ, ¯ µ ¯ + κp (¯ ¯ µ) κ(h) (λ) ¯)2 = κ(λ µ, λ)¯ ¯ µ for λ ¯ ∈ Λ. ¯ One ¯ ∈ Λ. ¯ So we deﬁne τ¯ : Λ¯ → Λ¯ by τ¯(λ) ¯ =λ ¯ + κp (¯ µ, λ)¯ for λ 2 (h) checks that τ¯ = 1, so τ¯ is an isometry of κ onto κ. Hence, our conclusion follows from Proposition 11.4(iii).

Theorem 11.10. Suppose that (A, ι) is an associative Λ-torus with involution and (A , ι ) is an associative Λ -torus with involution. Let L(A, ι) = ssp2r (A, ι) and L(A , ι ) = ssp2r (A , ι ). Then, the following are equivalent: (a) (A, ι) is isograded-isomorphic to (A , ι ). (b) L(A, ι) bi L(A , ι ). (c) L(A, ι) ∼ L(A , ι ). Proof. “(a)⇒(b)” is clear, and “(b)⇒(c)” is trivial. Finally, suppose that (c) holds. Thus, L(A , ι ) bi L(A, ι) by Propositions 11.8 and 11.9, so there exists a bi-isomorphism ϕ : L(A , ι ) → L(A, ι). Since the automorphism group of ∆ equals the Weyl group of ∆, we can, by Lemma 3.8, assume that ϕr = 1. (a) now follows from Proposition 11.8 (with s = 0 and h = 1).

40

Bruce Allison and John Faulkner

12 Some concluding remarks Putting together the theorems quoted and proved in this article, we obtain a very good understanding of the structure of EALAs of several types. Type A1 : By Theorems 8.1 and Theorem 8.6, the TKK construction induces a 1–1 correspondence between isotopy classes of Jordan Zn -tori and isotopy classes of centreless Lie Zn -tori of type A1 . Composing this with Neher’s EALA construction (Construction 4.1), we obtain, by the results of §6, an explicit 1-1 correspondence between isotopy classes of Jordan Zn -tori and families of EALAs of type A1 and nullity n up to bijective isomorphism. Type Ar (r ≥ 2): By the results of §’s 6, 9 and 10, if r = 2 (resp. r ≥ 3) the 3×3-projective elementary construction (resp. the (r+1)×(r+1)-special linear construction) followed by Neher’s construction gives us an explicit 1–1 correspondence between isograded-isomorphism classes of alternative (resp. associative) Zn -tori and families of EALAs of type Ar and nullity n up to bijective isomorphism. Type Cr (r ≥ 4): The structural information is particularly sharp in this case. Indeed, by Theorem 11.1 and Theorem 11.10 (again along with the results of §6), the construction of an associative torus with involution from a mod2 quadratic form (see Proposition 11.4(ii)), followed by the 2r × 2r-special symplectic construction, followed by Neher’s construction, gives an explicit 1–1 correspondence between isometry classes of quadratic forms on an ndimensional vector space over Z2 and families of EALAs of type Cr and nullity n up to bijective isomorphism. Type Dr (r ≥ 4), E6 , E7 , E8 : Let Xr be one of the indicated types. According to [BGK, Thm. 1.37] and [Y2, §5], any centreless Lie Λ-torus of type Xr is isograded-isomorphic to the untwisted Lie torus g ⊗ F[Λ], where g is the split simple Lie algebra of type Xr . Thus, there is only one bi-isomorphism class and hence only one isotopy class of centreless Lie Zn -tori of type Xr . Therefore, for a given nullity, there is only one family of EALAs of type Xr up to bijective isomorphism. Other types: We have omitted discussion of families of EALAs of types C3 , Br (r ≥ 2) and BCr (r ≥ 1). Type C3 can be handled using the methods of §11, as there is only one centreless Lie Zn -torus of type C3 that is not covered by the 6 × 6-special symplectic construction ([AG, Thm. 4.87], [BY, Thm. 5.9]). (This exceptional example exists only if n ≥ 3.) In types Br (r ≥ 2) and BCr (r ≥ 1), we expect that isotopes and isotopy of coordinate algebras will play a key role not only in the arguments but also in the statements of the results (as in type A1 ). Finally, we note that when F is algebraically closed, there is an alternative method (replacing coordinatization) to construct centreless Lie Zn -tori. This method, which has been investigated recently in [ABFP1] and [ABFP2], constructs centreless Lie tori as multiloop algebras, just (as in nullity 1) aﬃne Kac–Moody Lie algebras are constructed using loop algebras. To be somewhat more precise, suppose that σ = (σ1 , . . . , σn ) is an n-tuple of commuting

Isotopy for EALAs and Lie Tori

41

ﬁnite order automorphisms of a ﬁnite dimensional simple Lie algebra s. Let g = sσ be the ﬁxed point algebra of { σi }ni=1 in s, let h be a Cartan subalgebra of g, let ∆ = ∆(s, h), and let Q = spanZ ∆. We give the tensor product T = s ⊗ F[z1±1 , . . . , zn±1 ] a Q × Zn -grading using the natural Q-grading of the ﬁrst factor and the natural Zn -grading of the second factor. Next, for 1 ≤ i ≤ n, we ﬁx a primitive mth i root of unity ζmi for each i, where mi = |σi |, δ and we deﬁne an automorphism ηi of F[z1±1 , . . . , zn±1 ] by ηi (zj ) = ζmiji zj . Now −1 n let LT(s, σ, h) be the ﬁxed point algebra of { σi ⊗ ηi }i=1 in T , in which case LT(s, σ, h) is a Q × Zn -graded subalgebra of T . If σ satisﬁes some additional conditions that are listed in [ABFP2, Prop. 3.2.5] (the most important being that g is simple), then ∆ is a root system in h∗ and LT(s, σ, h) is a centreless Lie Zn -torus of type ∆, which is called a multiloop Lie torus. Also it is shown in [ABFP2, Thm. 3.3.1] that a centreless Lie torus L is bi-isomorphic to some multiloop Lie torus if and only if L is a ﬁnitely generated as a module over its centroid. Furthermore, necessary and suﬃcient conditions are given there for two multiloop Lie tori to be bi-isomorphic or to be isotopic. The reader can consult [ABFP1] and [ABFP2] for details about these results. The two approaches, using coordinatization and using multiloops, which have been used to construct and to determine isotopy of centreless Lie tori, each have signiﬁcant complimentary advantages. It is our view that both are needed in order to gain a full understanding of families of EALAs.

References [A]

A. Albert, Non-associative algebras. I. Fundamental concepts and isotopy, Ann. of Math. 43 (1942), 685–707. [AABGP] B. Allison, S. Azam, S. Berman, Y. Gao and A. Pianzola, Extended aﬃne Lie algebras and their root systems, Mem. Amer. Math. Soc. 126 (1997), no. 603, x+122 pp. [AB] B. Allison and G. Benkart, Unitary Lie algebras and Lie tori of type BCr , r ≥ 3, Proceedings of the Conference on Quantum Aﬃne Algebras, Extended Aﬃne Lie Algebras, and Applications, Banﬀ, Canada, 2008, Contemp. Math., 506, 1–47, Amer. Math. Soc., Providence, RI, 2010. [ABFP1] B. Allison, S. Berman, J. Faulkner, A. Pianzola, Realization of gradedsimple algebras as loop algebras, Forum Math. 20 (2008), 395–432. [ABFP2] B. Allison, S. Berman, J. Faulkner and A. Pianzola, Multiloop realization of extended aﬃne Lie algebras and Lie tori, Trans. Amer. Math. Soc. 361 (2009), 4807–4842. [ABGP] B. Allison, S. Berman, Y. Gao and A. Pianzola, A characterization of aﬃne Kac-Moody Lie algebras, Comm. Math. Phys. 185 (1997), 671–688. [AFY] B. Allison, J. Faulkner and Y. Yoshii, Structurable tori, Comm. Algebra 36 (2008), 2265–2332. [AG] B. Allison and Y. Gao, The root system and the core of an extended aﬃne Lie algebra, Selecta Math. (N.S.) 7 (2001), 149–212. [BN] G. Benkart and E. Neher, The centroid of extended aﬃne and root graded Lie algebras, J. Pure Appl. Algebra 205 (2006), 117–145.

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Y. Billig and K.-H. Neeb, On the cohomology of vector ﬁelds on parallelizable manifolds, Ann. Inst. Fourier (Grenoble) 58 (2008), 1937–1982. [BY] S. Benkart and Y. Yoshii, Lie G-tori of symplectic type, Q. J. Math. 57 (2006), 425–448. [BGK] S. Berman, Y. Gao and Y. Krylyuk, Quantum tori and the structure of elliptic quasi-simple Lie algebras, J. Funct. Anal. 135 (1996), 339–389. [BGKN] S. Berman, Y. Gao, Y. Krylyuk and E. Neher, The alternative torus and the structure of elliptic quasi-simple Lie algebras of type A2 , Trans. Amer. Math. Soc. 347 (1995), 4315–4363. [B1] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 1–3, Translated from the French. Reprint of the 1975 edition. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1989. [B2] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 4–6, Translated from the French, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. [B3] N. Bourbaki, Lie Groups and Lie Algebras. Chapters 7–9, Translated from the French, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. [D] J. Dieudonn´e, La g´eom´etrie des groupes classiques, Troisi`eme ´edition. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5. SpringerVerlag, Berlin-New York, 1971. [F1] J. Faulkner, Barbilian planes, Geom. Dedicata 30 (1989), 125–181. [F2] J. Faulkner, Lie tori of type BC2 and structurable quasitori, Comm. Algebra 36 (2008), 2593–2618. [GN] E. Garcia and E. Neher, Semiprime, prime and simple Jordan superpairs covered by grids, J. Algebra 273 (2004), 1–32. [H] J.T. Hartwig, Locally ﬁnite simple weight modules over twisted generalized Weyl algebras, J. Algebra 303 (2006), 42–76. [HT] R. Høegh-Krohn and B. Torresani, Classiﬁcation and construction of quasi-simple Lie algebras, J. Funct. Anal. 89 (1990), 106–136. [Mc1] K. McCrimmon, Homotopes of alternative algebras, Math. Ann. 191, 253–262. [Mc2] K. McCrimmon, A Taste of Jordan Algebras, Springer, New York, 2004. [MRY] R. Moody, S.E. Rao and T. Yokonuma, Toroidal Lie algebras and vertex representations. Geom. Dedicata 35 (1990), 283–307. [Ne] K.-H. Neeb, On the classiﬁcation of rational quantum tori and the structure of their automorphism groups. Canad. Math. Bull. 51 (2008), 261–282. [N1] E. Neher, Lie tori, C.R. Math. Acad. Sci. Soc. R. Can., 26 (2004), 84–89. [N2] E. Neher, Extended aﬃne Lie algebras, C.R. Math. Acad. Sci. Soc. R. Can., 26 (2004), 90–96. [S] R.D. Schafer, An Introduction to Nonassociative Algebras. Pure and Applied Mathematics, Vol. 22 Academic Press, New York-London 1966. [Y1] Y. Yoshii, Coordinate algebras of extended aﬃne Lie algebras of type A1 , J. Algebra 234 (2000), 128–168. [Y2] Y. Yoshii, Root-graded Lie algebras with compatible grading, Comm. Algebra 29 (2001), 3365–3391. [Y3] Y. Yoshii, Classiﬁcation of quantum tori with involution, Canad. Math. Bull. 45 (2002), 711–731. [Y4] Y. Yoshii, Classiﬁcation of Zn -graded alternative algebras, J. Alg. 256 (2002), 28–50.

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Y. Yoshii, Root systems extended by an abelian group and their Lie algebras, J. Lie Theory 14 (2004), 371–394. Y. Yoshii, Lie tori—A simple characterization of extended aﬃne Lie algebras, Publ. Res. Inst. Math. Sci. 42 (2006), 739–762. Y. Yoshii, Cayley polynomials, Algebra Logika 47 (2008), 54–70; translation in Algebra Logic 47 (2008), 32–41.

Remarks on the Isotriviality of Multiloop Algebras Philippe Gille1 and Arturo Pianzola2,3∗ 1 2 3

´ UMR 8553 du CNRS, Ecole Normale Sup´erieure, 45 rue d’Ulm, 75005 Paris, France, [email protected] Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada Instituto Argentino de Matem´ atica, Saavedra 15, (1083) Buenos Aires, Argentina, [email protected]

Summary. For a given ﬁnite dimensional simple Lie algebra g over an algebraically closed ﬁeld of characteristic 0, we establish the existence of a positive integer d(g) with the property that any multiloop algebra L(g, σ1 , . . . , σn ) based on g is split by ±1 a ﬁnite ´etale cover of degree d(g) of the base ring k[t±1 1 , . . . , tn ]. Key words: isotriviality, torsor, multiloop algebra. 2000 Mathematics Subject Classiﬁcations: Primary 14L15. Secondary 17B67, 20G15.

1 Introduction ±1 Let Rn = k[t±1 1 , . . . , tn ] denote the Laurent polynomial ring in n variables over an algebraically closed ﬁeld k of characteristic 0. Given a ﬁnite dimensional simple Lie algebra g over k, consider the class of Lie algebras L over Rn with the property that

L ⊗Rn S S−Lie g ⊗k S

(1)

for some faithfully ﬂat and ﬁnitely presented extension S of Rn . This interesting class of algebras has deep connections to the centreless cores of Extended Aﬃne Lie Algebras (see [ABFP], [ABFP2] and [GP2] for details and further references). For n = 1 for example, the algebras in question turn out to be the derived aﬃne Kac–Moody Lie algebras modulo their centres (see [P1] and [P2] for details). The algebras as in (1) are the forms of g for the f ppf -topology on Spec(Rn ). They can be classiﬁed by non-abelian cohomology (see §1.1 below for references). The quintessential example of forms—and the ones that are of ∗

Supported by the NSERC Discovery Grant program and by CONICET.

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_2, © Springer Basel AG 2011

45

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Philippe Gille and Arturo Pianzola

most interest to inﬁnite dimensional Lie theory—are the multiloop algebras based on g (see deﬁnition below).2 One of the major results of [GP1] is that any algebra L as in (1) is isotrivial, i.e., trivialized by a finite ´etale cover of Rn . This result has important implications for the classiﬁcation of Extended Aﬃne Lie Algebras by cohomological methods ([GP2, Theorem 5.13]). Relevant to the present work is the fact that the Isotriviality Theorem is of an abstract nature: It does not give any information on the degree of the ﬁnite ´etale cover needed to trivialize a given form. In this paper we show the existence of a degree which depends only on g with the property that any multiloop algebra based on g is trivialized by a ﬁnite ´etale cover of Rn of this degree.

1.1 Notation and conventions Throughout, k denotes an algebraically closed ﬁeld of characteristic 0. We ﬁx once and for all a set (ζn )n≥1 of compatible primitive n-roots of unity of k, i.e., ζn = ζn . The category of commutative associative unital k-algebras is denoted by k–alg. For a given R in k–alg, by an R-group we will understand a group scheme ˇ over Spec(R). If G is an R-group, the pointed set of non-abelian Cech cohomology on the f ppf and ´etale sites of X = Spec(R) with coeﬃcients in G, wil be denoted by Hf1ppf (R, G) and He´1t (R, G), respectively. These pointed sets measure the isomorphisms of sheaf torsors (with respect to the chosen topologies) over R under G (see [SGA3] Exp. IV.6, [DG] III §4, and Ch. IV §1 of [M] for basic deﬁnitions and references). If G is aﬃne, ﬂat and (locally) of ﬁnite presentation over R, then by faithfully ﬂat descent all of our sheaf torsors are representable (by aﬃne Rschemes). They are thus torsors in the usual sense. Furthermore, given that G is necessarily smooth3 all torsors are locally trivial for the ´etale topology. In particular, He´1t (R, G) = Hf1ppf (R, G). These assumption on G are present in all of the situations that arise in our work. Given an R-group G and a morphisms R → S in k–alg, we let GS denote the S-group G ×Spec(R) Spec(S) obtained by base change. For convenience, we will under these circumstances denote He´1t (S, GS ) simply by He´1t (S, G). To a given ﬁnite (abstract) group F corresponds a ﬁnite constant R-group that we denote by FR , or simply by F if R is understood from the context. 2

3

In nullity 1 all forms are loop algebras [P2]. An example of B. Margaux shows that already in nullity 2 there exists forms which are not multiloop algebras. See [GP2] §3.6. Since R is of characteristic zero, the geometric ﬁbers of G are smooth by Cartier’s theorem [DG], II.6.1.1. The smoothness criterion on ﬁbers ([EGA IV] §17.8.1) shows that G is smooth over R.

Remarks on the Isotriviality of Multiloop Algebras

47

The following notation will be extensively used throughout this work. ±1 Rn = k[t±1 1 , . . . , tn ],

±1

±1

±1

Rn,d = k[t1 d , t2 d , . . . , tn d ],

Rn,∞ = −→ lim Rn,d , d

and Fn = k((t1 ))((t2 )) . . . ((tn )),

±1

±1

±1

Fn,d = k((t1 d )), ((t2 d )), . . . , ((tn d )).

1.2 Rn -torsors under finite constant groups The extension Rn,d /Rn is ﬁnite Galois. Its Galois group Γn,d is henceforth identiﬁed with (Z/dZ)n acting naturally on Rn,d via our ﬁxed choice of com1/d 1/d patible roots of unity; namely ¯e (ti ) = ζdei ti for all e = (e1 , . . . , en ) ∈ Zn , n n where as usual : Z → (Z/dZ) is the canonical map. We ﬁx once and for all an algebraic closure Fn of Fn containing all of the ﬁelds Fn,d . This deﬁnes a geometric point a of X = Spec(Rn ), and we consider the corresponding algebraic fundamental group π1 (Rn , a) (see [SGA1] V.7). Since X is connected, normal and noetherian, we have an explicit realization, henceforth denoted by X sc , for the universal covering of X at a (ibid. V.8.2). In particular, we have a canonical homomorphism Gal(Fn ) → π1 (Rn , a). We recall for convenience the following fundamental fact (see [GP3] corollary . 2.10 for details). Proposition 1.1. 1. Spec(Rn,∞ ) is simply connected (i.e., it does not have any nontrivial finite ´etale covers), and it is naturally isomorphic to the simply connected covering X sc of Spec(Rn ) at the geometric point a. 2. π1 (Rn , a) is the Galois group of the pro-covering Rn,∞ /Rn , namely n . lim Gal(Rn,d /Rn ) = ←− lim (Z/dZ)n = Z π1 (Rn , a) = ←− d

d

3. The canonical map Gal(Fn ) → π1 (Rn , a) is an isomorphism.

The next basic lemma—especially its proof—gives a complete description of Rn -torsors under a ﬁnite constant group. Lemma 1.2. Let F be a finite group, and let d(F) be the maximum of all orders of abelian subgroups of F. If X is an Rn -torsor under F, then there exists a positive integer d(X) with the following properties: (a) d(X) ≤ d(F) (b) d(X) divides |F| (c) Xis trivialized by the base change Rn → Rn,d(X) . In particular, the base change Rn → Rn,|F| also trivializes X.

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Proof. The classes of such torsors are parametrized by He´1t (Rn , F). This same pointed set classiﬁes the Galois covers of Rn with a given action of the ﬁnite group F. We have natural correspondences (see [SGA1] XI.5) 1 1 n He´1t (Rn , F) Hct π1 (R, a), F Hct (Z , F) n , F)/inner automorphisms of F. Homct (Z n → F repreConsider a cocycle (i.e., a continuous homomorphism) u : Z n ). Then the senting an Rn -torsor X under F. Set K = ker (u) and F0 = u(Z n → F0 corresponds to a connected Galois extension induced cocycle u0 : Z n /K. By deﬁnition |F0 | ≤ d(F). X0 of Rn with Galois group F0 Z All connected components of X are isomorphic to X0 . There are (F : F0 ) similar connected components and F acts transitively on them according to the way that F permutes the right cosets of F0 . We summarize this by writing K X = (F:F0 ) X0 . The torsor X0 is trivialized by the extension S0 = Rn,∞ ; more precisely X0 Spec(S0 ) and S0 ⊗Rn S0 S0 × · · · × S0 under the map x ⊗ y → (σ(x)y)σ∈F0 . n ⊂ K. Since the subgroup |F0 |Z n of Z n corresponds to Clearly |F0 |Z the Galois extension Rn,|F0 | we see that X0 is trivialized bythe extension Rn,|F0 | /Rn . We summarize this by writing X0 ×Rn Rn,|F0 | = F0 Rn,|F0 | . The isomorphisms X ×Rn Rn,|F0 | X0 ×Rn Rn,|F0 | Rn,|F0 | FRn,|F0 | (F:F0 )

(F:F0 )

F0

are compatible with the action of F. This shows that the Rn -torsor X is trivialized by Rn,|F0 | , hence also by Rn,|F| (since |F0 | divides |F| and therefore Rn,|F| is an extension of Rn,|F0 | ).

2 Bounds on the isotriviality of multiloop algebras Let R ∈ k–alg. Recall that by an R-form of g we understand an algebra L over R for which there exists a faithfully ﬂat and ﬁnitely presented extension S/R in k–alg such that L ⊗R S S−Lie g ⊗k S (isomorphism of S-Lie algebras). Thus, by deﬁnition, forms are trivialized by some f ppf cover S of R. In the case of Laurent polynomials, we have a very precise control over the trivializing base change. Theorem 2.1 (Isotriviality, [GP1] and [GP3]). Every Rn -form L of g is isotrivial (i.e., trivialized by a finite ´etale cover of Rn ). More precisely, there exist a positive integer d = d(L), such that L ⊗Rn Rn,d Rn,d g ⊗k Rn,d .

Remarks on the Isotriviality of Multiloop Algebras

49

At ﬁrst, it does not seem reasonable to expect the existence of a “universal” d in the Isotriviality Theorem that splits all forms of g. The purpose of this note is to show that such a d thus indeed exists for multiloop algebras. More precisely, we have Proposition 2.2. Let d(g) be the maximum of all orders of abelian subgroups of the (finite) group of automorphisms of the root system corresponding to g. Any multiloop algebra L(g, σ1 , . . . , σn ) is trivialized by the Galois extension Rn,d(g) of Rn . In particular, every multiloop algebra based on g is trivialized by a finite ´etale cover of degree d(g)n . We begin by recalling the deﬁnition of multiloop algebras. Let σ = (σ1 , . . . , σn ) be a commuting family of ﬁnite order automorphisms of the kalgebra g. Let d be a common period of the σi , i.e., σid = 1. For each (i1 , . . . , in ) ∈ Zn , consider the simultaneous eigenspaces i

gi1 ...in := {x ∈ g : σj (x) = ζdj x for all 1 ≤ j ≤ n} (which of course depend only on the ij modulo d). The multiloop algebra associated to this data is the k-Lie subalgebra L of g ⊗k Rn,∞ deﬁned as follows: L = L(g, σ ) =

⊕

i /d

(i1 ,...,in )∈Zn

gi1 ...in ⊗t11

. . . tinn /d ⊂ g⊗k Rn,d ⊂ g⊗k Rn,∞ . (2)

Observe that L does not depend on the choice of period d. Observe also that L is in fact an Rn -submodule of g ⊗k Rn,∞ . In particular, L has a natural Rn -Lie algebra structure. One easily veriﬁes ([ABP2.5] Lemma 4.7) that L ⊗Rn Rn,d Rn,d g ⊗k Rn,d .

(3)

Since Rn,d /Rn is free of ﬁnite rank (hence f ppf ), L is an Rn -form of g which is trivialized by the extension Rn,d /Rn . Note that the integer d can be arbitrarily large (there exist automorphisms of any given ﬁnite order). Proposition 2.2 implies the existence of a ﬁxed d that splits all multiloop algebras based on a given g. From (3) it follows that to a multiloop algebra L corresponds an Rn -torsor XL under the group Aut(g).4 By descent theory, XL is representable by an aﬃne R n -scheme. The isomorphism class of the Rn -torsor XL will be denoted by XL . Thus, XL ∈ He´1t Rn,d /Rn , Aut(g) ⊂ He´1t Rn , Aut(g) . (4) is ﬁnite As mentioned above the extension Rn,d /Rn Galois, and its Galois group Γn,d can be identiﬁed with (Z/dZ)n acting naturally on Rn,d via our ﬁxed choice of compatible roots of unity; namely 4

Strictly speaking, the structure group is the aﬃne Rn -group Aut(A ⊗k Rn ). As already mentioned, this harmless slight abuse of notation is used throughout.

50

Philippe Gille and Arturo Pianzola

(ti ) = ζdei ti for all e = (e1 , . . . , en ) ∈ Zn . If we now let Γn,d act on Aut(g)(Rn,d ) = AutRn,d −alg (g ⊗ Rn,d ) by conjugation, i.e., γ σ = (1 ⊗ 1 γ)σ(1 ⊗ (−γ)), we have a natural correspondence H Γn,d , Aut(g)(Rn,d ) He´1t (Rn,d /Rn , Aut(g) , where the H 1 on the left-hand side is the usual nonabelian cohomology. In terms of H 1 (Γn,d , Aut(g)(Rn,d )), the loop torsor XL corresponds to the cocycle αL ∈ Z 1 Γn,d , Aut(g)(Rn,d ) given by (αL )e¯ = σ1−e1 . . . σn−en ⊗ id ∈ AutRn,d -alg (g ⊗ Rn,d ). Recall that we have an exact sequence of Rn -groups ¯ e

1/d

1/d

1 → Gad → Aut(g) → Out(g) → 1,

(5)

where Gad comes by base change from the Chevalley Z-group of g of adjoint type, and Out(g) is the ﬁnite constant group corresponding to the (abstract) ﬁnite group Out(g) of automorphisms of the Dynkin diagram of g ([SGA3] Exp. XXIV Th´eor`eme 1.3 and Proposition 7.3.1). The sequence (5) is split. The choice of section Out(g) → Aut(g) identiﬁes Out(g) with a subgroup of Aut(g) so that Aut(g) = Gad Out(g). (6) Proof of Proposition 2.2: By the main theorem of [BM] there exists a maximal torus T of Gad that is stable under the action of all the σi . Let Φ = Φ(G, T) be the corresponding root system, and W = W(Φ) its Weyl group. Let B be a Borel subgroup of G containing T. Then Aut(Gad , B, T)/T Out(g) ([SGA3] Exp. XXIV Prop. 2.1), and therefore NAut(g) (T) = NGad (T) Out(g). Thus NAut(g) (T)/T NG (T)/T Out(g) W Out(g) Aut(Φ), where Aut(Φ) is the ﬁnite group of automorphisms of the root system Φ. The foregoing discussion shows that the torsor XL corresponding to L belongs to the image of the canonical map He´1t Rn , NAut(g) (T) → He´1t Rn , Aut(g) . Consider the exact sequence of pointed sets π He´1t (Rn , T) → He´1t Rn , NAut(g) (T ) → He´1t Rn , Aut(Φ) . By the previous Lemma after the base change Rn → Rn,d(g) the torsor XL is in the kernel of π. Since He´1t (Rn,d(g) , T) = Pic(Rn,d(g) )rk(g) = 1 the Rn,d(g) torsor XL ⊗Rn Rn,d(g) under NAut(g) (T) ⊂ Aut(g) is trivial.

Remarks on the Isotriviality of Multiloop Algebras

51

Remark 2.3. If all σi ∈ T, then XL ∈ im H 1 (Rn , T) → H 1 Rn , Aut(g) = 1. Thus L(g, σ) is isomorphic as an Rn -Lie algebra, and a fortiori also as a k-Lie algebra, to g⊗k Rn . The “interesting” multiloop algebras are thus to be found by considering n-tuples of commuting automorphisms of g that do not belong to a torus.

References [ABFP]

B. Allison, S. Berman, J. Faulkner, and A. Pianzola, Realization of graded-simple algebras as loop algebras, Forum Mathematicum 20:3 (2008), 395–432. [ABFP2] B. Allison, S. Berman, J.Faulkner, and A. Pianzola, Multiloop realization of extended aﬃne Lie algebras and Lie tori, Transaction of the American Mathematical Society 361 (2009) 4807–4842. [ABP2.5] B.N. Allison, S. Berman and A. Pianzola, Iterated loop algebras, Paciﬁc Jour. Math. 227 (2006), 1–41. [BM] A. Borel and G.D Mostow, On semi-simple automorphisms of Lie algebras, Ann. Math. 61 (1955), 389–405. [DG] M. Demazure et P. Gabriel, Groupes alg´ebriques, Masson, 1970. [GP1] P. Gille and A. Pianzola, Isotriviality of torsors over Laurent polynomials rings, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 725–729. [GP2] P. Gille and A. Pianzola, Galois cohomology and forms of algebras over Laurent polynomial rings, Math. Annalen 338 (2007), 497–543. [GP3] P. Gille and A. Pianzola, Isotriviality and ´ etale cohomology of Laurent polynomials rings, Journal of Pure and Applied Algebra 212 (2008), 780–800. [EGA IV] A. Grothendieck (avec la collaboration de J. Dieudonn´e) El´ements de g´eom´etrie alg´ebrique IV, Pub del’ IHES. no 20, 24, 28 and 32 1964–1967. ´ [M] J.S. Milne, Etale Cohomology, Princeton University Press, 1980. [P1] A. Pianzola, Aﬃne Kac-Moody Lie algebras as torsors over the punctured line, Indagationes Mathematicae N.S. 13(2) (2002), 249–257. [P2] A. Pianzola, Vanishing of H 1 for Dedekind rings and applications to loop algebras, C. R. Acad. Sci. Paris, Ser. I 340 (2005), 633–638. [SGA1] S´eminaire de g´eom´etrie alg´ebrique de IHES., Revˆetements ´etales et groupe fondamental, dirig´ e par A. Grothendieck, Lecture Notes in Math. 224, Springer, 1971. [SGA3] S´eminaire de g´eom´etrie alg´ebrique de IHES., 1963–1964, sch´ emas en groupes, dirig´e par M. Demazure et A. Grothendieck, Lecture Notes in Math. 151–153. Springer, 1970.

Extended Aﬃne Lie Algebras and Other Generalizations of Aﬃne Lie Algebras – A Survey Erhard Neher∗

2

Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada, email: [email protected] Summary. This is a survey on extended aﬃne Lie algebras and related types of Lie algebras, which generalize aﬃne Lie algebras. Key words: aﬃne Lie algebras, extended aﬃne Lie algebra, Lie torus. 2000 Mathematics Subject Classiﬁcations: Primary 17B65, 17B67. Secondary 16W50, 17B70.

1 Introduction Motivation. The theory of aﬃne (Kac–Moody) Lie algebras has been a tremendous success story. Not only has one been able to generalize essentially all of the well-developed theory of ﬁnite-dimensional simple Lie algebras and their associated groups to the setting of aﬃne Lie algebras, but these algebras have found many striking applications in other parts of mathematics. It is natural to ask why? What is so special about aﬃne Lie algebras? What really makes them “tick”? One way to understand this is to generalize aﬃne Lie algebras and see where things go wrong or don’t. After all, it is conceivable that there is a whole theory of Lie algebras, ready to be discovered, for which aﬃne Lie algebras are just one example. Of course, one immediately thinks about Kac–Moody Lie algebras, for which ﬁnite-dimensional simple and aﬃne Lie algebras are the basic examples. But is it the right generalization? This is not such a sacriligious question as it seems. After all, one the founders and main contributor to the theory does not seem to be convinced himself ([Kac3]). So where to go? There are several directions that have been pursued: toroidal algebras, Borcherds algebras, GIM-algebras, root-graded algebras, to name just several. ∗

Support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_3, © Springer Basel AG 2011

53

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Erhard Neher

The ﬁrst generalization. The generalizations that are of interest for this paper arose in the work of Saito and Slodowy on elliptic singularities and in the paper by the physicists Høegh-Krohn and Torr´esani [HT] on Lie algebras of interest to quantum gauge ﬁeld theory. The latter paper, written by physicists, did however not stand up to mathematical scrutiny. Fortunately, a mathematically sound foundation of the theory was later provided by Allison, Azam, Berman, Gao, and Pianzola in their AMS memoirs [AABGP]. They also describe the type of root systems appearing in these algebras and give many examples. Also, they coined the term extended aﬃne Lie algebras for this new class of algebras. Before we give the deﬁnition, it is convenient to introduce the basic concept of a toral pair over K. Throughout this paper K denotes a ﬁeld of characteristic 0. A toral pair (E, T ) over K consists of a Lie algebra E over K and a subspace T ⊂ E acting ad-diagonalizably on E, i.e., putting Eα = {e ∈ E : [t, e] = α(t)e for all t ∈ T } for α ∈ T ∗ and R = {α ∈ T ∗ : Eα = 0}, we have E = α∈R Eα . It turns out that T is abelian, so T ⊂ E0 and hence 0 ∈ R (unless T = {0} = E). In general T is a proper subspace of E0 . If, however, T = E0 we will call T a splitting Cartan subalgebra and, following common usage, will use the letter H instead of T . The following deﬁnition is essentially the one given in [AABGP]: An extended aﬃne Lie algebra, or EALA for short, is a toral pair (E, H) over K = C satisfying the conditions (E1)–(E6) below: (E1) E has an invariant nondegenerate symmetric bilinear form (·|·). (E2) H is a nontrivial, ﬁnite-dimensional splitting Cartan subalgebra. Since H = E0 , it is immediate that the restriction of (·|·) to H is nondegenerate. We can therefore represent any α ∈ H ∗ by a unique vector tα ∈ H via α(h) = (tα |h). Hence R = R0 ∪ Ran (1) for R0 = {α ∈ R : (tα |tα ) = 0} (null roots) and Ran = {α ∈ R : (tα |tα ) = 0} (anisotropic roots). We can now state the remaining axioms (E3)–(E6): (E3) For any α ∈ Ran and x ∈ Eα , the adjoint map ad x is locally nilpotent on E. (E4) Ran is connected in the sense that any partition Ran = R1 ∪ R2 with (R1 |R2 ) = 0 either has R1 = ∅ or R2 = ∅. (E5) R0 ⊂ Ran + Ran . (E6) R is a discrete subset of H ∗ , equipped with the natural topology. Let (E, H) be an EALA. By (E6), the Z-span of R0 in H ∗ is free of ﬁnite rank, called the nullity of (E, H) (it turns out that this is independent of

EALAs and Generalizations

55

(·|·)). The core of (E, H) is the ideal Ec of E generated by all subspaces Eα , α ∈ Ran . We then have a Lie algebra homomorphism E → Der Ec , given by e → ad e | Ec , and one says that (E, H) is tame if the kernel of this map lies in Ec . The idea of tameness is that one can recover E from Ec and its derivations. Although there are examples of non-tame EALAs, it is natural to assume tameness for the purpose of classiﬁcation. We can now convince the reader that extended aﬃne Lie algebras indeed do what they are supposed to do: A tame EALA of nullity 0 is the same as a ﬁnite-dimensional simple Lie algebra (exercise), and a tame EALA of nullity 1 is the same as an aﬃne Lie algebra, proven by Allison–Berman–Gao–Pianzola in [ABGP]. Tame EALAs of nullity 2 are closely related to the Lie algebras studied by Saito and Slodowy, and toriodal Lie algebras provide examples of tame EALAs of arbitrarily high nullity. What makes EALAs an interesting class of Lie algebras is that there are other, nontoroidal examples requiring noncommutative coordinates. The revisions. The system of axioms (E1)–(E6) has been very successful. It describes an important and accessible class of Lie algebras, about which much is known. But there are drawbacks. Foremost among them is the restriction to the ground ﬁeld C, which is necessary for the axiom (E6). Looking at the basic examples of EALAs given in [AABGP], one sees that they make sense over arbitrary ﬁelds K. Hence, in [Neh7] the author proposed a new deﬁnition of EALAs, valid for Lie algebras over K. It used the same set of axioms, but replaced (E6) with its essence: (E6 ) the Z-span of R is free of ﬁnite rank. (To be more precise, only tame EALAs were considered in [Neh7].) It turned out that one can develop a satisfactory theory for EALAs over K, partially described in [Neh7] and [Neh6]. Some drawbacks however still remained: The theory only covers the “split case” because of the condition in (E2) that H be a splitting Cartan subalgebra, and it does not include ﬁnite-dimensional semisimple Lie algebras because of (E4), or inﬁnite rank aﬃne Lie algebras because of the requirement in (E2) that H be ﬁnite-dimensional. Some of these issues were addressed in papers by Azam [Az4], Azam–Khalili–Yousofzadeh [AKY] and Morita–Yoshii [MY] (see 6.17–6.19). A radically new look at the system of axioms seems to be called for: Delete the less important axioms (E4)–(E6) and modify (E2) so that all the aforementioned examples can be included. The purpose of this survey and research announcement is to propose such an approach by deﬁning a new class of Lie algebras and to explain its main features. The new deﬁnition. An invariant aﬃne reﬂection algebra, an IARA for short, is a toral pair (E, T ) over K satisfying the axioms (IA1)–(IA3) below (recall E = α∈R Eα ):

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Erhard Neher

(IA1) E has an invariant nondegenerate symmetric bilinear form (·|·) whose restriction to T is nondegenerate and which represents every root α ∈ R: There exists tα ∈ T such that α(t) = (tα |t) for all t ∈ T . (IA2) For every α ∈ R, α = 0, there are e±α ∈ E±α such that 0 = [eα , e−α ] ∈ T. (IA3) = (E3) For every α ∈ R with (tα |tα ) = 0 and for all xα ∈ Eα the adjoint map ad xα is locally nilpotent on E. The following are examples of invariant aﬃne reﬂection algebras: ﬁnite-dimensional reductive Lie algebras, inﬁnite-rank aﬃne Lie algebras, Lie algebras satisfying (E1)–(E3), in particular EALAs, and all the generalizations of EALAs that had been considered before. To describe the structure of IARAs, we will use the inclusions of categories aﬃne Lie algebras

⊂

EALAs

⊂

IARAs

(2)

as a leitmotif. We will ﬁrst discuss the combinatorics, viz. the roots appearing in IARAs, and then describe a procedure to determine their algebra structure. The roots. The roots for the algebras in (2) form, respectively, an aﬃne root sytem

⊂

EARS

⊂

ARS

where the aﬃne root systems are the roots appearing in aﬃne Lie algebras, EARS is the acronym for an extended aﬃne root system, the roots of EALAs, and ARS stands for aﬃne reﬂection system (3.6), of which the roots of an IARA are an example. To explain the general structure of an ARS, recall (2.12) that to any aﬃne root system R one can associate a ﬁnite irreducible, but not necessarily reduced root system S, a torsion-free abelian group Λ and a family (Λα : α ∈ S) of subsets Λα ⊂ Λ such that (3) R = ξ∈S (ξ ⊕ Λξ ) ⊂ spanK (R) = spanK (S) ⊕ spanK (Λ). Indeed, Λ = Zδ where δ is the basic null root. Now (3) also holds for EARS if one takes Λ to be free of ﬁnite rank. Since the toral subalgebra of an IARA may be inﬁnite-dimensional, it is not surprising that root systems of inﬁnite rank will occur. Indeed, if one allows S to be a locally ﬁnite root system (3.6) and Λ to be any torsion-free abelian group, then (3) also holds for ARSs. Moreover, the basic relations between the subsets Λξ , ξ ∈ S, developed in [AABGP, Ch.II] for EARS, actually also hold in the more general setting of aﬃne reﬂection systems. Hence, their theory is about as easy or complicated as the theory of EARS. The core and centreless core. Recall that the core of an EALA (E, H) is the ideal generated by all Eα , α ∈ Ran . It is a perfect Lie algebra. Hence Ecc = Ec / Z(Ec ) has centre {0} and is justiﬁably called the centreless core. By (IA1) the decomposition R = R0 ∪ Ran is also available for IARAs, and we

EALAs and Generalizations

57

can therefore deﬁne Ec and Ecc as in the EALA-case. Hence, from any IARA E we can go down to its centreless core Ecc : Ec central quotient

Ecc

ideal

/E w7 w7 7 w 7w 7w 7w 7w w 7 w 7w

This diagram describes the fundamental approach to the study of the Lie algebras (E, T ) in (2): (A) Describe the class C of Lie algebras that arise as centreless cores Ecc in a way which is independent of (E, T ), (B) for every Lie algebra L in C provide a construction which yields all the Lie algebras E with centreless core L, and (C) classify the Lie algebras in C. That this approach has good chances of success can already be witnessed by looking at aﬃne Lie algebras. For them, the core and the centreless core are the basic ingredients of the structure theory: Ec is the derived algebra of the aﬃne Lie algebra E and Ecc is the (twisted or untwisted) loop algebra that one uses to construct E from Ecc . The centreless cores of EALAs were described by Allison–Gao [AG] and then put into an axiomatic framework as centreless Lie Zn -tori by Yoshii [Yo5] and Neher [Neh6]. One can show that the centreless core of an IARA is a certain type of root-graded Lie algebra, a so-called invariant predivision-(S, Λ)-graded Lie algebra for (S, Λ) as above: S is a locally ﬁnite root system and Λ is a torsion-free abelian group. Conversely, every such Lie algebra arises as the centreless core of some IARA (5.1, 6.8, 6.10). Thus, for the Lie algebras in (2) the classes C are loop algebras

⊂

centreless Lie Zn -tori

⊂

invariant predivision-root-graded Lie algebras, Λ torsion-free. This solves problem (A). As for (B), we describe in 6.9 a construction, which starts with any invariant predivision-root-graded Lie algebra L with a torsion-free Λ and produces an invariant aﬃne reﬂection algebra (E, T ) with centreless core L. It maintains the main features of the construction of an aﬃne Lie algebra from a loop algebra, i.e., take a central extension and then add on some derivations: add derivations / L⊕C L ⊕5 C ⊕ D = E 5u 5u 5u u 5 u 5 5u central 5u 5u 5u 5u u 5 extension u5 u5 5u u5 u5 L

58

Erhard Neher

If L is a Lie torus this construction yields all EALAs with centreless core L, and hence settles problem (B) for them. Finally, the eﬀorts of many people have now lead to a complete solution of problem (C) for Lie tori, see the survey by Allison–Faulkner in [AF, §12]. The classiﬁcation of predivision-root-graded Lie algebras is not known in general, but in several important cases, see the review in 5.10. What certainly is fascinating about this class and root-graded Lie algebras in general is that there are many examples requiring noncommutative coordinates. For example, for S = Al , l ≥ 3, one has to consider central quotients of sll+1 (A) where A is an associative algebra, a crossed product algebra to be more precise. Moreover, all the important classes of non-associative algebras appear as coordinates: alternative algebras for S = A2 , Jordan algebras for A1 , structurable algebras for BC1 and BC2 . In the end, what have we gained? What is emerging, is the picture that EALAs and IARAs share many structural features with aﬃne Lie algebras, for example they all have concrete realizations (unlike arbitrary Kac–Moody algebras). But they allow many more possibilities and hence possible applications than aﬃne Lie algebras. We are just at the beginning of this new theory and many problems have not been solved (representation theory, quantization, to name just two). Contents. The theory of aﬃne reﬂection systems is described in §3 after we have laid the background for more general reﬂection systems in §2. Both sections are based on the author’s joint paper with Ottmar Loos [LN2]. We then move to graded algebras. In §4 we introduce what is needed from the theory of arbitrary graded algebras, e.g., the centroid and centroidal derivations. Following this, we describe in §5 the necessary background from the theory of root-graded Lie algebras and the subclasses of predivision-root-graded algebras and Lie tori. These two sections contain few new results. The new theory of invariant aﬃne reﬂection algebras and even more general algebras is presented in §6. Finally, in §7 we consider the example sln (A) in detail. In particular, we describe central extensions and derivations, the important ingredients of the EALA- and IARA-construction. Acknowledgements: The author thanks Bruce Allison for stimulating discussions on the topics of this paper and the referee for helpful comments on a preliminary version of this article.

2 Root systems and other types of reﬂection systems In this section and the following Section 3 we will describe the combinatorial background for the sets of roots that appear in the theory of extended aﬃne Lie algebras, Lie tori and their generalizations. See 2.14 for references. Throughout, we work in vector spaces over a ﬁeld K of characteristic 0. For a vector space X we denote by GL(X) the general linear group of X, by

EALAs and Generalizations

59

IdX the identity map on X, by X ∗ = Hom(X, K) the dual space of X and by

·, · : X × X ∗ → K the canonical pairing given by x, λ = λ(x). 2.1 Pre-reﬂection and reﬂection systems. A pre-reﬂection system is a triple (R, X, s) where R is a subset of a vector space X and s : R → GL(X), α → sα is a map satisfying the following axioms: (ReS0) 0 ∈ R, R spans X and for all α ∈ R we have codim{x ∈ X : sα x = x} ≤ 1 and s2α = IdX . Thus, either sα = IdX or sα is a reﬂection with ﬁxed point space of codimension 1, leading to a partition R = Rre ∪ Rim where Rre = {α ∈ R : sα = IdX }

(reﬂective or real roots)

Rim = {α ∈ R : sα = IdX }

(imaginary roots).

(ReS1) If α ∈ Rre then sα (α) = −α = α. (ReS2) For all α ∈ R we have sα (Rre ) = Rre and sα (Rim ) = Rim . A reﬂection system is a pre-reﬂection system (R, X, s) for which, in addition to (ReS0)–(ReS2), we also have (ReS3) scα = sα whenever 0 = c ∈ K and both α and cα belong to Rre . (ReS4) sα sβ sα = ssα β for all α, β ∈ R. Some explanations are in order. First, following recent practice (e.g., [AABGP] and [LN1]) we assume that 0 is a root, in fact 0 ∈ Rim because of (ReS1). While this is not really important, it will turn out to be convenient later on. The terminology of “real” and “imaginary” roots is of course borrowed from the theory of Kac–Moody algebras, whose roots provide an example of a reﬂection system. It is immediate from (ReS1) and (ReS2) that Rre = −Rre

and sα (R) = R

for all α ∈ R.

But we do not assume R = −R (this would exclude some examples in Lie superalgebras, see 2.11). 2.2 Some basic concepts. Let (R, X, s) be a pre-reﬂection system, often just denoted (R, X) or even simply R. For all pre-reﬂection systems we will use the same letter s to denote the reﬂections. Let α ∈ Rre . Because of (ReS1) there exists a unique α∨ ∈ X ∗ such that sα is given by the usual formula sα (x) = x − x, α∨ α

for x ∈ X.

(2.2.1)

Indeed, α∨ is determined by α∨ (α) = 2 and Ker α∨ = {x ∈ X : sα x = x}. So that (2.2.1) holds for all α ∈ R, we put α∨ = 0 for α ∈ Rim . We thus get a map ∨ : R → X ∗ which uniquely determines the map s : R → GL(X).

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One obtains a category of pre-reﬂection systems, and a fortiori a full subcategory of reﬂection systems, by deﬁning a morphism f : (R, X, s) → (S, Y, s) of pre-reﬂection systems as a K-linear map f : X → Y such that f (R) ⊂ S and f (sα β) = sf (α) f (β) hold for all α, β ∈ R. It is immediate that any morphism f satisﬁes f (Rim ) ⊂ S im and f (Rre ) ⊂ S re ∪ {0}. (2.2.2) Note that the axiom (ReS4) of reﬂection systems is equivalent to the condition that sα be an automorphism for every α ∈ R. The subgroup of GL(X) generated by all reﬂections sα is called the Weyl group of R and denoted W (R). A subsystem of R is a subset S ⊂ R with 0 ∈ S and sγ δ ∈ S for all γ, δ ∈ S. It follows that (S, span(S)) together with the obvious restriction of s is a pre-reﬂection system such that the inclusion S → R is a morphism. For any subsystem S we have S im = S ∩ Rim and S re = S ∩ Rre . An example of a subsystem is Re(R) = Rre ∪ {0}, called the real part of R. Also, for any subspace Y of X the intersection R ∩ Y is a subsystem of R. A pre-reﬂection system R is called (i) reduced if α ∈ Rre , 0 = c ∈ K and cα ∈ Rre imply c = ±1 (note that we do not require this condition for roots in Rim ); (ii) integral if R, R∨ ⊂ Z; (iii) nondegenerate if α∈R Ker(α∨ ) = {0}; (iv) symmetric if R = −R, equivalently, Rim = −Rim ; (v) coherent if for α, β ∈ Rre we have α, β ∨ = 0 ⇐⇒ β, α∨ = 0, (vi) tame if Rim ⊂ Rre + Rre . 2.3 Direct sums, connectedness. Let(Ri , Xi )i∈I be a family of pre reﬂection systems. Put R = i∈I Ri , X = i∈I Xi and deﬁne s : R → GL(X) by extending each sαi , αi ∈ Ri to a reﬂection of X by sαi |Xj = Id for i = j. Then (R, X, s) is a pre-reﬂection system called of the pre the direct sum reﬂection systems (Ri , Xi ). Obviously, Rre = i Rire and Rim = i Riim . We will say that a pre-reﬂection system with a nonempty set of real roots is indecomposable if it is not isomorphic to a direct sum of two pre-reﬂection systems, each of which has a nonempty set of real roots. Suppose R is coherent. Two roots α, β ∈ Rre are called connected if there exist ﬁnitely many roots α = α0 , α1 , . . . , αn = β such that αi , α∨i−1 = 0 for i = 1, . . . , n, in particular all αi ∈ Rre . It is easily seen that connectedness is an equivalence relation on Rre . A connected component of Re(R) is by deﬁnition the union of {0} and an equivalence class of Rre , and Re(R) is called connected if Re(R) has only one connected component. Equivalently, Re(R) is connected if and only if for any decomposition Rre = R1 ∪ R2 with

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R1 , R2∨ = 0 necessarily R1 = ∅ or R2 = ∅. Each connected component of Re(R) is a subsystem of R. 2.4 Lemma. If (R, X, s) is a nondegenerate coherent pre-reﬂection system, the subsystem Re(R) is the direct sum of its connected components. In particular, Re(R) is connected if and only if it is indecomposable. 2.5 Invariant bilinear forms. Let (R, X) be a pre-reﬂection system. A symmetric bilinear form b : X × X → K is called invariant if b(wx, wy) = b(x, y) holds for all w ∈ W (R) and x, y ∈ X. It is of course equivalent to require b(sα x, sα y) = b(x, y) for all x, y ∈ X and α ∈ Rre . Moreover, invariance of b is also equivalent to 2 b(x, α) = x, α∨ b(α, α)

for all x ∈ X and α ∈ Rre .

(2.5.1)

We will call an invariant bilinear form strictly invariant if (2.5.1) not only holds for α ∈ Rre but for all α ∈ R. It easily follows that an invariant symmetric bilinear form b is strictly invariant if and only if Rim ⊂ {x ∈ X : b(x, X) = 0} =: Rad b, the radical of b. For such a bilinear form b we have R ∩ Rad b = {α ∈ R : b(α, α) = 0} and Ker α∨ ⊂ Rad b. (2.5.2) α∈R

We note that we need not distinguish between invariant and strictly invariant forms if R = Re(R), for example if R is a locally ﬁnite root system as deﬁned in §2.7. 2.6 Root strings. Let R be a pre-reﬂection system, let α ∈ Rre , β ∈ R, and assume that a := − β, α∨ ∈ Z. The α-string through β is deﬁned by S(β, α) := (β + Zα) ∩ R. The reﬂection sα leaves S(β, α) invariant and corresponds to the reﬂection i → a − i of Z(β, α) := {i ∈ Z : β + iα ∈ R} about the point a/2. The set Z(β, α) is bounded if and only if it is bounded on one side. In this case, we put −q = min Z(β, α) and p = max Z(β, α) and then have p, q ∈ N and p − q = a = − β, α∨ . Moreover, the following conditions are equivalent: (i) the root string S(β, α) is unbroken, i.e., Z(β, α) is a ﬁnite interval in Z or equals Z, (ii) for all γ ∈ S(β, α) and all integers i between 0 and γ, α∨ we have γ − iα ∈ S(β, α), (iii) for all γ ∈ S(β, α), γ, α∨ > 0 implies γ − α ∈ R and γ, α∨ < 0 implies γ + α ∈ R. We will now present some examples of (pre-)reﬂection systems.

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2.7 Examples: Finite and locally ﬁnite root systems. A root system a la Bourbaki ([Bo2, VI, §1.1]) with 0 added will be called here a ﬁnite root ` system. Using the terminology introduced above, a ﬁnite root system is the same as a pre-reﬂection system (R, X) which is integral, coincides with its real part Re(R), and moreover satisﬁes the ﬁniteness condition that R be a ﬁnite set. Replacing the ﬁniteness condition by the local-ﬁniteness condition deﬁnes a locally ﬁnite root system. Thus, a locally ﬁnite root system is a pre-reﬂection system (R, X) which is integral, coincides with its real part Re(R), and is locally-ﬁnite in the sense that |R∩Y | < ∞ for any ﬁnite-dimensional subspace Y of X. We mention some special features of locally ﬁnite root systems, most of them well-known from the theory of ﬁnite root systems. By deﬁnition, a locally ﬁnite root system R is integral and symmetric. One can show that it is also a reﬂection system whose root strings are unbroken and which is nondegenerate and coherent, but not necessarily reduced. Moreover, R is the direct sum of its connected components, see §2.4. In particular, R is connected if and only if R is indecomposable, in which case R is traditionally called irreducible. A root α ∈ R is called divisible or indivisible according to whether α/2 is a root or not. We put Rind = {0} ∪ {α ∈ R : α indivisible}

and Rdiv = {α ∈ R : α divisible}. (2.7.1) Both Rind and Rdiv are subsystems of R. A root basis of a locally ﬁnite root system R is a linearly independent set B ⊂ R such that every α ∈ R is a Z-linear combination of B with coeﬃcients all of the same sign. A root system R has a root basis if and only if all irreducible components of R are countable ([LN1, 6.7, 6.9]). The set R∨ = {α∨ ∈ X ∗ : α ∈ R} is a locally ﬁnite root system in ∨ X = spanK (R∨ ), called the coroot system of R. The root systems R and R∨∨ are canonically isomorphic ([LN1, Th. 4.9]). 2.8 Classiﬁcation of locally ﬁnite root systems. ([LN1, Th. 8.4]) Examples of possibly inﬁnite locally ﬁnite root systems are the classical root systems ˙ I – BCI , deﬁned as follows. Let I be an arbitrary set and let XI = A i∈I K i be the vector space with basis { i : i ∈ I}. Then ˙ I = { i − j : i, j ∈ I}, A BI = {0} ∪ {± i : i ∈ I} ∪ {± i ± j : i, j ∈ I, i = j} = {± i : i ∈ I} ∪ DI , CI = {± i ± j : i, j ∈ I} = {±2 i : i ∈ I} ∪ DI , DI = {0} ∪ {± i ± j : i, j ∈ I, i = j}, and BCI = {± i ± j : i, j ∈ I} ∪ {± i : i ∈ I} = BI ∪ CI ˙ I which only spans are locally ﬁnite root systems which span XI , except for A ˙ ˙ a subspace XI of codimension 1 in XI . The notation A is supposed to indicate

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this fact. For ﬁnite I, say |I| = n ∈ N, we will use the usual notation An = ˙ I , . . . , DI are ˙ {0,...,n} , but Bn , Cn , Dn , BCn for |I| = n. The root systems A A reduced, but R = BCI is not. Any locally ﬁnite root system R is the direct limit of its ﬁnite subsystems, which are ﬁnite root systems, and if R is irreducible it is a direct limit of irreducible ﬁnite subsystems. It is then not too surprising that an irreducible locally ﬁnite root system is isomorphic to exactly one of (i) the ﬁnite exceptional root systems E6 , E7 , E8 , F4 , G2 or ˙ I (|I| ≥ 1), BI (|I| ≥ 2), CI (|I| ≥ 3), DI (ii) one of the root systems A (|I| ≥ 4) or BCI (|I| ≥ 1). Conversely, all the root systems listed in (i) and (ii) are irreducible. The classiﬁcation of locally ﬁnite root systems is independent of the base ﬁeld K, which could therefore be taken to be Q or R. But since the root systems which we will later encounter naturally “live” in K-vector spaces, ﬁxing the base ﬁeld is not convenient. Locally ﬁnite root systems can also be deﬁned via invariant nondegenerate symmetric bilinear forms: 2.9 Proposition ([LN2, 2.10]) For an integral pre-reﬂection system (R, X) the following conditions are equivalent: (i) R is a locally ﬁnite root system; (ii) there exists a nondegenerate strictly invariant form on (R, X), and for every α ∈ R the set R, α∨ is bounded as a subset of Z. In this case, (R, X) has a unique invariant bilinear form (·|·) which is normalized in the sense that 2 ∈ {(α|α) : 0 = α ∈ C} ⊂ {2, 4, 6, 8} for every connected component C of R. The normalized form is nondegenerate in general and positive deﬁnite for K = R. 2.10 Example: Reﬂection systems associated to bilinear forms. Let R be a spanning set of a vector space X containing 0, and let (·|·) be a symmetric bilinear form on X. For α ∈ X we denote the linear form x → (α|x) by α . Let Φ ⊂ {α ∈ R : (α|α) = 0}, deﬁne ∨ : R → X ∗ by 2α if α ∈ Φ, ∨ α = (α|α) (2.10.1) 0 otherwise and then deﬁne sα by 2.2.1. Thus sα is the orthogonal reﬂection in the hyperplane α⊥ if α ∈ Φ, and the identity otherwise. If sα (R) ⊂ R and sα (Φ) ⊂ Φ for all α ∈ Φ, then (R, X, s) is a coherent reﬂection system with the given subset Φ as set of reﬂective roots. The bilinear form (·|·) is invariant, and it is strictly invariant if and only if Rim = R ∩ Rad (·|·), i.e., (·|·) is aﬃne in the sense of 3.8.

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By 2.9 every locally ﬁnite root system is of this type. But also the not necessarily crystallographic ﬁnite root systems, see for example [Hu, 1.2], arise in this way. The latter are not necessarily integral reﬂection systems. 2.11 Example: Roots in Lie algebras with toral subalgebras. Let L be a Lie algebra over a ring k containing 12 (this generality will be used later). Following [Bo1, Ch.VIII, §11] we will call a nonzero triple (e, h, f ) of elements of L an sl2 -triple if [e, f ] = −h,

[h, e] = 2e

For example, in sl2 (k) = the elements esl2 =

01 , 00

and [h, f ] = −2f.

a b : a, b, c ∈ k c −a

hsl2 =

1 0 , 0 −1

0 0 , −1 0

fsl2 =

(2.11.1)

form an sl2 -triple in sl2 (k). In general, for any sl2 -triple (e, h, f ) in a Lie algebra L there exists a unique Lie algebra homomorphism ϕ : sl2 (k) → L mapping the matrices in (1) onto the corresponding elements (e, h, f ) in L. Now let k = K be a ﬁeld of characteristic 0, and let L be a Lie algebra over K and T ⊂ L a subspace. We call T ⊂ L a toral subalgebra, sometimes also called an ad-diagonalizable subalgebra, if L = α∈T ∗ Lα (T ) where Lα (T ) = {x ∈ L : [t, x] = α(t)x for all t ∈ T } (2.11.2) for any α ∈ T ∗ . In this case, (2.11.2) is referred to as the root space decomposition of (L, T ) and R = {α ∈ T ∗ : Lα (T ) = 0} as the set of roots of (L, T ). We will usually abbreviate Lα = Lα (T ). Any toral subalgebra T is abelian, thus T ⊂ L0 and 0 ∈ R unless T = {0} = E (which is allowed but not interesting). We will say that a toral T is a splitting Cartan subalgebra if T = L0 . We denote by Rint the subset of integrable roots of (L, T ), i.e., those α ∈ R for which there exists an sl2 -triple (eα , hα , fα ) ∈ Lα × T × L−α such that the adjoint maps ad eα and ad fα are locally nilpotent. Let X = spanK (R) ⊂ T ∗ and for α ∈ R deﬁne sα ∈ GL(X) by x − x(hα )α, α ∈ Rint , sα (x) = x otherwise. It is a straightforward exercise in sl2 -representation theory to verify that with respect to the reﬂections sα deﬁned above, (R, X) is an integral pre-reﬂection system, which is reduced in case T is a splitting Cartan subalgebra. A priori, sα will depend on hα , but this is not so for the Lie algebras which are of main interest here, the invariant aﬃne reﬂection algebras of §6.

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We note that several important classes of pre-reﬂection systems arise in this way from Lie algebras, e.g., locally ﬁnite root systems [Neh3], Kac–Moody root systems [Kac2], [MP] and extended aﬃne root systems [AABGP], see 3.11. An example of the latter is described in detail in 2.12. We point out that in general R is not symmetric. For example, this is so for the set of roots of classical Lie superalgebras [Kac1] (they can be viewed in the setting above by forgetting the multiplication of the odd part). 2.12 Example: Aﬃne root systems. Let S be an irreducible locally ﬁnite root system. By 2.9, S has a unique normalized invariant form, with respect to which we can introduce short roots Ssh = {α ∈ S : (α|α) = 2} and long roots Slg = {α ∈ S : (α|α) = 4 or 6} = S \ (Ssh ∪ Sdiv ), where Sdiv are the divisible roots of 2.7. Both Ssh ∪ {0} and Slg ∪ {0} are subsystems of S and hence locally ﬁnite root systems. Note that S = {0} ∪ Ssh if S is simply-laced, (Sdiv \ {0}) = ∅ ⇔ S = BCI , and if Slg = ∅ the set of long roots is given by Slg = {α ∈ S : (α|α) = k(S)} for 2, S = F4 or of type BI , CI , BCI , |I| ≥ 2, (2.12.1) k(S) = 3 S = G2 . Let X = span(S) ⊕ Kδ and let (·|·) be the symmetric bilinear form which restricted to span(S) is the normalized invariant form of S and which satisﬁes (X | Kδ) = 0. Furthermore, choose a tier number t(S) = 1 or t(S) = k(S), with t(S) = 1 in case S = BCI , |I| ≥ 2. Finally, put

Φ= α ⊕ t(S)Zδ ∪ α + (1 + 2Z)δ , (α ⊕ Zδ) ∪ α∈Ssh

α∈Slg

0=α∈Sdiv

R = Zδ ∪ Φ. Of course, if Slg = ∅ or Sdiv = {0} the corresponding union above is to be interpreted as the empty set. Note (β|β) = 0 for any β ∈ Φ. One can easily verify that every reﬂection sβ deﬁned in 2.10 leaves Φ and Zδ invariant, hence R = R(S, t(S)) is a symmetric reﬂection system, called the aﬃne root system associated to S and t(S). The following table lists all possibilities for (S, t(S)) where S is locally ﬁnite. For ﬁnite S of type XI , say |I| = l, we also include the corresponding labels of the aﬃne root system R(S, t(S)) as deﬁned [MP] (third column) and [Kac2] (fourth column). If S is reduced, then R = S ⊕ Zδ, i.e., t(S) = 1 in the deﬁnition of Φ, is a called an untwisted aﬃne root system. 2.13 Aﬃne Lie algebras. Every aﬃne root system appears as the set of roots of a Lie algebra E with a splitting Cartan subalgebra H in the sense of 2.11 (the letters (E, H) are chosen in anticipation of the deﬁnition of an extended aﬃne Lie algebra 6.11). This is easily veriﬁed in the untwisted case.

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S

t(S)

aﬃne label [MP]

aﬃne label [Kac2]

reduced

1

S (1)

S (1)

BI (|I| ≥ 2)

2

Bl

(2)

Dl+1

CI (|I| ≥ 3)

2

Cl

(2)

A2l−1

F4

2

F4

G2

3

G2

BC1

−

BC1

BCI (|I| ≥ 2)

1

BCl

(2)

(2)

(2)

E6

(2)

(3)

D4

(3)

(2)

A2

(2)

(2)

A2l

(2)

Indeed, let g be a ﬁnite-dimensional split simple Lie K-algebra, e.g., a ﬁnite-dimensional simple Lie algebra over an algebraically closed ﬁeld K, or let g be a centreless inﬁnite rank aﬃne algebra [Kac2, 7.11]. Equivalently, g is a locally ﬁnite split simple Lie algebra, classiﬁed in [NS] and [Stu], or g is the Tits–Kantor–Koecher algebra of a Jordan pair spanned by a connected grid, whose classiﬁcation is immediate from [Neh3]. In any case, g contains a splitting Cartan subalgebra h such that the set of roots of (g, h) is an irreducible locally ﬁnite root system S, which is ﬁnite if and only if g is ﬁnite dimensional. Moreover, g carries an invariant nondegenerate symmetric bilinear form κ, unique to a scalar (in the ﬁnite-dimensional case κ can be taken to be the Killing form). The Lie algebra E will be constructed in three steps: (I) L = g⊗K[t±1] is the untwisted loop algebra of g. Its Lie algebra product [·, ·]L is given by [x ⊗ p, y ⊗ q]L = [x, y]g ⊗ pq for x, y ∈ g and p, q ∈ K[t±1 ]. Although L could be viewed as a Lie algebra over K[t±1 ], we will view L as a Lie algebra over K. (II) K = L ⊕ Kc with Lie algebra product [., .]K given by [l1 ⊕ s1 c, l2 ⊕ s2 c]K = [l1 , l2 ]L ⊕ σ(l1 , l2 )c for li ∈ L, si ∈ K and σ : L × L → K the K-bilinear map determined by σ(x1 ⊗ tm , x2 ⊗ tn ) = δm+n,0 m κ(x1 , x2 ). (III) E = K ⊕ Kd. Here d is the derivation of K given by d(x ⊗ tm ) = mx ⊗ tm , d(c) = 0, and E is the semidirect product of K and Kd, i.e., [k1 ⊕ s1 d, k2 ⊕ s2 d]E = [k1 , k2 ]K + s2 d(k2 ) − s1 d(k1 ). One calls E the untwisted aﬃne Lie algebra associated to g.

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Denote the root spaces of (g, h) by gα , so that h = g0 . Then H = (h ⊗ K1) ⊕ Kc ⊕ Kd is a toral subalgebra of E whose set of roots in E is the untwisted aﬃne root system associated to the root system S of (g, h): E0 = H, Emδ = h ⊗ Ktm for 0 = m ∈ Z and Eα⊕mδ = gα ⊗ tm for 0 = α ∈ S and m ∈ Z. One can also realize the twisted aﬃne root systems, i.e., those with t(S) > 1, by replacing the untwisted loop algebra L in (I) by a twisted version. To deﬁne it in the case of a ﬁnite-dimensional g, let σ be a nontrivial diagram automorphism of g of order r. Hence g is of type Al (l ≥ 2, r = 2), Dl (l ≥ 4, r = 2), D4 (r = 3) or E6 (r = 2). For an inﬁnite-dimensional g (where diagrams need not exist) we take as σ the obvious inﬁnite-dimensional analogue of the corresponding matrix realization of σ for a ﬁnite-dimensional g, e.g., x → −xt for type A. The twisted loop algebra L is the special case n = 1 of the multi-loop algebra Mr (g, σ) deﬁned in 4.7. The construction of K and E are the same as in the untwisted case. Replacing the untwisted h in the deﬁnition of H above by the subspace hσ of ﬁxed points under σ deﬁnes a splitting Cartan subalgebra of E in the twisted case, whose set of roots is a twisted aﬃne root systems and all twisted aﬃne root systems arise in this way, see [Kac2, 8.3] for details. In the later sections we will generalize aﬃne root systems and aﬃne Lie algebras. Aﬃne root systems will turn out to be examples of extended aﬃne root systems and locally extended aﬃne root systems 3.11, namely those of nullity 1. Extended and locally extended aﬃne root systems are in turn special types of aﬃne reﬂection systems, which we will describe in the next section. In the same vein, untwisted aﬃne Lie algebras are examples of (locally) extended aﬃne Lie algebras, which in turn are examples of aﬃne reﬂection Lie algebras which we will study in §6. From the point of view of extended aﬃne Lie algebras, §6.11, the Lie algebras K and L are the core and centreless core of the extended aﬃne Lie algebra (E, H). As Lie algebras, they are Lie tori, deﬁned in 5.1. The construction from L to E described above is a special case of the construction in 6.13, see the example in 6.14. 2.14 Notes. With the exceptions mentioned in the text and below, all results in this section are proven in [LN1] and [LN2]. Many of the concepts introduced here are well known from the theory of ﬁnite root systems. The notion of tameness (§2.2) goes back to [ABY]. It is equivalent to the requirement that R not have isolated roots, where β ∈ R is called an isolated root (with respect to Rre ), if α + β ∈ R for all α ∈ Rre . In the setting of the Lie algebras considered in §6, tameness of the root system is a consequence of tameness of the algebra, see 6.8. Locally ﬁnite root systems are studied in [LN1] over R. But as already mentioned in [LN1, 4.14] there is a canonical equivalence between the categories of locally ﬁnite root systems over R and over any ﬁeld K of characteristic

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0. In particular, the classiﬁcation is the “same” over any K. The classiﬁcation itself is proven in [LN1, Th. 8.4]. A substitute for the non-existence of root bases are grid bases, which exist for all inﬁnite reduced root systems ([Neh1]). Our deﬁnition of an sl2 -triple follows Bourbaki ([Bo1, Ch.VII, §11]). It diﬀers from the one used in other texts by a sign. Replacing f by −f shows that the two notions are equivalent. Bourbaki’s deﬁnition is more natural in the setting here and avoids some minus signs later. The concept of an integrable root (2.11) was introduced by Neeb in [Nee1]. For a ﬁnite S, aﬃne root systems (2.12) are determined in [Kac2, Prop. 6.3]. The description given in 2.12 is diﬀerent, but of course equivalent to the one in [Kac2]. It is adapted to viewing aﬃne root systems as extended aﬃne root systems of nullity 1. In fact, the description of aﬃne root systems in 2.12 is a special case of the Structure Theorem for extended aﬃne root systems [AABGP, II, Th. 2.37], keeping in mind that a semilattice of nullity 1 is a lattice by [AABGP, II, Cor. 1.7]. The table in 2.12 reproduces [ABGP, Table 1.24]. Other examples of reﬂection systems are the set of roots associated to a “root basis” in the sense of H´ee [He]. Or, let R be the root string closure of the real roots associated to root data `a la Moody–Pianzola [MP, Ch. 5]. Then R is a symmetric, reduced and integral reﬂection system. The Weyl group of a pre-reﬂection system is in general not a Coxeter group. For example, this already happens for locally ﬁnite root systems, see [LN1, 9.9]. As a substitute, one can give a presentation by conjugation, essentially the relation (ReS4), see [LN1, 5.12] for locally ﬁnite root systems and [Ho] for Weyl groups of extended aﬃne root systems (note that the Weyl groups in [Ho] and also in [AABGP] are deﬁned on a bigger space than the span of the roots and hence our Weyl groups are homomorphic images of the Weyl groups in [Ho] and [AABGP]).

3 Aﬃne reﬂection systems In this section we describe extensions of pre-reﬂection systems and use them to deﬁne aﬃne reﬂection systems, which are a generalization of extended aﬃne root systems. The roots of the Lie algebras, which we will be studying in §6, will turn out to be examples of aﬃne reﬂection systems. Hints to references are given in 3.12. 3.1 Partial sections. Let f : (R, X) → (S, Y ) be a morphism of prereﬂection systems with f (R) = S, and let S ⊂ S be a subsystem spanning Y . A partial section of f over S is a morphism g : (S , Y ) → (R, X) of prereﬂection systems such that f ◦ g = IdY . The name is (partially) justiﬁed by the fact that a partial section leads to a partial section of the canonical epimorphism W (R) → W (S), namely a section deﬁned over W (S ).

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Let f : (R, X) → (S, Y ) be a morphism of pre-reﬂection systems satisfying f (R) = S. In general, a partial section of f over all of S need not exist. However, partial sections always exist in the category of reﬂection systems. 3.2 Extensions. Recall 2.2: f (Rim ) ⊂ S im and f (Rre ) ⊂ {0} ∪ S re for any morphism f : (R, X) → (S, Y ) of pre-reﬂection systems. We call f an extension if f (Rim ) = S im and f (Rre ) = S re . We will say that R is an extension of S if there exists an extension f : R → S. We mention some properties of an extension f : R → S. By deﬁnition, f is surjective. Also, R is coherent if and only if S is so, and in this case f induces a bijection C → f (C) between the set of connected components of Re(R) and of Re(S). In particular, Re(R) is connected ⇐⇒ Re(S) is connected ⇐⇒ Re(R) is indecomposable. Moreover, R is integral if and only if S is so. Finally, f maps a root string S(β, α), β ∈ R, α ∈ Rre , injectively to S f (β), f (α) . If R is an extension of a nondegenerate S, e.g., a locally ﬁnite root system, then S is unique up to isomorphism and will be called the quotient pre-reﬂection system of R. On the other hand, if R is nondegenerate every extension f : R → S is injective, hence an isomorphism. In particular, a locally ﬁnite root system R does not arise as a nontrivial extension of a prereﬂection system S. But as we will see below, a locally ﬁnite root system does have many interesting extensions. A ﬁrst example of an extension is the canonical projection R → S for R an aﬃne root system associated to a locally ﬁnite irreducible root system S, 2.12. In this case, S is the quotient pre-reﬂection system = quotient root system of R. 3.3 Extension data. Let (S, Y ) be a pre-reﬂection system, let S be a subsystem of S with span(S ) = Y and let Z be a K-vector space. A family L = (Λξ )ξ∈S of nonempty subsets of Z is called an extension datum of type (S, S , Z) if (ED1) for all ξ, η ∈ S and all λ ∈ Λξ , µ ∈ Λη we have µ − η, ξ ∨ λ ∈ Λsξ (η) , (ED2) 0 ∈ Λξ for all ξ ∈ S , and (ED3) Z is spanned by the union of all Λξ , ξ ∈ S. The only condition on Λ0 is (ED2): 0 ∈ Λ0 . Moreover, Λ0 is related to the other Λξ , 0 = ξ ∈ S, only via the axiom (ED3). As (ED3) only serves to determine Z and can always be achieved by replacing Z by the span of the Λξ , ξ ∈ S, it follows that one can always modify a given extension datum by replacing Λ0 by some other set containing 0. However, as 3.6 shows, this may change the properties of the associated pre-reﬂection system. Any extension datum L has the following properties, where WS ⊂ W (S) is the subgroup generated by all sξ , ξ ∈ S .

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Λ−ξ = −Λξ

for all ξ ∈ S re ,

2Λξ − Λξ ⊂ Λξ for all ξ ∈ S re , Λη = Λw (η) for all η ∈ S and w ∈ WS , Λη − η, ξ ∨ Λξ ⊂ Λη for ξ ∈ S , η ∈ S Λξ = Λ−ξ for ξ ∈ S .

and

In particular, Λξ is constant on the WS -orbits of S. However, in general, the Λξ are not constant on all of S, see the examples in 3.7. One might wonder if the conditions (ED1)–(ED3) are strong enough to force the Λξ to be subgroups of (Z, +). But it turns out that this is not the case. For example, this already happens for the extension data of locally ﬁnite root systems, 3.7. On the other hand, if R is an integral pre-reﬂection system and Λ a subgroup of (Z, +) which spans Z, then Λξ ≡ Λ is an example of an extension datum, the so-called untwisted case, cf. the deﬁnition of an untwisted aﬃne root system in 2.12. The conditions above have appeared in the context of reﬂection spaces (not to be confused with a reﬂection system). Recall ([Loo]) that a reﬂection space is a set S with a map S × S → S : (s, t) → s · t satisfying s · s = s, s · (s · t) = t and s · (t · u) = (s · t) · (s · u) for all s, t, u ∈ S. A reﬂection subspace of a reﬂection space (S, ·) is a subset T ⊂ S such that t1 · t2 ∈ T for all t1 , t2 ∈ T . In this case, T is a reﬂection space with the induced operation. Any abelian group (Z, +) is a reﬂection space with respect to the operation x · y = 2x − y for x, y ∈ Z. Correspondingly, a reﬂection subspace of the reﬂection space (Z, ·) is a subset A ⊂ Z satisfying 2a − b ∈ A for all a, b ∈ A, symbolically 2A − A ⊂ A. Hence all Λξ , ξ ∈ S re , are reﬂection subspaces of (Z, ·). Moreover, for any subset A of Z it is easily seen that A − 2A ⊂ A ⇔ A = −A and 2A + A ⊂ A ⇔ A = −A and 2A − A ⊂ A.

(3.3.1)

A subset A satisfying (3.3.1) will be called a symmetric reﬂection subspace. We will consider 0 as the base point of the reﬂection space Z. Also, we denote by Z[A] the subgroup of (Z, +) generated by A ⊂ Z. Then the following are equivalent ([NY, 2.1]): (i) 0 ∈ A and A − 2A ⊂ A, (ii) 0 ∈ A and 2A − A ⊂ A, (iii) 2Z[A] ⊂ A and 2Z[A] − A ⊂ A, (iv) A is a union of cosets modulo 2Z[A], including the trivial coset 2Z[A]. In this case, A will be called a pointed reﬂection subspace. It is immediate from the above that every Λξ , ξ ∈ S re is a pointed reﬂection subspace. We note that pointed reﬂection subspaces are necessarily symmetric. It is obvious from (iv) above that a pointed reﬂection subspace is in general not a subgroup of (Z, +).

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The following theorem characterizes extensions in terms of extension data. 3.4 Theorem. Let (S, Y ) be a pre-reﬂection system. (a) Let L = (Λξ )ξ∈S be an extension datum of type (S, S , Z). Put X := Y ⊕ Z, denote by π : X → Y the projection with kernel Z, and deﬁne

R := ξ ⊕ Λξ ⊂ X and sα (x) := sξ (y) ⊕ z − y, ξ ∨ λ , (3.4.1) ξ∈S

for all α = ξ⊕λ ∈ ξ⊕Λξ ⊂ R and all x = y⊕z ∈ X. Then R is a pre-reﬂection system in X, denoted E = E(S, S , L). Moreover, π : (R, X) → (S, Y ) is an extension of pre-reﬂection systems, and the canonical injection ι : Y → X is a partial section of π over S . (b) Conversely, let f : (R, X) → (S, Y ) be an extension and let g : S → R be a partial section of f , cf. 3.1. For every ξ ∈ S deﬁne Rξ ⊂ R and Λξ ⊂ Z := Ker(f ) by Rξ = R ∩ f −1 (ξ) = g(ξ) ⊕ Λξ . (3.4.2) Then L = (Λξ )ξ∈S is an extension datum of type (S, S , Z), and the vector space isomorphism ϕ : Y ⊕ Z ∼ = X sending y ⊕ z to g(y) ⊕ z is an isomorphism E(S, S , L) ∼ = R of pre-reﬂection systems making the following commutative diagram: S E EE g ι yyy EE EE yy y " |y ϕ /R. E EE y EE y y E yy π EE " |yy f S (c) In the setting of (b), the following are equivalent for g ∈ HomK (Y, Z): (i) g : S → R is another partial section of f , (ii) there exists ϕ ∈ HomK (Y, Z) such that g = g + ϕ and ϕ(ξ ) ∈ Λξ for all ξ ∈ S . In this case, the extension datum L = (Λξ )ξ∈S deﬁned by (3.4.2) with respect to g is related to the extension datum L by Λξ = Λξ − ϕ(ξ)

for ξ ∈ S.

(3.4.3)

In the context of root-graded Lie algebras, the partial sections g and g in (c) lead to isotopic Lie algebras, see 5.2 and 6.4. 3.5 Corollary. Let (S, Y ) be a pre-reﬂection system and let L = (Λξ )ξ∈S be an extension datum of type (S, S , Z). Let R = E(S, S , Z) be the pre-reﬂection system deﬁned in 3.4(a).

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1. R is reduced if and only if, for all 0 = ξ ∈ S, ξ, cξ ∈ S for c ∈ K \ {0, ±1}

=⇒

Λcξ ∩ cΛξ = ∅.

2. R is symmetric if and only if S is symmetric and Λ−ξ = −Λξ for all ξ ∈ S im . 3. R is a reﬂection system if and only if S is a reﬂection system. 3.6 Aﬃne reﬂection systems. A pre-reﬂection system is called an aﬃne reﬂection system if it is an extension of a locally ﬁnite root system. A morphism between aﬃne reﬂection systems is a morphism of the underlying pre-reﬂection systems. The name aﬃne reﬂection system, as opposed to aﬃne pre-reﬂection system, is justiﬁed in view of 3.5(c), since a locally ﬁnite root system is a reﬂection system and hence so is any extension of it. A morphism between aﬃne reﬂection systems is a morphism of the underlying reﬂection systems (2.2). Isomorphisms have the following characterization. A vector space isomorphism f : X → X is an isomorphism of the reﬂection systems (R, X) and (R , X ) if and only if f (Rim ) = Rim and f (Rre ) = Rre .

(3.6.1)

By deﬁnition, the nullity of an aﬃne reﬂection system R is the rank of the torsion-free abelian group Rim generated by Rim , i.e., nullity = dimQ ( Rim ⊗Z Q). It is clear from the deﬁnitions that a locally ﬁnite root system is an aﬃne reﬂection system of nullity 0. The aﬃne root systems R(S, t(S)) of 2.12 are aﬃne reﬂection systems of nullity 1, see also 3.11(a). In the following let (R, X) be an aﬃne reﬂection system and let f : (R, X) → (S, Y ) be an extension where (S, Y ) is a locally ﬁnite root system. As mentioned in 3.2, R is then coherent and integral. Also, nondegeneracy of S implies that S is unique, up to a unique isomorphism. We will call S the quotient root system of R in this context and refer to f as the canonical projection. Let Z = Ker(f ). One can show that the extension f has a partial section g over Sind . Let L = (Λξ )ξ∈S be the extension datum of type (S, Sind , Z) associated to f and g in Th. 3.4(b). Thus, up to an isomorphism we may assume that R is given by (3.4.1). The extension datum L appearing there has some special properties besides the ones mentioned in 3.5. Namely, for 0 = ξ, η ∈ S and w ∈ W (S) we have Λξ = Λ−ξ = −Λξ = Λw(ξ) for all w ∈ W (S), Λξ ⊃ Λξ − ξ, η ∨ Λη , and Λξ ⊃ Λ2ξ whenever 2ξ ∈ S.

Moreover, deﬁne Λdiﬀ := 0=ξ∈S Λξ − Λξ . Then ZΛdiﬀ = Λdiﬀ and we have: (a) R is symmetric if and only if Λ0 = −Λ0 .

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(b) All root strings S(β, α), β ∈ R, α ∈ Rre , are unbroken if and only if Λdiﬀ ⊂ Λ0 . (c) R is tame (cf. 2.2) if and only if Λ0 ⊂ Λdiﬀ . (d) |S(β, α)| ≤ 5 for all β ∈ R and α ∈ Rre . We now describe extension data in the irreducible case in more detail. 3.7 Extension data for irreducible locally ﬁnite root systems. Let S be an irreducible locally ﬁnite root system. Recall the decomposition S = Ssh ∪ Slg ∪ Sdiv of 2.12. Let L = (Λξ )ξ∈S be an extension datum of type (S, Sind , Z). One knows that W (S) operates transitively on the roots of the same length ([LN1, 5.6]). By 3.6 we can therefore deﬁne Λsh , Λlg and Λdiv by ⎧ ⎪ ⎨Λsh for α ∈ Ssh , Λα = Λlg (3.7.1) for α ∈ Slg , ⎪ ⎩ Λdiv for 0 = α ∈ Sdiv . By convention, Λlg and Λdiv are only deﬁned if the corresponding set of roots Slg and Sdiv \ {0} are not empty (we always have Ssh = ∅). To streamline the presentation, this will not be speciﬁed in the following. It will (hopefully) always be clear what is meant from the context. (a) We have already seen in 3.3 that in general Λsh and Λlg are pointed reﬂection subspaces and that Λdiv is a symmetric reﬂection subspace of Λ. In the setting of this subsection we have in addition the following: (i) Λsh is a subgroup of Z, if Ssh ∪ {0} contains a subsystem of type A2 , (ii) Λlg is a subgroup of Z, if Slg ∪ {0} contains a subsystem of type A2 , and (iii) the following relations hold for S as indicated and k = k(S) deﬁned in 2.12: Λsh + Λlg ⊂ Λsh , Λsh + Λdiv ⊂ Λsh , Λlg + Λdiv ⊂ Λlg ,

Λlg + kΛsh ⊂ Λlg , Λdiv + 4Λsh ⊂ Λdiv , Λdiv + 2Λlg ⊂ Λdiv ,

(Slg = ∅) (S = BC1 ) (S = BCI , |I| ≥ 2).

(b) Conversely, given a subset Λ0 with 0 ∈ Λ0 , pointed reﬂection subspaces Λsh and Λlg (if Slg = ∅) and a symmetric reﬂection subspace Λdiv ⊂ Λ (if {0} Sdiv ) satisfying (i) – (iii) of (a), then (3.7.1) deﬁnes an extension datum of type (S, Sind , Z). Moreover, the following hold where, we recall k is deﬁned in (2.12.1): (i) Λsh is a subgroup of Z if S = A1 , BI or BCI for any I, (ii)’ Λlg is a subgroup of Z if S = BI or BCI and |I| ≥ 3, or if S = F4 or G2 , (iv) kΛsh ⊂ Λlg ⊂ Λsh , 4Λsh ⊂ Λdiv ⊂ Λsh and Λdiv ⊂ Λlg ⊂ Λsh . (v) The inclusions Λsh + Λdiv ⊂ Λsh and Λdiv + 4Λsh ⊂ Λdiv hold for all S = BCI . One therefore obtains the following description of L× := (Λsh , Λlg , Λdiv ) for the various types of irreducible root systems. Note that the only condition on Λ0 is 0 ∈ Λ0 because of (ED2).

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˙ I , DI (|I| ≥ 4), E6 , E7 or E8 : L× = (Λsh ), (I) S is simply laced, i.e., S = A where Λsh is a pointed reﬂection subspace for S = A1 and a subgroup of Z otherwise. (II) S = BI (|I| ≥ 2), CI (|I| ≥ 3) or F4 : L× = (Λsh , Λlg ), where Λsh and Λlg are pointed reﬂection subspaces satisfying Λsh + Λlg ⊂ Λsh and Λlg + 2Λsh ⊂ Λlg . Moreover, Λsh is a subgroup of Z if S = CI or F4 , while Λlg is a subgroup if S = BI , |I| ≥ 3 or F4 . (III) S = G2 : L× = (Λsh , Λlg ), where Λsh and Λlg are subgroups of Z satisfying Λsh + Λlg ⊂ Λsh and Λlg + 3Λsh ⊂ Λlg . (IV) S = BC1 : L× = (Λsh , Λdiv ), where Λsh is a pointed reﬂection subspace, Λdiv is a symmetric reﬂection subspace and Λsh + Λdiv ⊂ Λsh and Λdiv + 4Λsh ⊂ Λdiv . (V) S = BCI , |I| ≥ 2: L× = (Λsh , Λlg , Λdiv ) where Λsh and Λlg are pointed reﬂection subspaces, Λdiv is a symmetric reﬂection subspace and Λsh + Λlg ⊂ Λsh , Λlg + 2Λsh ⊂ Λlg , Λlg + Λdiv ⊂ Λlg , Λdiv + 2Λlg ⊂ Λdiv . Moreover, if |I| ≥ 3 we require that Λlg is a subgroup of Z. 3.8 Aﬃne forms. Our deﬁnition of aﬃne reﬂection systems follows the approach of [Bo2] where root systems are deﬁned without reference to a bilinear form. In the literature, it is customary to deﬁne aﬃne root systems and their generalizations, the extended aﬃne root systems (EARS), in real vector spaces using positive semideﬁnite forms. We will therefore give a characterization of aﬃne reﬂections systems in terms of aﬃne invariant forms where, by deﬁnition, an aﬃne form of a pre-reﬂection system (R, X) over K is an invariant form b satisfying Rim = R ∩ Rad b, cf. (2.5.2). In particular, aﬃne forms are strictly invariant in the sense of 2.5. As an example, the forms used in the theory of EARS are aﬃne forms in our sense. In particular, the form (·|·) in 2.12 is an aﬃne form of the aﬃne root system R. 3.9 Theorem. Let (R, X) be a pre-reﬂection system. Then (R, X) is an aﬃne reﬂection system if and and only if it satisﬁes the following conditions: (i) (R, X) is integral, (ii) (R, X) has an aﬃne form, and (iii) R, α∨ is bounded for every α ∈ Rre . In this case: (a) Let b be an aﬃne form for (R, X) and let f : X → X/ Rad b be the canonical map. Then (S, Y ) = (f (R), X/ Rad b) is the quotient root system of R and f its canonical projection. Moreover, Re(R) is connected if and only if S is irreducible. (b) There exists a unique aﬃne form (·|·)a on (R, X) that is normalized in the sense of 2.9, i.e., for every connected component C of Re(R) we have 2 ∈ {(α|α)a : 0 = α ∈ C} ⊂ {2, 3, 4, 6, 8}.

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The form (·|·)a satisﬁes {(α|α)a : 0 = α ∈ C} ∈ {2}, {2, 4}, {2, 6}, {2, 8}, {2, 4, 8} . Its radical is Rad (·|·)a = Ker f . If K = R then (·|·)a is positive semideﬁnite. 3.10 Corollary. A pre-reﬂection system over the reals is an aﬃne reﬂection system if and only if it is integral and has a positive semideﬁnite aﬃne form. 3.11 Special types of aﬃne reﬂection systems. As usual, the rank of a reﬂection system (R, X) is deﬁned as rank(R, X) = dim X. If R has ﬁnite rank and K = R, we will say that R is discrete if R is a discrete subset of X. (a) Let (R, X) be an aﬃne reﬂection system over K = R with the following properties: R has ﬁnite rank, Re(R) is connected, R = −R and R is discrete. Then R is called (i) an EARS, an abbreviation for extended aﬃne root system, if R is reduced, tame (see (3.6.c)), and all root strings are unbroken; (ii) a SEARS, an abbreviation of Saito’s extended aﬃne root system, if R = Re(R). It was shown in [Az3, Th. 18] that every reduced SEARS can be uniquely extended to an EARS and, conversely, the reﬂective roots of an EARS are the nonzero roots of a SEARS. This is now immediate. Indeed, by the results mentioned in 3.6, an aﬃne reﬂection system is tame and has unbroken root strings if and only Rim = Λdiﬀ . The aﬃne root systems R(S, t(S)) of 2.12 with S ﬁnite are precisely the EARSs of nullity 1 ([ABGP]). (b) In [MY], Morita and Yoshii deﬁne a LEARS, an abbreviation for locally extended aﬃne root system. In our terminology, this is a symmetric aﬃne reﬂection system R over K = R such that R = Re(R) is connected. The equivalence of this deﬁnition with the one in [MY] follows from 3.10. (c) In [Az4], Azam deﬁnes a GRRS, an abbreviation of a generalized reductive root system. In our terminology, this is a symmetric real reduced, discrete aﬃne reﬂection system R which has ﬁnite rank and unbroken root strings. 3.12 Notes. With the exceptions mentioned in the text and below, all results in this section are proven in [LN2]. If S is an integral reﬂection system, our deﬁnition of an extension datum in 3.3 makes sense for any abelian group Z instead of a K-vector space. This generality is however not needed for extension data arising from Lie algebras with toral subalgebras, 2.11, like the generalizations of aﬃne Lie algebras to be discussed later. But we note that the results on extension data stated in 3.6 and 3.7 are true for an abelian group Z. In this generality, but for S a ﬁnite irreducible root system, extension data without Λ0 were deﬁned by Yoshii in [Yo4] as “root systems of type S extended by Z”. The results stated in 3.7 extend [Yo4, Th. 2.4] to the setting of locally ﬁnite root systems.

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We point out that our deﬁnition of an EARS is equivalent to the one given by Azam, Allison, Berman, Gao, and Pianzola in [AABGP, II, Def. 2.1]. That our deﬁnition of a SEARS is the same as Saito’s deﬁnition of an “extended aﬃne root system” in [Sa], follows from 3.10. Our characterization of extensions in terms of extension data 3.4 generalizes the Structure Theorem for extended aﬃne root systems [AABGP, II, Th. 2.37] as well as [MY, Prop. 4.2] and — modulo the limitations mentioned above — the description of root systems extended by an abelian group in [Yo4, Th. 3.4]. The isomorphism criterion (3.6.1) is proven in [Yo6] for the case of LEARSs (3.11(c)). The extension data arising in the theory of extended aﬃne root systems are studied in detail in [AABGP, Ch. II] and [Az1]. These references are only a small portion of what is presently known on extended aﬃne root systems and their Weyl groups. Many of these results likely have generalizations to the setting of aﬃne reﬂection systems.

4 Graded algebras In the previous sections 2 and 3 we reviewed the “combinatorics” needed to describe extended aﬃne Lie algebras and their generalizations in §6. In this and the following section we will introduce the necessary algebraic concepts. All of them have to do with algebras graded by an arbitrary abelian group, usually denoted Λ and written additively. (The reader will notice that at least in the ﬁrst few subsections Λ need not be abelian.) 4.1 Algebras per se. Throughout, we will consider algebras A over a unital commutative associative ring of scalars k. Unless explicitly stated otherwise, we do not assume that A belongs to some special variety of algebras, like associative algebras or Lie algebras. Therefore, an algebra or a k-algebra if we want to be more precise, is simply a k-module A together with a k-bilinear map A × A → A, (a, b) → ab, the product of A. As is customary, if A happens to be a Lie algebra, its product will be denoted [a, b] or sometimes just [a b] instead of ab. Let A be an algebra. The k-module of all its k-linear derivations is always a Lie algebra, denoted Derk A. A symmetric bilinear form (·|·) : A × A → k is called invariant if (ab|c) = (a|bc) holds for all a, b, c ∈ A; it is nondegenerate if (a|A) = 0 implies a = 0. We put AA = spank {ab : a, b ∈ A}. The algebra A is perfect if A = AA, and is simple if AA = 0 and if {0} and A are the only ideals of A. Any unital algebra is perfect, and any simple algebra is perfect since AA is an ideal of A. 4.2 The centroid of an algebra. The centroid of an algebra A is the subalgebra Centk (A) of the endomorphism algebra Endk (A) consisting of the

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k-linear endomorphisms of A, which commute with the left and right multiplications of A. One always has k IdA ⊂ Centk (A), but in general this is a proper inclusion. If the canonical map k → Centk (A) is an isomorphism, A is called central , and one says that A is central-simple if A is central and simple, see §4.4 for a characterization of central-simple algebras. The centroid is always a unital associative algebra, but not necessarily commutative in general. For example, the centroid of the null algebra (all products are 0) is the full endomorphism algebra. On the other hand, the centroid of a perfect algebra is always commutative. For example, let A be a unital associative algebra and denote by Z(A) = {z ∈ A : za = az for all a ∈ A} its centre. Then Z(A) → Centk (A),

z → Lz

(A associative)

(4.2.1)

is an isomorphism of algebras, where Lz denotes the left multiplication by z. However, even if the names are quite similar, one should not confuse the centroid with the centre of A! A nonzero Lie algebra L may well be centreless in the sense that Z(L) = {0}, but always has 0 = k IdL ∈ Cent(L), see 4.12(a) for an example. Since Centk (A) is a subalgebra of Endk (A) one can consider A as a left module over Centk (A). This change of perspective is particularly useful if Centk (A) is commutative since then A is an algebra over Centk (A). In general, an algebra A is called fgc (for finitely generated over its centroid) if A is a ﬁnitely generated Centk (A)-module. Fgc algebras are more tractable than general algebras, and we will characterize various classes of fgc algebras throughout this paper, see 4.6, 4.7, 4.13 and 5.8. 4.3 Graded algebras. Recall that Λ is an abelian group. A Λ-graded algebra is an algebra A together with a family (Aλ : λ ∈ Λ) of submodules Aλ of A satisfying A = λ∈Λ Aλ and Aλ Aµ ⊂ Aλ+µ for all λ, µ ∈ Λ. We will usually indicate this by saying “Let A = λ Aλ be a Λ-graded algebra ...” or simply “Let A be graded algebra...”. We will encounter many example of graded algebras. But an immediate and important example is k[Λ], the group algebra of Λ over k. By deﬁnition, k[Λ] is a free k-module with a k-basis in bijection with Λ, say by λ → z λ , and product determined by z λ z µ = z λ+µ for λ, µ ∈ Λ. It is graded by (k[Λ])λ = k z λ and is a unital associative algebra. We will need some more terminology for a graded algebra A. The submodules Aλ of A are referred to as homogeneous spaces. We will occasionally use subscripts to describe them, in particular whenever we consider an algebra with two gradings, say A = λ∈Λ Aλ and A = q∈Q Aq . The two gradings are called compatible if, putting Aλq := Aλ ∩Aq , we have Aλ = q∈Q Aλq for all λ ∈ Λ. We will indicate compatible gradings by saying that A is (Q, Λ)-graded. The support of a Λ-graded algebra A is the set suppΛ A = {λ ∈ Λ : Aλ = 0}. In general suppΛ A is a proper subset of Λ. The subgroup of Λ generated

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by suppΛ A will be denoted suppΛ A. One says that A has full support if

suppΛ A = Λ. Since the grading of A only depends on suppΛ A, it is of course always possible, and sometimes even useful, to assume that a graded algebra has full support. Two graded algebras A and A with grading groups Λ and Λ , respectively, are called isograded-isomorphic if there exists an isomorphism f : A → A of the underlying k-algebras and a group isomorphism ϕ : suppΛ A →

suppΛ A satisfying f (Aλ ) = Aϕ(λ) for all λ ∈ suppΛ A. Note that ϕ is uniquely determined by f . A more restrictive concept is that of a gradedisomorphism between two Λ-graded algebras A and A , which by deﬁnition is an isomorphism f : A → A of the ungraded algebras satisfying f (Aλ ) = Aλ for all λ ∈ Λ. A Λ-graded algebra is called graded-simple if AA = 0 and {0} and A are the only Λ-graded ideals of A. For example, k[Λ] is graded-simple if and only if k is a ﬁeld. But of course in this case k[Λ] may have many nontrivial ungraded ideals. A bilinear form (·|·) : A × A → k on A is Λ-graded if (Aλ |Aµ ) = 0 for λ + µ = 0. For example, the Killing form of a ﬁnite-dimensional Λ-graded Lie algebra is Λ-graded. The radical of an invariant Λ-graded symmetric bilinear form (·|·) is a graded ideal of A, namely Rad(·|·) = {a ∈ A : (a|A) = 0}. A Λ-graded bilinear form (·|·) is nondegenerate, i.e. Rad(·|·) = {0}, if and only if for all λ ∈ Λ the restriction of (·|·) to Aλ × A−λ is a nondegenerate pairing. We denote by GIF(A, Λ) the k-module of all Λ-graded invariant symmetric bilinear forms on A. If L is a perfect Λ-graded Lie algebra, then Rad(·|·) ⊃ Z(L) = {z ∈ L : [z, L] = 0}, the centre of L. Since Z(L) is also Λ-graded, the quotient algebra inherits a canonical Λ-grading, and every (·|·) ∈ GIF(L, Λ) induces an invariant symmetric bilinear form on L/ Z(L) by (¯ x|¯ y ) = (x|y), where x → x ¯ denotes the canonical map. This gives rise to an isomorphism (Lie algebras). (4.3.1) GIF(L, Λ) ∼ = GIF(L Z(L), Λ) If A is a unital algebra, say with identity element 1, then necessarily 1 ∈ A0 . If in addition A is associative, then GIF(A, Λ) ∼ (associative algebras) (4.3.2) = (A0 [A, A]0 )∗ by assigning to any linear form ϕ of A0 [A, A]0 the bilinear form (a | b)ϕ = ϕ((ab)0 + [A, A]0 ). Here [A, A] = spank {ab − ba : a, b ∈ A}, and (ab)0 and [A, A]0 denote the 0-component of ab and [A, A]. For example, the group algebra k[Λ] has up to scalars just one invariant symmetric bilinear form, which assigns to (a, b) the z 0 -coeﬃcient of ab with respect to the canonical k-basis. An endomorphism f of the underlying k-module of A is said to have degree λ if f (Aµ ) ⊂ Aλ+µ for all µ ∈ Λ. The submodules (Endk A)λ , consisting of

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79

all endomorphisms of degree λ, are the homogeneous spaces of the Λ-graded subalgebra grEndk (A) = λ∈Λ (Endk A)λ of the associative algebra Endk (A). We put grDerk (A) = grEndk (A) ∩ Derk (A) =

grCentk (A) = grEndk (A) ∩ Centk (A) =

λ∈Λ (Derk

λ∈Λ

A)λ ,

and

(Centk A)λ .

Then (Derk A)λ and (Centk A)λ are, respectively, the derivations and centroidal transformations of A of degree λ. In general, grDerk (A) or grCentk (A) are proper subalgebras of Derk (A) and Centk (A). However, grDerk (A) = Derk (A) if A is a ﬁnitely generated algebra ([Fa, Prop. 1]). Similarly, one has that grCentk (A) = Centk (A) if A is ﬁnitely generated as ideal. For example, any unital algebra is ﬁnitely generated as ideal (namely by 1 ∈ A). We will discuss some examples of (gr) Derk A and (gr) Centk A later, see 4.12, 4.13, 7.9 and 7.12. We call A graded-central if the canonical map k → (Centk A)0 is an isomorphism, and graded-central-simple if A is graded-central and graded-simple. It is well known that an (ungraded) algebra A over a ﬁeld k is central-simple if and only if A ⊗k K is simple for every extension ﬁeld K of k. The following theorem extends this to the case of graded algebras. 4.4 Theorem. Let A be a graded-simple k-algebra, where k is a ﬁeld. Then the following are equivalent: (i) A is graded-central-simple, (ii) for any extension ﬁeld K of k the base ﬁeld extension A⊗k K is a gradedcentral-simple K-algebra, (iii) for any extension ﬁeld K of k the algebra A ⊗k K is a graded-simple K-algebra, (iii) A ⊗k (Centk A)0 is graded-simple. In this theorem, “base ﬁeld extension” means that we consider A ⊗k K as the K-algebra with product (a1 ⊗ x1 ) (a2 ⊗ x2 ) = (a1 a2 ) ⊗ (x1 x1 ) for ai ∈ A and xi ∈ K and the Λ-grading determined by (A ⊗k K)λ = Aλ ⊗k K. We will describe the centroids of graded-simple algebras in 4.6, after we have introduced special classes of associative graded algebras. 4.5 Associative (pre)division-graded algebras and tori. A Λ-graded unital associative algebra A = λ∈Λ Aλ is called • predivision-graded , if every nonzero homogeneous space contains an invertible element; • division-graded if every nonzero homogenous element is invertible; • an associative torus if A is predivision-graded, k is a ﬁeld and dimk Aλ ≤ 1.

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For example, any group algebra k[Λ] is predivision-graded. Moreover, k[Λ] is division-graded if and only if k is a ﬁeld, and in this case k[Λ] is a torus. Group algebras cover only a small part of all possible predivision-graded algebras. If A is such an algebra, it is easily seen that the support of A is a subgroup of the grading group Λ. To simplify the notation, we assume suppΛ A = Λ in this subsection. Let (uλ : λ ∈ Λ) be a family of invertible elements uλ ∈ Aλ and put B = A0 . Then (uλ ) is a B-basis of the left B-module A, and the product of A is uniquely determined by the two rules uλ uµ = τ (λ, µ) uλ+µ

and uλ b = bσ(λ) uλ

(b ∈ B)

(4.5.1)

where τ : Λ × Λ → B × and σ : Λ → Autk (B) are functions. Associativity of A leads to the two identities τ (λ, µ)σ(ν) τ (ν, λ + µ) = τ (ν, λ) τ (νλ, µ), and bσ(λ) )σ(ν) τ (ν, λ) = τ (ν, λ) bσ(ν+λ)

(4.5.2)

for ν, λ, µ ∈ Λ and b ∈ B. Conversely, given any unital associative k-algebra B and functions τ , σ as above satisfying (4.5.2), one can deﬁne a Λ-graded kalgebra by (4.5.1). It turns out to be an associative predivision-graded algebra. Algebras arising in this way are called crossed product algebras and denoted (B, (uλ ), τ, σ). To summarize, any associative predivision-graded algebra A with support Λ is graded-isomorphic to a crossed product algebra (B, (uλ ), τ, σ). A is division-graded iﬀ B is a division algebra, and A is a torus iﬀ B = k is a ﬁeld.

(4.5.3)

The description of predivision-graded algebras as crossed product algebras may not be very illuminating, except that it (hopefully) demonstrates how general this class is. The reader may be more comfortable with the subclass of twisted group algebras deﬁned by the condition that σ is trivial, i.e., σ(λ) = IdB for all λ ∈ Λ, and denoted B t [Λ]. For example, any commutative crossed product algebra is a twisted group algebra. Or, up to graded isomorphism, a Λ-torus with full support is the same as a twisted group algebra k t [Λ]. In the torus case, (4.5.2) says that τ is a 2-cocycle of the group Λ with values in k × . We will describe the special case of associative Zn -tori in 4.13. Twisted group algebras arise naturally as centroids of graded-simple algebras. 4.6 Proposition. Let A = λ∈Λ Aλ be a graded-simple k-algebra. (a) Then Centk (A) is a commutative associative division-graded k-algebra,

EALAs and Generalizations

Centk (A) = grCentk (A) =

γ∈Γ

(Centk A)γ

where

81

(4.6.1)

Γ = suppΛ Centk A is a subgroup of Λ, called the central grading group of A. Hence Centk (A) is a twisted group algebra over the ﬁeld K = (Centk A)0 . (b) A is a graded-central-simple K-algebra. (c) A is a free Centk (A)-module with Γ + suppΛ A ⊂ suppΛ A. Moreover, A has ﬁnite rank as Centk (A)-module, i.e., A is fgc, if and only if [suppΛ A : Γ ] < ∞

and

dimK (Aλ ) < ∞ for all λ ∈ Λ.

(4.6.2)

The interest in algebras which are graded-central-simple and also fgc comes from the following important realization theorem. 4.7 Theorem. ([ABFP1, Cor. 8.3.5]) Let k be an algebraically closed ﬁeld of characteristic 0 and let Λ be free abelian of ﬁnite rank. Then the following are equivalent for a Λ-graded k-algebra B: (i) B is graded-central-simple, fgc and [Λ : Γ ] < ∞ where Γ is the central grading group of B; (ii) B is isograded-isomorphic to a multiloop algebra Mm (A, σ) for some ﬁnite-dimensional simple k-algebra A, described below. The multiloop algebras referred to in (ii) provide an interesting class of examples of graded algebras. They are deﬁned as follows (A can be arbitrary, but we assume that k is a ﬁeld with enough roots of unity). We let m = (m1 , . . . , mn ) ∈ Nn+ be an n-tuple of positive integers and let π : Zn → Z/(m1 ) ⊕ · · · ⊕ Z/(mn ) =: Ξ be the canonical map. Furthermore, σ = (σ1 , . . . , σn ) is a family of n pairwise commuting automorphisms of the (ungraded) algebra A such that σimi = Id for 1 ≤ i ≤ n. For λ = (λ1 , . . . , λn ) ∈ Zn we deﬁne λi Aπ(λ) = {a ∈ A : σi (a) = ζm a for 1 ≤ i ≤ n}, i

where for each l ∈ {m1 , . . . , mn } we have chosen a primitive lth -root of unity ζl ∈ k. Then A = ξ∈Ξ Aξ is a Ξ-grading of A. Denoting by k[z1±1 , . . . , zn±1 ] the Laurent polynomial algebra over k in the variables z1 , . . . , zn , Mm (A, σ) = (λ1 ,...,λn )∈Zn Aπ(λ1 ,...,λn ) ⊗ k z1λ1 · · · znλn is a Zn -graded algebra, called the multiloop algebra of σ based on A and relative to m. Thus, the theorem in particular says that Mm (A, σ) is graded-centralsimple. In fact, the centroid can be determined precisely.

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4.8 Proposition. ([ABP3, Cor. 6.6]) The centroid of a multiloop algebra Mm (A, σ) based on a ﬁnite-dimensional central-simple algebra A is gradedisomorphic to the Laurent polynomial ring k[z1±m1 , . . . , zn±mn ]. 4.9 Centroidal derivations. There is a close connection between the centroid and the derivations of an algebra A: [d, χ] ∈ Centk (A) for d ∈ Derk (A) and χ ∈ Centk (A). In other words, Centk (A) is a submodule of the Derk (A)module Endk (A). In the graded case one can deﬁne special derivations using the centroid. Let A = λ∈Λ Aλ be a Λ-graded algebra. We abbreviate C = Centk (A), λ grC = grCentk (A) and Cλ = (Centk A)λ , thus grC = λ C . We denote by Hom(Λ, C) ∼ = HomC (Λ ⊗Z C, C) the (left) C-module of all abelian group homomorphisms θ : Λ → (C, +). Every θ ∈ Hom(Λ, C) gives rise to a so-called centroidal derivation, deﬁned by ∂θ (aλ ) = θ(λ) (aλ )

for aλ ∈ Aλ .

(4.9.1)

We put CDer(A, Λ) = {∂θ : θ ∈ Hom(Λ, C)}, sometimes also denoted CDer(A) if Λ is clear from the context. We emphasize that CDer(A, Λ) not only depends on C but also on Λ. For example, any algebra A can be graded by Λ = {0} in which case CDer(A, Λ) = {0}. We will also consider the Λ-graded submodule grCDer(A, Λ) of CDer A, grCDer(A, Λ) = λ∈Λ (CDer A)λ where (CDer A)λ = CDer(A) ∩ (Endk A)λ = { ∂θ : θ ∈ Hom(Λ, Cλ )}. Suppose now that Centk (A) is commutative, e.g., that A is perfect. Then any ∂θ is C0 -linear and CDer(A) is a C0 -subalgebra of the Lie algebra DerC0 (A) of all C0 -linear derivations of A. Indeed, [∂θ , ∂ψ ] = ∂θ∗ψ−ψ∗θ for θ ∗ ψ ∈ Hom(Λ, C) deﬁned by (θ ∗ ψ) (λ) = µ∈Λ θ(µ) ψ(λ)µ , (4.9.2) where ψ(λ)µ denotes the µ-component of ψ(λ) ∈ C. The sum in (4.9.2) converges in the ﬁnite topology. In particular, for θ ∈ Hom(Λ, Cλ ) and ψ ∈ Hom(Λ, Cµ ) one has [∂θ , ∂ψ ] = θ(µ)∂ψ − ψ(λ)∂θ ∈ (CDer A)λ+µ ;

(4.9.3)

in particular (CDer A)0 is abelian. Suppose A is perfect and (·|·) is a Λ-graded invariant nondegenerate symmetric bilinear form on A. We denote by grSCDer(A, Λ) = λ∈Λ (SCDer A)λ

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83

the graded skew centroidal derivations of A, i.e., the graded subalgebra of grCDer(A, Λ) consisting of those graded centroidal derivations of A which are skew-symmetric with respect to (·|·). Since A is perfect, any centroidal transformation is symmetric with respect to (·|·). This implies that (SCDer A)λ = {∂θ ∈ (CDer A)λ : θ(λ) = 0}, in particular (SCDer A)0 = (CDer A)0 . Because of (4.9.3) we have [(SCDer A)λ , (SCDer A)−λ ] = 0 for any λ ∈ Λ. Hence grSCDer(A, Λ) is the semidirect product of the ideal spanned by the nonzero homogeneous spaces and the abelian subalgebra (CDer A)0 . 4.10 Degree derivations. Again let A = λ Aλ be a Λ-graded k-algebra. Recall that k IdA ⊂ (Centk A)0 . We can therefore consider the submodule D = D(A, Λ) = {∂θ : θ ∈ Hom(Λ, k IdA )} ⊂ (CDer A)0 of so-called degree derivations of (A, Λ). For example, if Λ = Zn we have D = spank {∂i : 1 ≤ i ≤ n} where ∂i (aλ ) = λi aλ for λ = (λ1 , . . . , λn ) ∈ Zn . Observe that [∂θ , ∂ψ ] = θ(µ) ∂ψ for ∂θ ∈ D and ψ ∈ Hom(Λ, Centk (A)µ ). So the action of D on grCDer A is diagonalizable. In order to actually get a toral action, we will assume in this subsection and in 4.11 that k is a ﬁeld and that A has full support. In this case, we can identify Hom(Λ, k IdA ) ≡ HomZ (Λ, k). Moreover, the canonical map Hom(Λ, k) → D = {∂θ : θ ∈ Hom(Λ, k)} is an isomorphism. We therefore have an evaluation map ev : Λ → D∗ , deﬁned by evλ (∂θ ) = θ(λ). Observe that (4.9.1) says that Aλ ⊂ {a ∈ A : d(a) = evλ (d) a for all d ∈ D}.

(4.10.1)

The following lemma speciﬁes a condition under which (4.10.1) becomes an equality. 4.11 Lemma. As in 4.10 we suppose that k is a ﬁeld and A is a Λ-graded k-algebra with full support. We let T ⊂ D be a subspace satisfying the condition that the restricted evaluation map Λ → T ∗ , λ → evλ |T , is injective.

(4.11.1)

Then Aλ = {a ∈ A : d(a) = evλ (t) for all t ∈ T }, in particular equality holds in (4.10.1). Moreover, if Centk (A) is commutative, T is a toral subalgebra of

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grCDerk (A), and if A is perfect and has a Λ-graded invariant nondegenerate symmetric bilinear form, then T is also a toral subalgebra of grSCDerk (A). We mention that (4.11.1) always holds for T = D, Λ torsion-free and k a ﬁeld of characteristic 0. 4.12 Examples. (a) Let g be a ﬁnite-dimensional split simple Lie algebra over a ﬁeld K of characteristic 0, e.g., a ﬁnite-dimensional simple Lie algebra over an algebraically closed K. Hence g has a root space decomposition g = α∈R gα with respect to some splitting Cartan subalgebra h = g0 of g, where R ⊂ h∗ is the root system of (g, h). We consider this decomposition as a Q(R)-grading. It is well known that g is central (hence central-simple): Any centroidal transformation χ leaves the root space decomposition of g invariant and, by simplicity, is uniquely determined by χ|gα for some 0 = α ∈ R. For any θ ∈ Hom(Q(R), K) ∼ = HomK (h∗ , K) there exists a unique hθ ∈ h such that α(hθ ) = θ(α). Therefore ∂θ = ad hθ and CDer(g, Q(R)) = grCDer(g, Q(R)) = grSCDer(g, Q(R)) = ad h. (b) Let g be as in (a) and let C be a unital commutative associative Λ-graded K-algebra. We view L = g ⊗ C as a Λ-graded Lie algebra over K with product [x ⊗ c, x ⊗ c ] = [x, x ] ⊗ cc for x, x ∈ g and c, c ∈ C and with homogeneous spaces Lλ = g ⊗ C λ (here and in the following all tensor products are over K). The centroid of L is determined in [ABP3, Lemma 2.3] or [BN, Cor. 2.23]: CentK (L) = K Idg ⊗C. (4.12.1) (We note in passing that (4.12.2) also holds for the inﬁnite-dimensional versions of g deﬁned in 2.13, as well as for the version of g over rings which we will introduce in 5.5.) By [Bl2, Th. 7.1] or [BM, Th. 1.1], the derivation algebra of L is

DerK (L) = Der(g)⊗C ⊕ K Idg ⊗ DerK (C) = IDer(L)⊕ K Idg ⊗ DerK (C) (4.12.2) where IDer L is the ideal of inner derivations of L. It is then immediate that CDer(L) and grCDer(L) are given by the formulas (gr)CDer(L, Λ) = K Idg ⊗(gr)CDer(C, Λ), where here and in the following (gr) indicates that the formula is true for the graded as well as the ungraded case. Let κ be the Killing form of g and suppose that is a Λ-graded invariant nondegenerate symmetric bilinear form on C. The bilinear form κ ⊗ of L, deﬁned by (κ ⊗ )(x ⊗ c, x ⊗ c ) = κ(x, x ) (c, c ), is Λ-graded, invariant, nondegenerate and symmetric (if g is simple, any such form arises in this way

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85

by [Be, Th. 4.2]). With respect to κ ⊗ , the skew-symmetric derivations and skew-symmetric centroidal derivations are

(gr)SDerK (L) = IDer L ⊕ K Idg ⊗(gr)SDerK (C) , (gr)SCDerK (L) = K Idg ⊗(gr)SCDerK (C). In particular, let C = K[t±1 ] be the K-algebra of Laurent polynomials over K, which we view as Z-graded algebra with C i = Kti . Its derivation algebra is (isomorphic to) the Witt algebra,

DerK K[t±1 ] = i∈Z Kd(i) , where d(i) acts on K[t±1 ] by d(i) (tj ) = jti+j . Hence d(i) is the centroidal derivation given by the homomorphism θ(i) : Z → C i , θ(i) (j) = jti . It follows that

Der K[t±1 ] = grDer K[t±1 ] = CDer K[t±1 ] = grCDer K[t±1 ] . Up to scalars, the algebra K[t±1 ] has a unique Z-graded invariant nondegenerate symmetric given by (ti , tj ) = δi+j,0 . With respect to this

bilinear form ±1 (0) form SCDer K[t ] = Kd and hence the skew derivations of the untwisted loop algebra L = g ⊗ K[t±1 ] are SCDer(g ⊗ K[t±1 ]) = Kd(0). It is important for later that SCDer(L) are precisely the derivations needed to construct the untwisted aﬃne Lie algebra in terms of L. (c) Let E be an aﬃne Lie algebra (see 2.13). Then (b) applies to L = [E, E] Z([E, E]) ∼ = g ⊗ K[t±1 ] (the centreless core when E is considered as an extended aﬃne Lie algebra, see 6.11). In particular, L has an inﬁnite-dimensional centroid. But the centroid of the core K := [E, E] is 1-dimensional: Cent(K) = K IdK , while the centroid of E is 2-dimensional, namely isomorphic to K IdE ⊕ HomK (Kc, Kd) where Kc is the centre of the derived algebra K and E = K Kd ([BN, Cor. 3.5]). 4.13 Quantum tori. In short, a quantum torus is an associative Zn -torus. As in 4.5 we may assume that a quantum torus has full support. In this case it can be described as follows. Let k = F be a ﬁeld and let q = (qij ) ∈ Matn (F ) be a quantum matrix , i.e., qij qji = 1 = qii for all 1 ≤ i, j ≤ n. The quantum torus associated to q is the associative F -algebra Fq presented by generators ti , ti −1 and relations ti ti −1 = 1 = ti −1 ti

and ti tj = qij tj ti

for 1 ≤ i, j ≤ n.

±1 Observe that if all qij = 1, then Fq = F [t±1 1 , . . . , tn ] is the Laurent polynomial ring in n variables. Hence, for a general q, a quantum torus is a

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Erhard Neher

non commutative version (a quantization) of the Laurent polynomial algebra ±1 F [t±1 1 , . . . , tn ], which is the coordinate ring of an n-dimensional algebraic torus. This explains the name “quantum torus” for Fq . Let A = Fq be a quantum torus. It is immediate that Fq =

λ∈Zn

F tλ ,

for tλ = tλ1 1 · · · tλnn .

The multiplication of Fq is determined by the 2-cocycle τ : Zn × Zn → F × where λ µ tλ tµ = τ (λ, µ) tλ+µ , τ (λ, µ) = 1≤j
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87

(iii) Fq is fgc (4.2). (d) ([BGK, Remark 2.45]) Up to scalars, Fq has a unique Zn -graded invariant symmetric bilinear form (·|·), given by (tλ |tµ ) = π(tλ tµ ) where π : Fq → F t0 is the canonical projection. The form (·|·) is nondegenerate. (e) ([OP, Cor. 2.3], [BGK, 2.48–2.56]) The derivation algebra of Fq is the semidirect product of the ideal IDer(Fq ) = {ad a : a ∈ Fq } of inner derivations, where (ad a)(b) = ab − ba, and the subalgebra CDer(Fq ) of centroidal derivations: DerF (Fq ) = IDer(Fq ) CDer(Fq ). Both IDer(Fq ) and CDer(Fq ) are graded subalgebras of DerF (Fq ): IDer(Fq ) = λ∈Γ (DerF (Fq ))λ ∼ = [Fq , Fq ] = λ∈Γ (F tλ ), CDer(Fq ) = λ∈Γ (Der(Fq ))λ = Z(Fq ) · D, where D = spanF {∂1 , . . . , ∂n } are the degree derivations of the Zn -graded algebra Fq with ∂i operating as ∂i (tλ ) = λi tλ . 4.14 Notes. The notion of the centroid of an algebra arose in the theory of forms of algebras, see e.g., [Ja, Ch.X, §1] or [Mc, Ch.II, §1.6]. The centroids of not necessarily split ﬁnite-dimensional simple Lie algebras over a ﬁeld of characteristic 0 were determined in [Se, V.1]. Some recent references regarding the centroids of Lie algebras are the papers [ABP3], [ABFP1] and [BN]. Theorem 4.4 is a standard result in the ungraded case, see for example, [Ja, Ch.X, §1] or [Mc, Ch.II, §1.6, Th. 1.7.1]. The graded case can be proven by a graded version of the Density Theorem. The implication (i) ⇒ (ii) is mentioned in [ABFP1, Remark 4.3.2(ii)]. For more information about crossed product algebras, see for example, the book [Pa]. Prop. 4.6(a) and (b) are proven in [BN, Prop. 2.16]. Part (c) is shown in [ABFP1, Prop. 4.4.5]; it was independently discovered by the author and used in the proof of [Neh6, Th. 7(b)]. Theorem 4.7 is only one of several realization theorems proven in [ABFP1]. The special case of Lie-Zn -tori is developed in detail in [ABFP2]. Multiloop algebras are studied in detail in [ABP1, ABP2, ABP3]. Their derivations are determined in [Az5]. A cohomological approach to the classiﬁcation of multiloop algebras is developed in [GiP1] and [GiP2]. Quantum tori are the algebraic counterparts of the so-called rotation algebras, a special class of C ∗ -algebras which has a very well-developed theory. The terminology “quantum torus” seems to go back to Manin [Ma, Ch. 4, §6]. Much more is known about quantum tori than mentioned in 4.13. For example, the structure of quantum tori Fq for which all qij roots of unity is elucidated in [ABFP1, Prop. 9.3.1], [Ha, Th. 4.8] and [Nee3, Th. III.4]. The involutions of quantum tori are determined in [Yo3].

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5 Lie algebras graded by root systems While the previous Section 4 provided some background material for arbitrary graded algebras, in this section we consider Lie algebras with a special type of grading by the root lattice of a root system, called root-graded Lie algebras. We will only describe that part of the theory which is of importance for the following sections. Throughout, and unless speciﬁed otherwise, all algebras are deﬁned over a unital commutative associative base ring k in which 2 and 3 are invertible. As in §4, Λ denotes an abelian group, written additively. 5.1 Root-graded Lie algebras. Let R be a locally ﬁnite root system. It is convenient to put × R× = R \ {0} and Rind = R× ∩ Rind ,

see 2.7. We denote the Z-span of R by Q(R). Let L be a (Q(R), Λ)-graded Lie algebra. We will always use subscripts to indicate the Q(R)-grading and superscripts for the Λ-grading: L = q∈Q(R) Lq and L = λ∈Λ Lλ . By compatibility we therefore have L=

q∈Q(R), λ∈Λ

Lλq

for Lλq = Lq ∩ Lλ .

To stress the analogy between the deﬁnitions in 4.5 and the ones below, we will ﬁrst deﬁne invertible elements of L. A nonzero element e ∈ Lλα for α ∈ R× is called invertible, more precisely, an invertible element of the (Q(R), Λ)graded Lie algebra L, if there exists f ∈ L−λ −α such that h = [f, e] acts on Lq , q ∈ Q(R), by [h, xq ] = q, α∨ xq (xq ∈ Lq ). (5.1.1) In this case, (e, h, f ) is an sl2 -triple. If L3α = {0} (a condition which will always be fulﬁlled later), then f ∈ L−λ −α is uniquely determined by (5.1.1). We will therefore refer to f as the inverse of e. λ A (Q(R), Λ)-graded Lie algebra L = q∈Q(R), λ∈Λ Lq is called (R, Λ)graded if it satisﬁes the axioms (RG1)–(RG3) below: (RG1) suppQ(R) L ⊂ R, whence L = α∈R Lλα , × (RG2) every L0α , α ∈ Rind , contains an invertible element of L, and (RG3) L0 = 0=α∈R [Lα , L−α ].

An (R, Λ)-graded Lie algebra L is called • •

predivision-(R, Λ)-graded if every nonzero Lλα , α ∈ R× , contains an invertible element; division-(R, Λ)-graded if every nonzero element of Lλα , α ∈ R× , is invertible;

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• •

89

a Lie-(R, Λ)-torus if L is predivision-graded, deﬁned over a ﬁeld k and dimk Lλα ≤ 1 for all α ∈ R× ; invariant if L has a Λ-graded invariant nondegenerate symmetric bilinear form (·|·) such that for some choice of invertible elements eα ∈ L0α , α ∈ × Rind , with inverses fα the restriction of (·|·) to spank {[eα , fα ] : 0 = α ∈ Rind } is nondegenerate as well.

Let L be (R, Λ)-graded. By (RG1) and (RG2) we have Rind ⊂ suppQ(R) L = {α ∈ R : Lα = 0} ⊂ R, whence L has full Q(R)-support: suppQ(R) L = Q(R). One can show that suppQ(R) L is a subsystem of R. We can therefore always assume that suppQ(R) L = R, if we so wish. Analogously, we can always assume that L has full Λ-support in the sense that Λ = suppΛ L. If in the deﬁnitions above Λ = {0}, we will speak of an R-graded or a (pre)division-R-graded Lie algebra. Lie algebras that are R-graded for some locally ﬁnite root system R are simply called root-graded . Similarly, a predivision-root-graded Lie algebra is a Lie algebra which is predivision-(S, Λ)graded for some (S, Λ), and a Lie torus is a Lie-(R, Λ)-torus for a suitable pair (R, Λ). We will present examples of root-graded Lie algebras in 5.5 and in §7. But we mention here right away the following examples: the derived algebra of an aﬃne Kac–Moody algebra (see 2.13), toroidal Lie algebras, Slodowy’s intersection matrix algebras and ﬁnite-dimensional simple Lie algebras over ﬁelds of characteristic 0 which contain an sl2 -subalgebra. Moreover, predivision-graded Lie algebras will turn out to be the basic building blocks of the generalizations of aﬃne Lie algebras that we will be describing in Section 6. 5.2 Isomorphisms and isotopy. An (R, Λ)-graded Lie algebra L is called isograded-isomorphic to an (R , Λ )-graded Lie algebra L if there exists an isomorphism ϕa : L → L of Lie k-algebras, an isomorphism ϕr : R → R of locally ﬁnite root systems and an isomorphism ϕe : Λ → Λ of abelian groups satisfying ϕ (λ) ϕa (Lλα ) = Lϕer (α) for all α ∈ R and λ ∈ Λ. We say L and L are graded-isomorphic if there exists an isograded isomorphism with ϕr = Id and ϕe = Id. Let ι : Q(R) → Λ be a group homomorphism. We can then deﬁne a new (Q(R), Λ)-graded Lie algebra L(ι) by

The following are equivalent: (i) L(ι) is (R, Λ)-graded,

L(ι)

λ α

= Lλ+ι(α) . α

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Erhard Neher ι(α)

× (ii) Lα contains an invertible element for all α ∈ Rind . 1 1 1 If R has a root basis and, in addition to 2 , 3 and 5 ∈ k, then (i) and (ii) are equivalent to ι(α)

(iii) Lα

contains an invertible element for all α ∈ B.

In this case, we say that L(ι) is an isotope of L. An (R , Λ )-graded Lie algebra L is called isotopic to L if some isotope of L is isograded-isomorphic to L . Being isotopic is an equivalence relation on the class of all root-graded Lie algebras. Isotopy preserves the classes of (pre)division-graded Lie algebras and Lie tori. For these cases, condition (ii) can be replaced by Lι(α) = 0 for all α ∈ × Rind , and similarly for (iii). 5.3 Coverings and central quotients. Let Ξ be an abelian group and let L be a Ξ-graded Lie algebra. By deﬁnition, a Ξ-graded central extension of L is a Ξ-graded Lie algebra K together with a graded epimorphism f : K → L with Ker f contained in the centre of K. A Ξ-graded central extension f : K → L is called a Ξ-covering if K is perfect. We will use the terms central extension and covering in case Ξ = {0}. A central extension u : L → L is called a universal central extension if for any central extension f : K → L there exists a unique homomorphism f : L → k such that u = f ◦ f: f! L _? _ _ _ _ _ _/ K ?? ~ ?? ~~ ~ ~ u ?? ~~ f L

It is obvious from this universal property that two universal central extensions of L are isomorphic in the sense that one has a commutative triangle as above with the horizontal map being an isomorphism. A Lie algebra L has a universal central extension, say u : uce(L) → L, if and only if L is perfect. In this case, uce(L) is perfect and is a universal central extension of all covering algebras of L as well as all central quotients. In particular, it is a universal central extension of the centreless algebra L/ Z(L). If L is Ξ-graded, then u : uce(L) → L is a Ξ-covering. One calls L simply connected if L is a universal central extension of itself. For example, a universal central extension is always simply connected. As an example we mention that a universal central extension of the untwisted and twisted loop algebra L of 2.13 is the Lie algebra K deﬁned there, [Ga, Wi]. Now let L be an (R, Λ)-graded Lie algebra. We will apply the concepts above to the Ξ-grading of L where Ξ = Q(R) ⊕ Λ. We observe ﬁrst that the centre of L is Λ-graded and contained in L0 ,

EALAs and Generalizations

Z(L) =

λ∈Λ

(Z(L) ∩ Lλ0 ),

91

(5.3.1)

whence Z(L) is also Ξ-graded. 5.4 Proposition. Let f : K → L be a Λ-covering of the Λ-graded Lie algebra L. Then K is (R, Λ)-graded if and only if L is (R, Λ)-graded. In this case suppQ(R) K = suppQ(R) L and f |Kα is bijective for all α ∈ R× . Moreover, K is predivision- or division-graded or a Lie torus if and only if L has the corresponding property. As a consequence of this proposition, the classiﬁcation of root-graded Lie algebras and their subclasses introduced above can be done in three steps: 1. Describe all centreless root-graded Lie algebras, 2. determine their universal central extensions and 3. characterize those that are (pre)division-graded or Lie tori. Note that any root-graded Lie algebra is perfect and therefore has a universal central extension. Examples of this approach are given in, §§5.5–5.7 and §7, see also, §5.10 for a short survey. 5.5 Example. Let gZ be a Chevalley order of a ﬁnite-dimensional semisimple complex Lie algebra gC of type R, see [Bo1, Ch. VIII, §12]. Thus, we have a Chevalley system (eα : 0 = α ∈ R) such that, putting hα = [e−α , eα ],

gZ = 0=α∈R Zeα ⊕ 0=α∈R Zhα is a Lie algebra over Z which is a Z-form of gC , i.e., gZ ⊗Z C ∼ = gC . Then gk = gZ ⊗Z k is a Q(R)-graded Lie k-algebra with homogeneous spaces gk, α = g Z,α ⊗Z k. For every unital commutative associative Λ-graded k-algebra C = λ λ∈Λ C , the Lie k-algebra L = gk ⊗ k C ∼ = gZ ⊗ Z C

(5.5.1)

is (Q(R), Λ)-graded with homogeneous spaces Lλα = gk,α ⊗k C λ . An element eα ⊗ c is invertible in L if and only if c ∈ C × . Hence, L is (R, Λ)-graded. It is (pre)division-(R, Λ)-graded if and only if C is (pre)division-graded. If k is a ﬁeld (recall of characteristic = 2, 3), then L is a Lie torus if and only if C is an associative torus. Let κ be the Killing form of gC and let h∨ be the dual Coxeter number. Then 2h1∨ κ is an invariant nondegenerate symmetric bilinear form on gZ ([SS, I.4.8], [GrN, §5]). The canonical extension (·|·) of this form to gk is then an invariant symmetric bilinear form on gk . It is nondegenerate if R is irreducible and not of type A. In this case L is invariant with respect to (·|·) ⊗ , where is any Λ-graded invariant nondegenerate symmetric bilinear form on C, see (4.3.2). If R is irreducible, the Lie algebra gk is simply connected ([Ste, 6.1], [vdK, Cor. 3.14], [Kas]). The universal extension u : uce(L) → L of an arbitrary

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L as in (5.5.1) is determined in [Kas, 3.8]. It turns out that the kernel of u is ΩC/k dC where ΩC/k is the module of K¨ ahler diﬀerentials of C over k and d : C → ΩC/k is the universal k-linear derivation. In view of (7.8.1) it is appropriate to point out that ΩC/k dC = HC1 (C), the ﬁrst cyclic homology group of C. For root systems of type D or E the algebras of this subsections are in fact all examples of R-graded Lie algebras. 5.6 Theorem. ([BeMo], [Neh3]) Suppose R is a ﬁnite root system of type Dl , l ≥ 4 or El , l = 6, 7 or 8, and that k is a ﬁeld of characteristic 0 in case of E8 . Then any (R, Λ)-graded Lie algebra is graded-isomorphic to a Λ-covering of a Lie algebra L as in (5.5.1) for a suitable C. If in case D one replaces g by the corresponding inﬁnite rank Lie algebra (2.13), the result remains true for (DI , Λ)-graded Lie algebras, |I| = ∞. But it is no longer true for the other types of root systems, see §7, in particular 7.6 ˙ I and 5.10 for a short summary of the classiﬁcation and 7.7, for R of type A results. The models used to describe R-graded Lie algebras are diﬀerent for every type of R and involve all important classes of nonassociative algebras (Jordan algebras, alternative algebras, structurable algebras). On the other hand, one has the following type-free approach for most root systems, not only ﬁnite ones. 5.7 Theorem. ([Neh3]) Let R be a locally ﬁnite root system without an irreducible component of type E8 , F4 or G2 . Then a Lie algebra L is R-graded if and only if L is a covering of the Tits–Kantor–Koecher algebra of a Jordan pair covered by a grid whose associated root system is R. 5.8 Centroids of root-graded Lie algebras. Let L = α∈R, λ∈Λ Lλα be an (R, Λ)-root-graded Lie algebra over k. (a) Since L is perfect, the centroid of L is a commutative (and of course also unital and associative) subalgebra of Endk (L). One can show that a centroidal transformation of L leaves every Lα , α ∈ R, invariant. If R has only ﬁnitely many irreducible components, Centk (L) is Λ-graded. As an example, we mention that the centroid of the Lie algebra L of (5.5.1) is isomorphic to C. (b) Suppose now that the Λ-graded algebra L is graded-simple. Then R is irreducible and L is also graded-simple with respect to the Q(R)-grading of L. Hence, the centroid of L is a twisted group algebra K t [Γ ] over the ﬁeld K = (Cent L)0 , as described in 4.6. The condition (4.6.2) characterizing fgc algebras now simpliﬁes to the following, where Λα = {λ ∈ Λ : Lλα = 0}: L is fgc if and only if (i) R is ﬁnite, (ii) [Λα : Γ ] < ∞ for all short roots α ∈ R, and (iii) dimK Lλα < ∞ for all short roots α and all λ ∈ Λα .

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93

If also 15 ∈ k, then for every w ∈ W (R) there exists a Cent(L)-linear automorphism of L which for all α ∈ R and λ ∈ Λ maps Lλα to Lλw(α) . Therefore, in this case it is suﬃcient to require (ii) and (iii) for one short root α. If L is in addition predivision-graded and Λ = suppΛ L is ﬁnitely generated, then the condition (ii) can be replaced by (ii ) [Λ : Γ ] < ∞. (c) Let L be division-graded with Z(L) = {0} and suppose that R is irreducible. Then L is Λ-graded-simple. So its centroid is described in (b). Moreover, if Λ is torsion-free, L is a prime algebra, i.e., [I, J] = 0 for ideals I, J of L implies I = 0 or J = 0. In this case, Centk (L) is an integral domain, ˜ ˜ = L ⊗ C, ˜ where C˜ is the quotient ﬁeld of and L embeds into the C-algebra L ˜ is central and prime ([EMO]). If the Lie algebra L Centk (L). The algebra L ˜ is ﬁnite-dimensional. Hence, if C has characteristic 0, then L ˜ is is fgc, then L ˜ a ﬁnite-dimensional central-simple C-algebra. (d) Let L be a centreless Lie-(R, Λ)-torus and suppose that R is irreducible. Then L is graded-central-simple by (c) and (b). In particular, if Λ = suppΛ L is ﬁnitely generated, then L is fgc if and only if R is ﬁnite and [Λ : Γ ] < ∞. See also 6.12. We will now consider root-graded Lie algebras in characteristic 0. It will turn out that the Q(R)-grading of a root-graded Lie algebra is in fact the root space decomposition with respect to a toral subalgebra, as we will explain below. 5.9 Proposition. Let L be an (R, Λ)-graded Lie algebra, deﬁned over a ﬁeld × K of characteristic 0. We suppose suppQ(R) L = R. For α ∈ Rind let eα ∈ L0α be an invertible element with inverse fα . We put hα = [fα , eα ], h0 = 0, h2α = 12 hα in case 2α ∈ R, and h = spanK {hα : α ∈ R}. (a) For every α ∈ R there exists a unique α ˜ ∈ h∗ deﬁned by α ˜ (hβ ) = α, β ∨ ˜ = {α for β ∈ R. The set R ˜ ∈ h∗ : α ∈ R} is a locally ﬁnite root system, isomorphic to R via the map α → α. ˜ (b) The set {hα : α ∈ R} is a locally ﬁnite root system in h, canonically isomorphic to the coroot system R∨ via the map α∨ → hα . (c) The subspace h is a toral subalgebra of L whose root spaces are the homogeneous subspaces Lα with corresponding linear form α ˜: Lα = {l ∈ L : [h, l] = α(h)l ˜ for all h ∈ h}. (d) Let R be ﬁnite and let B be a root basis of R. Then {eβ , fβ : β ∈ B} generates a ﬁnite-dimensional split semisimple subalgebra g of L with splitting ˜ ind . Cartan subalgebra h. The root system of (g, h) is R It follows immediately from this result that for K a ﬁeld of characteristic 0, an (R, Λ)-graded Lie K-algebra for R an irreducible root system and Λ

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Erhard Neher

an abelian group can be equivalently deﬁned as follows: L = Λ-graded Lie algebra over K satisfying (i)–(iv) below:

λ∈Λ

Lλ is a

(i) L0 contains as a subalgebra a ﬁnite-dimensional split simple Lie algebra g, called the grading subalgebra, with splitting Cartan subalgebra h, (ii) either R is reduced and equals the root system Rg ⊂ h∗ of (g, h), or R = BCI and Rg = A1 in case |I| = 1 or Rg = BI for |I| ≥ 2. (iii) h is a toral subalgebra of L, and the weights of L relative h are in R, whence L = α∈R Lα , (iv) L0 = 0=α∈R [Lα , L−α ]. This is the original deﬁnition of a root-graded Lie algebra, see 5.10. 5.10 Notes. Unless stated otherwise, the results mentioned in 5.1–5.9 are due to the author. Some of them have also been proven by others, as indicated below. The deﬁnition of a root-graded Lie algebra in 5.1 is in the spirit of the deﬁnition in [Neh3], which introduces root-graded Lie algebras over rings for reduced root systems. The deﬁnition given at the end of 5.9 is the original one of [BeMo], where R-graded Lie algebras were introduced R simply-laced. It is mentioned in [Neh3, Remark 2 of 2.1] that in characteristic 0 the two deﬁnitions are equivalent. A1 -graded Lie algebras had been studied much earlier in [Ti] and then in [Kan1, Kan2, Kan3] and [Ko1, Ko2]. The paper [BeMo] contains a classiﬁcation of centreless R-graded Lie algebras for R simply laced. The classiﬁcation for reduced, non-simply-laced R is given in [BZ]. The theory for a non-reduced R, i.e., R = BCI , is developed in [ABG2] for R = BCl , 2 ≤ l < ∞) and [BS] for R = BC1 . In this case, one can be more general by allowing the grading subalgebra g to have type C or D. This too can be done in the setting of 5.1, but we will not need this in the following and have therefore refrained from describing the more general setting. The papers [BeMo], [BZ], [ABG2] and [BS] work with ﬁnite root systems and Lie algebras over ﬁelds of characteristic 0. The extensions to locally ﬁnite root systems and Lie algebras over rings is given in [Neh3] for R reduced and 3-graded. As already mentioned in [GaN, 2.10], the classiﬁcation results in [Neh3], see 5.7, can easily be extended to the (R, Λ)-graded and (R, Λ)-(pre)division-graded case. Finite-dimensional simple Lie algebras and aﬃne Lie algebras graded by ﬁnite, respectively, aﬃne root systems are described in [Ner1] and [Ner2]. At present, predivision-(R, Λ)-graded Lie algebras have been classiﬁed in the centreless case only for certain root systems and base rings k: R ﬁnite of type Al , (l ≥ 3), D or E, Λ = Zn , k = K a ﬁeld of characteristic 0 in [Yo1]; R = Br , 3 ≤ r < ∞, L division-graded, k = K a ﬁeld of characteristic 0 in [Yo4]; R = Cr , 2 ≤ r < ∞, L division-graded over k = K a ﬁeld of characteristic 0 in [BY]; R = B2 = C2 , Λ torsion-free, L graded-simple (e.g.,

EALAs and Generalizations

95

division-graded), k arbitrary in [NT]. Regarding the classiﬁcation of Lie tori, the situation the reader is referred to the survey in [AF, §12]. The concept of isotopy was introduced in [AF] for Lie tori over ﬁelds of characteristic 0 and related to isotopy of their coordinate algebras. In the context of Lie tori in characteristic 0, the equivalence of (i)–(iii) in 5.2 is proven in [ABFP2, Prop. 2.2.3]. Universal central extensions of Lie algebras over rings were described in [vdK, §1] (this reference also describes the universal central extensions of the Lie algebras gk of 5.5, where k not necessarily contains 12 and 13 ). Later references on universal central extensions are [Ga, §1], [MP, 1.9], [We, 7.9] or [Neh4, ] (the list is incomplete). In characteristic 0, the universal central extensions of root-graded Lie algebras are determined in [ABG1], [ABG2], and [BS]. Proposition. 5.4 is standard, proven in [ABG2, Th. 5.36], [BS, §5], [BZ, Prop. 1.5–1.8] and [BeMo, Prop. 1.29] for the root-graded Lie algebras studied in these papers, viz. Lie algebras over ﬁelds of characteristic 0, where one can use that the grading subalgebra g is simply connected. One can prove 5.4 for arbitrary k using the model of the universal central extension from [vdK]. More on Example 5.5: We give some more information about this example in case k = F is a ﬁeld (recall of characteristic = 2, 3). It follows as in Example 5.5 earlier that gF is central. Moreover, the description of DerF L given there remains true, except that DerF (gF ) need not be equal to IDer gF and hence ˙ L is centreless, see 7.4 for the IDer L = DerF (gF ) ⊗ C in general. If R = A, ˙ case R = A. Theorem. 5.6 is proven in [BeMo] for k a ﬁeld of characteristic 0 and in [Neh3, 5.4, 7.2, 7.3] for arbitrary k (of course containing 12 and 13 ) and R = E8 . The methods of [Neh3] do not permit us to treat the case E8 , but 1 the result is probably also true in this case for any k containing 30 . Similarly, Theorem. 5.7 can be generalized to cover all locally ﬁnite root systems by using Kantor pairs instead of Jordan pairs. With this method one can likely extend the classiﬁcation results to cover all root-graded Lie algebras (R not 3-graded and Lie algebras over rings). More on Section 5.8: That a centreless division-(R, Λ)-graded Lie algebra with R irreducible is graded-simple is proven in [Yo4, Lemma 4.4] for ﬁnite R (the proof easily generalizes). The characterization of fgc Lie tori in 5.8(d) is proven in [ABFP2, Prop. 1.4.2]. For R a ﬁnite root system, R = BC1 or BC2 , the centroid of an R-graded Lie algebra over a ﬁeld of characteristic 0 is described in [BN, §5] in terms of the centre of the coordinate system associated to an R-graded Lie algebra. For 3-graded R, Prop. 5.9(a) was shown in [Neh2, 3.2]. It is also proven in [ABFP2, Prop. 1.2.2] for Lie tori. Invariant forms and derivations of root-graded Lie algebras over ﬁelds of characteristic 0 are described in [Be], [ABG2], and [BS].

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6 Extended aﬃne Lie algebras and generalizations In this section we will introduce Lie algebras whose set of roots are aﬃne reﬂection systems (6.2). We will then discuss special types, most importantly invariant aﬃne reﬂection algebras (6.7) and extended aﬃne Lie algebras (6.11). Unless speciﬁed otherwise, in Section 6 all algebras are deﬁned over a ﬁeld K of characteristic 0. Throughout, (E, T ) will be a toral pair , i.e., a non-zero Lie algebra E with a toral subalgebra T . We will denote the set of roots of (E, T ) by R and put X = spanK (R) ⊂ T ∗ , cf. 2.11. 6.1 Core and tameness. Let (E, T ) be a toral pair. Recall that 0 ∈ R if T = 0 and that E has a root space decomposition (6.1.1) E = α∈R Eα . If (R, X) is a pre-reﬂection system with respect to some set of real roots Rre ⊂ R, we deﬁne the core of (E, T ) as the subalgebra Ec of E generated by {Eα : α ∈ Rre }. It is a graded subalgebra with respect to the root space decomposition (6.1.1),

Ec = (6.1.2) α∈Rre Eα ⊕ β∈Rim (Ec ∩ Eβ ) , and also with respect to any Λ-grading of E which is compatible with the root space decomposition (6.1.1). If Ec is perfect (which will be the case from 6.2 on), the quotient algebra Ecc = Ec / Z(Ec ) (6.1.3) is centreless and is called the centreless core. We call (E, T ) tame if CE (Ec ) ⊂ Ec . Tameness of (E, T ) is not the same as tameness of R as deﬁned in 2.2, cf. 6.5, 6.8 and 6.10. 6.2 Aﬃne reﬂection Lie algebras. Let α be an integrable root of (E, T ). Hence, by deﬁnition, there exists an integrable sl2 -triple (eα , hα , fα ) ⊂ Eα × T × E−α of (E, T ). It gives rise to a reﬂection sα of (R, X), deﬁned by sα (x) = x − x(hα )α,

whence x, α∨ = x(hα ).

(6.2.1)

We will call (E, T, Rre ) or simply (E, T ) an aﬃne reﬂection Lie algebra (abbreviated ARLA) if (R, X) is an aﬃne reﬂection system with Rre ⊂ Rint and the reﬂections sα , α ∈ Rre , deﬁned by (1). Thus Rre ⊂ Rint , but not necessarily Rre = Rint . Because of (3.6.1), other choices of integrable sl2 -triples for α ∈ Rre lead to the same reﬂection sα . We will therefore not consider the family (eα : α ∈ Rre ) as part of the structure of an aﬃne reﬂection Lie algebra. Following this point of view, two aﬃne reﬂection Lie algebras (E, T, Rre ) and (E , T , Rre ) are called isomorphic if there exists a Lie algebra isomorphism ϕ : E → E with ϕ(T ) = T and ϕr (Rre ) = Rre where ϕr (α) = α ◦ (ϕ|T )−1 .

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97

Recall from 3.2 and 3.6 that any aﬃne reﬂection system is integral, coherent and has ﬁnite root strings. The aﬃne reﬂection systems occurring as roots in an aﬃne reﬂection Lie algebra have the following additional properties: • All root strings S(β, α) for β ∈ R, α ∈ Rre , are unbroken. • R is reduced in case T is a splitting Cartan subalgebra. By deﬁnition, the nullity of (E, T, Rre ) is the nullity of the aﬃne reﬂection system R, i.e., the rank of the subgroup Λ of (T ∗ , +) generated by Rim , cf. 3.6. We always have

nullity ≥ dimK spanK (Rim ) . (6.2.2) In general this is a strict inequality, see 6.17. 6.3 The core of an aﬃne reﬂection Lie algebra. To study the core of an aﬃne reﬂection Lie algebra (E, T, Rre ) we need some more notation: • (S, Y ) is the quotient root system of (R, X) (3.6); • f : (R, X) → (S, Y ) is the corresponding extension (3.2); • Z = Ker(f ) = spanK (Rim ); • g : Y → X, y → g(y) =: y, ˙ is a partial section over Sind (3.1); • L = (Λξ )ξ∈S is the extension datum associated to f and g, see 3.3–3.7; thus f −1 (ξ) ∩ R = g(ξ) + Λξ and the set R can be decomposed as

˙ R = ξ∈S ξ˙ ⊕Λξ ⊂ X = g(Y )⊕Z with Rim = 0⊕Λ 0 = Λ0 . (6.3.1) • Λ is the subgroup of (Z, +) generated by

ξ∈S

Λξ ,

• K = Ec is the core of (E, T, Rre ), and Kξλ ⊂ Eg(ξ)⊕λ , Kξ and K λ are deﬁned for ξ ∈ S and λ ∈ Λ by Kξλ = Eξ⊕λ , (ξ = 0), ˙ K0λ = τ ∈S × , µ∈Λ [Eτ˙ ⊕µ , E−τ˙ ⊕(λ−µ) ] , Kξ = λ∈Λ Kξλ , (ξ ∈ S), λ λ K = ξ∈S Kξ , (λ ∈ Λ). • To distinguish the Λ-support sets of K and the extension datum (Λξ )ξ∈S , we put λ K ΛK = spanZ {ΛK ξ = {λ ∈ Λ : Kξ = 0} and Λ ξ : ξ ∈ S}.

The data S, f , Z and the Lie algebra K are invariants of (E, T, Rre ). Moreover, since Kξ = f (α)=ξ Eα for ξ ∈ S × , also the subspaces Kξ , ξ ∈ S, are uniquely determined by (E, T, Rre ). But this is not so for the partial section g and hence for the family (Λξ )ξ∈S . Part (d) of the following Theorem 6.4 says that another choice of g leads to an isotope of K, as deﬁned in 5.2.

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6.4 Proposition. We use the notation of 6.3. The core K of an aﬃne reﬂection Lie algebra (E, T, Rre ) is a perfect ideal of E and a predivision-(S, ΛK )graded Lie algebra with respect to the decomposition K = ξ∈S, λ∈Λ Kξλ . Moreover, K has the following properties. × (a) The root system S embeds into h∗K where hK = spanK {hg(ξ) : ξ ∈ Sind }, such that K = ξ∈S Kξ is the root space decomposition of K with respect to the toral subalgebra hK of K. The set of roots of (K, hK ) is S, unless K = {0} and thus R = Rim and S = {0}. (b) The family LK = (ΛK ξ : ξ ∈ S) of Λ-supports of K is an extension datum of type (S, Sind , spanK (ΛK )) with Λξ = ΛK ξ for 0 = ξ ∈ S and Λ0 ⊃ ΛK 0 = suppΛ K =

ξ∈S ×

(Λξ + Λξ ),

hence Λ0 = Λ ⊃ ΛK = ΛK 0 . (c) The following are equivalent: (i) K is a Lie torus; (ii) dimK Eα ≤ 1 for all α ∈ Rre ; (iii) [Eα , E−α ] ⊂ T for all α ∈ Rre . In particular, if T is a splitting Cartan subalgebra, then K and the centreless core Ecc are Lie tori. (d) Suppose (E, T ) is isomorphic to the aﬃne reﬂection Lie algebra (E , T ). Denote by R , S , Λ , and K the data for (E , T ) corresponding to the ones introduced in 6.3 for (E, T ). Then R and R are isomorphic as reﬂection systems, and the (S, Λ)-graded Lie algebra K is isotopic to the (S , Λ )graded Lie algebra K . We have excluded K = {0} in (a) because, by deﬁnition in 2.11, the set of roots of (K, hK ) consists of those α ∈ h∗ for which the corresponding root space Kα (h) = 0. Of course, K = {0} can still be viewed as a root space decomposition. 6.5 A construction of an aﬃne reﬂection Lie algebra. We have seen in 6.4 that the core of an aﬃne reﬂection Lie algebra (E, T ) is a predivision(S, ΛK )-graded Lie algebra, where S is the quotient root system of the aﬃne reﬂection system of (E, T, Rre ) and where ΛK is a torsion-free abelian group. Changing notation (replacing ΛK by Λ) we will now show that, conversely, every predivision-(S, Λ)-graded Lie algebra, where S is any locally ﬁnite root system and where Λ is any torsion-free abelian group, is the core of an aﬃne reﬂection Lie algebra.

(6.5.1)

Our construction uses K and D, where K is a predivision-(S, Λ)-graded Lie algebra over K with Λ a torsion-free abelian group and where D is described

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below. Since suppQ(S) K is a subsystem of S, we can assume suppQ(S) K = S. Also, the family L = (Λξ : ξ ∈ S), Λξ = {λ ∈ Λ : Kξλ = 0}, is an extension datum of type S satisfying suppΛ K = Λ0 . Again it does no harm to assume that Λ is spanned by suppΛ K. Let (eλξ ∈ Kξλ : 0 = ξ ∈ S, λ ∈ Λξ ) be a family of invertible elements of the λ (Q(S), Λ)-graded Lie algebra K. We denote by fξλ ∈ L−λ −ξ the inverse of eξ , put hλξ = [eλξ , fξλ ] ∈ K00 and deﬁne × } ⊂ TK = spanK {hλξ : ξ ∈ S × , λ ∈ Λξ }. h = spanK {h0ξ : ξ ∈ Sind

Then h is a toral subalgebra of K and S canonically embeds into the dual space h∗ such that ξ, τ ∨ = ξ(h0τ ) for σ, τ ∈ Sind , τ = 0, and the root spaces of (K, h) are the subspaces Kξ , ξ ∈ S. Since TK ⊂ h + Z(K), TK is also a toral subalgebra of K with the same root spaces as h. We view S (non-canonically) ∗ embedded into TK . Let D = {∂θ : θ ∈ HomZ (Λ, K)} be the space of degree derivations of the Λ-graded Lie algebra K, 4.10. Since Λ is torsion-free, there exists an embedding of the abelian group Λ into D∗ mapping λ to the linear form evλ given by evλ (∂θ ) = θ(λ), 4.11. Let D ⊂ D be a subspace such that λ → evλ |D is injective, e.g., D = D. For such a D, the subspaces K λ ⊂ K are determined by D, namely K λ = {x ∈ K : [d, x] = evλ (d) for all d ∈ D}. It is now clear that T = TK ⊕ D is a toral subalgebra of the semidirect product E = K D whose root spaces are E0 = K00 ⊕ D

and Eξ⊕λ = Kξλ for (ξ, λ) = (0, 0),

where for ξ ∈ S and λ ∈ Λ the linear form ξ ⊕ λ ∈ T ∗ is given by (ξ ⊕ λ) (tK ⊕ d) = ξ(tK ) + evλ (d) for tK ∈ TK and d ∈ D. Thus the set of roots of (E, T ) is R = ξ∈S (ξ ⊕ Λξ ) ⊂ T ∗ . We let Rre = {ξ ⊕ λ : ξ ∈ S × , λ ∈ Λξ } and observe that Rre ⊂ Rint . The corresponding reﬂection sα , α = ξ ⊕ λ, maps β = τ ⊕ µ ∈ R, µ ∈ Λτ , to sα (β) = β − β(hλξ )α = β − τ (hλξ )α = β − τ, ξ ∨ α. Hence R with the reﬂections sα , α ∈ Rre , is the aﬃne reﬂection system associated to S and the extension datum (Λξ : ξ ∈ S) in 3.4. Thus, (E, T ) is an aﬃne reﬂection Lie algebra. Moreover, since K is R-graded, it equals the core of (E, T ). This proves (6.5.1). Note that Λ0 = ΛK 0 in the notation of 6.2. This implies that the aﬃne reﬂection system R of (E, T, Rre ) is tame. However, (E, T ) is tame if and only if IDer K ∩ D = {0}. We will now give a suﬃcient criterion for a toral pair (E, T ) to be an aﬃne reﬂection Lie algebra.

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6.6 Theorem. Suppose (E, T ) satisﬁes the conditions (AR1)–(AR3) below. (AR1) There exists a symmetric bilinear form (·|·) on X = spanK (R), (AR2) for every α ∈ Ran = {α ∈ R : (α|α) = 0} there exists an integrable sl2 -triple (eα , hα , fα ) of (E, T ) such that (β|α) =

(α|α) 2

β(hα )

holds for all β ∈ R, and (AR3) for every α ∈ R0 = {α ∈ R : (α|α) = 0} there exists a triple (eα , tα , fα ) ∈ Eα × T × E−α such that [eα , fα ] = tα

and

(β|α) = β(tα )

holds for all β ∈ R. Then, with respect to the orthogonal reﬂections sα , α ∈ Ran , of 2.10, (R, X) is an aﬃne reﬂection system with Rre = Ran ⊂ Rint , Rim = R0 , and the bilinear form (·|·) of (AR1) as aﬃne form. In particular, (E, T, Ran ) is an aﬃne reﬂection Lie algebra. It follows that there exists an aﬃne form which is positive-semideﬁnite on the rational span of R and hence on the real span of R if R ⊂ K (Kac’s conjecture). We also note that the triple (eα , tα , fα ) in (AR3) spans a 2-dimensional abelian or a 3-dimensional Heisenberg algebra, depending on tα = 0 or tα = 0. 6.7 Invariant aﬃne reﬂection algebras. Recall that any bilinear form (·|·) on a vector space V gives rise to a linear map : V → V ∗ : v → v , where v is deﬁned by v (u) = (v|u) for v, u ∈ V . The map is injective if and only if (·|·) is nondegenerate. In this case, is an isomorphism if and only if V is ﬁnite dimensional. We will call (E, T ) an invariant aﬃne reﬂection algebra, or IARA for short, if the axioms (IA1)–(IA3) below hold. (IA1) E has an invariant nondegenerate symmetric bilinear form (·|·) such that the restriction (·|·)T of the form to T × T is nondegenerate. (IA2) For every 0 = α ∈ R there exists e±α ∈ E±α such that 0 = [eα , e−α ] ∈ T . Suppose (E, T ) satisﬁes (IA1) and (IA2). Then for every α ∈ R there exists a unique tα ∈ T such that (tα ) = α, i.e., (tα | t)T = α(t) for all α ∈ R and t ∈ T . We use : T → T ∗ to transport the bilinear form (·|·)T to a symmetric bilinear form (·|·)X on X = spanK (R). Thus, by deﬁnition (α | β)X = (tα | tβ )T = α(tβ ) for α, β ∈ R. We deﬁne the null and anisotropic roots as R0 = {α ∈ R : (α|α)X = 0} and Ran = {α ∈ R : (α|α)X = 0}.

(6.7.1)

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We can now introduce the remaining axiom (IA3): (IA3) ad xα is locally nilpotent for any α ∈ Ran and xα ∈ Eα . Remarks. (a) Of particular interest are invariant aﬃne reﬂection algebras, whose toral subalgebras are ﬁnite-dimensional splitting Cartan subalgebras. They have the following characterization: A toral pair (E, T ) is an invariant aﬃne reﬂection algebra with T a ﬁnite-dimensional splitting Cartan subalgebra if and only if • E has an invariant nondegenerate symmetric bilinear form, • T = E0 is ﬁnite-dimensional, and • (IA3) holds. (b) We point out that an IARA may well have an empty set of anisotropic roots. For example, the extended Heisenberg algebra, constructed in [Kac2, 2.9] and denoted g(0) there, is an IARA with respect to the subalgebra h of loc. cit. with Ran = ∅. 6.8 Theorem. Suppose (E, T ) is an invariant aﬃne reﬂection algebra. Then the conditions (AR1)–(AR3) of 6.6 are satisﬁed for the form (·|·)X deﬁned in 6.7, and (R, X) is a symmetric aﬃne reﬂection system with Rre = Ran = Rint and R0 = Rim . Moreover, the following hold. (a) If (E, T ) is tame, then R is tame. (b) The centreless core L = Ecc = K/ Z(K) is a predivision-(S, Λ)graded Lie algebra, which is invariant with respect to the form induced from (·|·) | K × K by the canonical epimorphism K → L. This result and the examples below indicate that the bilinear form (·|·) on an invariant aﬃne reﬂection algebra E plays only a secondary role. We therefore deﬁne an isomorphism of IARAs as an isomorphism of the underlying aﬃne reﬂection algebras, i.e., two IARAs (E, T ) and (E , T ) are isomorphic if there exists a Lie algebra isomorphism ϕ : E → E such that ϕ(T ) = T . It then follows that ϕr (Ran ) = Ran . 6.9 A construction of invariant aﬃne reﬂection algebras. We have seen in 6.8 that the centreless core of an invariant aﬃne reﬂection algebra is an invariant predivision-(S, Λ)-graded Lie algebra with Λ a torsion-free abelian group. We will now describe a construction of invariant aﬃne reﬂection algebras which starts from any invariant predivision-root-graded Lie algebra. In particular, it will follow from 6.10 that every invariant predivision-(S, Λ)-graded Lie algebra L with S a locally ﬁnite root system and Λ a torsion-free abelian group arises as the centreless core of an invariant aﬃne reﬂection algebra.

(6.9.1)

Our construction uses data (L, D, TD , C, TC , τ ) as described below in (IAa) – (IAf).

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(IAa) L = λ∈Λ, ξ∈S Lλξ is an invariant predivision-(S, Λ)-graded Lie algebra with Λ torsion-free. As explained in 6.5, it does no harm to assume that suppQ(S) L = S and that Λ is spanned by suppΛ L = ξ∈S Λξ where Λξ = {λ ∈ Λ : Lλξ = 0}. Since L is predivision-graded, we can choose families (eλξ , hλξ , fξλ ) ∈ Lλξ × L00 × L−λ −ξ , ξ ∈ S and λ ∈ Λξ , such that eλξ for ξ = 0 is invertible with inverse fξλ , in particular (eλξ , hλξ , fξλ ) is an integrable sl2 -triple, and such that (eλ0 |f0λ )L = 0 but [f0λ , eλ0 ] = 0 = hλ0 . We put × TL = h = spanK {h0ξ : ξ ∈ Sind }.

λ We denote by grSCDer L = λ∈Λ (SCDer L) the Λ-graded subalgebra of centroidal derivations which are skew-symmetric with respect to (·|·)L , see 4.9. One knows that D = {∂θ : θ ∈ HomZ (Λ, K)}, the space of Λ-degree derivations of L (4.10), is contained in (SCDer L)0 . As in 6.5 every λ ∈ Λ gives rise to a well-deﬁned linear form evλ ∈ D∗ . We can now describe the remaining data (D, TD , C, TC , τ ). (IAb) D = λ∈Λ Dλ is a Λ-graded subalgebra of grSCDer L. (IAc) TD ⊂ D0 ∩ D is a subspace such that the restricted evaluation map ∗ Λ → TD , given by λ → evλ |TD , is injective. (IAd) C = λ∈Λ C λ is a Λ-graded subspace of Dgr∗ , which is invariant under the contragredient action of D on Dgr∗ and contains spanK {σD (l1 , l2 ) : li ∈ L}, where σD : L × L → Dgr∗

is the central 2-cocycle deﬁned by σD (l1 , l2 )(d) = d(l1 ) | l2 L . ∗ (IAe) TC ⊂ C 0 is a subspace such that the restriction map TC → TD , λ λ tC → tC |TD , is injective and such that σD (eξ , fξ ) ∈ TC for all ξ ∈ S and λ ∈ Λξ .

(IAf) τ : D × D → C is an invariant toral 2-cocycle, i.e., τ is a bilinear map τ : D × D → C such that for d, d1 , d2 , d3 ∈ D

τ (d, d) = 0, τ ([d1 d2 ], d3 ) = d1 · τ (d2 , d3 ), τ (d1 , d2 )(d3 ) = τ (d2 , d3 )(d1 ) τ (TD , D) = 0.

Here

and

indicates the sum over all cyclic permutations of (1, 2, 3).

Now assume that (L, TL , D, TD , C, TC , τ ) satisfy (IAa) – (IAf). Then E = C ⊕ L ⊕ D becomes a Lie algebra with respect to the product

EALAs and Generalizations

[c1 ⊕ l1 ⊕ d1 , c2 ⊕ l2 ⊕ d2 ] =

103

σD (l1 , l2 ) + d1 · c2 − d2 · c1 + τ (d1 , d2 )

⊕ [l1 , l2 ]L + d1 · l2 − d2 · l1 ⊕ [d1 , d2 ]D ,

where ci ∈ C, li ∈ L, di ∈ D, and [., .]L and [., .]D denote the Lie algebra product of L and D respectively. Moreover, T = TC ⊕ h ⊕ TD is a toral subalgebra of E such that (IA1.i) holds for (E, T ) with respect to the bilinear form (·|·)E given by ( c1 ⊕ l1 ⊕ d1 | c2 ⊕ l2 ⊕ d2 )E = c1 (d2 ) + c2 (d1 ) + (l1 | l2 )L .

(6.9.2)

To describe the roots of (E, T ), recall that S uniquely embeds into h∗ . For ξ ∈ S and λ ∈ Λξ we deﬁne a linear form ξ ⊕ λ ∈ T ∗ by (ξ ⊕ λ) (tC ⊕ h ⊕ tD ) = ξ(h) + evλ (tD ) (tC ∈ TC , h ∈ h, and tD ∈ TD ). The root spaces of (E, T ) then are C λ ⊕ Lλ0 ⊕ Dλ , ξ = 0, λ ∈ Λ0 , Eξ⊕λ = Lλξ , ξ = 0, λ ∈ Λξ . Hence the set of roots of (E, T ) is R = {ξ ⊕ λ ∈ T ∗ : ξ ∈ S, λ ∈ Λξ }. Examples. Suppose L satisﬁes (IAa). Of course, D = grSCDer L or D = D are always possible choices for D. By 4.9, grSCDer L is the semidirect product of the abelian subalgebra (CDer L)0 and the ideal λ=0 (SCDer L)λ , which indicates that there are lots of possibilities for D. An example for C is Cmin = spanK {σD (l1 , l2 ) : li ∈ L}. In general, any D-submodule between Cmin and Cmax = (grSCDer L)gr∗ is a possible choice for C. The conditions on TD and TC are interrelated. Since K has characteristic 0 and Λ is torsion-free, TD = D fulﬁlls the conditions in (IAc), see 4.11. Then TC,min = spanK {σD (eλξ , fξλ ) : ξ ∈ S, λ ∈ Λξ } satisﬁes (IAe), and hence any bigger space does as well. For τ , we can of course always take τ = 0. But there are in general many more interesting and nontrivial cocycles. For example, this already happens in the case where D = grSCDer L is a generalized Witt algebra and C = Dgr∗ , see [ERM]. In particular, the choices (D, TD , C, TC ) = (D, D, Cmin , Cmin ) and (D, TD , C, TC ) = (grSCDer L, D, (grSCDer L)gr∗ , Dgr∗ ) fulﬁll all conditions (IAb)–(IAe). 6.10 Theorem. The pair (E, T ) constructed in 6.9 is an invariant aﬃne reﬂection algebra with respect to the form (6.9.2). Its centreless core is L. Moreover: (a) The core Ec of E is always contained in Cmin ⊕ L, and (E, T ) is tame if and only if Ec = C ⊕ L, which is in turn equivalent to C ⊕ L being perfect.

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(b) If D = TD = D and C = TC = Cmin then (E, T ) is tame. (c) The root system R of (E, T ) is always tame. (d) E0 = T if and only if L is an invariant Lie torus, TD = D0 and TC = C 0 . 6.11 Extended aﬃne Lie algebras. These are special types of invariant aﬃne reﬂection algebras and are deﬁned in the introduction. However, since the goal here is a description of all extended aﬃne Lie algebras in terms of their centreless cores (6.14), we will only consider tame extended aﬃne Lie algebras in this section. To simplify the presentation, we will include tameness in the deﬁnition of an EALA. Hence, an EALA here is the same as a tame EALA in the introduction. For the convenience of the reader, who is solely interested in extended aﬃne Lie algebras, and also for easier comparison with [Neh7] and other papers on extended aﬃne Lie algebras, we list the complete set of axioms, following [Neh7]. Moreover, following the tradition in EALA theory, we will denote the toral subalgebra of E by H (and not by T ). This is “justiﬁed” since (6.11.1) below shows that H is a splitting Cartan subalgebra of E. And now, without any further ado, an extended aﬃne Lie algebra or EALA for short, is a pair (E, H) consisting of a Lie algebra E over a ﬁeld K of characteristic 0 and a toral subalgebra H satisfying the following axioms (EA1)– (EA6). We denote by R ⊂ H ∗ the set of roots of (E, H), hence E has a root space decomposition E = α∈R Eα with respect to H. (EA1) E has an invariant nondegenerate symmetric bilinear form (·|·). (EA2) H is nontrivial, ﬁnite-dimensional and self-centralizing subalgebra E0 = H.

(6.11.1)

Before we can state the other axioms, we need to draw some consequences of (EA1) and (EA2). The condition (6.11.1) implies that the restriction of the form (·|·) to H × H is nondegenerate. Hence (IA1) is satisﬁed. As in 6.7 we can therefore deﬁne tα ∈ H for all α ∈ H ∗ , transport the restricted form (·|·) | H × H to a symmetric bilinear form (·|·)X on X = H ∗ and deﬁne isotropic roots R0 = {α ∈ R : (α|α)X = 0} and anisotropic roots Ran = {α ∈ R : (α|α)X = 0}. Welet K = Ec be the core of (E, H), i.e., the subalgebra of E generated by {Eα : α ∈ Ran }. We can now state the remaining axioms. (EA3) For every α ∈ Ran and xα ∈ Eα , the operator ad xα is locally nilpotent on E. (EA4) Ran is connected in the sense that for any decomposition Ran = R1 ∪ R2 with (R1 | R2 ) = 0 we have R1 = ∅ or R2 = ∅. (EA5) (E, H) is tame.

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(EA6) The subgroup Λ = R0 ⊂ H ∗ generated by R0 in (H ∗ , +) is a free abelian group of ﬁnite rank. Suppose the pair (E, H) satisﬁes (EA1)–(EA3). Then (IA1) holds, and because of (EA3) also (IA2) and (IA3) hold. It therefore follows that an EALA is an IARA (E, H) with the following four additional properties: H is a ﬁnite-dimensional splitting Cartan subalgebra and (EA4)–(EA6) hold. In particular, we will deﬁne an isomorphism of EALAs as an isomorphism of their underlying IARAs. Since any pair satisfying (EA1)–(EA3) is an IARA, Th. 6.8 implies that (R, X) is an aﬃne reﬂection system. It therefore follows from 3.2 that (EA4) is equivalent to (EA4 ) The quotient root system (S, Y ) of the aﬃne reﬂection system (R, X) is an irreducible ﬁnite root system. Without axiom (EA6), Λ is always a torsion-free abelian group. Because a torsion-free ﬁnitely generated abelian group is free of ﬁnite rank, the axiom (EA6) is equivalent to Λ being ﬁnitely generated. As we mentioned earlier, Λ is an invariant of an EALA whose rank is called the nullity of E. Since an EALA (E, H) is in particular an aﬃne reﬂection algebra, the root system of (E, H) and its core are described in 6.3 and 6.4. Because of (1), the core and the centreless core are invariant Lie-(S, Λ)-tori. We will next describe some properties of this class of Lie tori, without assuming that they are the centreless core of an EALA. This is important since in 6.13 we will specialize the construction 6.9 to obtain EALAs starting with any invariant Lie-(S, Λ)-torus. Because of the special properties of Lie tori we will in fact obtain a greatly simpliﬁed version of 6.9. 6.12 Some properties of invariant Lie tori. Let L be an invariant Lie(S, Λ)-torus where, as in 6.11, S is a ﬁnite irreducible root system and Λ is a free abelian group of ﬁnite rank. Thus L has a (Q(S), Λ)-grading L = ξ∈S, λ∈Λ Lλξ . (6.12.1) λ (i) (5.9) The decomposition L = ξ∈S Lξ , where Lξ = λ∈Λ Lξ , is the root space decomposition with respect to the toral subalgebra hL = × spanK {hα : α ∈ Sind }. We have L00 = hL . (ii) L is ﬁnitely generated as Lie algebra and has bounded dimension with respect to the (Q(S), Λ)-grading (1), i.e., there exists a constant C such that dimK Lλξ ≤ C for all ξ ∈ S and λ ∈ Λ (this is of course only of interest for ξ = 0).

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Let t be the maximum cardinality of minimal generating sets of the ﬁnitely generated reﬂection spaces Λξ , ξ ∈ Sind . Then L can be generated by ≤ 2t rank S elements and dimK Lλ0 ≤ 4(t rank S)2 . (iii) Because of (ii), DerK L is (Q(S), Λ)-graded: (Der L)λξ , DerK L = ξ∈S, λ∈Λ

where (DerK L)λξ consists of those derivations mapping Lµτ to Lµ+λ τ +ξ . Moreover, the dimension of the homogeneous subspaces (DerK L)λξ is bounded. The analogous statements also hold for SCDer L. (iv) (5.8(c)) The Λ-graded algebra L is graded-simple and prime. The ˜ ˜ centroid of L is an integral domain, and L embeds into the C-algebra L⊗C˜ = L ˜ (tensor product over CentK (L)), where C is the quotient ﬁeld of CentK (L). (v) (5.8(b)) Γ = suppΛ CentK (L) is a subgroup of Λ, and CentK (L) is isomorphic to the group algebra K[Γ ] over K, hence to a Laurent polynomial ring in m variables, 0 ≤ m ≤ rank Λ (all possibilities occur). Because of (CentK L)0 ∼ = K · Id, it follows that (SCDer L)0 = D. (vi) (5.8(b)) L is free as CentK L-module; the following are equivalent: (a) L is a ﬁnite rank Centk (L)-module; (b) L is fgc; (c) the number [Λα : Γ ] of cosets of Λα in Γ is ﬁnite for some (and hence for every) short root α; (d) [ suppΛ L : Γ ] < ∞. ˜ is ﬁnite dimensional and central-simple over C. ˜ In this case, the Lie algebra L (vii) L is fgc, if S is not of type A. (viii) Der L is the semidirect product of the ideal IDer L of all inner derivations and the subalgebra CDer L of centroidal derivations: Der L = IDer L CDer L. Consequently, SDerL = IDer L SCDer L, where SDerL denotes the subalgebra of skew-symmetric derivations of L. 6.13 A construction of EALAs. We specialize the construction and the data (L, D, TD , C, TC ) of 6.9 with the aim of obtaining an EALA (E, H). As explained in 6.10, L must therefore be an invariant Lie-(S, Λ)-torus where S is a ﬁnite irreducible root system and Λ is a free abelian group of ﬁnite rank. Also, since H is a splitting Cartan subalgebra, we must take TD = D0 and TC = C 0 , see 6.10. By 6.12, we have grSCDer L = SCDer L with (SCDer L)0 = D. Moreover SCDer L and hence any Λ-graded subalgebra have ﬁnite homogenous dimension. Since C and D are nondegenerately paired, it follows that C = Dgr∗ . Therefore, specializing 6.9, we obtain the following construction, based on data (L, D, τ ), where

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(EAa) L is an invariant Lie-(S, Λ)-torus with (S, Λ) as above, (EAb) D is a graded subalgebra of SCDer L such that the canonical map Λ → (D0 )gr∗ , λ → evλ |D0 , is injective, and (EAc) τ : D × D → Dgr∗ is an invariant toral 2-cocycle, as in (IAf). We denote the resulting invariant aﬃne reﬂection algebra by E(L, D, τ ). By construction, H = (D0 )∗ ⊕ h ⊕ D0 is a splitting Cartan subalgebra and (EA4) and (EA6) hold. One can also show that Dgr∗ ⊕ L is perfect, so that (E, H) is tame. Thus we have the ﬁrst part of the following theorem, describing the structure of EALAs in terms of Lie tori. 6.14 Theorem. The invariant aﬃne reﬂection algebra E(L, D, τ ) constructed in 6.13 is an EALA. Conversely, let E be an EALA and let L be its centreless core. Then there exist unique data (D, τ ) as in 6.13 such that E = E(L, D, τ ). Since D = D or D = SCDer L and τ = 0 always satisfy the conditions (EAb) and (EAc), any invariant Lie torus arises as the centreless core of an EALA. Example. If L = g ⊗K K[t±1 ] is the untwisted loop algebra as in 2.13 and 4.12(b), then grSCDer L = SCDer L = K d(0) . Hence D = K d(0) is the only possible choice for D and then necessarily τ = 0. Thus, in this case E(L, D, τ ) is the Lie algebra E constructed in 2.13(III), and this is the only extended aﬃne Lie algebra based on L = g ⊗ K[t±1 ]. In the remainder of this section we will discuss other classes of invariant aﬃne reﬂection algebras. 6.15 Discrete extended aﬃne Lie algebras. Let E be a Lie algebra over K = C and let H be a toral subalgebra of E. We call (E, H) a discrete extended aﬃne Lie algebra if (E, H) satisﬁes the axioms (EA1)–(EA5) above and in addition (DE) R is a discrete subset of H ∗ . In (DE) “discrete” refers to the natural topology of H ∗ . Suppose (E, H) satisﬁes (EA1)–(EA5). Then tameness of (E, H), which is axiom (EA5), implies tameness of R. Also, the quotient root system S is irreducible by (EA4). One can easily show that (DE) holds iﬀ R0 is discrete iﬀ Λ is discrete. Since a discrete subgroup of H ∗ is always free of ﬁnite rank, (DE) implies (EA6). Hence, a discrete EALA is in particular an EALA, but not every EALA over K = C or R is discrete, see 6.17. In the construction 6.13, E(L, D, τ ) is a discrete EALA if the evaluation map Λ → (D0 )∗ has a discrete image. Conversely, any discrete EALA arises in this way. For example, the embedding Λ → D∗ always has a discrete image.

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6.16 Reductive Lie algebras. Let E be a ﬁnite-dimensional reductive Lie algebra over K, and let T be a maximal toral subalgebra of the semisimple subalgebra [E, E]. Then (E, T ) is an invariant aﬃne reﬂection algebra of nullity 0. To discuss the core and tameness of (E, T ) we ﬁrst recall that a ﬁnitedimensional simple Lie algebra s is called anisotropic if s does not contain a nonzero toral subalgebra. Otherwise, s is said to be isotropic. Coming back to the case of a reductive Lie algebra E with a maximal toral subalgebra T of [E, E], we observe that the core of (E, T ) is the ideal of [E, E] spanned by all isotropic components of [E, E]. It equals the centreless core and is division-graded. Moreover, (E, T ) is tame if and only if E is semisimple and every simple component of E is isotropic. In this case, E equals its core and centreless core. One can characterize the class of semisimple Lie algebras with all simple components being isotropic as those tame IARAs of nullity 0 whose set of roots is ﬁnite and whose core is division-graded with ﬁnite-dimensional root spaces. In particular, (E, H) is an EALA of nullity 0 if and only if E is ﬁnitedimensional split simple with splitting Cartan subalgebra H. 6.17 Locally extended aﬃne Lie algebras. A toral pair (E, H) with set of roots R is called a locally extended aﬃne Lie algebra or LEALA for short ([MY]), if (E, H) satisﬁes the following axioms: (LEA1) H = E0 , (LEA2) E has an invariant nondegenerate symmetric bilinear form (·|·), (LEA3) adE x is locally nilpotent for x ∈ Eα and α ∈ Ran = {α ∈ R : (α|α) = 0}. (LEA4) Ran is connected. Let (E, H) be such a Lie algebra. We point out that Ran = ∅ is allowed. Since (·|·) | H × H is nondegenerate by invariance of (·|·) and (LEA1), the axiom (IA1.i) is fulﬁlled. Observe (IA1.ii) = (LEA3) and (IA3) = (LEA4). Because also (IA2) holds, we have a LEALA is the same structure as an invariant aﬃne reﬂection algebra (E, H) with a splitting Cartan subalgebra H and a connected set of anisotropic roots. In this case, the core K of (E, H) and hence also the centreless core L are Lie tori, 6.4. Moreover, by 6.8, L is an invariant Lie torus. We note that LEALAs generalize EALAs: A LEALA (E, H) is an EALA if and only if H is ﬁnite dimensional, (E, H) is tame and R0 is ﬁnitely generated.

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Examples. (a) By 6.10(e), the invariant aﬃne reﬂection algebras constructed in 6.9 are LEALAs if and only if TD = D0 , TC = C 0 and L is an invariant Lie torus, say of type (S, Λ) where S is an irreducible locally ﬁnite root system. For example, the IARAs (L, D, D, Cmin (D), Cmin (D), τ ) are LEALAs. (b) Let g be a ﬁnite-dimensional split simple Lie K-algebra or a centreless inﬁnite rank aﬃne algebra, see 2.12. Let Λ be a subgroup of (K, +) generated by a set B of Q-linearly independent elements, and let K[Λ] be the group algebra of Λ. Then L = g ⊗K K[Λ] is a Lie-(S, Λ)-torus which is an invariant with respect to the bilinear form κ ⊗ , where κ is an invariant nondegenerate symmetric bilinear form on g and is the canonical trace form of K[Λ]. Let θ : Λ → K be given by θ(λ) = λ and denote by ∂θ the corresponding degree derivation. Then D = K∂θ = TD and C = D∗ = TC satisfy (IAb)–(IAe), and hence the construction 6.9 produces a LEALA E, e.g., by taking τ = 0. It is tame. Note that for this example the aﬃne root system is R = {ξ ⊕ λ ∈ h∗ ⊕ (K∂θ )∗ : ξ ∈ S, λ ∈ Λ} with isotropic roots R0 ∼ = Λ ⊂ (K∂θ )∗ . Hence, the nullity of (E, H) is the cardinality |B|, while dimK spanK (Rim ) = 1. In particular, this shows that 6.2(2) is in general a strict inequality. It also follows that this LEALA is an EALA if and only if S and B are ﬁnite. Assume this to be the case and let K = C. Then Λ = R0 is a discrete subset of D ∼ = K, i.e., E is a discrete EALA, if and only if B is linearly √ independent over R. For example, the EALA corresponding to B = {1, 2} is not discrete. 6.18 Generalized reductive Lie algebras. By deﬁnition [Az4], these are the invariant aﬃne reﬂection algebras (E, H) over K = C with the following properties: (GRLA1) H is a nontrivial ﬁnite-dimensional splitting Cartan subalgebra, (GRLA2) the set of roots R of (E, H) is a discrete subset of H ∗ . We point out that Ran is not necessarily connected, i.e., the quotient root system of the corresponding aﬃne reﬂection system is not necessarily irreducible. It follows from Theorem. 6.4(c), 6.8 and the axioms (GRLA1), (GRLA2) that the aﬃne reﬂection system of such an algebra has ﬁnite rank, unbroken root strings and is symmetric, discrete and reduced. In other words, R is a generalized reductive root system in the sense of 3.11. As in 6.17, the core and centreless core of such an algebra are invariant Lie tori, say of type (S, Λ). Here S is a ﬁnite, not necessarily irreducible root system and Λ is a free abelian group of ﬁnite rank. The construction 6.9 provides examples of this type of IARAs, assuming in (IAa) that L is an invariant Lie-(S, Λ)-torus with S a ﬁnite root system and Λ ∼ = Zn and in (IAc) and (IAe) that TD = D0 and TC = C 0 .

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6.19 Toral type extended aﬃne Lie algebras. By deﬁnition [AKY], these are invariant aﬃne reﬂection algebras (E, T ) which also satisfy the conditions (TEA1)–(TEA4): (TEA1) T is nontrivial and ﬁnite-dimensional, (TEA2) the aﬃne root system R of (E, T ) is tame and Ran is connected, (TEA3) (E, T ) has ﬁnite nullity, and (TEA4) R is a discrete subset of spanQ (R) ⊗ R. It follows from Th. 6.4(c), 6.8 and the axioms (TEA2)–(TEA4) that the aﬃne reﬂection system of such an algebra has ﬁnite rank, a connected Re(R), unbroken root strings and is symmetric, discrete and tame. In other words, R is an extended aﬃne root system as deﬁned in 3.11 with the exception that R need not be reduced. 6.20 Kac-Moody algebras. Concerning Kac–Moody algebras we will follow the deﬁnitions and terminology of [Kac2]. We have the following characterization of invariant aﬃne reﬂection algebras and extended aﬃne Lie algebras among the Kac–Moody algebras: Let g(A) be a Kac–Moody algebra for some generalized Cartan matrix A and let h be its standard Cartan subalgebra. Then the following are equivalent: (i) (g(A), h) is an invariant aﬃne reﬂection algebra, (ii) A is symmetrizable and every imaginary root is a null root, (iii) every connected component of the Dynkin diagram of A is either of ﬁnite or of aﬃne type. In this case, the anisotropic roots of the associated aﬃne reﬂection system equals the set of real roots in the terminology of [Kac2], and the core and the centreless core of g(A) are Lie tori. In particular, (g(A), h) is an EALA if and only if A is of ﬁnite or aﬃne type, in which case the nullity of g(A) is, respectively, 0 or 1, and g(A) is discrete. In fact, one has a much stronger result: 6.21 Theorem. ([ABGP]) A complex Lie algebra E is (isomorphic to) a discrete extended aﬃne Lie algebra of nullity 1 if and only if E is (isomorphic to) an aﬃne Kac–Moody algebra. 6.22 Notes. The results in this section are due to the author, unless indicated otherwise in the text or below. Some results have been independently proven by others, as indicated below. The importance of isotopy for the theory of extended aﬃne Lie algebras is explained in [AF]. Assuming the Structure Theorem for extended aﬃne Lie algebras, Th. 6.4(d) is in fact proven in [AF, Th. 6.1] for the case of extended aﬃne Lie algebras.

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The notion of “nullity” goes back to the early papers [BGK] and [BGKN] on EALAs, where it was implicitly assumed that (6.2.2) is an equality. This was later corrected in [Ga2]. Beware that in the paper [MY] the term “null rank” is used for the nullity as deﬁned here, while nullity has another meaning. Kac’s conjecture (6.6) for EALAs was proven in [AABGP, I, §2]. It was shown for LEALAs in [MY, Th. 3.10] and for GRLAs in [Az4, §1]. The proof of 6.6 generalizes techniques found in these papers. Theorem. 6.8(a) is proven in [ABP1, Lemma 3.62] for EALAs. The deﬁnition of an extended aﬃne Lie algebra in 6.11 is due to the author ([Neh7]). It extends the previous deﬁnition from [AABGP], which only made sense for Lie algebras over C or R. Tame extended aﬃne Lie algebras in the sense of [AABGP] are here called discrete extended aﬃne Lie algebras (see 6.11). To describe the structure of the core and the centreless core of an EALA E, it would be enough to require that R be tame. We have included tameness of (E, H) in the deﬁnition of an EALA since there seems to be little hope of classifying non-tame EALAs, as the examples in 6.10 hopefully demonstrate. The reader can ﬁnd another example at the beginning of §3 of [BGK]. That a tame extended aﬃne Lie algebra has a tame root system, see 6.8(b), is proven in [ABP1, 3.62]. The structure of the root system and the core of an EALA is determined in [AG, §1]. Some of the properties of Lie tori summarized in 6.12, the construction 6.13 and Theorem. 6.14 were announced in [Neh7]. The characterization of fgc Lie tori in 6.12(vi) is also proven in [ABFP2, Prop. 1.4.2]. The interest in fgc Lie tori comes from the multiloop realization described in 4.7. This description is made more precise in [ABFP2]. In this context it is of importance to know 6.12(vii). Part (viii) is crucial for the Structure Theorem 6.14. Yoshii has shown in [Yo5] that, for S a ﬁnite root system, every centreless complex Lie-(S, Zn )-torus automatically has a graded invariant nondegenerate symmetric bilinear form (·|·)L with respect to which (L, h) is invariant. Such a form is unique, up to scalars. Hence, in this case one could leave out the quantiﬁer “invariant”. Yoshii’s proof uses the classiﬁcation of Lie tori of type (A1 , Zn ) and hence ultimately Zelmanov’s Structure Theorem for strongly prime Jordan algebras ([Z], [MZ]). The construction 6.13 with D = D and τ = 0 appears in [AABGP, III] ˙ l , (l ≥ 3), D and a modiﬁed (twisted) version of it in [Az2]. For S of type A or E, the construction with D = SCDer L and arbitrary τ can be found in [BGK]. The fact that the Moody–Rao cocycle provides a nontrivial example of an invariant toral 2-cocycle is also mentioned there. Finally, for S = A2 the general cosntruction is given in [BGKN], modulo the mistake mentioned above. The two papers [BGK] and [BGKN] rely heavily on a concrete realization of the corresponding Lie torus L, and do not present the construction in a way suitable for generalization to the case of an arbitrary S.

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LEALAs (6.17) were introduced in [MY]. The Example(b) of 6.17 appears there. This paper also contains examples of LEALAs with Ran = ∅ and a classiﬁcation of LEALAs of nullity 0. It is shown in [Az4] that the set of roots of a generalized reductive Lie algebra is a generalized root system and that its core and centreless core are Lie tori. A proof that the set of roots of a toral type extended aﬃne Lie algebra is an extended aﬃne root system is given in [AKY]. This paper also gives an example of a toral type EALA of type BC with a non-reduced set of roots.

7 Example: slI (A) for A associative We describe the various concepts introduced in the previous sections for the special linear Lie algebra slI (A) and its central quotient pslI (A), where A is an associative algebra. Unless speciﬁed otherwise, k is a commutative associative ring with unit element 1k such that 2·1k and 3·1k are invertible in k. All algebraic structures are deﬁned over k. Throughout, I is a not necessarily ﬁnite set of cardinality |I| ≥ 3 and Λ denotes an abelian group, written additively. 7.1 Basic deﬁnitions. Let A be a unital associative k-algebra with identity element 1A . We recall that [a, b] := ab−ba for a, b ∈ A denotes the commutator in A. We put [A, A] = spank {[a, b] : a, b ∈ A} and we denote by Z(A) = {z ∈ A : [z, a] = 0} the centre of A. The underlying k-module of A becomes a Lie algebra with respect to the commutator product [., .], denoted A− . We denote by MatI (A) the set of all ﬁnitary I × I-matrices with entries from A. By deﬁnition, an element of MatI (A) is a square matrix x = (xij )i,j∈I with all xij ∈ A and xij = 0 for only ﬁnitely many indices (ij) ∈ I × I. With respect to the usual addition and multiplication of matrices, MatI (A) is an associative k-algebra. For i, j ∈ I we let Eij ∈ MatI (A) be the matrix with entry 1A at the position (ij) and 0 elsewhere. Then any x ∈ MatI (A) can be uniquely written as x = i,j xij Eij , xij ∈ A, with almost all xij = 0. We denote by glI (A) the Lie algebra MatI (A)− and by slI (A) = [glI (A), glI (A)] its derived algebra. One easily veriﬁes that slI (A) is the subalgebra of glI (A) generated by all oﬀ-diagonal matrices AEij , i = j and that slI (A) can also be described as slI (A) = {x ∈ glI (A) : tr(x) ∈ [A, A]} (7.1.1) where tr(x) = i xii is the trace of x. Since slI (A) ∼ = slJ (A) if |I| = |J| we put sln (A) = slI (A) if |I| = n ∈ N.

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It is important that the algebra A can be recovered from the Lie algebra slI (A) by the product formula (7.1.2) ab Eij = [aEij , Ejl ], Eli , bEij where a, b ∈ A and i, j, l ∈ I are distinct. What makes slI (A) somewhat delicate are the diagonals of its elements. For a commutative A the description (7.1.1) of slI (A) shows that slI (A) is the “usual” Lie algebra: slI (A) = {x ∈ glI (A) : tr(x) = 0} ∼ (A commutative). = slI (k) ⊗k A (7.1.3) But (7.1.3) is no longer true for a noncommutative A. Rather, if n · 1 ∈ k × , then any x ∈ sln (A) can be uniquely written in the form x = aEn + x with a ∈ [A, A] and x ∈ Matn (A) with tr(x ) = 0, namely 1 1 x = tr(x)En + (x − tr(x)En ). n n Hence, as a k-module, sln (A) = [A, A]En ⊕ (sln (k) ⊗k A)

(n · 1k ∈ k × ),

(7.1.4)

where En is the n×n-identity matrix. One can describe the Lie algebra product with respect to this decomposition. To do so, let us abbreviate a ◦ b = ab + ba for a, b in an associative algebra B, like B = A or B = MatI (A) (the experts will recognize a ◦ b as the “circle product” of the Jordan algebra associated to B). Then, for x, y ∈ sln (k), a, b ∈ A and c, d ∈ [A, A],

[x ⊗ a, y ⊗ b] = ( n1 tr(xy) [a, b] En ) ⊕ [x, y] ⊗ 12 (a ◦ b) +( 12 (x ◦ y) − n1 tr(xy)En ) ⊗ [a, b] , [cEm , dEn ] = [c, d] En , [cEn , x ⊗ a] = x ⊗ [ca]. ˙ I -graded Lie algebra. Recall the irreducible root system 7.2 slI (A) as A ˙ R = AI = { i − j : i, j ∈ I} deﬁned in 2.8. For α = i − j = 0 and β = m − n ∈ R the integers β, α∨ are m − n , ( i − j )∨ = δmi −δmj −δni +δnj , where δ∗∗ is the Kronecker delta. ˙ I )-grading with support A ˙ I, The Lie algebra slI (A) has a natural Q(A whose homogeneous spaces are given by α = i − j = 0, AEij , slI (A)α = (7.2.1) {x ∈ slI (A) : x diagonal}, α = 0. Because of our assumption that 2 and 3 are invertible in k, the space slI (A)α , α = i − j = 0, has the description slI (A)α = {x ∈ slI (A) : [Eii − Ejj , x] = 2x}

(7.2.2)

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An element aEij ∈ slI (A)α , α = i − j = 0, is an invertible element of the ˙ I )-graded Lie algebra slI (A), cf. 5.1, if and only if a is invertible in A. In Q(A this case, −a−1 Eji is the inverse of aEij and (aEij , Eii − Ejj , −a−1 Eji ) is an sl2 -triple. In particular, every matrix unit Eij , i = j, is invertible in slI (A). Since by construction slI (A)0 = 0=α∈R [slI (A)α , slI (A)−α ] ˙ I -graded Lie algebra in the sense of 5.1. Hence, it follows that slI (A) is an A either by 5.4 or by an immediate veriﬁcation, pslI (A) = slI (A)/ Z(slI (A)) ˙ I -graded Lie algebras. For rank A ˙ I ≥ 3 these and any covering of pslI (A) are A ˙ are in fact all AI -graded Lie algebras. ˙ I -graded if and only if 7.3 Theorem. Let |I| ≥ 4. Then a Lie algebra L is A L is a covering of pslI (A) for a unital associative algebra A. In light of this theorem and the deﬁnition of pslI (A) it is of interest to identify the centre of slI (A), which will be done in 7.4, and the coverings of pslI (A). Since any covering of a perfect Lie algebra P is a quotient of its universal central extension uce(P ) by a central ideal, it is enough to describe the universal central extension uce(pslI (A)) → pslI (A). Because slI (A) is a covering of pslI (A), the Lie algebras uce(pslI (A)) and uce(slI (A)) are the same (in any case, in many interesting cases slI (A) = pslI (A), see 7.4 and 7.7). It is therefore suﬃcient to identify uce(slI (A)), which we will do in 7.8 after we introduce gradings in 7.5. 7.4 Centre of slI (A). The centre of slI (A) has the following description, cf. (5.3.1):

{0}, if |I| = ∞, (7.4.1) Z slI (A) = {zEn ∈ slI (A) : z ∈ Z(A)}, if |I| = n ∈ N. In particular slI (A) = pslI (A) for |I| = ∞, while

Z sln (A) = {zEn : z ∈ Z(A), nz ∈ [A, A]}

for n ∈ N,

(7.4.2)

and therefore also sln (A) ∼ = psln (A) if, for example, k is a ﬁeld of characteristic 0 and A is commutative. Since Z(slI (A)) is diagonal, the homogeneous spaces pslI (A)α are given by the obviously modiﬁed formula (7.2.1). In particular, we can identify AEij = pslI (A)α for 0 = α = i − j and then also have (7.2.2) and the product formula (7.1.2) available in pslI (A).

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7.5 Graded coeﬃcient algebras. Let A = λ∈Λ Aλ be a Λ-grading of A and deﬁne slI (A)λ as the submodule of matrices in slI (A) whose entries all lie in Aλ . Then slI (A) = λ∈Λ slI (A)λ (7.5.1) ˙ I )-grading in (7.2.1). is a Λ-grading of slI (A) which is compatible with the Q(A The homogeneous spaces slI (A)λα = slI (A)α ∩ slI (A)λ are given by slI (A)λα = Aλ Eij , α = i − j = 0, while slI (A)λ0 consists of the diagonal matrices whose entries all lie in Aλ . Since 1A ∈ A0 , it follows that slI (A)0α , α = 0, contains an invertible element. Thus ˙ I , Λ)-graded Lie algebra. slI (A) = α∈A˙ I , λ∈Λ slI (A)λα is an (A (7.5.2) It is immediate from the product formula (7.1.2) that, conversely, any Λ˙ I -grading is obtained from a grading of slI (A) which is compatible with the A Λ-grading of A as described above. Since the centre of A and [A, A] are graded submodules, it follows from ˙ I ), Λ)-graded with (7.1.2) that Z(slI (A)) is Λ-graded. Hence pslI (A) is (Q(A homogenous spaces ˙ I and pslI (A)λα ∼ = slI (A)λα for 0 = α ∈ A λ λ λ pslI (A)0 = slI (A)0 Z(slI (A)) . This is of course a special case of 5.4. The graded version of 7.3 is the following theorem. ˙ I , Λ)-graded if and 7.6 Theorem. Let |I| ≥ 4. Then a Lie algebra L is (A only if L is a Λ-covering of pslI (A) for some unital associative Λ-graded algebra A. Combining this with the description of invertible elements in slI (A) immediately shows that slI (A) and pslI (A) are predivision- or division˙ I , Λ)-graded Lie algebras if and only if A is, respectively, a predivision(A ˙ I , Λ) or a division-Λ-graded algebra. Similarly, slI (A) is a Lie torus of type (A if and only if A is an associative Λ-torus. Hence we get the following corollary (a special case of 5.4). 7.7 Corollary. Let |I| ≥ 4. Then L is a predivision- or a division-(A˙ I , Λ)graded Lie algebra if and only if L is a Λ-covering of pslI (A) where A is, respectively, a predivision- or a division-Λ-graded algebra. If k is a ﬁeld, then ˙ I , Λ) if and only if L is a Λ-covering of pslI (A) L is a Lie torus of type (A where A is a Λ-torus. We refer to 4.5 for a discussion of (pre)division-graded associative algebras. Concerning the structure of Lie tori, we note that A = Z(A) ⊕ [A, A]

(7.7.1)

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for every associative Λ-torus A. Hence, by (7.4.1) and (7.4.2), we have slI (A) = pslI (A) for a Lie torus if |I| = ∞ or if A does not have |I|-torsion. In particular, the corollary says that for K a ﬁeld of characteristic 0, a Lie algebra L over K ˙ I , Zn )-torus if and only if L is a Zn -covering of slI (Kq ) where Kq is a Lie-(A is a quantum torus (see 4.13). 7.8 The universal central extension of slI (A) and pslI (A). Let A, A = (A ∧ A) B where B = spanK {ab ⊗ c + bc ⊗ a + ca ⊗ b : a, b, c ∈ A}. It follows that A, A is spanned by elements a, b = a ∧ b + B, a, b ∈ A, and that

a, b → [a, b] is a well-deﬁned map A, A → [A, A]. The kernel of this map is HC1 (A), the ﬁrst cyclic homology group of A. We also need to recall the Steinberg Lie algebra stI (A) associated to A. It is the Lie k-algebra presented by generators Xij (a), where i, j ∈ I, i = j, and a ∈ A, and relations (st1)–(st3): (st1) The map a → Xij (a) is k-linear, (st2) [Xij (a), Xjl (b)] = Xil (ab) for distinct i, j, l ∈ I, and (st3) [Xij (a), Xlm (b)] = 0 for i = m and j = l. Under our assumptions on k and I, the universal central extension of slI (A) is the map u : stI (A) → slI (A),

Xij (a) → aEij . Its kernel is HC1 (A),

(7.8.1)

˙ I -grading of stI (A) is determined by stI (A)α = Xij (A) for cf. 5.5. The A 0 = α = i − j . Any invertible element aEij ∈ slI (A) lifts to an invertible ˙ I -graded, as stated in element Xij (a) ∈ stI (A). It then follows that stI (A) is A ˙ I ) ⊕ Λ)-graded Theorem. 7.3. Similarly, if A is Λ-graded, then stI (A) is (Q(A with homogeneous spaces stI (A)λα = Xij (Aλ ) for α = i − j = 0. This explains Theorem. 7.6 and Corollary. 7.7, at least for stI (A), but then the general case easily follows. If the extension u is split on the level of k-modules, e.g., if slI (A) is projective as k-module (for example k is a ﬁeld), we can identify slI (A) with a k-submodule of stI (A) such that stI (A) = HC1 (A) ⊕ slI (A) as k-modules and the Lie algebra product [. , .]st of stI (A) is given in terms of the Lie algebra product [. , .] of slI (A) by [HC1 (A), stI (A)]s t = 0 and [x, y]st = i,j xij , yji ⊕ [x, y] (7.8.2) for x = (xij ) and y = (yij ) ∈ slI (A). In case n · 1k ∈ k × , one can give a description of uce(slI (A)) in the spirit of 7.1: We deﬁne a Lie algebra on

uce sln (A) = A, A ⊕ (sln (k) ⊗k A) (7.8.3)

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by modifying the product formulas in 7.1 as follows: [x ⊗ a, y ⊗ b] = ( n1 tr(xy) a, b) ⊕ ( . . . as in 7.1 . . .) [ c1 , c2 , d1 , d2 ] = [c1 , c2 ], [d1 , d2 ] , [ c1 , c2 , x ⊗ a] = x ⊗ [c1 , c2 ], a

for x, y ∈ sln (k) and a, b, ci , di ∈ A. The obvious map uce slI (A) → slI (A) is then a universal central extension. For example, let A = K[t±1 ] be the Laurent polynomial ring in one variable t, where K is a ﬁeld of characteristic 0. Since HC1 (A) is 1-dimensional, the Lie algebra K = slI (K[t±1 ]) ⊕ Kc constructed in 2.13 is a universal central extension of slI (A). 7.9 The centroid of slI (A). It is convenient here and in the description of derivations of slI (A) to use the following notation. Any endomorphism f ∈ Endk (A), which leaves [A, A] invariant, induces an endomorphism of slI (A), denoted slI (f ), by applying f to every entry of a matrix in slI (A): slI (f ) (x) = (f (xij )) for x = (xij ) ∈ slI (A).

(7.9.1)

The map slI (.) : {f ∈ Endk (A) : f ([A, A]) ⊂ [A, A]} → Endk (slI (A)) is a monomorphism of k-algebras. If f ∈ Endk (A) also leaves the centre Z(A) of A invariant, the map slI (f ) descends to an endomorphism of pslI (A), naturally denoted by pslI (f ). Recall (4.2) that Z(A) → Cent(A), z → Lz , is an isomorphism of kalgebras. For z ∈ Z(A) ∼ = Centk (A) the endomorphism χz = slI (Lz ) of slI (A), mapping x = (xij ) to χz (x) = (zxij ), belongs to the centroid of slI (A). The map

Z(A) → Cent slI (A) , z → χz is an isomorphism of k-algebras. Moreover, z → pslI (Lz ) is also an isomorphism between Z(A) and Cent(pslI (A)). If A is Λ-graded, then so is the centre Z(A) and hence also Cent(slI (A)). For the description of slI (A) as a multiloop algebra ([ABFP1]) it is important to know when is slI (A) ﬁnitely generated over its centroid. This is the case if and only if I is ﬁnite and A is ﬁnitely generated over Z(A). For example, if A = kq , q = (qij ) ∈ Matn (k), is a quantum torus over a ﬁeld k, A is a ﬁnitely generated module over its centre if and only all qij ∈ k are roots of unity, cf. 4.13. 7.10 Invariant bilinear forms. For a Ξ-graded algebra X, Ξ an abelian group, we denote by GIF(X, Ξ), the k-module of Ξ-graded invariant symmetric bilinear forms on X, see 4.3. In this subsection we will determine

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˙ I ) ⊕ Λ) for a Λ-graded algebra A. For simpler notation we GIF(slI (A), Q(A ˙ put Ξ = Q(AI ) ⊕ Λ. To (·|·)A ∈ GIF(A, Λ) we associate the bilinear form (·|·)sl deﬁned on slI (A) by

(7.10.1) i,j xij Eij | p,q ypq Epq sl = i,j (xij | yji )A . It is an easy exercise to show that (·|·)sl is invariant. Since it is obviously also Ξ-graded, we have (·|·)sl ∈ GIF(slI (A), Ξ). Observe (aEij | bEji )sl = (a | b)A

(7.10.2)

for a, b ∈ A and distinct i, j ∈ I. Conversely, this formula assigns to every (·|·)sl ∈ GIF(slI (A), Ξ) a graded bilinear form (·|·)A on A. It is immediate, using (7.1.2), that (·|·)A is invariant. Since every invariant form on slI (A) is uniquely determined by its restriction to AEij × AEji , i = j, we have an isomorphism of k-modules ∼ =

GIF(slI (A), Ξ) → GIF(A, Λ), (·|·)sl → (·|·)A . (7.10.3) Moreover, by (4.3.2), GIF(A, Λ) ∼ = (A0 [A, A]0 )∗ . Combining these two isomorphisms, it follows that any Λ-graded invariant bilinear form b of slI (A) can be uniquely written in the form b(x, y) = ϕ(tr(xy) + [A, A]) for ϕ ∈ (A0 /[A, A]0 )∗ .

(7.10.4)

Since slI (A) is perfect, GIF(slI (A)) ∼ = GIF(pslI (A)) by (4.3.1). Combining (7.10.3) with this isomorphism implies GIF(pslI (A)) ∼ = GIF(A, Λ) ∼ = (A0 /[A, A]0 )∗ .

(7.10.5)

We now turn to the description of nondegenerate invariant forms. Since the centre of a perfect Lie algebra is contained in the radical of any invariant form, Z(slI (A)) = {0} is a necessary condition for the existence of such a form on slI (A). But rather than assuming Z(slI (A)) = {0} we will work with the centreless Lie algebra pslI (A) and the isomorphism (7.10.5), written as (·|·)p sl → (·|·)A . It is easily seen that (·|·)p sl is nondegenerate

⇐⇒

(·|·)A is nondegenerate.

(7.10.6)

Example. Let A be a Λ-torus over a ﬁeld F . Since [A, A]0 = [A, A] ∩ A0 = {0} by (7.7.1), GIF(pslI (A)) ∼ = (A0 )∗ is a 1-dimensional space. Moreover, any nonzero graded invariant bilinear form on pslI (A) is nondegenerate. Indeed, any nonzero bilinear form (·|·) on A is of type (a1 | a2 ) = ψ(a1 a2 ) for a linear form ψ ∈ A∗ with Ker ψ = 0=λ Aλ . Since Rad(·|·) is graded and 1 ∈ zA for every nonzero homogeneous z ∈ A, it follows that Rad(·|·) = {z ∈ A : zA ⊂ Ker ψ} = {0}.

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I (A) the space of all row- and column7.11 Derivations. We denote by Mat ﬁnite matrices with entries from A. This is an associative k-algebra with respect to the usual matrix multiplication. Hence it becomes a Lie algebra, I (A), with respect to the commutator. It is immediate that denoted by gl (A) → [glI (A), slI (A)] ⊂ slI (A). We obtain a Lie algebra homomorphism gl I Derk slI (A), x ˆ → ad x ˆ|slI (A) with kernel {zE : z ∈ Z(A)} where E is the I (A). Its image will be denoted by identity matrix of gl ˆ (A)} ∼ (A) {zE : z ∈ Z(A)}. slI (A) = {ad x ˆ|slI (A) : x ˆ ∈ gl IDer = gl I I slI (A) = IDer slI (A) if I is ﬁnite. Since [ˆ x, Z(slI (A))] = 0 for Of course, IDer x ˆ ∈ glI (A), ad x ˆ descends to a derivation of pslI (A). Another class of derivations of slI (A) and pslI (A) are given by the maps slI (d) and pslI (d) for d ∈ Derk (A), cf. (7.9.1). The map slI : Derk (A) → Derk slI (A), d → slI (d) is clearly a monomorphism of Lie algebras. We denote by slI (Derk A) its image. One checks that ˆ I (A). [slI (d), ad x ˆ] = ad d(ˆ x) for xˆ ∈ gl 7.12 Theorem. (a) The ungraded derivation algebras are

slI (A) + slI (Derk A) with Derk slI (A) = IDer ∼ IDer A, slI (A) ∩ slI (Derk A) = slI (IDer A) = IDer

∼ CDer slI (A) = slI (CDer A) = CDer A,

slI (A) + slI (SDerA), SDer slI (A) = IDer where in the last formula the derivations in SDer are skew-symmetric with respect to a bilinear form (·|·)sl and the corresponding bilinear form (·|·)A , cf. (7.10.2) (b) Suppose A is Λ-graded. Then the homogeneous subspaces (Der slI (A))λα of derivations of degree α ⊕ λ are given by λ

α = i − j = 0, Der slI (A) α = ad slI (A)λα ) = {ad x : x ∈ Aλ Eij },

λ I (A) diagonal, all xii ∈ Aλ } ⊕ slI (Der A)λ Der slI (A) 0 = {ad x ˆ : xˆ ∈ gl In particular,

grSCDer (slI (A)) = slI grSCDer(A) ∼ = grSCDer A. (c) All statements in (a) and (b) hold, mutatis mutandis, for pslI (A). 7.13 The standard toral subalgebra. In this subsection we suppose that k = K is a ﬁeld of characteristic 0. We put h = spanK {Eii − Ejj : i, j ∈ I}

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which is the space of diagonal trace-0-matrices over K. We write an element h ∈ h as h = diag(hi )i∈I where hi ∈ K is the entry of the matrix h at the ˙ I into h∗ by position (ii). We embed the root system A ( i − j )(h) = hi − hj

for h = diag(hi ).

˙ I. Then h is a toral subalgebra of slI (A) with set of roots equal to A λ 0 ∗ Suppose A = λ∈Λ A is Λ-graded and let ψ ∈ (A ) with [A A]0 ⊂ Ker ψ. Hence ψ gives rise to a Λ-graded invariant symmetric bilinear form (·|·)A, ψ on A, given by (a1 |a2 )A, ψ = ψ(a1 a2 ). By (7.10.1) we can extend (·|·)A, ψ to ˙ I ) ⊕ Λ)-graded symmetric bilinear form (·|·)sl, ψ on slI (A). an invariant (Q(A Note that by (7.10.4) for h = diag(hi ) and h = diag(hi ) we get

(h|h )sl, ψ = i∈I ψ(hi hi ) = i∈I hi hi ψ(1). This implies (·|·)sl, ψ | h × h is nondegenerate if and only if ψ(1) = 0. Indeed, ψ(1) = 0 is obviously necessary for nondegeneracy. Conversely, if (h|h)sl, ψ = 0 for h = diag(hi ), then hi = hj for all i, j ∈ I, whence h = 0 in case I is inﬁnite, but this also holds in case of a ﬁnite I, say |I| = n, since then h = h1 En has trace nh1 = 0. ˙ I ) ⊕ Λ)As we have seen in (7.10.6), the form (·|·)sl, ψ descends to a (Q(A graded invariant symmetric bilinear form (·|·)psl, ψ on pslI (A). By (7.10.6), ˙ I , Λ)-graded Lie algebra if pslI (A) is an invariant (A and only if A has a Λ-graded invariant nondegenerate symmetric bilinear form, say (·|·)A, ψ , and ψ(1) = 0. In particular, if A is a Λ-torus the Lie torus pslI (A) is invariant with respect to any form (·|·)psl, ψ with ψ = 0. 7.14 Notes. The deﬁnition of slI (A) is standard. Some mathematicians call it the elementary Lie algebra and denote it by eI (A). One can of course also deﬁne slI (A) for |I| = 2 (recall our convention |I| ≥ 3) but then the claims in this section, e.g., (7.1.1), are no longer true. Some results on sln (A), n ∈ N, are proven in [BGK], e.g., the formula in 7.1 for the Lie algebra product of sln (A) can be found in [BGK, (2.27)–(2.29)]. In the generality stated, Theorem. 7.3 is proven in [Neh3, (3.5)]. For k a ˙ I ﬁnite, it was proven earlier by ﬁeld of characteristic 0 and I ﬁnite, i.e., A Berman–Moody [BeMo, 0.7]. Their proof shows at the same time that stn (A) is a universal central extension of any An−1 -graded Lie algebra. As stated, Theorem. 7.6 is an immediate consequence of [GaN, 2.11 and 3.4] and [Neh5, 4.7]. Corollary. 7.7 is proven in [Yo1, Prop. 2.13] for I ﬁnite and k a ﬁeld of characteristic 0. This paper also contains a construction of

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crossed products B ∗Zn ([Yo1, §3]). The description of Lie tori of type (Al , Zn ) is given in [BGK]. The formula (7.7.1) is proven in [NY, Prop. 3.3]. The case Λ = Zn is already contained in [BGK, Prop. 2.44]. That (7.8.1) is a universal central extension is proven in [Bl1] for A commutative and |I| ≥ 5, for A arbitrary and again |I| ≥ 5 in [KL] and for 3 ≤ |I| ≤ 4 in [Ga1, 2.63] (recall our assumption that 16 ∈ k). The universal central 2cocycle (7.8.2) is also given in these papers. The results in 7.8 are no longer true if 2 or 3 are not invertible in k, see [GS] for the case of an algebra A that is free as k-module. If slI (A) is a free k-module or a ﬁnitely generated and projective k-module, the kernel of the universal central extension is the second ho mology with trivial coeﬃcients, i.e., H2 slI (A) = HC1 (A). That the Lie algebra (7.8.3) is the universal central extension of sln (A) is shown in [BGK, page 359–360]. A lot is known about the cyclic homology groups HC1 (A), see [Lod]. 1 For example, if A is commutative, then HC1 (A) ∼ dA [Lod, 2.1.14]. For = ΩA|k a complex quantum torus HC1 (Cq ) is described in [BGK, Prop. 3.19] (the result is true for any ﬁeld). In particular, it follows that dim HC1 (K[t±1 ]) = 1. The formula (7.10.4) is proven in [BGK, Lemma 2.8]. The description of Der slI (A) in 7.12(a) is proven for ﬁnite I and k a ﬁeld (with A arbitrary non-associative) in [BO, Th. 4.8 and Cor. 4.9]. The proof can easily be adapted to yield the result as stated. For k = A a ﬁeld of characteristic 0 one has Derk (k) = {0}, slI (k) = pslI (k) and

slI (k), which is proven in [Nee2, Th. I.3] with a hence Derk slI (k) = IDer diﬀerent method. The derivation algebra of Cq is determined in [BGK, Lemma 2.48]. The astute reader will have noticed that we have excluded the A2 -case ˙ I , Λ)-graded Lie algebras in all statements referring to the description of (A (7.3, 7.6 and 7.7). The reason for this is that not all (A2 , Λ)-graded Lie algebra are Λ-coverings of psl3 (A) for A a Λ-graded associative algebra. Rather, one has to allow alternative algebras A. Various models of A2 -graded Lie algebras have been given, see e.g., [BeMo, 3.3], [BGKN, §2], [Neh3, 3.3] or [AF, §9]. Once a replacement of psl3 (A) has been deﬁned, the Theorems 7.3, 7.6 and 7.7 hold, mutatis mutandis, by replacing the associative algebra A with an alternative algebra A. For example, for Λ = Zn and Lie algebras over ﬁelds of characteristic 0, this has been worked out in [Yo2, §6], while the classiﬁcation of the corresponding coordinate algebras, i.e., division-Zn -graded alternative algebras, is given in [Yo2, Th. 5.7] and is valid in characteristic = 2. For arbitrary Λ, special types of alternative division Λ-graded algebras, so-called quasi-algebras, are described in [AEP]. An alternative nonassociative Zn -torus over a ﬁeld F of characteristic = 2 is graded-isomorphic to an octonion torus [BGKN, Th. 1.25], [Yo2, Cor. 5.13], see [AF, Example 9.2] for a presentation.

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Tensor Representations of Classical Locally Finite Lie Algebras Ivan Penkov and Konstantin Styrkas Jacobs University Bremen, Bremen 28759, Germany [email protected], [email protected] Summary. The structure of tensor representations of the classical finite-dimensional Lie algebras was described by H. Weyl. In this paper we extend Weyl’s results to the classical infinite-dimensional locally finite Lie algebras gl∞ , sl∞ , sp∞ and so∞ , and study important new features specific to the infinite-dimensional setting. Let g be one of the above locally finite Lie algebras and let V be the natural representation of g. The tensor representations of g have the form V⊗d for the cases g = sp∞ , so∞ , and the form V⊗p ⊗ V∗⊗q for the cases g = gl∞ , sl∞ , where V∗ is the restricted dual of V. In contrast with the finite-dimensional case, these tensor representations are not semisimple. We explicitly describe their Jordan–H¨ older constituents, socle filtrations, and indecomposable direct summands. Key words: tensor representation, Schur–Weyl duality, socle filtration, rootreductive Lie algebra. 2000 Mathematics Subject Classifications: Primary 17B65, 17B10.

Introduction One of H. Weyl’s beautiful results is the description of the structure of the space of tensors of given rank as a module over the various classical groups of transformations of the underlying space. In this paper we extend Weyl’s constructions to the locally-ﬁnite Lie algebras gl∞ , sl∞ , so∞ and sp∞ , and highlight some important diﬀerences and new features arising in the inﬁnitedimensional case. Each of the Lie algebras gl∞ , sl∞ , so∞ and sp∞ can be deﬁned as the union of the respective series of ﬁnite-dimensional Lie algebras under the obvious inclusions. We refer to gl∞ , sl∞ , so∞ and sp∞ as the classical inﬁnitedimensional locally ﬁnite Lie algebras. The structure of these Lie algebras and their representations have been studied actively in recent years; an incomplete list of references for this ﬁeld is: [B, BB, BS, DP1, DP2, DPS, DPW, N, NS]. Nevertheless, the structure of the tensor representations of the classical K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_4, © Springer Basel AG 2011

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inﬁnite-dimensional locally ﬁnite Lie algebras has not yet been suﬃciently explored. Recall that for the simple classical Lie groups SLn , SOn and Sp2n every irreducible ﬁnite-dimensional module can be realized inside the covariant tensor algebra of the natural representation V . In particular, for the dualrepren−1 sentation V ∗ we have V ∗ ∼ V = V in the cases of SOn and Sp2n , and V ∗ ∼ = in the case of SLn . However, for the reductive Lie algebra GLn not every ﬁnitedimensional irreducible representation occurs in the tensor algebra T(V ). For example, the dual representation V ∗ does not appear in T(V ). Thus, in order to obtain all irreducible ﬁnite-dimensional GLn -modules, we must consider mixed tensors, i.e. the tensor algebra T(V ⊕ V ∗ ). A similar phenomenon occurs for the simple Lie algebra sl∞ . Let V denote the natural representation of sl∞ . One can show that the (restricted) dual representation V∗ does not occur in the tensor algebra T(V), and we study the space of mixed tensors T(V ⊕ V∗ ). The main new feature of these tensor representations is their failure to be completely reducible. For so∞ and sp∞ the natural representation V is self-dual, and we consider just the tensor algebra T(V), which also fails to be a completely reducible so∞ - or sp∞ module. The main purpose of this paper is to describe the structure of the tensor algebras T(V ⊕ V∗ ) and T(V) as explicitly as possible. First, we decompose the tensor algebra into indecomposable submodules of ﬁnite length. The answer is very simple: for g = gl∞ , sl∞ the indecomposable submodules are of the form Γλ;0 ⊗ Γ0;µ , where Γλ;0 and Γ0;µ are the simple irreducible sl∞ submodules of the tensor algebras T(V) and T(V∗ ) respectively, and for g = so∞ , sp∞ the indecomposable submodules are the gl∞ -modules Γλ;0 , considered as g-modules. We then explicitly describe the socle ﬁltration of Γλ;0 ⊗ Γ0;µ as an sl∞ module, and the socle ﬁltrations of Γλ;0 both as an so∞ - and an sp∞ -module. In particular, we prove that each indecomposable submodule has a simple socle. In the last section of the paper we extend our description of tensor rep+ m, where the ideal k is a resentations to a wider class of Lie algebras g = k⊂ classical simple locally-ﬁnite Lie algebra, and m satisﬁes a certain technical condition. This class of Lie algebras includes, in particular, the root-reductive Lie algebras. The main result in this case states that the natural representation V of k can be equipped with a g-module structure, and the socle ﬁltration of any tensor representation of g coincides with its socle ﬁltration as a k-module.

1 Preliminaries By N we denote the set of positive integers. In this paper we work with a ﬁeld k of characteristic zero. All vector spaces, Lie algebras, and their modules are assumed to be deﬁned over k. Tensor products are also over k. All algebras

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and modules are assumed to be at most countable dimensional over k. The + stands for semidirect sum of Lie algebras: in k⊂ + m the Lie algebra k sign ⊂ is an ideal. Let V be a ﬁnite length module over some algebra. The socle of V , denoted socV , is the unique maximal semisimple submodule of V . Equivalently, socV is the sum of all simple submodules of V . The socle ﬁltration of V is deﬁned inductively by soc(0) V = 0,

soc(i+1) V /soc(i) V = soc(V /soc(i) V ),

i ≥ 0.

The smallest positive integer l such that soc(l) V = V is called the Loewy def length of V . The semisimple modules soc(i) V = soc(i) V /soc(i−1) V are called the layers of the socle ﬁltration. A partition λ is by deﬁnition a ﬁnite nonstrictly decreasing sequence of positive integers: λ = (λ1 ≥ λ2 ≥ · · · ≥ λk ). def

For any partition λ we set |λ| = λ1 + · · · + λk , and use the convention that λi = 0 if i > k. The empty partition is denoted by 0. For any partition λ we denote by 2λ the partition determined by (2λ)i = 2λi for all i, and by λ the partition determined by (λ )i = #{j | λj ≥ i} for all i. The Young diagram for λ is obtained by transposing the Young diagram for λ. Irreducible representations of the symmetric group Sd are parameterized by partitions λ satisfying |λ| = d, and can be realized inside the regular representation k[Sd ]. We denote def

Hλ = k[Sd ] cλ , where cλ ∈ k[Sd ] is the Young projector corresponding to the standard Young tableau of shape λ (see e.g., [FH] for details). The left regular action makes Hλ an irreducible Sd -module, and any irreducible representation of Sd can be obtained in this way. For any vector space V , the symmetric group Sd acts in V ⊗d by permuting the tensor factors. For any partition λ with |λ| = d we denote ⊗d ⊗d Sλ V = im cλ : V . →V The correspondence V Sλ V is called the Schur functor corresponding to λ. The Littlewood–Richardson coeﬃcients Nνλ,µ are nonnegative integers, ν determined for any partitions λ, µ, ν by the relation Sλ Sµ = ν Nλ,µ Sν , where Sλ denotes the Schur symmetric polynomial corresponding to the partition λ. In particular, Nνλ,µ = 0 unless |ν| = |λ| + |µ|.

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2 Tensor representations of gl∞ and sl∞ Let V, V∗ be countable dimensional vector spaces, and let ·, · : V ⊗ V∗ → k be a nondegenerate pairing. The Lie algebra gl∞ is deﬁned as the space V⊗V∗ , equipped with the Lie bracket [u ⊗ u∗, v ⊗ v ∗ ] = u∗ , v u ⊗ v ∗ − v ∗ , u v ⊗ u∗,

u, v ∈ V, u∗ , v ∗ ∈ V∗ . (1)

The kernel of the map ·, · is a Lie subalgebra of gl∞ , which we denote sl∞ . As observed by G. Mackey [M], there always exist dual bases {ξi }i∈I of V and {ξi∗ }i∈I of V∗ , indexed by a countable set I, so that we have ξj∗ , ξi = δi,j for i, j ∈ I. This gives another, more straightforward coordinate deﬁnition of gl∞ as the Lie algebra with a linear basis {Ei,j = ξi ⊗ ξj∗ }i,j∈I satisfying the usual commutation relations [Ei,j , Ek,l ] = δj,k Ei,l − δi,l Ek,j . We call V the natural representation of gl∞ and sl∞ , and V∗ its restricted dual. For any nonnegative integers p, q we deﬁne the tensor representation V⊗(p,q) as the vector space V⊗p ⊗ V∗⊗q , equipped with the following gl∞ module structure: (u ⊗ u∗ ) · (v1 ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vq∗ ) =

p

u∗ , vi v1 ⊗ . . . ⊗ vi−1 ⊗ u ⊗ vi+1 ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vq∗

i=1

−

q

∗ vj∗ , u v1 ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vj−1 ⊗ u∗ ⊗ vj+1 ⊗ . . . ⊗ vq∗ ,

j=1

for u, v1 , . . . , vp ∈ V and u∗ , v1∗ , . . . , vq∗ ∈ V∗ . The product of symmetric groups Sp × Sq acts in V⊗(p,q) by permuting the factors, and this action commutes with the action of gl∞ . We express this by saying that V⊗(p,q) is a (gl∞ , Sp × Sq )-module, and use similar notation throughout the paper. Our main goal in this section is to reveal the structure of the tensor representations as a gl∞ -module, and in particular to identify the Jordan–H¨ older constituents of V⊗(p,q) . We describe these modules explicitly as appropriately deﬁned highest Consider the direct sum decomposition weight gl∞ -modules. gl gl∞ = hgl ⊕ α∈∆gl k Xα , where hgl =

i∈I

k Ei,i ,

∆gl

= εi − εj i, j ∈ I, i = j ,

Xεgli −εj = Ei,j ,

and εi denotes the functional on hgl determined by εi (Ej,j ) = δi,j . We have [H, Xαgl ] = α(H)Xαgl for any α ∈ ∆gl and H ∈ hgl . As usual, we refer to elements of ∆gl as roots, and to functionals on hgl as weights. Remark 1. For an algebraically closed ﬁeld k the general notion of a Cartan subalgebra of gl∞ has been deﬁned and studied in [NP, DPS]. The subalgebra

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hgl is an example of a splitting Cartan subalgebra, and all splitting Cartan subalgebras of gl∞ are conjugated. From now on we identify the index set I with Z \ {0}, and consider the + polarization of the root system ∆gl = ∆+ −∆+ gl gl , where the set ∆gl of positive roots is given by ∆+ {εi − εj | i < j < 0} {εi − εj | j < 0 < i}. gl = {εi − εj | 0 < i < j} gα and deﬁne bgl as the Lie subalgebra of gl∞ , We denote ngl = α∈∆+ gl generated by ngl and hgl . It is clear that ngl is a Lie subalgebra of gl∞ , that + h , and that [b , b ] = n . bgl = ngl ⊂ gl gl gl gl Let V be a gl∞ -module, and let v ∈ V . We say that v is a highest weight vector if it generates a one-dimensional bgl -module. Any such v must satisfy ngl v = 0,

H v = χ(H) v

∀H ∈ hgl ,

for some χ ∈ h∗gl . We say that V is a highest weight module if V is generated by a highest weight vector v as above; the functional χ is then called the highest weight of V . As in the ﬁnite-dimensional case, it is easy to prove that for each χ ∈ h∗gl there exists a unique irreducible highest weight gl∞ -module with highest weight χ. Remark 2. The subalgebra bgl is an example of a splitting Borel subalgebra of gl∞ , see [DP2] for a classiﬁcation of the splitting Borel subalgebras in gl∞ over algebraically closed ﬁelds. In this paper we only consider the notion of a highest weight module, associated with bgl . This choice has the important property that all Jordan–H¨ older constituents of all tensor representations are highest weight modules. In particular, the natural representation V is a highest weight module with highest weight ε1 , generated by the highest weight vector ξ1 ; similarly, V∗ is a highest weight module with highest weight −ε−1 , generated by the highest ∗ weight vector ξ−1 . We now describe the Weyl construction for gl∞ . For any pair of indices I = (i, j) with i ∈ {1, 2, . . . , p} and j ∈ {1, 2, . . . , q}, deﬁne the contraction ΦI : V⊗(p,q) → V⊗(p−1,q−1) ,

v1 ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vq∗ →

→ vj∗ , vi v1 ⊗ . . . ⊗ vˆi ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vˆj∗ ⊗ . . . ⊗ vq∗ , and consider the (gl∞ , Sp × Sq )-submodule V{p,q} of V⊗(p,q) , def ker ΦI : V⊗(p,q) → V⊗(p−1,q−1) . V{p,q} = I

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Set also V{p,0} = V⊗p and V{0,q} = V∗⊗q . For any partitions λ, µ such that |λ| = p and |µ| = q, deﬁne the gl∞ -submodule of V⊗(p,q) def

def

Γλ;µ = V{p,q} ∩ (Sλ V ⊗ Sµ V∗ ). def

Theorem 2.1 For any p, q there is an isomorphism of (gl∞ , Sp × Sq )modules V{p,q} ∼ Γλ;µ ⊗ (Hλ ⊗ Hµ ). (2) = |λ|=p |µ|=q

For any partitions λ, µ, the gl∞ -module Γλ;µ is an irreducible highest weight def module with highest weight χ = i∈N λi εi − i∈N µi ε−i . Furthermore, Γλ;µ is irreducible when regarded by restriction as an sl∞ -module. Proof. Pick an enumeration I = {k1 , k2 , k3 , . . . } of the index set I = Z \ {0}, such that ki = i for 1 ≤ i ≤ p, and kp+i = −i for 1 ≤ i ≤ q. For each n denote by Vn the subspace of V spanned by ξk1 , . . . , ξkn , and by Vn∗ the subspace of V∗ spanned by ξk∗1 , . . . , ξk∗n ; the pairing between V and V∗ restricts to a nondegenerate pairing between Vn and Vn∗ . Denote by gn the Lie subalgebra of gl∞ generated by {Eki ,kj }1≤i,j≤n , def def and set bn = bgl ∩ gn , hn = hgl ∩ gn . It is clear that gn ∼ = gln , and that hn (respectively, bn ) is a Cartan (respectively, Borel) subalgebra of gn . Moreover, the inclusion gn → gn+1 restricts to inclusions hn → hn+1 and bn → bn+1 . The tensor representations (Vn )⊗(p,q) and the contractions (n) ΦI : (Vn )⊗(p,q) → (Vn )⊗(p−1,q−1) are deﬁned by analogy with their inﬁnitedimensional counterparts. We set def (n) (n) def (Vn ){p,q} = ker ΦI , Γλ;µ = (Vn ){p,q} ∩ (Sλ Vn ⊗ Sµ Vn∗ ). I

A version of the ﬁnite-dimensional Weyl construction (see Appendix for (n) more details) implies that for n ≥ p + q the gln -module Γλ;µ is an irreducible highest weight module with highest weight χ, regarded by restriction as a functional on hn . Futhermore, we have (gn , Sp × Sq )-module isomorphisms (n) Γλ;µ ⊗ (Hλ ⊗ Hµ ). (3) (Vn ){p,q} ∼ = |λ|=p |µ|=q

It is clear from the deﬁnitions that we have the following diagram of inclusions: (Vp+q ){p,q} ⊂ (Vp+q+1 ){p,q} ⊂ . . . ⊂ (Vn ){p,q} ⊂ . . . ⊂ ∪ ∪ ∪ (p+q) (p+q+1) . . . ⊂ Γ ⊂ ⊂ Γ(n) ⊂ . . . ⊂ Γ λ;µ

λ;µ

λ;µ

{p,q} n∈N (Vn )

∪

(n) n∈N Γλ;µ

= V{p,q} ∪ = Γλ;µ . (4)

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133

(n)

The irreducibility of the gln -module Γλ;µ for each n implies the irreducibility of the gl∞ -module Γλ;µ . Similarly, the isomorphism (2) follows from the isomorphisms (3), and it remains to show that Γλ;µ is a highest weight gl∞ module with highest weight χ. (n) For each n ≥ p + q the highest weight subspace Γλ;µ [χ] is one-dimensional, and therefore (n) (p+q) (p+q+1) (n) Γλ;µ [χ] = Γλ;µ [χ] = · · · = Γλ;µ [χ] = · · · = Γλ;µ [χ] = Γλ;µ [χ]. n∈N

We also have bgl · Γλ;µ [χ] =

n∈N

(n) (n) bn · Γλ;µ [χ] = n∈N Γλ;µ [χ] = Γλ;µ [χ],

which means that any nonzero v ∈ Γλ;µ [χ] is a highest weight vector generating the gl∞ -module Γλ;µ [χ]. This completes the proof of the theorem. Applying Theorem 2.1 to the special cases p = 0 and q = 0, we obtain a version of Schur–Weyl duality for gl∞ : Γλ;0 ⊗ Hλ , V∗⊗q ∼ Γ0;µ ⊗ Hµ . (5) V⊗p ∼ = = |λ|=p

|µ|=q

Next, we study the structure of the gl∞ -module V⊗(p,q) . The main feature and the crucial diﬀerence from the ﬁnite-dimensional case is that V⊗(p,q) fails to be completely reducible. Theorem 2.2 Let p, q be nonnegative integers, and let = min(p, q). Then the Loewy length of the gl∞ -module V⊗(p,q) equals + 1, and (r) ⊗(p,q) ⊗(p,q) ⊗(p−r,q−r) soc V , r = 1, . . . , . = ker ΦI1 ,...,Ir : V →V I1 ,...,Ir

(6) Moreover, the socle ﬁltration of V⊗(p,q) , regarded as an sl∞ -module, coincides with (6). Proof. Denote by F(r) the subspaces of V⊗(p,q) on the right side of (6), and for each n set (r) ⊗(p,q) ⊗(p−r,q−r) , r = 1, . . . , , Fn = ker Φ{I1 ,...,Ir } : (Vn ) → (Vn ) I1 ,...,Ir

where {Vn }n∈N is the exhaustion of V used in the proof of Theorem 2.1. Then we have the commutative diagram of (gl∞ , Sp × Sq )-modules in which the vertical arrows are obtained as restrictions of the inclusions in the rightmost column, and yield exhaustions of F(r) . Denote for convenience (0) (+1) F(0) = 0, F(+1) = V⊗(p,q) , and similarly Fn = 0, Fn = (Vn )⊗(p,q) for all (r+1) (r) (r+1) (r) n. It is easy to check that the induced maps Fn /Fn → Fn+1 /Fn+1 are

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injective for all n, r, and therefore for each r = 0, 1, . . . , the layer F(r+1) /F(r) (r+1) (r) is the union of quotients Fn /Fn .

(1) Fn

⊂

(2) Fn

⊂

...

⊂

() Fn

⊂

(Vn )⊗(p,q)

(1) Fn+1

⊂

(2) Fn+1

⊂

...

⊂

() Fn+1

⊂

(Vn+1 )⊗(p,q)

F(1)

⊂

F(2)

⊂

...

⊂

F()

⊂

V⊗(p,q) ,

It is a standard exercise (see Appendix for more details) to show that for each r there exists a (gln , Sp × Sq )-module isomorphism (n) Γλ;µ ⊗ H(λ, µ; r), (7) Fn(r+1) /Fn(r) ∼ = |λ|=p−r |µ|=q−r

where H(λ, µ; r) are some Sp × Sq -modules. As in the proof of Theorem 2.1, it follows that F(r+1) /F(r) ∼ Γλ;µ ⊗ H(λ, µ; r). = |λ|=p−r |µ|=q−r

older series In particular, this shows that V⊗(p,q) has a ﬁnite Jordan–H¨ with irreducible constituents of the form Γλ;µ for appropriate λ, µ. Moreover, F(r) is characterized as the unique submodule of V⊗(p,q) such that for any λ, µ [V⊗(p,q) : Γλ;µ ] if |λ| > p − r and |µ| > q − r, (r) [F : Γλ;µ ] = 0 otherwise . We use induction on r to prove the main statement of the theorem: F(r) = soc(r) V⊗(p,q) . The base of induction r = 0 is trivial since F(0) = soc(0) V⊗(p,q) = 0. Suppose that F(r) = soc(r) V⊗(p,q) for some r. The quotient F(r+1) /F(r) is a semisimple gl∞ -module, hence F(r+1) ⊂ soc(r+1) V⊗(p,q) . Now let U be a simple submodule of V⊗(p,q) /F(r) ; then U ∼ = Γλ;µ with λ, µ satisfying |λ| = p − s and |µ| = q − s for some s ≥ r. In particular U ⊂ F(s+1) /F(r) . Our goal is to show that in fact s = r; indeed, we would then conclude that soc(r+1) V⊗(p,q) ⊂ F(r+1) , proving the induction step. Fix a vector u ∈ V⊗(p,q) of weight χ = i λi εi − i µi ε−i , such that the image of u under the projection V⊗(p,q) → V⊗(p,q) /F(r) generates U. Fix a

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135

large enough m ∈ N such that u ∈ (Vm )⊗(p,q) . We may further assume without loss of generality that u generates a glm -submodule of (Vm )⊗(p,q) isomorphic (m) to Γλ;µ . (s+1)

(n)

→ Γλ;µ ⊗ H(λ, µ; s) the proFor each n > m denote by πn : Fn jection corresponding to the decomposition (7). The map πn can be described explicitly, and we show in Proposition 6.3 in the appendix that (s−1) u − πn (u) ∈ / Fn for inﬁnitely many n. On the other hand, u generates (s+1) (r) (n) (r) a submodule of Fn /Fn isomorphic to Γλ;µ , therefore u − πn (u) ∈ Fn for all n. We conclude that s ≤ r, which implies that in fact s = r. The induction is now complete, and thus F(r) = soc(r) V⊗(p,q) for all r. Finally, we need to verify that the ﬁltration {F(r) } is also the socle ﬁltration of V⊗(p,q) regarded as an sl∞ -module. The irreducibility of Γλ;µ as an sl∞ -module implies that the layers of the ﬁltration F(r) remain semisimple. Therefore F(r) ⊂ soc(r) V⊗(p,q) for all r. The proof of the opposite inclusion, given above for gl∞ , works without any alterations for sl∞ as well, and this completes the proof of the theorem. Next, we describe the indecomposable constituents of the tensor representations of gl∞ . Theorem 2.3 For any partitions λ, µ the gl∞ -module Γλ;0 ⊗ Γ0;µ is indecomposable, and ⎛ ⎞ ⎝ soc(r+1) (Γλ;0 ⊗ Γ0;µ ) ∼ Nλλ ,γ Nµµ ,γ ⎠ Γλ ;µ . (8) = λ ,µ

|γ|=r

The same applies to Γλ;0 ⊗ Γ0;µ regarded as an sl∞ -module. Proof. Schur–Weyl Duality (5) for gl∞ implies that (Γλ;0 ⊗ Γ0;µ ) ⊗ (Hλ ⊗ Hµ ). V⊗(p,q) ∼ = |λ|=p |µ|=q

Hence the gl∞ -module Γλ;0 ⊗ Γ0;µ is realized as the direct summand (Sλ V ⊗ Sµ V∗ ) of the gl∞ -module V⊗(p,q) . Therefore soc(r) Γλ;µ = Γλ;µ ∩ soc(r) V⊗d . It is known that for all partitions λ, µ, λ , µ we have [Γλ;0 ⊗ Γ0;µ : Γλ ;µ ] = λ µ γ Nλ ,γ Nµ ,γ provided n is large enough, see e.g. [HTW]. This yields (n)

[Γλ;0 ⊗ Γ0;µ : Γλ ;µ ] =

γ

Nλλ ,γ Nµµ ,γ ,

(n)

(n)

(9)

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Ivan Penkov and Konstantin Styrkas

and combining (9) with the description of the socle ﬁltration of V⊗(p,q) , we obtain (8). In particular, soc(Γλ;0 ⊗ Γ0;µ ) ∼ = Γλ;µ , and the simplicity of the socle implies the indecomposability of the gl∞ -module Γλ;0 ⊗ Γ0;µ . Corollary 2.4 The decomposition of V⊗(p,q) into indecomposable gl∞ -modules is given by the isomorphism V⊗(p,q) ∼ (dim Hλ dim Hµ ) Γλ;0 ⊗ Γ0;µ . = |λ|=p |µ|=q

Examples. We begin with the description of tensors of rank 2. Purely covariant and purely contravariant tensor representations of gl∞ and sl∞ are completely reducible, and are decomposed by the Schur–Weyl duality (2). For tensors of rank 2 these decompositions correspond to the isomorphisms V⊗V∼ = S 2 V ⊕ Λ2 V,

V∗ ⊗ V∗ ∼ = S 2 V∗ ⊕ Λ2 V∗ .

The tensor representation of mixed type V⊗V∗ is the adjoint representation of gl∞ , for which we have the non-splitting short exact sequence of gl∞ -modules 0 → sl∞ → gl∞ → k → 0. In other words, the socle ﬁltration of the adjoint gl∞ -module has length 2, with a simple socle isomorphic to sl∞ , and a simple top isomorphic to k. We summarize the structure of these modules graphically by V⊗(2,0) ∼ Γ(2);(0) ⊕ Γ(1,1);(0) , V(1,1) ∼

Γ(0);(0) , Γ(1);(1)

V⊗(0,2) ∼ Γ(0);(2) ⊕ Γ(0);(1,1) . In these diagrams each tower represents an indecomposable direct summand of a module, and the vertical arrangement of boxes in each tower represents the structure of the layers of the socle ﬁltration, with the bottom box corresponding to the socle. Similarly, Theorem 2.3 yields the following diagrams for tensor representations of rank 3: V⊗(3,0) ∼ Γ(3);(0) ⊕ 2 Γ(2,1);(0) ⊕ Γ(1,1,1);(0) , V⊗(2,1) ∼

Γ(1);(0) Γ(1);(0) ⊕ , Γ(2);(1) Γ(1,1);(1)

V⊗(0,3) ∼ Γ(0);(3) ⊕ 2 Γ(0);(2,1) ⊕ Γ(0);(1,1,1) , and for tensor representations of rank 4:

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137

V⊗(4,0) ∼ Γ(4);(0) ⊕ 3 Γ(3,1);(0) ⊕ 2 Γ(2,2);(0) ⊕ 3 Γ(2,1,1);(0) ⊕ Γ(1,1,1,1);(0) , V⊗(3,1) ∼

Γ(2);(0) Γ ⊕ Γ(1,1);(0) Γ(1,1);(0) ⊕ 2 (2);(0) ⊕ , Γ(3);(1) Γ(2,1);(1) Γ(1,1,1);(1)

Γ(0);(0) Γ(0);(0) V⊗(2,2) ∼ Γ(1);(1) ⊕ Γ(1);(1) ⊕ Γ(1);(1) ⊕ Γ(1);(1) , Γ(2);(1,1) Γ(1,1);(2) Γ(2);(2) Γ(1,1);(1,1) V⊗(1,3) ∼

Γ(0);(2) Γ ⊕ Γ(0);(1,1) Γ(0);(1,1) ⊕ 2 (0);(2) ⊕ , Γ(1);(3) Γ(1);(2,1) Γ(1);(1,1,1)

V⊗(0,4) ∼ Γ(0);(4) ⊕ 3 Γ(0);(3,1) ⊕ 2 Γ(0);(2,2) ⊕ 3 Γ(0);(2,1,1) ⊕ Γ(0);(1,1,1,1) .

3 Tensor representations of sp∞ Let V be a countable dimensional vector space, and let Ω : V ⊗ V → k be a non-degenerate anti-symmetric bilinear form on V. We realize the Lie algebra gl∞ as in Section 2 by taking V∗ = V and Ω as the pairing ·, · . The Lie algebra sp∞ is deﬁned as the Lie subalgebra of gl∞ , which preserves the form Ω, i.e.,

sp∞ = X ∈ gl∞ Ω(Xu, v) + Ω(u, Xv) = 0 for all u, v ∈ V . It is always possible to pick a basis {ξi } of V, indexed as before by the set I = Z\{0}, such that Ω(ξi , ξj ) = sign(i) δi+j,0 . In the coordinate realization of gl∞ , the Lie algebra sp∞ has a linear basis {sign(j)Ei,j − sign(i)E−j,−i }i,j∈I . Since the dual basis {ξi∗ }i∈I is given by ξi∗ = sign(i) ξ−i , it follows that sp∞ = S 2 V, and the Lie bracket is induced by (1). We call V, regarded as an sp∞ -module by restriction, the natural representation of sp∞ . It is easy to see that the sp∞ -action on the gl∞ -module V⊗(p,q) coincides with the sp∞ -action on the gl∞ -module V⊗(p+q) . Therefore it suﬃces to study the tensor representations V⊗d . To deﬁne the notion of a highest weight sp∞ -module, we consider the sp direct sum decomposition sp∞ = hsp ⊕ α∈∆sp k Xα , where hsp =

k (Ei,i − E−i,−i ),

i∈I

∆sp = {±(εi + εj ) | i, j ∈ N}

{εi − εj | i, j ∈ N and i = j},

sp = Ei,−j + Ej,−i , X−ε = E−i,j + E−j,i , Xεsp i +εj i −εj

Xεsp = Ei,j − E−j,−i . i −εj

as before using the polarizaThe subalgebra bsp and its ideal nsp

are deﬁned tion of the root system ∆sp = ∆+ −∆+ sp sp , where {εi − εj | i, j ∈ N and i < j}. ∆+ sp = {εi + εj | i, j ∈ N and i ≤ j}

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An sp∞ -module V is called a highest weight module with highest weight χ ∈ h∗sp , if it is generated by a vector v ∈ V satisfying nsp v = 0 and H v = χ(H) v for all H ∈ hsp . In particular, the natural representation V is a highest weight sp∞ -module with highest weight ε1 , generated by the highest weight vector ξ1 . For any pair I = (i, j) of integers such that 1 ≤ i < j ≤ d, the form Ω determines a contraction ΦI : V⊗d → V⊗(d−2) , v1 ⊗ . . . ⊗ vd → Ω(vi , vj ) v1 ⊗ . . . ⊗ vˆi ⊗ . . . ⊗ vˆj ⊗ . . . ⊗ vd . def

def

We set V0 = k, V1 = V and def Vd = ker ΦI : V⊗d → V⊗(d−2) I

for d ≥ 2. For any partition λ with |λ| = d we deﬁne the sp∞ -submodule Γλ of V⊗d , def Γλ = Vd ∩ Sλ V. Theorem 3.1 For any nonnegative integer d there is an isomorphism of (sp∞ , Sd )-modules Vd ∼ Γλ ⊗ Hλ . (10) = |λ|=d

Γλ is an irreducible highest weight For every partition λ, the sp∞ -module module with highest weight ω = i∈N λi εi . Proof. For each n, denote by Vn the 2n-dimensional subspace of V spanned by ξ±1 , . . . , ξ±n , and identify the Lie subalgebra of End(Vn ) which preserves the restriction of the form Ω to Vn with the Lie algebra sp2n . The tensor rep(n) resentations (Vn )⊗d of sp2n and the contractions ΦI : (Vn )⊗d → (Vn )⊗(d−2) are deﬁned as before. We set def (n) (n) def (Vn )d = ker ΦI , Γλ = (Vn )d ∩ Sλ Vn . I

The Weyl construction for sp2n implies that there are isomorphisms of (sp2n , Sd )-modules (n) (Vn )d ∼ Γλ ⊗ Hλ , (11) = |λ|=d

(n)

and that for n ≥ d the module Γλ is an irreducible highest weight module with highest weight i∈N λi εi . As in the proof of Theorem 2.1, the diagram of inclusions

Tensor Representations of Classical Locally Finite Lie Algebras

(Vd )d ⊂ (Vd+1 )d ⊂ . . . ⊂ (Vn )d ⊂ . . . ⊂ ∪ ∪ ∪ (d) (d+1) (n) ⊂ . . . ⊂ Γλ ⊂ ... ⊂ Γλ ⊂ Γλ

d n∈N (Vn )

∪ (n) n∈N Γλ

139

= Vd ∪ = Γλ

(12) establishes both the irreducibility of the sp∞ -module Γλ and the isomorphism (10), and the equalities (n) (d) (d+1) (n) Γλ [χ] = Γλ [χ] = · · · = Γλ [χ] = · · · = Γλ [χ] = Γλ [χ] n∈N

yield the characterization of Γλ [χ] as a highest weight sp∞ -module.

Next, we describe the socle ﬁltration of the tensor representations of sp∞ . Theorem 3.2 For any nonnegative integer d the gl∞ -module V⊗d , regarded d as an sp∞ -module, has Loewy length 2 + 1, and d . r = 1, . . . , ker ΦI1 ,...,Ir : V⊗d → V⊗(d−2r) , soc(r) V⊗d = 2 I1 ,...,Ir

(13) Proof. The proof is obtained as a minor variation of the proof of Theorem 2.2. Denoting by F(r) the module on the right side of (13), we argue as before that Γλ ⊗ H(λ; r) F(r+1) /F(r) ∼ = |λ|=p−2r

for some Sd -modules H(λ; r). The semisimplicity of these layers shows that F(r) ⊂ soc(r) V⊗d for all r, and the opposite inclusion is proved by an obvious modiﬁcation of Corollary 6.3. Theorem 3.3 For any partition λ the sp∞ -module Γλ;0 is indecomposable, and ⎛ ⎞ d (r+1) λ ⎝ ⎠ . (14) soc Γλ;0 = Nµ, (2γ) Γµ , r = 1, . . . , 2 µ |γ|=r

Proof. By construction, the gl∞ -module Γλ;0 is realized as the direct summand Sλ V of the gl∞ -module V⊗d . It remains a direct summand when V⊗d is regarded as an sp∞ -module, therefore soc(r) Γλ;0 = Γλ;0 ∩ soc(r) V⊗d . (2n) (n) It is known that for any partitions λ, µ we have [Γλ : Γµ ] = γ Nλµ, (2γ) provided n is large enough, see e.g. [HTW]. This implies that [Γλ;0 : Γµ ] = λ γ Nµ, (2γ) , and combining it with the description of the socle ﬁltration of ⊗d V , we get (14). In particular, socΓλ;0 ∼ = Γλ , and the simplicity of the socle implies the indecomposability of Γλ;0 as an sp∞ -module.

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Examples. We begin by describing the structure of V⊗V = Λ2 V⊕S 2 V as an sp∞ -module. The symmetric square S 2 V = Γ(2);(0) is the irreducible adjoint sp∞ -module, isomorphic to Γ(2) . For the exterior square Λ2 V = Γ(1,1);(0) one has the short exact sequence of sp∞ -modules Ω

0 → Γ(1,1) → Λ2 V → k → 0 which does not split. Therefore the structure of V ⊗ V is graphically represented as Γ V⊗V ∼ ⊕ (0) . Γ(2) Γ(1,1) Similarly, the structure of the tensor representation of rank 3 is represented as Γ Γ(1) V⊗3 ∼ ⊕ 2 (1) ⊕ , Γ(3) Γ(2,1) Γ(1,1,1) and the structure of the tensor representation of rank 4 as V⊗4

∼ Γ(4)

Γ(0) Γ(0) ⊕3 Γ(2) ⊕2 Γ(1,1) ⊕3 Γ(2) ⊕ Γ[(1,1)] ⊕ Γ(1,1) . Γ(3,1) Γ(2,1,1) Γ(2,2) Γ(1,1,1,1)

4 Tensor representations of so∞ Let V be a countable dimensional vector space, and let Q : V ⊗ V → k be a nondegenerate symmetric bilinear form on V. We realize the Lie algebra gl∞ as in Section 2 by taking V∗ = V and Q as the pairing ·, · . The Lie algebra (Q) so∞ is deﬁned as the Lie subalgebra of this gl∞ , which preserves the form Q, i.e.,

so(Q) Q(Xu, v) + Q(u, Xv) = 0 for all u, v ∈ V . = X ∈ gl ∞ ∞ It is customary when dealing with the orthogonal groups to work over algebraically closed ﬁelds. Throughout this section we assume that k is algebraically closed. Then all nondegenerate forms Q on V are equivalent (e.g., they can all be transformed to the standard sum of squares), and as a (Q) consequence the corresponding Lie algebras so∞ are isomorphic. Hence we drop Q from the notation, and denote our Lie algebra simply by so∞ . It is convenient to pick a basis {ξi } of V, indexed as before by the set I = Z \ {0}, such that Q(ξi , ξj ) = δi+j,0 . In the coordinate realization of gl∞ , the Lie algebra so∞ has a linear basis {Ei,j − E−j,−i }i,j∈I . Since the dual 2 basis {ξi∗ }i∈I is given by ξi∗ = ξ−i , it follows that so∞ = V, and the Lie bracket is induced by (1).

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141

We call V, regarded as an so∞ -module by restriction, the natural representation of so∞ . It is easy to see that the so∞ -action on the gl∞ -module V⊗(p,q) coincides with the so∞ -action on the gl∞ -module V⊗(p+q) . Therefore it suﬃces to study the tensor representations V⊗d . To deﬁne the notion of a highest weight so∞ -module, we consider the so direct sum decomposition so∞ = hso ⊕ , where k X α α∈∆sp hso =

k (Ei,i − E−i,−i ),

∆so = {±εi ± εj ) | i, j ∈ N and i = j},

i∈I so = E−j,i − E−i,j . Xεsoi +εj = Ei,−j − Ej,−i , Xεsoi −εj = Ei,j − E−j,−i , X−ε i −εj

The subalgebras bso and its ideal nso deﬁned as before using the polariza are + tion of the root system ∆so = ∆+ −∆ so so , where ∆+ so = {εi ± εj | i, j ∈ N and i < j}. An so∞ -module V is called a highest weight module with highest weight χ ∈ h∗so , if it is generated by a vector v ∈ V satisfying nso v = 0 and H v = χ(H) v for all H ∈ hso . In particular, the natural representation V is a highest weight module with highest weight ε1 , generated by the highest weight vector ξ1 . Remark 3. Our choice of a splitting Cartan subalgebra hso leads to a root decomposition of type D∞ , corresponding to the “even” inﬁnite orthogonal series of Lie algebras so2n . It is also possible to pick another splitting Cartan ˜so which leads to a root decomposition of type B∞ , corresponding subalgebra h to the “odd” inﬁnite orthogonal series so2n+1 . These two types of splitting Cartan subalgebras are clearly not conjugated, and in [DPS] it is proved that any splitting Cartan subalgebra of so∞ is either “even” or “odd”. For any pair I = (i, j) of integers, satisfying 1 ≤ i < j ≤ d, deﬁne the contraction Φ[I] : V⊗d → V⊗(d−2) , v1 ⊗ . . . ⊗ vd → Q(vi , vj ) v1 ⊗ . . . ⊗ vˆi ⊗ . . . ⊗ vˆj ⊗ . . . ⊗ vd . def

(15)

def

We set V[0] = k, V[1] = V and V

[d] def

=

⊗d ⊗(d−2) ker Φ[I] : V → V

I

for d ≥ 2. For any partition λ, deﬁne the so∞ -submodule Γ[λ] of V⊗d , def

Γ[λ] = V[d] ∩ Sλ V.

(16)

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Theorem 4.1 For any d ∈ N there is an isomorphism of (so∞ , Sd )-modules Γ[λ] ⊗ Hλ . (17) V[d] ∼ = |λ|=d

For every partition λ, the so∞ -module Γ[λ] is an irreducible highest weight module with highest weight ω = i∈N λi εi . Proof. For each n denote by Vn the 2n-dimensional subspace of V spanned by ξ±1 , . . . , ξ±n . Let gn be the Lie subalgebra of End(Vn ) which preserves the restriction of Q to Vn . It is clear that gn ∼ = so2n . The tensor representations (n) (Vn )⊗d of gn and the contractions Φ[I] : (Vn )⊗d) → (Vn )⊗(d−2) are deﬁned as before. We set def (n) (n) def ker Φ[I] , Γ[λ] = (Vn )[d] ∩ Sλ Vn . (Vn )[d] = I

The Weyl construction for so2n implies that there are isomorphisms of (so2n , Sd )-modules (n) (Vn )[d] ∼ Γ[λ] ⊗ Hλ , (18) = |λ|=d (n)

and that for n ≥ d the module Γ[λ] is an irreducible highest weight module with highest weight i∈N λi εi . As in the proof of Theorem 2.1, the diagram of inclusions (V1 )[d] ⊂ (V2 )[d] ⊂ . . . ⊂ (Vn )[d] ⊂ . . . ⊂ n∈N (Vn )[d] = V[d] ∪

∪

(1) Γ[λ]

(2) Γ[λ]

⊂

∪ ⊂ ... ⊂

(n) Γ[λ]

⊂ ... ⊂

∪

(n) n∈N Γ[λ]

∪

(19)

= Γ[λ]

establishes both the irreducibility of the so∞ -module Γλ and the isomorphism (17), and the equalities (n) (d) (d+1) (n) Γλ [χ] = Γλ [χ] = · · · = Γλ [χ] = · · · = Γλ [χ] = Γλ [χ] n∈N

yield the characterization of Γ[λ] [χ] as a highest weight so∞ -module.

⊗d Theorem 4.2 For any nonnegative d integer d the Loewy length of V , regarded as an so∞ -module, equals 2 + 1, and d . r = 1, . . . , ker Φ[I1 ,...,Ir ] : V⊗d → V⊗(d−2r) , soc(r) V⊗d = 2 I1 ,...,Ir

(20)

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143

Proof. The proof is a minor variation of the proof of Theorem 2.2. Denoting by F(r) the module on the right side of (20), we argue as before that Γ[λ] ⊗ H(λ; r) F(r+1) /F(r) ∼ = |λ|=p−2r

for some Sd -modules H(λ; r). The semisimplicity of these layers shows that F(r) ⊂ soc(r) V⊗d for all r, and the opposite inclusion is proved by an obvious modiﬁcation of Corollary 6.3.

Theorem 4.3 Let λ be a partition of a positive integer d. Then the so∞ module Γλ;0 is indecomposable, and the layers of its socle ﬁltration satisfy ⎛ ⎞ ⎝ soc(r+1) Γλ;0 ∼ Nλµ, 2γ ⎠Γ[µ] . (21) = µ

|γ|=r

Proof. By construction, the gl∞ -module Γλ;0 is realized as the direct summand Sλ V of the gl∞ -module V⊗d . It remains a direct summand when V⊗d is regarded as an so∞ -module, therefore soc(r) Γλ;0 = Γλ;0 ∩ soc(r) V⊗d . (2n) (n) It is known that [Γλ : Γ[µ] ] = γ Nλµ, 2γ for any partitions λ, µ, provided λ n is large enough, see e.g., [HTW]. Hence [Γλ;0 : Γ[µ] ] = γ Nµ, 2γ , and combining this with the description of the socle ﬁltration of V⊗d , we get (21). In particular, socΓλ;0 ∼ = Γ[λ] , and simplicity of the socle implies the indecomposability of Γλ;0 as an so∞ -module. Corollary 4.4 The decomposition of V⊗d into indecomposable so∞ -modules is given by the isomorphism V⊗d ∼ (dim Hλ ) Γλ;0 . = |λ|=d

Examples. We begin by describing the structure of V ⊗ V = Λ2 V ⊕ S 2 V as an so∞ -module. The exterior square Λ2 V = Γ(1,1);(0) is the irreducible adjoint so∞ -module, isomorphic to Γ[(1,1)] . For the symmetric square S 2 V = Γ(2);(0) one has the short exact sequence of so∞ -modules Q

0 → Γ[(2)] → S 2 V → k → 0 which does not split. Therefore, the structure of V ⊗ V is graphically represented as Γ[(0)] V⊗V ∼ ⊕ . Γ[(1,1)] Γ[(2)]

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Similarly, the structure of the tensor representation of rank 3 is represented as V⊗3

∼

Γ[(1)] Γ ⊕ 2 [(1)] ⊕ , Γ[(1,1,1)] Γ[(3)] Γ[(2,1)]

and the structure of the tensor representation of rank 4 as V⊗4

∼

Γ[(0)] Γ[(0)] Γ[(2)] ⊕3 Γ[(2)] ⊕ Γ[(1,1)] ⊕2 Γ[(2)] ⊕3 Γ[(1,1)] ⊕ Γ[(1,1,1,1)] . Γ[(3,1)] Γ[(2,1,1)] Γ[(4)] Γ[(2,2)]

5 Tensor representations of root-reductive Lie algebras It is known that over an algebraically closed ﬁeld any inﬁnite-dimensional simple locally ﬁnite Lie algebra which admits a root decomposition is classical, i.e., isomorphic to sl∞ , sp∞ or so∞ , see [PS, NS]. Here we discuss a generalization of our results for sl∞ , so∞ and sp∞ to a more general class of inﬁnite-dimensional Lie algebras. Let k be one of the Lie algebras sl∞ , sp∞ or so∞ , and let hk denote its splitting Cartan subalgebra introduced in previous sections. Let V denote the natural representation of k. Suppose that a Lie algebra g is the semidirect sum of k and some Lie algebra m, + m, g = k⊂ and suppose furthermore that g has a subalgebra h such that h = hk ⊕ m. Theorem 5.1 The socle ﬁltration of the tensor representation of V⊗(p,q) as a g-module coincides with the socle ﬁltration of V⊗(p,q) as a k-module. Proof. Since m commutes with hk , the action of m preserves the hk -weight subspaces of any k-module. The hk -weight subspaces of V are one-dimensional, hence any element of m acts in any V[χ] as a scalar. Thus V admits a hweight subspace decomposition, and the same is true for the weight subspaces of tensor representations V⊗(p,q) . This shows that each k-submodule of V⊗(p,q) is automatically a g-module, and thus the socle ﬁltration of V⊗(p,q) as a k-module is a ﬁltration by gsubmodules. Moreover, the layers of the socle ﬁltration for k remain semisimple as g-modules, and the statement follows. Our main application of Theorem 5.1 is to the class of inﬁnite-dimensional root-reductive Lie algebras, studied in [DP1, PS]. Recall the following structural theorem from [DP1].

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Theorem 5.2 Let g be a root reductive Lie algebra. Set s = [g, g] and a = g/[g, g]. Then s= s(i) , i∈I

where each s(i) is isomorphic either to sl∞ , so∞ , sp∞ , or to a simple ﬁnitedimensional Lie algebra, and I is an at most countable index set. Moreover, the short exact sequence of Lie algebras 0→s→g→a→0

(22)

+ a. splits. In other words, g ∼ = s⊂

To apply Theorem 5.1 to a root-reductive Lie algebra g, we identify g + a, pick an inﬁnite-dimensional direct summand with a semidirect sum g ∼ = s⊂ (j) (i) + k = s of s, and set m = ⊂ a. The Lie algebra g acts in the i =j s natural representation V of s(j) : s(i) annihilate V for i = j, and a acts by scalars in each hk -weight subspace. In contrast with the ﬁnite-dimensional reductive Lie algebras, there exist root-reductive Lie algebras g such that g s ⊕ a. For example, for g = gl∞ one has s = sl∞ and a = k, but gl∞ sl∞ ⊕ k. Another interesting example is the Lie algebra ˜ g, constructed via the following root injections ⎛ Tr(A) ⎞ gln → gln+2 ,

A → ⎝

n

A ⎠. 0

Then g˜ is not isomorphic to gl∞ , although it can still be included in a short exact sequence of Lie algebras 0 → sl∞ → ˜ g → k → 0, see [DPS]. However, Theorem 5.1 still applies and describes the socle ﬁltration of the tensor representations of ˜ g.

6 Appendix For completeness, we discuss the details of Weyl’s duality approach, see [W, FH]. Let p, q, n be nonnegative integers such that n > p + q. Let Vn = kn be the natural representation of the Lie algebra gln . For partitions λ, µ such (n) that |λ| = p and |µ| = q, we denote by Γλ;µ the irreducible highest weight gln -module with highest weight ω = (λ1 , . . . , λp , 0, . . . , 0, −µq , . . . , −µ1 ). Proposition 6.1 For any n > p + q there is an isomorphism (3) (n) (Vn ){p,q} ∼ Γλ;µ ⊗ (Hλ ⊗ Hµ ). = |λ|=p |µ|=q

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Proof. For any I = (i, j) with i ∈ {1, 2, . . . , p} and j ∈ {1, 2, . . . , q}, we deﬁne the inclusion (n)

ΨI

∗ : (Vn )⊗(p−1,q−1) → (Vn )⊗(p,q) , v1 ⊗ . . . ⊗ vp−1 ⊗ v1∗ ⊗ . . . ⊗ vq−1 → n 1 ∗ ∗ → . . . ⊗ vi−1 ⊗ ζk ⊗ vi+1 ⊗ . . . ⊗ vj−1 ⊗ ζk∗ ⊗ vj+1 ⊗ ..., n k=1

where {ζk } and {ζk∗ } are any dual bases of Vn and Vn∗ respectively. Set θI = (n) (n) (n) (n) ΨI ΦI . The operators θI are idempotent, and (Vn ){p,q} = I ker θI . ⊗(p,q) Let A denote the subalgebra of endomorphisms of (Vn ) , generated by the images of elements of gn , and let B denote its commutator subalgebra, i.e., the set of all endomorphisms of (Vn )⊗(p,q) , commuting with the action (n) of gn . From invariant theory it is known that B is generated by θI for various I, and by permutation maps corresponding to elements from Sp × Sq . A general result from the theory of semisimple ﬁnite-dimensional algebras In other words, implies that, conversely, A is the commutator subalgebra of B. (n) ⊗(p,q) any endomorphism of (Vn ) , commuting with all θI and all permutations from Sp × Sq , must lie in A. Let A denote the subalgebra of endomorphisms of (Vn ){p,q} , generated by the images of elements of gn , and let B denote the subalgebra of endomorphisms of (Vn ){p,q} , generated by the permutations from Sp × Sq. We claim that A and B are each other’s commutator subalgebras in End (Vn ){p,q} . Indeed, suppose that L lies in the commutator subalgebra of B. Using the (gn , Sp × Sq )-module isomorphism (see e.g., [FH] for details) (n) (Vn )⊗(p,q) = (Vn ){p,q} ⊕ im θI , (n)

I

of (Vn )⊗(p,q) by extending L trivially on the we construct an endomorphism L commutes with all permutations second direct summand. It is clear that L (n) from Sp × Sq , and also with the operators θI , all of which act on (Vn ){p,q} belongs to the commutator subalgebra of B, i.e., L ∈ A, by zero. Hence L and by restriction L ∈ A. Thus A is the commutator subalgebra of B, and it follows that B is also the commutator subalgebra of A. The general theory of dual pairs and the fact that {Hλ ⊗ Hµ }|λ|=p,|µ|=q is a complete list of irreducible Sp × Sq -modules imply the existence of an isomorphism (Vn ){p,q} ∼ Γ (λ, µ) ⊗ (Hλ ⊗ Hµ ) = |λ|=p |µ|=q

for some irreducible gln -modules Γ (λ, µ). To identify these modules explicitly, we note that Schur–Weyl Duality yields (n) (n) (Vn )⊗p ∼ Γλ;0 ⊗ Hλ , (Vn∗ )⊗q ∼ Γ0;µ ⊗ Hµ , = = |λ|=p

|µ|=q

Tensor Representations of Classical Locally Finite Lie Algebras (n)

147

(n)

and therefore Γ (λ, µ) must be a submodule of Γλ;0 ⊗ Γ0;µ . On the other hand, (n) Γλ;µ

of this tensor product does not occur as a submodule of the submodule (Vn )⊗(p−1,q−1) , and thus lies in the kernel of all operators ΦI . We conclude that (n) Γλ;µ ⊂ Γ (λ, n), and the irreducibility of Γ (λ, µ) yields the desired statement. Finally, to prove the technical statement used in the proof of Theorem 2.2, we need a preparatory lemma. Deﬁne the contractions ΦI1 ,...,Ir : V⊗(p,q) → V⊗(p−r,q−r) as the r-fold convolutions between copies of V and V∗ indicated by the pairwise disjoint collection of index pairs I1 , . . . , Ir . For r = 1, . . . , , deﬁne the inclu(n) sions ΨI1 ,...,Ir : (Vn )⊗(p−r,q−r) → (Vn )⊗(p,q) , by analogy with ΦI1 ,...,Ir , as the r-fold insertions of the canonical element of Vn ⊗ Vn∗ into positions speciﬁed by disjoint pair of indices I1 , . . . , Ir . Set also (n) (Vn ){p,q} = im ΨI1 ,...,Ir : (Vn )⊗(p−r,q−r) → (Vn )⊗(p,q) . r I1 ,...,Ir

It is a standard exercise to show that for each n one has the direct sum decomposition {p,q}

(Vn )⊗(p,q) = (Vn )0

{p,q}

⊕ (Vn )1

{p,q}

⊕ (Vn )2

{p,q}

⊕ · · · ⊕ (Vn )

.

For any I = (i, j) we consider the linear map (n)

ΞI

: (Vn )⊗(p,q) → (Vn )⊗(p−1,q−1) ,

v1 ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vq∗ →

→ n ξn∗ , vi vj∗ , ξn v1 ⊗ . . . ⊗ vˆi ⊗ . . . ⊗ vp ⊗ v1∗ ⊗ . . . ⊗ vˆj∗ ⊗ . . . ⊗ vq∗ . Lemma 6.2 For any v ∈ V{p,q} we have v, if {I1 , . . . , Ir } = {J1 , . . . , Jr } as sets (n) (n) (n) . lim Ξ ΦJ2 ,...,Jr ΨI1 ,...,Ir v = n→∞ J1 0, otherwise. Proof. We use induction on s. The base of induction s = 1 states that v, if I = J (n) (n) lim Ξ ΨI v = n→∞ J 0, otherwise, which is clear from the deﬁnition. Assume now that r ≥ 2, and let Jr = (i, j). Case 1. There exists k such that Ik = (i, j); we may assume that k = r. Then (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

ΞJ1 ΦJ2 ,...,Jr ΨI1 ,...,Ir v = ΞJ1 ΦJ2 ,...,Jr−1 ΨI1 ,...,Ir−1 ΦIr ΨJr v = ΞJ1 ΦJ2 ,...,Jr−1 ΨI1 ,...,Ir−1 v, and the desired statement follows from the induction hypothesis.

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Case 2. There exist k, l such that Ik = (i, a) and Il = (b, j); we may assume that k = r and l = r − 1. Setting I = (b, a) and using the identity (n) (n) (n) Φ(i,j) Ψ(i,a),(b,j) = n1 Ψb,a , we get (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

ΞJ1 ΦJ2 ,...,Jr ΨI1 ,...,Ir v = ΞJ1 ΦJ2 ,...,Jr−1 ΨI1 ,...,Ir−2 ΦJr ΨIr−1 ,Ir v 1 (n) (n) (n) Ξ Φ Ψ v. n J1 J2 ,...,Jr−1 I1 ,...,Ir−2 ,I

=

Applying the induction hypothesis, in both cases we obtain (n)

(n)

(n)

lim ΞJ1 ΦJ2 ,...,Jr ΨI1 ,...,Ir v = 0.

n→∞

Case 3. There exists k such that Ik = (i, a), but j never occurs in the second position of any Il ; we may assume that k = r. Using the identity (n) (n) (m) (m) Φ(i,j) Ψ(i,a) v = m n Φ(i,j) Ψ(i,a) v, we get (n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

lim ΞJ1 ΦJ2 ,...,Jr ΨI1 ,...,Ir v = lim ΞJ1 ΦJ2 ,...,Jr−1 ΨI1 ,...,Ir−1 ΦJr ΨIr v n→∞ m (n) (n) (n) (m) (m) ΞJ1 ΦJ2 ,...,Jr−1 ΨI1 ,...,Ir−1 ΦJr ΨIr v . = lim n→∞ n

n→∞

Applying the induction hypothesis to ΦJr ΨIr v ∈ V{p,q} , we see that the desired statement holds. Case 4. The index i never occurs in the ﬁrst position of any Ik , and j never occurs in the second position of any Il . Then for all n we have (m)

(n)

(n)

(n)

(n)

(n)

(m)

(n)

ΞJ1 ΦJ2 ,...,Jr ΨI1 ,...,Ir v = ΞJ1 ΦJ2 ,...,Jr−1 ΨI1 ,...,Ir ΦIr v = 0.

We are now ready to prove the following assertion, used in the proof of Theorem 2.2. (s−1)

/ Fn Proposition 6.3 There exist inﬁnitely many n such that u − πn (u) ∈ (s−1)

.

for all n m. Proof. Assume that, on the contrary, u − πn (u) ∈ Fn Let J1 , . . . , Js be any collection of pairwise disjoint indices. Since for all n ΦJ2 ,...,Jr (u − πn (u)) = 0, we obtain (n)

(n)

ΞJ1 ΦJ2 ,...,Jr πn (u) = ΞJ1 ΦJ2 ,...,Jr u = 0. On the other hand, the vector πn (u) can be represented as πn (u) = ΨI1 ,...,Is ζI1 ,...,Is I1 ,...,Is

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for some collection {ζI1 ,...,Is } of vectors from (Vn ){p,q} , and according to Lemma 6.2 (n) lim ΞJ1 ΦJ2 ,...,Jr πn (u) = ζJ1 ,...,Jn . n→∞

It follows that ζJ1 ,...,Jn = 0, and thus πn (u) = 0. This contradicts the assump(s+1) (r) (n) tion that πn (u) generates a submodule of Fn /Fn isomorphic to Γλ;µ .

References [B]

A. Baranov, Finitary simple Lie algebras. J. Algebra 219 (1999), no. 1, 299–329. [BB] Y. Bakhturin, G. Benkart, Weight modules of direct limit Lie algebras, Comm. Algebra 27 (1998), 2249–2766. [BS] Y. Bakhturin, H. Strade, Some examples of locally finite simple Lie algebras. Arch. Math. (Basel) 65 (1995), no. 1, 23–26. [DP1] I. Dimitrov, I. Penkov, Weight modules of direct limit Lie algebras. Internat. Math. Res. Notices 1999, no. 5, 223–249. [DP2] I. Dimitrov, I. Penkov, Borel subalgebras of gl(∞). Resenhas 6 (2004), no. 2-3, 153–163. [DPW] I. Dimitrov, I. Penkov, J. Wolf, A Bott-Borel-Weil theory for direct limits of algebraic groups. Amer. J. Math. 124 (2002), no. 5, 955–998. [DPS] E. Dan-Cohen, I. Penkov, N. Snyder, Cartan subalgebras of root-reductive Lie algebras. J. Algebra 308 (2007), no. 2, 583–611. [FH] W. Fulton, J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. [HTW] R. Howe, E. Tan, J. Willenbring, Stable branching rules for classical symmetric pairs. Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601–1626. [M] G. Mackey, On infinite-dimensional linear spaces. Trans. Amer. Math. Soc. 57 (1945), 155–207. [N] K.-H. Neeb, Holomorphic highest weight representations of infinitedimensional complex classical groups, J. Reine Angew. Math, 497 (1998), 171–222. [NP] K.-H. Neeb, I. Penkov, Cartan subalgebras of gl∞ , Canad. Math. Bull. 46 (2003), no. 4, 597–616. [NS] K.-H. Neeb, N. Stumme, The classification of locally finite split simple Lie algebras. J. Reine Angew. Math. 533 (2001), 25–53. [Na] L. Natarajan, Unitary highest weight-modules of inductive limit Lie algebras and groups, J. Algebra 167 (1994), no. 1, 9–28. [O1] G. Olshanskii, Unitary representations of the group SO0 (∞, ∞) as limits of unitary representations of the groups SO0 (n, ∞) as n → ∞. Funct. Anal. Appl. 20 (1998), 292–301. [O2] G. Olshanskii, Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians. In: Topics in Representation Theory (A. A. Kirillov, ed.). Advances in Soviet Math., vol. 2. Amer. Math, Soc., Providence, R.I., 1991, 1–66. [PS] I. Penkov, H. Strade, Locally finite Lie algebras with root decomposition, Archiv Math 80 (2003), 478–485.

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[PZ]

I. Penkov, G. Zuckerman, A construction of generalized Harish-Chandra modules for locally reductive Lie algebras, Transformation Groups 13 (2008), 799–817. H. Weyl, The Classical Groups: Their Invariants and Representations, Princeton, 1939.

[W]

Lie Algebras, Vertex Algebras, and Automorphic Forms Nils R. Scheithauer Fachbereich Mathematik, Technische Universit¨ at Darmstadt, Schloßgartenstraße 7, 64289 Darmstadt, Deutschland, [email protected] Summary. Generalized Kac–Moody algebras are natural generalizations of the finite dimensional simple Lie algebras. In important cases their denominator identities are automorphic forms on orthogonal groups. The generalized Kac–Moody algebras with this property can probably be classified and realized as strings moving on suitable spacetimes. In this paper we describe these ideas in more detail. Key words: generalized Kac–Moody algebras, vertex algebras, automorphic forms. 2000 Mathematics Subject Classifications: 17B67, 17B68, 11F55.

1 Introduction The Fock space V of a bosonic string moving on a 26-dimensional torus carries a rich algebraic structure, i.e., V is a vertex algebra. There is a BRST-operator Q with Q2 = 0 acting on V . The physical states of the bosonic string are described by the cohomology group g of Q. The vertex algebra structure on V induces a Lie bracket on g. The Lie algebra g is called the fake monster algebra. It is almost a Kac–Moody algebra but not quite. The Lie algebra g has imaginary simple roots and therefore is a generalized Kac–Moody algebra. The theory of generalized Kac–Moody algebras is still similar to the Kac–Moody case. In particular there is a character formula and a denominator identity. The denominator identity of g is ∞ ρ α [1/∆](−α2 /2) ρ mρ 24 e . (1 − e ) = det(w) w e (1 − e ) + α∈II25,1

w∈W

m=1

Borcherds showed that the inﬁnite product deﬁnes an automorphic form for an orthogonal group. The automorphism group of the Leech lattice acts naturally on g. Since the denominator identity is a cohomological identity this action gives twisted denominator identities. One of the nicest conjectures in this ﬁeld is that these identities are also automorphic forms on orthogonal groups. K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_5, © Springer Basel AG 2011

151

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Nils R. Scheithauer

This example shows that generalized Kac–Moody algebras, vertex algebras and automorphic forms on orthogonal groups are closely connected. However we are far away from a good understanding of this relation. In the following sections we explain the above terms and ideas in more detail and state some of the open problems.

2 Generalized Kac–Moody algebras Generalized Kac–Moody algebras are natural generalizations of the ﬁnite dimensional simple Lie algebras. In this section we motivate the deﬁnition of these Lie algebras and sketch some of their properties. Let g be a ﬁnite-dimensional simple complex Lie algebra with Cartan subalgebra h and Killing form ( , ). Then g decomposes as g=h⊕ gα , α∈∆

where ∆ is the set of roots and the gα are 1-dimensional root spaces. We can choose simple roots α1 , . . . , αn , n = dim h. They have the property that each root α can be written as a linear combination of the simple roots αi with integral coeﬃcients which are either all nonnegative or nonpositive. The symmetrized Cartan matrix A = (aij ) of g is deﬁned by aij = (αi , αj ) . The entries of A are rational. Furthermore A has the following properties: 1. 2. 3. 4. 5.

aii > 0 aij = aji aij ≤ 0 for i = j 2aij /aii ∈ Z A is positive deﬁnite.

Serre found out that g can be recovered from its Cartan matrix in the following way. The Lie algebra with generators {ei , hi , fi | i = 1, . . . , n} and relations 1. 2. 3.

[ei , fj ] = δij hi [hi , ej ] = aij ej , [hi , fj ] = −aij fj ad(ei )1−2aij /aii ej = ad(fi )1−2aij /aii fj = 0 for i = j

is isomorphic to g. Killing and Cartan classiﬁed the ﬁnite-dimensional simple Lie algebras over the complex numbers. There are 4 inﬁnite families: the classical Lie algebras

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153

An = sln+1 (C), Bn = so2n+1 (C), Cn = sp2n (C) and Dn = so2n (C), and 5 exceptional Lie algebras G2 , F4 , E6 , E7 and E8 . Let g be a ﬁnite-dimensional simple Lie algebra. Then the ﬁnite-dimensional representations of g decompose into irreducible components. Their characters can be calculated by Weyl’s character formula. A special case of this formula is the denominator identity of g eρ (1 − eα ) = det(w)w(eρ ) . α>0

w∈W

Outer automorphisms of g give twisted denominator identities. Serre’s construction can also be applied to matrices which are not positive deﬁnite. In this way we obtain Kac–Moody algebras [K1]. These Lie algebras are in general no longer ﬁnite dimensional but their theory is still similar to the ﬁnite-dimensional theory. Let A be an indecomposable real n × n-matrix satisfying the conditions 1.-4. above. Vinberg has shown that A is either positive deﬁnite or positive semideﬁnite and of rank n − 1 or indeﬁnite. We say that A is of ﬁnite, aﬃne or indeﬁnite type. The Kac–Moody algebra corresponding to A is then ﬁnitedimensional simple, aﬃne or indeﬁnite. Let A be of aﬃne type. Then Vinberg’s trichotomy implies that any proper principal submatrix of A decomposes into a sum of matrices of ﬁnite type. Using this result it is easy to classify the aﬃne Kac–Moody algebras. The outcome is that to each ﬁnite dimensional simple Lie algebra corresponds an untwisted aﬃne Kac–Moody algebra, and furthermore there are 6 twisted aﬃne Kac–Moody algebras. The aﬃne Kac–Moody algebras have become very important in many areas of mathematics. The main reason for this is that they admit realizations as central extensions of current algebras. Let g be a ﬁnite dimensional simple Lie algebra and ˆ g the corresponding untwisted aﬃne Kac–Moody algebra. Then ˆ g = g ⊗ C[t, t−1 ] ⊕ Ck where C[t, t−1 ] is the algebra of Laurent polynomials, k a central element and [ x ⊗ tm, y ⊗ tn ] = [x, y] ⊗ tm+n + mδm+n (x, y)k . This realization implies for example that the denominator identities of the aﬃne Kac–Moody algebras are Jacobi forms and that their vacuum modules carry a vertex algebra structure. Borcherds found out that the conditions on the Cartan matrix can be weakened further. We can also drop the condition that the elements on the diagonal of A are positive, i.e., allow imaginary simple roots, and still get nice Lie algebras from Serre’s construction. These Lie algebras are called generalized Kac–Moody algebras [B1], [B3], [B4], [J]. We describe them in more detail.

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Nils R. Scheithauer

Let A = (aij )i,j∈I be a real square matrix, where I is some ﬁnite or countable index set, satisfying the following conditions. 1. 2. 3.

aij = aji aij ≤ 0 for i = j 2aij /aii ∈ Z if aii > 0

Let ˆ g be the Lie algebra with generators {ei , hij , fi | i ∈ I} and relations 1. 2. 3. 4.

[ei , fj ] = hij [hij , ek ] = δij aik ek , [hij , fk ] = −δij aik fk ad(ei )1−2aij /aii ej = ad(fi )1−2aij /aii fj = 0 [ei , ej ] = [fi , fj ] = 0 if aij = 0.

if aii > 0 and i = j

Then ˆ g has the following properties. The elements hij are 0 unless the i-th and j-th columns of A are equal. The nonzero hij are linearly independent and span a commutative subalgebra ˆ of ˆ h g. The elements hii are usually denoted hi . Every nonzero ideal of ˆg ˆ The center of ˆg is in h ˆ and contains all the has nonzero intersection with h. elements hij with i = j. ˆ of gˆ is the free abelian group generated by elements The root lattice Q αi , i ∈ I, with the bilinear form given by (αi , αj ) = aij . The elements αi are called simple roots. The Lie algebra ˆ g is graded by the root lattice if we deﬁne the degree of ei as αi and the degree of fi as −αi . We have the usual deﬁnitions of roots and root spaces. A root is called real if it has positive norm and imaginary otherwise. There is a unique invariant symmetric bilinear form on ˆg satisfying ˆ to (hi , hj ) = aij . We have a natural homomorphism of abelian groups from Q ˆ sending αi to hi . h There is also a character formula. The denominator identity takes the following form eρ (1 − eα )mult(α) = det(w) w eρ ε(α)eα α>0

w∈W

α

where ε(α), for α in the root lattice, is (−1)n if α is the sum of n pairwise orthogonal imaginary simple roots and 0 otherwise. A Lie algebra g is a generalized Kac–Moody algebra if g is isomorphic to ˆ g/c ⊕ d, where c is a subspace of the center of ˆ g and d an abelian subalgebra of g such that the elements ei and fi are eigenvectors of d and [d, hij ] = 0. ˆ ⊕ d is an abelian subalgebra of g called Cartan subalgebra. We Then h/c ˆ to h ˆ to transfer notations from ˆg to g. use the natural homomorphism from Q The denominator identity of g is usually given as a specialization of the denominator identity of ˆ g.

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155

Borcherds showed [B5] that a real Lie algebra g satisfying the following conditions is a generalized Kac–Moody algebra. 1. g has a nondegenerate invariant symmetric bilinear form. 2. g has a selfcentralizing subalgebra h such that g decomposes into ﬁnite dimensional eigenspaces of h. 3. The bilinear form is Lorentzian on h. 4. The norms of the roots are bounded above. 5. If two roots are positive multiples of the same norm 0 vector then their root spaces commute. This characterization applies in most cases of interest. We will see that similar to the aﬃne case in Kac–Moody theory there seems to be a class of generalized Kac–Moody algebras which can be classiﬁed and admit realizations.

3 Vertex algebras Vertex algebras give a mathematically rigorous deﬁnition of 2-dimensional quantum ﬁeld theories. In this section we give a short introduction into the theory of these algebras. Nice references are [K2] and [FB]. A vertex algebra is a vector space V with a state-ﬁeld correspondence which associates to each state a ∈ V a ﬁeld a(z) = n∈Z an z −n−1 satisfying certain conditions. The precise deﬁnition is as follows. Let V be a vector space and V → End(V )[[z, z −1 ]] an z −n−1 a → a(z) = n∈Z

a state-ﬁeld correspondence, i.e., an b = 0 for n suﬃciently large. V is a vertex algebra if it satisﬁes the following conditions. 1. There is an element 1 ∈ V such that 1(z)a = a and a(z)1|z=0 = a. 2. The operator D on V deﬁned by Da = a−2 1 satisﬁes [D, a(z)] = ∂a(z). 3. The locality condition (z − w)n [a(z), b(w)] = 0 holds for n suﬃciently large. V is called a vertex operator algebra of conformal weight c if in addition V contains an element ω such that the operators Lm = ωm+1 give a representation of the Virasoro algebra of central charge c, i.e. [Lm , Ln ] = (m − n)Lm+n + L−1 = D and V = of L0 and V0 = C.

n≥0

m3 − m δm+n c , 12

Vn , where the Vn are ﬁnite-dimensional eigenspaces

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The above deﬁnitions easily carry over to the Z2 -graded case. The most important property of vertex operators is Borcherds’ identity which follows from locality. For k, m, n ∈ Z we have ∞ m (−1)j am+k−j bn+j − (−1)m bm+n−j ak+j j j=0

=

∞ k (am+j b)k+n−j . j j=0

A consequence of this identity is that the space V /DV is a Lie algebra under [a, b] = a0 b. We will see that a slight modiﬁcation of this construction relates vertex algebras to generalized Kac–Moody algebras. The simplest and best understood vertex algebras are lattice vertex algebras and Wess–Zumino–Witten models. A lattice vertex algebra describes the Fock space of a quantized chiral bosonic string moving on a torus. We give a short sketch of the construction. Let L be an even lattice, h = L ⊗ C the underlying complex vector space and ˆ = h ⊗ C[t, t−1 ] ⊕ Ck h the bosonic Heisenberg algebra with central element k and commutation relations [ x ⊗ tm, y ⊗ tn ] = mδm+n (x, y)k . ˆ− = h ⊗ t−1 C[t−1 ] is an abelian subalgebra of h. ˆ We denote by S(h ˆ− ) Then h − ˆ the symmetric algebra of h . ˆ →L→0 The lattice L has a unique central extension 0 → {±1} → L by a group of order 2 such that the commutator of the inverse images of α, β in L is (−1)(α,β) . Let ε : L × L → {±1} be the corresponding 2-cocycle and C[L]ε the twisted group algebra with basis {eα | α ∈ L} and products eα eβ = ε(α, β)eα+β . Then the vector space ˆ− ) ⊗ C[L]ε V = S(h carries a natural action of the Heisenberg algebra which can be used to deﬁne a vertex algebra structure. Now let ˆ g be an untwisted aﬃne Kac–Moody algebra. Then the irreducible highest weight module LkΛ0 , where k is a positive integer, has a vertex operator algebra structure. These vertex algebras are called WZW models. They describe strings moving on compact Lie group manifolds. Modules of vertex algebras and vertex operator algebras are deﬁned in a natural way. A vertex operator algebra is rational if it has only ﬁnitely many nonisomorphic simple modules and every module decomposes into a ﬁnite direct sum of simple modules. Examples of such vertex operator algebras are

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vertex algebras of positive deﬁnite lattices and WZW models. A vertex operator algebra is called meromorphic if it has only one simple module, namely itself. Let V be a rational vertex operator algebra and M a simple V -module of conformal weight h. Then the character χM (τ ) = q −c/24 tr q L0 = q h−c/24

∞

dim(Mn )q n

n=0

is well deﬁned. Zhu showed that the characters of a rational vertex operator algebra satisfying suitable ﬁniteness conditions are holomorphic on the upper halfplane and the vector space generated by the characters is invariant under SL2 (Z). For example the character of a meromorphic vertex operator algebra of central charge 24 satisfying the ﬁniteness conditions is modular invariant and therefore equal to j(τ ) = q −1 + 744 + 196884q + 21493760q 2 + 864299970q 3 + . . . up to an additive constant. Zhu’s result holds for vertex algebras of positive deﬁnite lattices and WZW models. Let V be a vertex operator algebra. Then V1 is a ﬁnite dimensional subalgebra of the Lie algebra V /DV . Schellekens shows in [ANS] that if V is a meromorphic vertex operator algebra of central charge 24 and nonzero V1 , then either dim V1 = 24 and V1 is commutative or dim V1 > 24 and V1 is semisimple. In the ﬁrst case V is the vertex operator algebra of the Leech lattice. In the second case V can be written as a sum of modules over the aﬃnization Vˆ1 of V1 and Schellekens shows that there are at most 69 possibilities for V1 . For each of these possibilities he ﬁnds a modular invariant partition function and describes V explicitly as a module over Vˆ1 . If the monster vertex operator algebra is unique and for each of the 69 candidates there exists a unique vertex operator algebra, Schellekens’ result implies that there are 71 meromorphic vertex operator algebras of central charge 24. Up to now these conjectures are open.

4 Automorphic forms on orthogonal groups Borcherds’ singular theta correspondence is a map from modular forms transforming under the Weil representation of SL2 (Z) to automorphic forms on orthogonal groups. In this section we give a short description of this lifting. References are [B6], [Br], [S3], and [S4]. Let L be an even lattice of even rank and L the dual lattice of L. The discriminant form of L is the ﬁnite abelian group D = L /L with quadratic

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form γ 2 /2 mod 1. The level of D is the smallest positive integer N such that N γ 2 /2 = 0 mod 1 for all γ ∈ D. We deﬁne a scalar product on the group ring C[D] which is linear in the ﬁrst and antilinear in the second variable by (eγ , eβ ) = δ γβ . Then there is a unitary action of SL2 (Z) on C[D] deﬁned by ρD (T ) eγ = e(−γ 2 /2) eγ ρD (S) eγ =

e(sign(D)/8)

e(γβ) eβ , |D| β∈D

and T = ( 10 11 ) are the standard generators of SL2 (Z). This where S = 01 −1 0 representation is called the Weil representation of SL2 (Z) corresponding to D. A holomorphic function F (τ ) = Fγ (τ )eγ γ∈D

on the upper halfplane with values in C[D] is called a modular form for ρD of weight k if F (M τ ) = (cτ + d)k ρD (M )F (τ ) a b for all M = c d ∈ SL2 (Z) and F is meromorphic at i∞. Classical examples of modular forms transforming under the Weil representation are theta functions. Suppose L is a positive deﬁnite even lattice of even rank 2k. For γ ∈ D deﬁne 2 θγ+L (τ ) = q α /2 . α∈γ+L

Then θ(τ ) =

θγ+L (τ )eγ

γ∈D

is a modular form for the dual Weil representation of weight k which is holomorphic at i∞. Suppose D has level N . Then we can construct modular forms transforming under the Weil representation by lifting scalar valued modular forms on Γ0 (N ). The identity e0 in C[D] is up to a character value invariant under Γ0 (N ). Hence if f is a modular form on Γ0 (N ) of weight k and character χD then F (τ ) = f |M (τ ) ρD (M −1 )e0 M∈Γ0 (N )\SL2 (Z)

is a modular form for ρD of weight k. We can use the map for example to construct Eisenstein series for the Weil representation by lifting scalar valued Eisenstein series on Γ0 (N ).

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Now we assume that L has signature (n, 2) with n ≥ 2. Let V = L ⊗ R and G be the Grassmannian of 2-dimensional negative deﬁnite subspaces of V . Then the orthogonal group O(V ) acts naturally on G. The Grassmannian G can be realized as a tube domain H ⊂ Cn as follows. Let z be a primitive norm 0 vector in L and z ∈ L such that (z, z ) = 1. Deﬁne K = L ∩ z ⊥ ∩ z ⊥ . Then V = K ⊗ R ⊕ Rz ⊕ Rz. Since K is a Lorentzian lattice the vector space K ⊗ R has two cones of negative norm vectors. We choose one of these cones and denote it by C. Let Z = X + iY ∈ K ⊗ C with Y ∈ C. Then ZL = Z + z − (Z 2/2 + z 2/2)z = XL + iYL is a norm 0 vector in V ⊗ C and {XL , YL } is an orthogonal basis of a negative deﬁnite subspace of V . This map gives a bijection between K ⊗ R + iC and G. The natural action of O+ (V ) on G induces an action by fractional linear transformations on H. A meromorphic function f on H is called an automorphic form of weight k for a subgroup Γ of ﬁnite index in O+ (L) if f (M Z) = j(M, Z)k f (Z) for all M ∈ Γ. Here j(M, Z) is an automorphy factor for O+ (V ). If n > 2 and f is holomorphic on H then by the Koecher principle f is also holomorphic at the cusps. We say that f has singular weight if f has weight n/2 − 1. If n > 2 this is the smallest possible weight of a holomorphic automorphic form on H. If f is a holomorphic automorphic form of singular weight and n > 2 then the only nonzero Fourier coeﬃcients of f correspond to norm 0 vectors of K. Borcherds’ singular theta correspondence is a map from modular forms with poles at cusps transforming under the Weil representation of SL2 (Z) to automorphic forms on orthogonal groups. Since these automorphic forms have nice product expansions they are called automorphic products. We describe Borcherds’ construction in more detail. Let F be a modular form for ρD of weight 1 − n/2. The Siegel theta function θ(Z, τ ) of L is a function that is invariant under O+ (L) in Z ∈ H and transforms under the dual Weil representation in the second variable. Then the integral

dx dy Φ(Z) = F (τ )θ(Z, τ )y 2 , y F where F is the standard fundamental domain of SL2 (Z) on the upper halfplane, is formally invariant under the subgroup O+ (L, F ) preserving F . However the integral does not converge because F has a pole for τ → i∞. It can be regularized by taking the constant term in the Laurent expansion at s = 0 of the meromorphic continuation in s of

dx dy lim F (τ )θ(Z, τ )y 1+s 2 u→∞ F y u which converges for suﬃciently large real part of s. Here Fu = {τ ∈ F | y ≤ u} denotes the truncated fundamental domain.

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The automorphic product Ψ corresponding to F is obtained by exponentiating Φ. It is an automorphic form for O+ (L, F ). The zeros and poles of Ψ lie on rational quadratic divisors λ⊥ = {Z ∈ H | aZ 2/2−(Z, λk )−az 2/2−b = 0 }, where λ = λK + az + bz is a primitive vector of positive norm in L, and their orders are determined by the principal part of F . In the theory of generalized Kac–Moody algebras reﬂective automorphic products [S3] play an important role. A root of L is a primitive vector α in L of positive norm such that the reﬂection σα (x) = x − 2(x, α)α/α2 is an automorphism of L. If α is a vector in L of norm 2, then α is a root. In general L also has roots of norm greater than 2. We say that Ψ is reﬂective if its divisors are orthogonal to roots of L and are zeros of order one. The automorphic product Ψ is called symmetric reﬂective if Ψ is reﬂective and all roots of L of a given norm correspond to zeros of order one, and Ψ is completely reﬂective if Ψ is reﬂective and all roots of L give zeros of order one.

5 Moonshine for Conway’s group The largest sporadic simple group, the monster, acts on the monster algebra. This is a generalized Kac–Moody algebra describing the physical states of a bosonic string moving on a 2-dimensional orbifold. Borcherds [B4] shows in his proof of Conway and Norton’s moonshine conjecture for the monster [CN], [G] that the twisted denominator identities under this operation are automorphic forms of weight 0 on O2,2 (R). Conway’s group Co0 acts by diagram automorphisms on the fake monster algebra. We conjecture that the corresponding twisted denominator identities are automorphic forms of singular weight on orthogonal groups. The vertex algebras in this section are deﬁned over the real numbers. The vertex algebra VII25,1 of the even unimodular Lorentzian lattice II25,1 describes a bosonic string moving on the torus R25,1 /II25,1 . In order to determine the physical states of the string we multiply VII25,1 with the vertex superalgebra VZσ of the b,c -ghost system. We obtain the vertex superalgebra V = VII25,1 ⊗ VZσ . Since VII25,1 has a conformal structure of central charge 26 and VZσ of central charge −26, the vertex superalgebra V carries an action of the Virasoro algebra of central charge 0. There is also a BRST-operator Q with Q2 = 0 acting on V and this operation commutes with the action of the Virasoro algebra. We deﬁne the ghost b = eσ and the ghost number operator j0N = σ(0). Then C = V ∩ ker b1 ∩ ker L0 is invariant under Q and graded by the ghost number Cαn . C= α∈II25,1 n∈Z

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We have a sequence Q

Q

Q

Q

. . . −→ Cαn−1 −→ Cαn −→ Cαn+1 −→ . . . with cohomology groups Hαn . Let H=

Hαn .

α∈II25,1 n∈Z

Then H 1 = α∈II25,1 Hα1 is the space of physical states of the compactiﬁed bosonic string. It is easy to see that Hα1 is isomorphic to II25,1 ⊗ R if α = 0. Frenkel et al. [FGZ] have shown that for α = 0 the cohomology groups Hαn are trivial unless n = 1. The Euler–Poincar´e principle implies dim Hα1 = [1/∆](−α2 /2) for α = 0, where 1/∆(τ ) =

∞ 1 1 = q −1 + 24 + 324q + 3200q 2 + . . . . q n=1 (1 − q n )24

For example if α has norm 2, then Hα1 is 1-dimensional. We deﬁne a product [LZ] on C by [u, v] = (b0 u)0 v. This product projects down to H and to g = H 1 . It deﬁnes a Lie bracket on g. It is not diﬃcult to see that g satisﬁes all the conditions of Borcherds’ characterization. Hence g is a generalized Kac–Moody algebra. Using the singular theta correspondence we can show that the denominator identity of g is given by ∞ 24 2 eρ . 1 − emρ (1 − eα )[1/∆](−α /2) = det(w) w eρ + α∈II25,1

w∈W

m=1

Here ρ is a primitive norm 0 vector in II25,1 corresponding to the Leech lattice and W is the reﬂection group of II25,1 . The real simple roots of g are the norm 2 vectors in II25,1 such that (ρ, α) = −1 and the imaginary simple roots are the positive multiples mρ of ρ, each with multiplicity 24. The Lie algebra g has been constructed ﬁrst in a slightly diﬀerent way in [B2]. It is now called the fake monster algebra. Conway’s group Co0 is the automorphism group of the Leech lattice Λ. The characteristic polynomial of an element g in O(Λ) n can be written of order as k|n (xk − 1)bk . The eta product ηg (τ ) = η(kτ )bk is a modular form, possibly with poles at cusps, for a group of level N . We call N the level of g. The Leech lattice has a unique central extension 0 → {±1} → Λˆ → Λ → 0 such that the commutator of the inverse images of α, β in Λ is (−1)(α,β) . The ˆ = 224 .O(Λ) of automorphisms preserving the inner product acts group O(Λ) naturally on the fake monster algebra [B4]. Since II25,1 = Λ ⊕ II1,1 we can

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write g =

e

ρ

ga with ga =

v∈Λ g(v,a) . Then for a = 0 the space ga is ˆ isomorphic as O(Λ)-module to the L0 -eigenspace VΛ,1−a2 /2 of degree 1 − a2 /2 of the vertex algebra of the Leech lattice. If a = 0 then ga is isomorphic to ˆ VΛ,1 ⊕ R2 as O(Λ)-module. The orthogonal decomposition Λ ⊗ R = Λg ⊗ R ⊕ Λg ⊥ ⊗ R gives a natural projection π : Λ ⊗ R → Λg ⊗ R which sends Λ to Λg . ˆ which acts trivially on the Each element g in O(Λ) has a lift gˆ to O(Λ) g inverse image of the ﬁxed point lattice Λ . If g has order n, then the order n ˆ of gˆ is either n or 2n. The denominator identity of the fake monster algebra can be written as a cohomological identity. Taking the trace of gˆ over this identity we obtain a∈II1,1

α mult(α)

(1 − e )

=

det(w) w e

w∈W g

α∈L+

ρ

∞

1−e

mkρ bk

k|n m=1

where mult(α) =

dk|((α,L),ˆ n)

v∈Λ π(v)=r/dk

µ(k) d tr gˆ VΛ,1−a2 /2d2 k2 ,v dk

for α = (r, a) in L = Λg ⊕ II1,1 . The vector ρ is a primitive norm 0 vector in II1,1 and W g is the subgroup of elements in W mapping L into L. This identity is independent of the choice of gˆ. It is the denominator identity of a generalized Kac–Moody superalgebra whose real simple roots are the simple roots of W g , i.e., the roots α of L such that (ρ, α) = −α2 /2, and imaginary simple roots are the positive multiples mρ of the Weyl vector with multiplicity mult(mρ) = k|(m,n) bk . We conjecture that this identity defines an automorphic product of singular weight k/2, where k = dim Λg . We have Theorem 5.1. The conjecture is true for elements of squarefree level. The assertion is proved in [S1] for elements of squarefree level and nontrivial ﬁxed point lattice and in [S2] for elements of squarefree level and trivial ﬁxed point lattice. We give an outline of the proof for nontrivial ﬁxed point lattices. Let g be an element in O(Λ) of squarefree level N and nontrivial ﬁxed point lattice Λg . Then g has order N . We lift the modular form fg = 1/ηg to a vector valued modular form Fg on the lattice Λg ⊕ II1,1 ⊕ II1,1 (N ) . Then we apply the singular theta correspondence to obtain an automorphic form Ψg . We can represent this by the following diagram

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163

g → 1/ηg → Fg → Ψg . Explicit calculation shows that Ψg has singular weight. The theta correspondence gives the product expansions of Ψg at the diﬀerent cusps. We can determine the corresponding sum expansions because Ψg has singular weight so that the nonzero Fourier coeﬃcients correspond to norm 0 vectors. The expansion of Ψg at the level N cusp is the twisted denominator identity of the fake monster algebra corresponding to g eρ

2

(1 − eα )[ck fg |Wk ](−α

k|N α∈(L∩kL )+

=

/2k)

∞ b 1 − emkρ k . det(w) w eρ k|N m=1

w∈W

Here L = Λg ⊕ II1,1 and ck is a constant such that ck fg |Wk has constant term bk . The conjecture is also proved for some elements which do not have squarefree level [S4]. Recall that the Mathieu group M23 acts on Λ. Theorem 5.2. Let g be an element in M23 of order N . Then the twisted denominator identity of g defines a reflective automorphic product Ψ of singular weight on Λg ⊕II1,1 ⊕II1,1 (N ). If N is squarefree then Ψ is completely reflective and the expansion of Ψ at any cusp is given by e((ρ, Z))

[1/ηg ](−α2 /2k) 1 − e((α, Z))

k|N α∈(L∩kL )+

=

det(w) ηg ((wρ, Z)),

w∈W

where W is the full reflection group of L = Λg ⊕ II1,1 . This identity is the denominator identity of a generalized Kac–Moody algebra whose real simple roots are the simple roots of W , which are the roots α of L with (ρ, α) = −α2 /2, and imaginary simple roots are the positive multiples mρ of the Weyl vector with multiplicity 24 σ0 ((N, m))/σ1 (N ). We remark that if g is an element in M23 of squarefree order N , then bk = 24/σ1 (N ) for all k|N . The theorem gives 10 generalized Kac–Moody algebras similar to the fake monster algebra. We describe them in more detail. We take simple roots αi in L according to their multiplicity as stated in the theorem and form the symmetrized Cartan matrix aij = (αi , αj ). Then we deﬁne ˆg by generators and relations as above. Let g be the quotient of ˆg by its center. Then g is a simple real generalized Kac–Moody algebra with Cartan subalgebra naturally isometric to L ⊗ R and root lattice L. The denominator identity of g is given in the theorem.

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6 Classification results In this section we describe some classiﬁcation results for automorphic products and generalized Kac–Moody algebras [S3]. Reﬂective automorphic products are theta lifts of reﬂective modular forms. These are modular forms transforming under the Weil representation which have very mild singularities corresponding to roots of the underlying lattice. We obtain a necessary condition for the existence of a reﬂective modular form by pairing it with an Eisenstein series. We describe this in more detail. Let an even lattice of even signature L be and discriminant form D. Let F = Fγ eγ be a modular form for the Weil representation ρD of weight 2 − k and E = Eγ eγ the Eisenstein series for ρD of weight k. Then by the unitarity of ρD the inner product F E = Fγ Eγ is a scalar valued modular form on SL2 (Z) of weight 2 with a pole at the cusp i∞. Hence F Edτ deﬁnes a meromorphic 1-form on the Riemann sphere with a pole at i∞. By the residue theorem its residue has to vanish. This implies that the constant term in the Fourier expansion of F E is 0. If L has squarefree level we can calculate this condition explicitly. The fast growth of the Bernoulli numbers implies the following result. Theorem 6.1. The number of symmetric reflective automorphic products of singular weight on lattices of signature (n, 2) with n > 2, squarefree level and p-ranks at most n + 1 is finite. In the theorem we consider symmetric automorphic products in order to exclude oldforms which are induced from sublattices. The condition on the p-ranks prevents that we have to discuss rescalings of lattices. We can work out a bound on the level and determine the solutions of the necessary condition by computer search. We ﬁnd that the necessary condition has very few solutions and that in the case of completely reﬂective forms they come from existing forms. Theorem 6.2. Let L be an even lattice of signature (n, 2) with n > 2 and squarefree level N . Suppose L splits II1,1 ⊕ II1,1 (N ). Let Ψ be a completely reflective automorphic product of singular weight on L. Then Ψ can be constructed from an element of order N in M23 . If L has even p-ranks np the equation parametrizing the completely reﬂective automorphic products is given by n /2 k 1 1 −1 p k−np /2 np /2 p − 2 =1 + p p k − 2 Bk pk − 1 p p|N

where k = 1+n/2 . . . and Bk is the k-th Bernoulli number. This is an equation in the indeterminates N, k, p and np . It has exactly 8 solutions. Since the automorphic products in Theorem 6.2 are the denominator identities of generalized Kac–Moody algebras we obtain the following classiﬁcation result.

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Theorem 6.3. Let L be an even lattice of signature (n, 2) with n > 2 and squarefree level N . Suppose L splits II1,1 ⊕ II1,1 (N ). Let g be a real generalized Kac–Moody algebra whose denominator identity is a completely reflective automorphic product of singular weight on L. Then g corresponds to an element of order N in M23 .

7 Construction as strings In this section we show that some of the above generalized Kac–Moody algebras describe strings moving on suitable orbifolds (cf. [HS], [CKS]). There are exactly 10 generalized Kac–Moody algebras whose denominator identities are completely reﬂective automorphic products of singular weight on lattices of squarefree level. These Lie algebras correspond naturally to the elements of squarefree order N in M23 . In the case N = 1 the generalized Kac–Moody algebra can be realized as the cohomology group of the BRSToperator Q acting on the vertex superalgebra VII25,1 ⊗ VZσ . We will see that a similar result also holds in 4 other cases. Recall that Schellekens has constructed 69 vector spaces as modules over aﬃne Kac–Moody algebras which are conjecturally meromorphic vertex operator algebras of central charge 24. Theorem 7.1. Suppose the meromorphic vertex operator algebra V of central charge 24 and spin-1 algebra Aˆrq,p , where p = 2, 3, 5 or 7, q = p − 1 and r = 48/q(p + 1), exists and has a real form. Then the cohomology group of the BRST-operator Q acting on V ⊗ VII1,1 ⊗ VZσ gives a natural realization of the generalized Kac–Moody algebra corresponding to the elements of order p in M23 . For p = 2 the existence of V and of a real form is proved. We sketch the proof of the theorem. As a vector space V is the sum of tensor products ⊗ri=1 VΛi where VΛi is an irreducible highest weight module over Aˆq of level p. The character of VΛi can be expressed in terms of string and theta functions. Using the transformation properties of the string functions under SL2 (Z) we show that the character of V as Aˆrq -module can be written as Fγ θγ . χV = F θ = γ∈N /N

Here N is the lattice of the largest possible minimal norm in the genus II2m,0 pp (m+2) with m = 24/(p + 1), θ = γ γ∈N /N Fγ e the lift of

γ∈N /N

f (τ ) =

θγ eγ the theta function of N and F =

1 η(τ )m η(pτ )m

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Nils R. Scheithauer

to N . If V has the structure of a vertex algebra and admits a real form, we can proceed analogously as in the construction of the fake monster algebra. We only have to replace the vertex algebra of the Leech lattice VΛ in VII25,1 = VΛ ⊗ VII1,1 by V . Then the cohomology group of ghost number one of the BRST-operator Q acting on V ⊗ VII1,1 ⊗ VZσ is a generalized Kac–Moody algebra which we denote by g. Let Λp be the sublattice of the Leech lattice ﬁxed by an element of order p in M23 . The Lie algebra g is graded by the lattice N ⊕ II1,1 which becomes isomorphic to L = Λp ⊕ II1,1 after rescaling by p. Using the singular theta correspondence we show that the denominator identity of g is given by eρ

2

(1 − eα )[f ](−α

α∈L+

/2)

α∈pL+

=

w∈W

2

(1 − eα )[f ](−α det(w) w e

ρ

/2p)

∞

nρ m

(1 − e ) (1 − e

pnρ m

)

n=1

so that g is the generalized Kac–Moody algebra corresponding to the elements of order p in M23 .

8 Open problems In this section we describe some open problems in the theory of generalized Kac–Moody algebras, vertex algebras and automorphic products. 1. One of the most important open problems in the theory of vertex algebras is to classify the meromorphic vertex operator algebras of central charge 24. There are conjecturally 71 such vertex operator algebras. This problem includes the proof that the monster vertex algebra is the only meromorphic vertex operator algebra of central charge 24 and trivial V1 and the veriﬁcation of the existence and uniqueness of the potential theories in [ANS]. For the 24 lattice vertex algebras corresponding to the 24 positive deﬁnite even unimodular lattices in dimension 24 the existence and uniqueness is proved and 15 other vertex algebras can be constructed as Z2 -orbifolds [DGM1], [DGM2]. The remaining candidates can possibly be constructed by using higher twists and further orbifolding of some of the twisted theories [M]. However a more natural construction would be desirable. 2. The automorphism group of the Leech lattice acts naturally on the fake monster algebra. Show that the corresponding twisted denominator identities are automorphic forms of singular weight on orthogonal groups. So far this is

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proved for the elements of squarefree level and some additional classes [S1], [S2], [S4]. 3. Classify reﬂective automorphic products, which are not assumed to be symmetric, of singular weight on arbitrary lattices. Then derive classiﬁcation results for generalized Kac–Moody algebras whose denominator identities are reﬂective automorphic products of singular weight. Both problems have been solved for completely reﬂective automorphic products on lattices of squarefree level [S3]. The pairing argument used there can also be applied to the general case. For some classiﬁcation results in signature one, see [GN]. 4. Can the generalized Kac–Moody algebras, whose denominator identities are reﬂective automorphic products of singular weight, be characterized by their root system? 5. Show that the generalized Kac–Moody algebras, whose denominator identities are reﬂective automorphic products of singular weight, can be realized as bosonic strings moving on suitable target spaces. This is proved in 2 cases [B2], [HS] and proved up to the existence of certain vertex operator algebras in 3 more cases [CKS]. 6. The denominator identities of some generalized Kac–Moody algebras are automorphic forms on orthogonal groups. Is the representation theory of these Lie algebras related to automorphic forms?

References [B1] [B2] [B3] [B4] [B5] [B6] [Br] [CKS] [CN] [DGM1]

R. E. Borcherds, Generalized Kac–Moody algebras, J. Algebra 115 (1988), 501–512. R. E. Borcherds, The monster Lie algebra, Adv. Math. 83 (1990), 30–47. R. E. Borcherds, Central extensions of generalized Kac–Moody algebras, J. Algebra 140 (1991), 330–335. R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444. R. E. Borcherds, A characterization of generalized Kac–Moody algebras, J. Algebra 174 (1995), 1073–1079. R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), 491–562. J. H. Bruinier, Borcherds products on O(2, l) and Chern classes of Heegner divisors, Lecture Notes in Mathematics, Vol.1780, Springer, Berlin, 2002. T. Creutzig, A. Klauer, N. R. Scheithauer, Natural constructions of some generalized Kac–Moody algebras as bosonic strings, Comm. Number Theory Phys. 1 (2007), 453–477. J. H. Conway and S. P. Norton, Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308–339. L. Dolan, P. Goddard, P. Montague, Conformal field theory of twisted vertex operators, Nuclear Phys. B 338 (1990), 529–601.

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[DGM2] [FB] [FGZ] [G] [GN] [HS] [J] [K1] [K2] [LZ] [M] [S1] [S2] [S3] [S4] [ANS]

L. Dolan, P. Goddard, P. Montague, Conformal field theory, triality and the Monster group, Phys. Lett. B 236 (1990), 165–172. E. Frenkel, D. Ben-Zvi, Vertex Algebras and Algebraic Curves, 2nd ed., Mathematical Surveys and Monographs 88, Amer. Math. Soc, 2004. I. B. Frenkel, H. Garland and G. Zuckerman, Semi-infinite cohomology and string theory, Proc. Natl. Acad. Sci. USA 83 (1986), 8442–8446. T. Gannon, Monstrous moonshine: the first twenty-five years, Bull. London Math. Soc. 38 (2006), 1–33. V. A. Gritsenko, V. V. Nikulin, On the classification of Lorentzian Kac– Moody algebras, Russian Math. Surveys 57 (2002), 921–979. G. H¨ ohn and N. R. Scheithauer, A natural construction of Borcherds’ Fake Baby Monster Lie Algebra, Am. J. Math. 125 (2003), 655–667. E. Jurisich, Generalized Kac–Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra, J. Pure Appl. Algebra 126 (1998), 233–266. V. Kac, Infinite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, 1990. V. Kac, Vertex Algebras for Beginners, 2nd ed., University Lecture Series 10, Amer. Math. Soc, 1998. B. H. Lian, G. J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Comm. Math. Phys. 154 (1993), 613–646. P. S. Montague, Conjectured Z2 -orbifold constructions of self-dual conformal field theories at central charge 24 – the neighborhood graph, Lett. Math. Phys. 44 (1998), 105–120. N. R. Scheithauer, Generalized Kac–Moody algebras, automorphic forms and Conway’s group I, Adv. Math. 183 (2004), 240–270. N. R. Scheithauer, Generalized Kac-Moody algebras, automorphic forms and Conway’s group II, J. Reine Angew. Math. 625 (2008), 125–154. N. R. Scheithauer, On the classification of automorphic products and generalized Kac–Moody algebras, Invent. Math. 164 (2006), 641–678. N. R. Scheithauer, The Weil representation of SL2 (Z) and some applications, Int. Math. Res. Not. 2009, no. 8, 1488–1545. A. N. Schellekens, Meromorphic c=24 conformal field theories, Comm. Math. Phys. 153 (1993), 159–185.

Kac–Moody Superalgebras and Integrability Vera Serganova Dept. of Mathematics, University of California at Berkeley, Berkeley, CA 94720 [email protected]

Summary. The ﬁrst part of this paper is a review and systematization of known results on (inﬁnite-dimensional) contragredient Lie superalgebras and their representations. In the second part, we obtain character formulae for integrable highest weight representations of sl (1|n) and osp (2|2n); these formulae were conjectured by Kac–Wakimoto. Key words: Kac–Moody superalgebras, aﬃne superalgebras, Cartan matrices, integrable representations, character formulae. 2000 Mathematics Subject Classiﬁcations: Primary 17B67. Secondary 17B20.

1 Introduction The principal goal of this paper is to study a special class of Lie superalgebras which, in our opinion, plays the same role in the theory of Lie superalgebras as the Kac–Moody Lie algebras play in the theory of Lie algebras. Since the terminology is not completely uniform even in the case of Lie algebras, we start with brief discussion of this case. Given an arbitrary matrix A, one can deﬁne a contragredient Lie algebra g (A) by generators and relations (see [2] or Section 2 for precise deﬁnition). People are usually interested in the subclass of so-called Kac–Moody Lie algebras; by deﬁnition those are contragredient Lie algebras whose matrices satisfy the conditions: aii = 2, aij ∈ Z≤0 , and aij = 0 implies aji = 0. The main property which distinguishes Kac–Moody Lie algebras among all contragredient Lie algebras is the local nilpotency of generators in the adjoint representation. That allows one to deﬁne the Weyl group and the notion of an integrable representation whose character has a large group of symmetries. Finite-dimensional semisimple Lie algebras and aﬃne Lie algebras can be characterized as the only Kac–Moody superalgebras which are of polynomial growth.

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_6, © Springer Basel AG 2011

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The ﬁrst indication of usefulness of the notion of a contragredient Lie superalgebra appeared in [1]: a signiﬁcant part of the list of ﬁnite-dimensional simple superalgebras is contragredient. Many properties of contragredient Lie algebras are shared by contragredient Lie superalgebras; one notable exception is the uniqueness of representation: contragredient Lie superalgebras g (A) and g (B) might be isomorphic even if A is essentially diﬀerent from B. Our understanding of contragredient Lie superalgebras advanced a lot in the last three decades. In [10] Kac classiﬁed ﬁnite growth contragredient superalgebras whose Cartan matrices do not have zeros on the diagonal. Superalgebras of this class have the Weyl group and nice character formulas. But this class does not cover all ﬁnite-dimensional simple contragredient Lie superalgebras, and that makes one look for further generalization. Van de Leur in [8] studied a class of contragredient superalgebras with symmetrizable matrices. The condition that A is symmetrizable is equivalent to the existence of an even invariant symmetric form on the superalgebra g (A). He classiﬁed such superalgebras of ﬁnite growth; as in the usual case, those superalgebras are either ﬁnite dimensional, or central extensions of (twisted) loop superalgebras (the latter are called aﬃne superalgebras). The representation theory of aﬃne superalgebras has interesting applications in physics and number theory (see [11] and [12]). (These applications are based on conjectural character formulae for highest-weight representations; we prove these formulae for algebras sl (1|n)(1) and osp (2|n)(1) .) In this paper we discuss some aspects of structure theory and representations of contragredient Lie superalgebras. In particular, we suggest a deﬁnition of a Kac–Moody Lie superalgebra as a contragredient superalgebra with locally nilpotent adXi and adYi for any choice of a set of contragredient generators Xi , Yi , hi . The observation that all ﬁnite-growth contragredient Lie superalgebras are in this class led to a classiﬁcation of ﬁnite-growth contragredient Lie superalgebras ([9]); this gives evidence that such a deﬁnition is reasonable. Deﬁned in this way, Kac–Moody superalgebras have a very interesting replacement of the notion of Weyl group; it is not a group but a groupoid which acts transitively on the set of all Cartan matrices describing the given algebra. (This groupoid was the tool used in [9] to classify contragredient superalgebras of ﬁnite-growth; also see Section 7.) [9] also incorporates the classiﬁcation of Kac–Moody superalgebras with indecomposable Cartan matrices satisfying certain nondegeneracy conditions; this classiﬁcation was obtained by C. Hoyt in her thesis [3]. In the ﬁrst part of the paper we review the classiﬁcation results mentioned above, and we develop the structure theory similar to [2]. The main new result of this part is Theorem 4.14, which provides an algorithm listing all Cartan matrices deﬁning the same Kac–Moody superalgebra. This result is parallel to the transitivity of the Weyl group action on the set of bases (see [2] Proposition 5.9). In Section 2 we deﬁne contragredient Lie superalgebras, introduce the notions of admissibility and type, and apply restrictions of quasisimplicity and

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regularity. The purpose of the restriction of quasisimplicity is, essentially, to simplify the discussion; on the one hand, the regularity is a key restriction required by the methods of this paper; on the other hand, in the examples we know, similar results seem to hold without this restriction. In Section 3 odd reﬂections and Weyl groupoid for a contragredient Lie superalgebra g are deﬁned. We use odd reﬂections to construct a “large contragredient Lie subalgebra” g (B) of g and to investigate the span of roots of g (B), as well as the cone spanned by positive roots of g (B). In Section 4 we introduce notions of Kac–Moody superalgebra, Weyl group, real and imaginary roots, and discuss some geometric properties of roots of Kac–Moody superalgebras. Next, we provide useful examples of quasisimple Kac–Moody superalgebras: in Section 5, of rank 2, and in Section 6, what would turn out to be all quasisimple regular Kac–Moody Lie superalgebras. In Section 7 we collect together all known classiﬁcation results for contragredient Lie superalgebras. Section 8 contains ampliﬁcations of several results from Sections 3 and 4. In Section 9 we revisit the examples and provide a more detailed description of their structure. In the rest of the paper we use the Weyl groupoid to study highest weight representations of Kac–Moody superalgebras. The main results there are Corollary 14.5 and Theorem 14.7, where we prove Kac–Wakimoto conjecture of [11, 12] for completely integrable highest weight representations. In Section 10 we deﬁne highest weight modules and integrable modules, and classify integrable highest weight modules over regular quasisimple Kac–Moody superalgebras. Section 11 collects results about highest weight representations which are true for any quasisimple regular Kac–Moody superalgebra. In Section 12 we study in detail highest weight representations for the Lie superalgebras sl (1|n)(1) , osp (2|2n)(1) , and S (1, 2, b). Those are the only inﬁnite-dimensional Kac–Moody superalgebras of ﬁnite growth which have nontrivial integrable highest weight representations and a nontrivial Weyl groupoid. The invariant symmetric form starts playing its role in Section 13; this excludes S (1, 2, b) from the discussion; here we obtain “the non-obvious” results on geometry of odd reﬂections. Finally, in Section 14 the collected information allows us to obtain character formulae for all highest weight (1) (1) integrable representations over sl (1|n) and osp (2|2n) . The ﬁrst part of the paper is mostly a review of results obtained somewhere else, we skip most technical proofs and refer the reader to the original papers. In the second part all proofs are written down as most of results there are new. The author thanks I. Zakharevich for careful and belligerent reading of a heap of versions of the manuscript and for providing innumerable useful suggestions and comments. The author also thanks M. Gorelik for pointing out defects in the initial version of the manuscript.

2 Basic deﬁnitions Our ground ﬁeld is C. By p (x) we denote the parity of an element x in a vector superspace V = V¯0 ⊕ V¯1 .

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Let I be a ﬁnite set of indices, p : I → Z2 , and A = (aij ) , i, j ∈ I, be a matrix. Fix an even vector space h of dimension 2|I| − rk (A). It was shown in [2] that one can choose linearly independent αi ∈ h∗ and hi ∈ h such that αj (hi ) = aij for all i, j ∈ I, and this choice is unique up to a linear transformation of h. Deﬁne a superalgebra ¯ g (A) by generators Xi , Yi , i ∈ I and h with relations [h, Xi ] = αi (h) Xi ,

[h, Yi ] = −αi (h) Yi ,

[Xi , Yj ] = δij hi ,

[h, h] = 0. (1)

Here we assume that h ∈ h, p (Xi ) = p (Yi ) = p (i). By g (A) (or g when the Cartan matrix is ﬁxed) denote the quotient of ¯ g (A) by the (unique) maximal ideal which intersects h trivially. It is clear that if B = DA for some invertible diagonal D, then g (B) ∼ = g (A). Indeed, an isomorphism can be obtained by mapping hi and Xi to dii hi and dii Xi respectively. Therefore, without loss of generality, we may assume that aii = 2 or 0. We call such matrices normalized. The action of h on the Lie superalgebra g = g (A) is diagonalizable and deﬁnes a root decomposition g=h⊕ gα , ∆ ⊂ h∗ ; α∈∆

by linear independence of αi , every root space gα is either purely even or purely odd. Therefore one can deﬁne p : ∆ → Z2 by putting p (α) = 0 or 1 whenever gα is even or odd, respectively. By ∆0 and ∆1 we denote the set of even and odd roots, respectively. The relations (1) imply that every root is a purely positive or purely negative integer linear combination i∈I mi αi . According to this, we call a root positive or negative, and have the decomposition ∆ = ∆+ ∪ ∆− . The triangular decomposition is by deﬁnition the decomposition g = n+ ⊕h⊕n− , where n± are subalgebras generated by Xi , i ∈ I, respectively, Yi , i ∈ I. The roots αi , i ∈ I, are called simple roots. Sometimes instead of aij we write aαβ with α = αi , β = αj ; likewise for Xα , Yα , hα instead of Xi , Yi , hi . Remark 2.1. Obviously, given a simple root α = αi , one of the following possibilities holds: 1. if aαα = 0 and p (α) = 0, then Xα , Yα and hα generate a Heisenberg subalgebra; 2. if aαα = 0 and p (α) = 1, then [Xα , Xα ] = [Yα , Yα ] = 0 and Xα , Yα , and hα generate a subalgebra isomorphic to sl (1|1); in this case Xα2 = Yα2 = 0 in any representation of g (A); 3. if aαα = 2 and p (α) = 0, then Xα , Yα , and hα generate a subalgebra isomorphic to sl (2); 4. if aαα = 2 and p (α) = 1, then [Xα , Xα ] = [Yα , Yα ] = 0, and Xα , Yα , and hα generate a (3|2)-dimensional subalgebra isomorphic to osp (1|2); in this case 2α ∈ ∆.

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Consider a simple root α. We call α isotropic iﬀ aαα = 0, otherwise α is called non-isotropic; α is called regular if for any other simple root β, aαβ = 0 implies aβα = 0, otherwise a simple root is called singular. A superalgebra g = g (A) has a natural Z-grading with 0-component being h, 1-component being i∈I gαi . We say that g is of ﬁnite growth if the dimension of n-component grows not faster than a polynomial in n. We say that A is admissible if adXi is locally nilpotent in g (A) for all i ∈ I. In this case adYi is also locally nilpotent. One can check (see, for instance, [9]) that A is admissible if and only if, after normalization, A satisﬁes the following conditions: 1. If aii = 0 and p (i) = 0, then aij = 0 for any j ∈ I; 2. If aii = 2 and p (i) = 0, then aij ∈ Z≤0 for any j ∈ I, i = j and aij = 0 ⇒ aji = 0; 3. If aii = 2 and p (i) = 1, then aij ∈ 2Z≤0 for any j ∈ I, i = j and aij = 0 ⇒ aji = 0; Note that if aii = 0 and p (i) = 1, there is no condition on the entries aij . We call g (A) quasisimple if for any ideal i ⊂ g (A) either i ⊂ h, or i + h = g (A). This is equivalent to saying that every ideal of g (A) is either in the center of g (A), or contains the commutator [g (A) , g (A)]. Recall that for usual Kac–Moody Lie algebras, g (A) is quasisimple iﬀ A is indecomposable; g (A) is simple if, in addition, A is nondegenerate. For ﬁnite-dimensional Lie algebras quasisimplicity is equivalent to simplicity, but in general this is not true. For example, aﬃne Lie algebras are quasisimple but not simple. In supercase even ﬁnite-dimensional contragredient superalgebras can be quasisimple but not simple (for example, gl (n|n), see Section 6). The following theorem is proven in [9]. Theorem 2.2. (Hoyt, Serganova) If g (A) is quasisimple and has ﬁnite growth, then A is admissible. One can describe an arbitrary contragredient Lie superalgebra as a sequence of extensions of quasisimple contragredient superalgebras and Heisenberg superalgebras (see [9] for details). Lemma 2.3. If g (A) is quasisimple, then A is indecomposable and does not have zero rows. In particular, for admissible A, aii = 0 implies p (i) = 1, so all simple isotropic roots are odd. Proof. If after suitable permutation of indices A = B ⊕ C is the sum of two nontrivial blocks, then g (A) is a direct sum of ideals g (B) ⊕ g (C). Hence A must be indecomposable. If A has a zero row, i.e., aij = 0 for some ﬁxed i ∈ I and all j ∈ I, then it is not hard to see that [Xi , b] + Chi is an ideal in g (A), here b = h ⊕ n+ . Hence A cannot have zero rows. We call A regular if for any i, j ∈ I, aij = 0 implies aji = 0. The following is a straightforward generalization of Proposition 1.7 in [2] to the supercase.

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Lemma 2.4. Let A be regular. Then g (A) is quasisimple iﬀ A is indecomposable and not zero. Denote by Q the lattice generated by ∆, and by Q0 the lattice generated by even roots ∆0 . Theorem 2.5. If g (A) is quasisimple and Q0 = Q, then either Q/Q0 ∼ = Z or Q/Q0 ∼ = Z2 . In the ﬁrst case, g (A) has a Z-grading g (A) = g−1 ⊕ g0 ⊕ g1 such that g0 is the even part of g (A). Proof. Consider the Q/Q0 grading g (A) = i∈Q/Q0 gi , where gi = s(α)=i gα , s : Q → Q/Q0 being the natural projection. Obviously, g0 = g¯0 , and gi ⊂ g¯1 for any i = 0. Choose i ∈ Q/Q0 , i = 0. Let l = gi ⊕ [gi , g−i ] ⊕ g−i . We claim that l is an ideal of g. Indeed, [gi , gi ] = [g−i , g−i ] vanish if 2i = 0, and coincide with [gi , g−i ] if 2i = 0. For j ∈ / {−i, 0, i}, [gj , g±i ] = 0, thus [gj , [gi , g−i ]] = 0 by the Jacobi identity; [g0 , [gi , g−i ]] ⊂ [gi , g−i ], again by the Jacobi identity. Now quasisimplicity of g (A) implies l + h = g (A). Therefore g (A) = gi ⊕ g0 ⊕ g−i ; hence i is a generator of Q/Q0 . If 2i = 0, then Q/Q0 ∼ = Z2 . If 2i = 0, then g (A) has Z-grading with gi being degree 1. Hence Q/Q0 must be Z.

Following [1], we call g (A) of type I, if Q/Q0 ∼ = Z, and of type II if Q/Q0 ∼ = Z2 . In the case when g (A) is ﬁnite dimensional and quasisimple, g (A) is of type I if it is isomorphic to sl (m|n) , m = n, gl (n|n), or osp (2|2n), and of type II if it is isomorphic to osp (m|2n) with m = 2, G3 or F4 . For more examples, see Corollary 7.4. Remark 2.6. It is useful to note that if g (A) is of type I, then all its odd simple roots are isotropic.

3 Odd reﬂections, Weyl groupoid, and principal roots A linearly independent set Σ of roots of a Lie superalgebra g (A) is a base if one can ﬁnd Xβ ∈ gβ and Yβ ∈ g−β for each β ∈ Σ such that Xβ , Yβ , β ∈ Σ, and h generate g (A), and for any β, γ ∈ Σ, β = γ [Xβ , Yγ ] = 0.

(2)

If we put hβ = [Xβ , Yβ ], then Xβ , Yβ and hβ satisfy the relations [h, Xβ ] = β (h) Xβ ,

[h, Yβ ] = −β (h) Yβ ,

[Xβ , Yγ ] = δβγ hβ .

(3)

¯ (AΣ ) → g (A), here AΣ = This induces a natural surjective mapping g (β (hγ )), β, γ ∈ Σ; since g (AΣ ) and g (A) have the same Cartan subalgebra h, the kernel must coincide with Ker ¯ g (AΣ ) → g (AΣ ). Therefore, g (A) is isomorphic to g (AΣ ). The matrix AΣ is called the Cartan matrix of a base Σ.

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The original set Π = {αi } is called a standard base. If g is a Kac–Moody Lie algebra, then every base can be obtained from the standard one by the Weyl group action (and, maybe, for inﬁnite roots systems, by multiplication by −1). It is crucial for our discussion that this is not true for superalgebras. Let Σ be a base, α ∈ Σ, aαα = 0 and p (α) = 1. For any β ∈ Σ, deﬁne rα (β) by rα (α) = −α, if α = β and aαβ = aβα = 0, rα (β) = β rα (β) = β + α

if aαβ = 0 or aβα = 0.

Lemma 3.1. The set rα (Σ) = {rα (β) | β ∈ Σ} is linearly independent. Moreover, if α is regular, then rα (Σ) is a base. Proof. The linear independence of rα (Σ) is obvious. To prove the second statement, set X−α = Yα ,

Yrα (β) = [Yα , Yβ ] (4) for any β ∈ Σ such that r (β) = β + α. First, we have to check that α Xrα (β) , Yrα (γ) = 0 if β, γ ∈ Σ and β = γ. If β, γ = α, then rα (β) − rα (γ) = β − γ or β − γ ± α ∈ / ∆, hence Xrα (β) , Yrα (γ) = 0. So assume that β = α. A simple calculation shows that if aαγ = aγα = 0, then [Yα , Yγ ] = [Xα , Xγ ] = 0. Therefore if rα (γ) = γ, then [X−α , Yγ ] = [Yα , Yγ ] = 0. If rα (γ) = γ + α, then

Y−α = Xα ,

Xrα (β) = [Xα , Xβ ] ,

1 X−α , Yrα (γ) = [Yα , [Yα , Yγ ]] = [[Yα , Yα ] , Yγ ] = 0, 2

since [Yα , Yα ] = 0. Similarly, one can deal with the case α = γ. Now we have to check that h, Xrα (β) and Yrα (β) generate g (A). Note that if rα (β) = α + β, then 1 1 [Yα , [Xα , Xβ ]] = X−α , Xrα (β) . β (hα ) β (hα ) Similarly, Yβ = β(h1 α ) Y−α , Yrα (β) . Hence every generator Xβ , Yβ , β ∈ Σ, can be expressed in terms of Xrα (β) , Yrα (β) . Xβ =

Remark 3.2. It is useful to write down the generators hrα (β) of the reﬂected base rα (Σ): hrα (β) = hβ if aαβ = aβα = 0, hrα (β) = aαβ hβ + aβα hα

if aαβ = 0 or aβα = 0.

If α is singular, then rα (Σ) generates a subalgebra in g (A) (see Section 6, examples D (2, 1; 0), S (1, 2, ±1)). We see from Lemma 3.1 that, given a base Σ with the Cartan matrix AΣ and a regular isotropic odd root α ∈ Σ, one can construct another base Σ

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with the Cartan matrix AΣ such that g (AΣ ) ∼ = g (AΣ ). Using (4), one can construct an isomorphism g (AΣ ) → g (AΣ ); denote it by the same symbol rα . We say that AΣ is obtained from AΣ and a base Σ is obtained from Σ by an odd reﬂection with respect to α. If Σ = rα (Σ) and Σ = rβ (Σ ) one can deﬁne a composition rβ rα : g (AΣ ) → g (AΣ ). It is clear from the deﬁnition of rα that r−α rα (Σ) = Σ. For any base Π denote by ∆± (Π) the set of positive (negative) roots with respect to the base Π. If α ∈ Π, then −α ∈ rα (Π) ⊂ ∆+ (rα (Π)). Lemma 3.3. Let Σ = rα (Π), where α ∈ Π is an odd isotropic regular root. Then ∆+ (Π) {α} = ∆+ (Σ) {−α}. Proof. Assume γ ∈ ∆+ (Π) {α}. Then γ = β∈Π mβ β for some nonnegative integers mβ ; moreover, mβ > 0 for at least one β = α. Note that rα (β) = β + nβ α for some integer nβ ; hence ⎛ ⎞ mβ rα (β) + ⎝mα − mβ n β ⎠ α γ= β=α

=

β=α

⎛ mβ rα (β) + ⎝

β=α

⎞

mβ nβ − mα ⎠ rα (α) .

β=α

At least one of mβ is positive, β = α, therefore γ ∈ ∆+ (Σ). Hence ∆+ (Π) {α} ⊂ ∆+ (Σ) {−α}, and the lemma follows by symmetry. Definition 3.4. g (A) is called a regular contragredient superalgebra if any A obtained from A by odd reﬂections is regular. Consider a category C, which has an object g (A) for every square matrix A. A C-morphism f : g (A) → g (A ) is by deﬁnition an isomorphism of superalgebras which maps a Cartan subalgebra of g (A) to the Cartan subalgebra of g (A ). A Weyl groupoid C (A) of a contragredient superalgebra g (A) is a connected component of C which contains g (A). Let g (A) be a regular superalgebra. With each base Σ of g (A) we associate a Dynkin graph ΓΣ whose vertices are elements of Σ, and α and β are connected iﬀ aαβ = 0. If one denotes even, odd non-isotropic, and odd isotropic roots by white, black and gray circles, respectively, and one writes aαβ and aβα above and under the edge joining α and β, then one can reconstruct the Cartan matrix of AΣ from ΓΣ . Definition 3.5. Call an even root α ∈ ∆0 principal if there exists a base Π obtained from a standard base Π by odd reﬂections such that α or α/2 belongs to Π . Let B denote the set of all principal roots. For each β ∈ B choose Xβ ∈ gβ , Yβ ∈ g−β , set hβ = [Xβ , Yβ ], and deﬁne the matrix B = bαβ = β (hα ), α, β ∈ B.

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Note that by Lemma 3.3 a positive even root remains positive for any base obtained from Π by odd reﬂections. Hence all principal roots are positive. Lemma 3.6. [Xα , Yβ ] = δαβ hβ for any α, β ∈ B. Proof. It is suﬃcient to check that if α = β, then α − β is not a root. Assume that γ = α − β is a root. Without loss of generality, one may assume that γ is positive. Then α = β + γ. Choose a base Π such that α or α/2 belongs to Π , then α can not be a sum of two positive roots. Generally speaking, B may be inﬁnite. We deﬁne a contragredient Lie algebra with inﬁnite Cartan matrix as an inductive limit of usual contragredient Lie algebras. Let g denote the Lie subalgebra of g¯0 (A) generated by Xβ , Yβ for all principal β. Lemma 3.7. There exist a subalgebra c in the center of g (B) and a homomorphism q : g → g (B)/c which maps Xβ to the corresponding generator of g (B). Im q = [g (B) , g (B)]/c. ¯ β , Y¯β be the generators of ¯ Proof. Let X g (B). Lemma 3.6 implies that there exists a surjective homomorphism s : [¯ g (B) , ¯g (B)] → g deﬁned by ¯ β = Xβ , s Y¯β = Yβ . Since all principal roots are positive, one can s X ﬁnd h in the Cartan subalgebra of g (A) such that β (h) > 0 for all β ∈ B. Hence g has an R-grading such that g0 = h ∩ g , [¯g (B) , ¯g (B)] has the similar grading and s is a homomorphism of R-graded Lie algebras. Then Ker s is a graded ideal, in particular, Ker s = m− ⊕m0 ⊕m+ , where m0 lies in the Cartan subalgebra of ¯ g (B), and m± are the positive (negative) graded components. + 0 Since simple roots are not weights of m , m is central and Y¯β , m+ ⊂ m+ ; ¯ β , m− ⊂ m− . Therefore, m− ⊕ m+ generates the ideal in ¯g (B) likewise X which intersects the Cartan subalgebra trivially. If p : ¯g (B) → g (B) denotes the natural projection, then m− ⊕ m+ ⊂ Ker p. Let c = p m0 , and p : ¯ g (B)/m0 → g (B)/c be the surjection induced by p. Then q : g → g (B)/c given by q (Xβ ) = p s−1 Xβ , q (Yβ ) = p s−1 Yβ is well deﬁned. It is straightforward to check that q satisﬁes the conditions of the lemma. Remark 3.8. The subalgebra g is the best approximation to “the largest almost contragredient Lie subalgebra” of g¯0 we can construct explicitly (compare with Lemma 3.10). In many examples q is an embedding and we suspect that is so in general. It would be nice to prove it. For any base Σ consider a closed convex cone

C + (Σ) = aα α | a α ≥ 0 . α∈Σ

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+ Denote by CΠ the intersection of C + (Σ) for all bases Σ obtained from Π by + odd reﬂections. By Lemma 3.3, CΠ contains all positive even roots.

Remark 3.9. Note that Lemma 3.1 used only the following relation of g (A) which is not a relation in ¯ g (A) : if aαγ = aγα = 0, then [Xα , Xγ ] = 0. Thus one could redo our discussion for any quotient of ¯g (A) in which these relations hold. Lemma 3.10. Assume that there are only ﬁnitely many Σ obtained from Π + by odd reﬂections. Then CΠ is the convex cone generated by principal roots. Proof. We prove the statement by induction on the number of simple roots n. + Note that CΠ is an intersection of simplicial cones, hence a closed convex set. + Let C + (B) be the cone generated by principal roots; obviously C + (B) ⊂ CΠ . + To prove inverse inclusion it suﬃces to show that the boundary of CΠ belongs + to C + (B). If γ is a point on the boundary of CΠ , then there is Σ such that γ lies on the boundary of C + (Σ); in other words, γ belongs to one of the faces of C + (Σ). Hence there exists α ∈ Σ such that γ = β∈Σ{α} mβ β for some nonnegative mβ . Let k be the contragredient subalgebra of g with simple roots Σ = Σ {α} (as proven in [9], the corresponding Xγ , Yγ , hγ generate a subalgebra k which is contragredient; alternatively, one could use Remark 3.9, and work with k directly). Since odd reﬂections of Σ are induced by those of + Σ, γ ∈ CΣ . By inductive assumption γ is a nonnegative linear combination of principal roots of k. Clearly, any principal root of k is a principal root of g. Thus γ ∈ C + (B). Remark 3.11. We suspect that Lemma 3.10 is true without any ﬁniteness assumption. Denote by Q the sublattice of Q0 generated by all principal roots. Lemma 3.12. For any regular contragredient superalgebra g (A) with n simple roots and indecomposable A, the rank of Q is at least n − 1. Proof. Choose a base Π of g (A), let Γ be its graph. We say that a subgraph α1 , . . . , αk is a string if it is connected, the vertices α1 and αk are of valence 1 and odd, and all vertices α2 , . . . , αk−1 are of valence 2 and even. We claim that, for each string α1 , . . . , αk , either 2 (α1 + · · · + αs ) is principal for some s < k, or α1 + · · · + αk is principal. Indeed, note that either α1 is non-isotropic, in which case 2α1 is principal, or α1 is isotropic, in which case rα1 (α2 ) = α1 +α2 is an odd root. Again either α1 + α2 is non-isotropic, in which case 2 (α1 + α2 ) is principal, or α1 + α2 is isotropic, in which case rα1 +α2 (α3 ) = α1 + α2 + α3 is an odd root. Proceeding in this manner, one gets either a principal root 2 (α1 + · · · + αs ) for some s < k, or a principal root α1 + · · · + αk . Since all even simple roots are principal, one can see that for any odd roots α and β ∈ Π joined by a string, at least one of 2α and α + β belongs to Q . Consider all odd roots α such that 2α ∈ / Q . Since Γ is connected, the sum or diﬀerence of any two of them is in Q . Therefore Q generates a subspace of dimension at least n − 1. Hence the rank of Q is at least n − 1.

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4 Kac–Moody superalgebras In this section we, ﬁnally, introduce the main character of our story. Definition 4.1. g (A) is a Kac–Moody Lie superalgebra if any A obtained from A by odd reﬂections is admissible. Lemma 4.2. Let g (A) be a regular Kac–Moody superalgebra. Then g (A) is a direct sum of quasisimple regular Kac–Moody superalgebras and even or odd Heisenberg superalgebras. Proof. Follows from Lemma 2.4. Remark 4.3. Note that the theory of highest-weight representations of Heisenberg (super)algebras can be formulated in a very similar way to what we do in this paper. For “typical” representations, the Weyl character formula (22) holds. For “atypical” representations, the character can be easily described. Therefore, although this paper concentrates on regular quasisimple Kac– Moody superalgebras, one could easily rewrite the theory for all regular Kac– Moody superalgebras. We start with deﬁning the Weyl group. Let g be a quasisimple Kac–Moody superalgebra. Then an even simple root is not isotropic. Therefore, for any principal root α, there exists an sl (2)-triple {Xα ∈ gα , hα ∈ h, Yα ∈ g−α }. Deﬁne an even reﬂection rα : h∗ → h∗ by rα (µ) = µ − µ (hα ) α. Since the sl (2)–subalgebra spanned by {Xα , hα , Yα } acts locally ﬁnitely on g (A), rα is dual to the restriction of the automorphism exp adXα exp ad−Yα exp adXα to h. Therefore rα permutes the roots and maps a base to a base. If β is an odd root such that 2β is principal, then we put rβ = rα . The group W generated by all even reﬂections is called the Weyl group of g (A). The main diﬀerence between even and odd reﬂections is that an even reﬂection depends only on a root and does not depend on a base to which the root belongs. The following straightforward identity is very important: wrα (Π) = rw(α) w (Π)

(5)

for a base Π, a regular isotropic α ∈ Π, and w ∈ W . Note that one can deﬁne exp adXα exp ad−Yα exp adXα for any even α such that adXα and adYα are locally nilpotent. We do not know any example where α is not in the W -orbit of a simple root (in which case the corresponding automorphism of h∗ is, obviously, in W ).

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Lemma 4.4. Let Σ = rα (Π), where α ∈ Π is any root (even or odd). Then ∆+ (Π) {α, 2α} = ∆+ (Σ) {−α, −2α}. Proof. The same as for Lemma 3.3. Corollary 4.5. Let ∆ be ﬁnite, Π and Π be bases. Then Π can be obtained from Π by even and odd reﬂections. Proof. Assume Π = Π . There exists α ∈ Π ∩ ∆− (Π ). Then |∆+ (rα (Π)) ∩ ∆− (Π ) | < |∆+ (Π) ∩ ∆− (Π ) |. Repeating this process, one can get Π such that ∆+ (Π ) ∩ ∆− (Π ) = ∅. But then Π = Π. Recall notations from Deﬁnition 3.5. Lemma 4.6. If g is a quasisimple Kac–Moody Lie algebra, then B is a Cartan matrix of a Kac–Moody Lie algebra. Proof. By quasisimplicity of g, α (hα ) = 0 for any α ∈ B. We normalize hα so that α (hα ) = 2. First, we have to show that α (hβ ) ∈ Z≤0 for any α, β ∈ B, α = β. By deﬁnition of a Kac–Moody superalgebra, the adjoint action of sl2 subalgebra spanned by Xβ ∈ gβ , hβ ∈ h, Yβ ∈ g−β on g is locally ﬁnite, and Xα is a lowest vector, hence α (hβ ) ∈ Z≤0 . Finally, α (hβ ) = 0 implies [Xβ , Xα ] = 0, therefore β (hα ) = 0. Remark 4.7. Recall that Serre’s relations in a Kac–Moody Lie algebra g (B) are the relations 1−bβγ 1−bβγ adXβ Xγ = 0, adYβ Yγ = 0. If B is symmetrizable, then Serre’s relations together with the “contragredient” relations [h, Xβ ] = β (h) Xβ ,

[h, Yβ ] = β (h) Yβ ,

[Xβ , Yγ ] = δβγ hβ ,

[h, h] = 0

deﬁne the Kac–Moody Lie algebra g (B). Since adXβ , adYβ , β ∈ B, are locally nilpotent operators in g, Serre’s relations hold in g . Thus, if B is symmetrizable, the homomorphism q constructed in Lemma 3.7 is injective. Remark 4.8. Later, in Section 9, we will see that for all quasisimple regular Kac–Moody superalgebras the set B of principal roots is ﬁnite. It would be interesting to obtain a proof avoiding the use of the classiﬁcation. Remark 4.9. As the preceding remark shows, the group W is manifestly ﬁnitely generated. Later the fact that we restrict our attention to (a ﬁnite set of) principal roots leads to an eﬀective criterion of integrability of a weight (see Theorem 10.5). On the other hand, the group W is also “suﬃciently large”. Indeed, there are the following indications:

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1. It induces the “even” Weyl group when reduced to the “even contragredient part” g (B) of g (Corollary 4.10); 2. When the character formula for integrable inﬁnite-dimensional highest weight irreducible representations is known for weights in general position, it coincides with the Weyl character formula (Corollary 14.5). Corollary 4.10. Let g be a quasisimple Kac–Moody superalgebra. Then the Weyl group W is isomorphic to the Weyl group of g (B). Deﬁne real and imaginary roots following [2]. A root α is real iﬀ there exists a base Π obtained from Π by even and odd reﬂections such that one of α or α/2 belongs to Π . By deﬁnition, a principal root is real. If α is not real, then we call it imaginary. Lemma 4.11. (a) The set of real roots is W-invariant. (b) If α is real, then gα is one-dimensional. (c) If α is a real odd isotropic root, then mα ∈ ∆ implies m = ±1. (d) If α is a real odd non-isotropic root, then mα ∈ ∆ implies m = ±1, ±2. (e) If α is an even real root, then mα ∈ ∆ implies m = ±1, ±1/2. Proof. W -invariance follows from the fact that if Π is a base, then w (Π) is a base. The other statements easily follow from the fact that α belongs to some base Π and Remark 2.1. + Denote by DΠ the intersection of C + (Σ) for all Σ obtained from Π by even and odd reﬂections. It follows from (5) that + + DΠ . = w CΠ w∈W + . Corollary 4.12. If α is an imaginary root, then α or −α belongs to DΠ + Proof. Assume that α is a positive root, and α ∈ / DΠ . By deﬁnition α ∈ + + C (Π), but since α ∈ / DΠ there exists some base Π obtained from Π by even and odd reﬂection such that α ∈ / C + (Π ). Write Π = r1 . . . rs (Π), and + choose i such that α ∈ C (ri+1 . . . rs (Π)) and α ∈ / C + (ri . . . rs (Π)). Then Lemma 4.4 implies ri = rα , hence α or α/2 ∈ ri+1 . . . rs (Π). Thus α is real. Similarly for α being a negative root.

Corollary 4.13. If ∆ is ﬁnite, then every root is real; therefore g is ﬁnitedimensional. Proof. By Corollary 4.5, every base can be obtained from a standard base Π by even and odd reﬂections. In particular, −Π can be obtained from Π in this + way. Therefore DΠ = {0}. Now the statement follows from Corollary 4.12. Theorem 4.14. Let g (A) and g (A ) be regular quasisimple Kac–Moody superalgebras which belong to the same connected component of C. Then A is obtained from A by a composition of odd reﬂections, permutation of indices, and multiplication by an invertible diagonal matrix. This theorem will be proven in Section 8. (See Remark 8.4.)

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5 Regular Kac–Moody superalgebras with two simple roots It is easy to classify rank 2 regular quasisimple Kac–Moody superalgebras. We assume that a superalgebra g (A) has at least one simple isotropic root; otherwise, by deﬁnition, there are no restrictions on a matrix A (except the admissibility conditions). Theorem 5.1. ([9]) Let g (A) be a regular quasisimple Kac–Moody superalgebra with 2 simple roots such that at least one root is isotropic. Then g is isomorphic to one of the following: 0 1 2 −1 ; with p (1) = 0, or 1. g ∼ sl (1|2), A = = 1 0 1 0 2 −2 with p (1) = 0, or p (1) = 1. 2. g ∼ = osp (3|2), A = 1 0 Proof. If g has a base with two roots, then, without loss of generality, isotropic 01 (since multiplication by a diagonal matrix one may assume that A = 10 does not change the algebra). The reﬂection with respect to α2 gives α + 1 = α1 2 −1 . α2 , α2 = −α2 , h1 = h1 + h2 , h2 = h2 (see Remark 3.2). Hence A = 1 0 Now assume that g has a base with one non-isotropic odd root α1 and one isotropic odd root α2 . One may assume without loss of generality that 2 2b for some negative integer b. Again by Remark 3.2, the reﬂection A= 1 0 1 2b rα2 gives α1 = α1 +α2 , α2 = −α2 , h1 = 1+2b h1 + 1+2b h2 (after normalization), −2b h2 = h2 . Since α2 (h1 ) ∈ Z≤0 , we obtain 1+2b ∈ Z≤0 , which is possible only if b = −1. Finally assume that g has a base with one even root and one isotropic root. In this case the odd reﬂection will move the base to one of the above cases. Hence theorem is proven.

The following hereditary principle is obvious but very important. If A is a Cartan matrix of some regular Kac–Moody superalgebra, then any main minor of A is also a Cartan matrix of some regular Kac–Moody superalgebra. Corollary 5.2. Let A be a Cartan matrix of a quasisimple regular Kac–Moody superalgebra. If α is an even simple root and β is an isotropic root, then aαβ = 0, −1 or −2. If α is an odd non-isotropic root and β is an isotropic root, then aαβ = 0 or −2.

6 Examples of regular quasisimple superalgebras Finite-dimensional superalgebras. Finite-dimensional quasisimple Kac–Moody superalgebras were classiﬁed in [1]. In most cases, they have nondegenerate

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Cartan matrices. However the superalgebra gl (n|n) is quasisimple, but not simple, and its Cartan matrix has corank 1. (To get the simple superalgebra psl (n|n) one should factor the commutator sl (n|n) = [gl (n|n) , gl (n|n)] by the center consisting of scalar matrices.) For example, the superalgebra gl (2|2) has the following Cartan matrices: ⎞ ⎞ ⎛ ⎞ ⎛ ⎛ 0 1 0 0 1 0 2 −1 0 ⎝−1 2 −1⎠ ⎝1 0 −1⎠ , ⎝1 0 −1⎠ , 0 1 0 0 1 0 0 −1 2 Another unusual example in this class is a family D (2, 1; a) depending on the parameter a. If we start with the matrix ⎞ ⎛ 2 −1 0 ⎝ 1 0 a⎠ 0 −1 2 with a = 0, −1, then reﬂection rα2 transforms it to the matrix ⎛ ⎞ 0 1 −1 − a ⎝ 1 0 a ⎠, −1 − a1 1 0 and the odd reﬂections of the latter matrix give the matrices of D (2, 1; −1 − a) and D 2, 1; −1 − a1 (after suitable permutations of indices). Thus, the group S3 generated by a → a1 , and a → −1 − a acts on the space of parameters, and the points on the same orbit of this action correspond to isomorphic superalgebras. Hence one can describe the moduli space of such algebras as CP1 {0, −1, ∞} modulo the above S3 -action. The cases a = 0, −1, ∞ correspond to a non-regular Kac–Moody superalgebra. For instance, if a = 0, then the singular odd reﬂection rα2 maps the generators into an ideal isomorphic to psl (2|2). It is not hard to show that D (2, 1; 0) is isomorphic to the algebra of ∼ all derivations of psl (2|2), and D (2, 1; 0)/ psl (2|2) = sl (2). Another 3-element 1 S3 -orbit a = 1, −2, − 2 corresponds to the algebra osp (4|2). All Cartan matrices of regular ﬁnite-dimensional quasisimple Kac–Moody superalgebras are symmetrizable, hence they have non-degenerate invariant bilinear symmetric forms. Furthermore, ∆ is a ﬁnite set, hence any root is real. Principal roots are the simple roots of g = [g¯0 , g¯0 ]. Aﬃne Kac–Moody superalgebras. Let s be a ﬁnite-dimensional simple Lie superalgebra from the previous class (so s = gl (n|n)), (·, ·) be a nondegenerate invariant symmetric form on s. Deﬁne an inﬁnite-dimensional superalgebra s(1) as s ⊗ C t, t−1 ⊕ CD ⊕ CK, here D, K are even elements and the bracket is deﬁned by X ⊗ tk , Y ⊗ tl = [X, Y ] ⊗ tk+l + kδk,−l (X, Y ) K, [D, K] = 0, D, X ⊗ tk = kX ⊗ tk .

(6)

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It is not diﬃcult to show (as in the Lie algebra case) that s(1) is a regular quasisimple Kac–Moody superalgebra. To construct the set of generators choose some generators X1 , . . . , Xn , Y1 , . . . , Yn of s, and add X0 = x ⊗ t, Y0 = y ⊗ t−1 , where x is a lowest weight vector, and y is a highest weight vector in the adjoint module. Quasisimplicity follows from simplicity of s. Regularity follows from existence of an invariant symmetric form deﬁned by X ⊗ tk , Y ⊗ tl = (X, Y ) δk,−l , (D, K) = 1, (7) k k K, X ⊗ t = D, X ⊗ t = (D, D) = (K, K) = 0. Indeed, existence of the form implies that every Cartan matrix is symmetrizable; therefore, there are no singular roots. Now let us deﬁne the aﬃnization of gl (n|n). Let s = psl (n|n) be the quotient of sl (n|n) by the center, for simplicity of s we need n ≥ 2. Then s(1) is deﬁned as s ⊗ C t, t−1 ⊕ CD ⊕ CK ⊕ CD ⊕ CK , the bracket with additional even elements D and K is given by p(X) D , X ⊗ tk = 1 − (−1) X ⊗ tk , K , X ⊗ tk = 0, X ⊗ tk , Y ⊗ tl = [X, Y ] ⊗ tk+l + kδk,−l (X, Y ) K + δk,−l tr [X, Y ] K ; here tr is the usual trace (not the supertrace), which is not zero only if both (1) X and Y are odd. The Cartan matrix of aﬃne superalgebra psl (n|n) has corank 2, therefore psl (n|n)(1) is a regular quasisimple Kac–Moody superalgebra with two dimensional center. For example, one of Cartan matrices of (1) psl (2|2) is ⎞ ⎛ 0 1 0 −1 ⎜1 0 −1 0⎟ ⎟. ⎜ ⎝0 1 0 −1⎠ 1 0 −1 0 What follows is applicable for any ﬁnite-dimensional simple s. To describe the roots of s(1) , deﬁne δ ∈ h∗ by conditions δ (hi ) = 0, δ (D) = 1. Denote by ∆◦ the roots of s. Then the roots of g are of the form α+kδ with k ∈ Z, α ∈ ∆◦ , or kδ with k ∈ Z {0}. The standard base Π = {α0 , . . . , αn } is obtained from the standard base {α1 , . . . , αn } of s by adding the root α0 = θ + δ, where θ is the lowest root of s. Twisted aﬃne superalgebras. Choose an automorphism φ of s of ﬁnite order p, which preserves the invariant form on s. Let ε be a p-th primitive root of 1. One can extend the action of φ on s(1) by φ X ⊗ tk = εk φ (X) ⊗ tk , φ (D) = D, φ (K) = K. Then sφ is deﬁned as the set of elements ﬁxed by φ. It was proven in [5] that the construction does not depend on ε, and if two automorphisms are

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in the same connected component of Aut s, then the corresponding algebras are isomorphic. Up to isomorphism, there are the following twisted aﬃne (2) (4) superalgebras: (p) sl (m|n) if mn is even, (m, n) = (2, 2); (p) sl (m|n) if (2) mn is odd; osp (2m|2n) ; the upper index denotes the minimal possible order of an automorphism. In the case of twisting of psl (n|n) the corank of a Cartan matrix is 1, and the center is one-dimensional. For example, ⎞ ⎛ 2 −2 0 ⎝−1 0 1⎠ 0 −2 2 with parity functions (0,1,0) or (1,1,1) is a Cartan matrix of psl (3|3)(4) . All twisted aﬃne superalgebras are regular quasisimple Kac–Moody superalgebras. A base in a twisted aﬃne superalgebra can be obtained by the following procedure. Deﬁne a Zp -grading s = s0 ⊕ s1 ⊕ · · · ⊕ sp−1 by sk = s ∈ s | φ (s) = εk s . In all cases φ can be chosen so that s0 is a simple ﬁnite-dimensional superalgebra, s1 is an irreducible s0 -module and s0 + s1 generates s. For example, for (p) sl (m|2n)(2) , s0 = osp (m|2n), s1 is a unique nontrivial irreducible subquotient of the symmetric square of the standard (m|2n)-dimensional s0 -module; (4) for (p) sl (2m + 1|2n + 1) , s0 = osp (2m + 1|2n), and s1 is the standard (2) module with inverted parity; for osp (2m|2n) , s0 = osp (2m − 1|2n), s1 is the standard module. Let ∆0 be the set of roots of s0 and ∆j denote the set of weights of sj with respect to a Cartan subalgebra of s0 . The roots of s(p) are of the form α + kδ, with α ∈ ∆j , k ∈ j + pZ, and kδ with k ∈ pZ {0}. The standard base Π = {α0 , . . . , αn } can be obtained from a base {α1 , . . . , αn } of s by adding the weight α0 = θ + δ, where θ is the lowest weight of s1 . Finally, let us mention that the condition that φ preserves the invariant form on For example, an automorphism π of psl (n|n) such s is nontrivial. DC AB does not preserve the invariant form on psl (n|n), = that π BA CD and psl (n|n)π is not contragredient. It also explains why the Lie superalgebra psl (2|2) does not have a twisted aﬃnization. The subgroup of automorphisms of psl (2|2) preserving the invariant form is connected due to existence of nontrivial derivations, see above example for D (2, 1; 0). A strange twisted aﬃne superalgebra. The Lie superalgebra s = q (n) ⊂ psl (n|n) is the subalgebra of all elements ﬁxed by the automorphism π deﬁned above. The superalgebra q (n) is simple for n ≥ 3. The involution φ such that φ (x) = (−1)p(x) x does not belong to the connected component of unity (1) in Aut q (n). Although the Lie superalgebras q (n) and q (n) are not contragredient, twisting by φ gives a regular quasisimple Kac–Moody superalgebra

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(2)

which we denote by q (n) . As a vector space q (n) is isomorphic to s¯0 ⊗ C t2 , t−2 ⊕ s¯1 ⊗ tC t2 , t−2 ⊕ CK ⊕ CD ∂ with D = t ∂t and K being a central element. For any x, y ∈ s, the commutator is deﬁned by the formula

[x ⊗ tm , y ⊗ tn ] = [x, y] ⊗ tm+n + δm,−n (1 − (−1)m ) tr (xy) K. A Cartan matrix of q (n)(2) has size n × n. Identify the set of indices with the abelian group Zn . Fix a parity function p : Zn → {0, 1} so that the number of odd indices is odd. Set aij = 0 if j = i or i ± 1 (modulo n); set aii = 2, aii = 0,

ai,i±1 = −1 if p (i) = 0; ai,i+1 = −1, ai,i−1 = 1 if p (i) = 1.

The graph of this Cartan matrix is a cycle with odd number of isotropic vertices. Any two such Cartan matrices are related by a chain of odd reﬂections; it is a simple exercise to check that they are not symmetrizable. The corank of a Cartan matrix is 1. (2) Roots of g = q (n) are of the form α + mδ, where α is a root of sl (n) and m ∈ Z or mδ with m ∈ Z {0}. The parity of a root equals the parity of m. Non-symmetrizable superalgebra S (1, 2; b). This superalgebra appears in the list of conformal superalgebras classiﬁed by Kac and van de Leur [7]. By deﬁnition, it is the Kac–Moody superalgebra with Cartan matrix ⎞ ⎛ 0 b 1−b ⎝ −b 0 1 + b⎠ −1 −1 2 and parity (1,1,0). Obviously, S (1, 2; b) ∼ = S (1, 2; −b). If b = 0, then the matrix is a matrix of gl (2|2); if b = −1, 1, the matrix has singular roots. In all other cases the odd reﬂection rα2 transforms the matrix of S (1, 2; b) to the matrix of S (1, 2; 1 + b). Hence an isomorphism S (1, 2; b) ∼ = S (1, 2; 1 + b). 1 The moduli space of S (1, 2; b) can be identiﬁed with CP /G where G is the 11 −1 0 . A Cartan matrix and subgroup of PGL (2, Z) generated by 01 0 1 of S (1, 2; b) has corank 1, hence the algebra has one-dimensional center. The Lie superalgebra S (1, 2; b) is a quasisimple Kac–Moody superalgebra. It is regular if b ∈ / Z. Realization of S (1, 2; b). Consider a supercommutative algebra R = C t, t−1 , ξ1 , ξ2 with an even generator t and two odd generators ξ1 , ξ2 . Denote by W (1, 2) the Lie superalgebra of derivations of R, in other words W (1, 2) is the superspace of all linear maps d : R → R such that d (f g) = d (f ) g + (−1)p(d)p(f ) f d (g) .

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An element d ∈ W (1, 2) can be written as d=f

∂ ∂ ∂ + f1 + f2 ∂t ∂ξ1 ∂ξ2

for some f, f1 , f2 ∈ R. It is easy to see that the subset of all d ∈ W (1, 2) satisfying the condition ∂f1 ∂f ∂f2 p(d) − (−1) bf t−1 + ≡ const, + ∂t ∂ξ1 ∂ξ2 form a subalgebra S¯b of W (1, 2). Let E = ξ1 ∂ξ∂ 1 + ξ2 ∂ξ∂2 . Then adE is a diagonalizable operator in S¯b which deﬁnes the grading ⊕ S¯b ⊕ S¯b ⊕ S¯b , S¯b = S¯b

such that S¯b

2

−1

0

1

2

∂ = 0 for b ∈ / Z, S¯b 2 = Cξ1 ξ2 t−b ∂t for b ∈ Z. Deﬁne Sb = S¯b −1 ⊕ S¯b 0 ⊕ S¯b 1 .

One can see that Sb is an ideal in S¯b when b ∈ Z. The commutator [Sb , Sb ] is simple and has codimension 1 in Sb . It consists of all derivations d ∈ Sb satisfying the condition ∂f1 ∂f ∂f2 p(d) −1 − (−1) = 0. + bf t + ∂t ∂ξ1 ∂ξ2 Set X1 =

∂ ∂ξ1 ,

∂ Y1 = (b + 1) ξ1 ξ2 ∂ξ∂2 + ξ1 t ∂t , ∂ h1 = (b + 1) ξ2 ∂ξ∂ 2 + t ∂t ,

∂ X2 = −bξ1 ξ2 t−1 ∂ξ∂ 1 + ξ2 ∂t ,

X3 = ξ1 ∂ξ∂ 2 ,

Y2 = t ∂ξ∂ 2 ,

Y3 = ξ2 ∂ξ∂1 ,

∂ h2 = bξ1 ∂ξ∂1 + ξ2 ∂ξ∂ 2 + t ∂t , h3 = ξ1 ∂ξ∂ 1 − ξ2 ∂ξ∂ 2 .

Then Xi , Yi , hi , i = 1, 2, 3, generate [Sb , Sb ] if b = 0. They satisfy the relations (1) with Cartan matrix S (1, 2; b). The contragredient Lie superalgebra S (1, 2; b) can be obtained from Sb by a suitable central extension. Divide the ﬁst row of the Cartan matrix S (1, 2; b) by b and the second by −b, let a = 1b . Then the renormalized matrix ⎞ ⎛ 0 1 −1 − a ⎝ 1 0 −1 + a⎠ −1 −1 2 (1)

clearly is a deformation of a Cartan matrix of the aﬃne superalgebra sl (1|2) . (1) Thus, S (1, 2, ∞) ∼ = sl (1|2) . The roots and the root multiplicities of S (1, 2; b) are the same as the roots and the root multiplicities of sl (1|2)(1) .

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The family Q± (l, m, n). This family was discovered by C. Hoyt (see [3]). To deﬁne it let us classify 3 × 3 matrices with zeros on the diagonal and nonzeros anywhere else, which deﬁne regular quasisimple Kac–Moody superalgebras. Using multiplication by a diagonal matrix, any such matrix can be reduced to the form ⎞ ⎛ 0 a 1 ⎝1 0 b ⎠ . c 1 0 The odd reﬂection rα1 transforms it to the matrix (parity= (1, 0, 0)) ⎛ ⎞ 0 a 1 ⎝−1 2 1 + b + a1 ⎠ . 1 −1 1 + a + c 2 The latter matrix must be admissible, therefore a+ 1c , b+ a1 ∈ Z<0 . Application of two other odd reﬂections produces one more condition c + 1b ∈ Z<0 . The matrix B is ⎛ ⎞ 2 1 + b + a1 1 + b + a1 ⎝1 + a + 1 2 1 + a + 1c ⎠ . c 1 1 1+c+ b 1+c+ b 2 If all non-diagonal entries of B are zero, g (A) belongs to the family D (2, 1; a) of ﬁnite-dimensional superalgebras deﬁned above. If all of them equal −1, A is (2) of type q (3) . It was shown in [3] that for any three k, l, m ∈ Z≥1 , klm > 1, there exist two solutions of the system 1+a+

1 = −k, c

1+b+

1 = −l, a

1+c+

1 = −m, b

one with −1 < a, b, c < 0 and one with a, b, c ≤ −1. We denote the corresponding superalgebras Q+ (k, l, m) and Q− (k, l, m), respectively. In conclusion, let us note that all superalgebras listed above are quasisimple and regular except D (2, 1; 0), which is not regular and not quasisimple, and S (1, 2; b) with b ∈ Z, which is quasisimple but not regular.

7 Classiﬁcation results Quasisimple ﬁnite-dimensional contragredient superalgebras were listed by V. Kac in [1]. The reader can ﬁnd there the description of root systems and Cartan matrices. The Kac–Moody superalgebras without isotropic simple roots and of ﬁnite growth were also classiﬁed by V. Kac in [10]. They are all aﬃne or twisted aﬃne superalgebras (automatically symmetrizable). This result was generalized by Van de Leur [8] as follows.

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Theorem 7.1. (Van de Leur) Any ﬁnite-growth contragredient superalgebra with an indecomposable symmetrizable Cartan matrix is isomorphic to a ﬁnitedimensional superalgebra, its aﬃnization, or its twisted aﬃnization. It is not diﬃcult to see that the property of a contragredient superalgebra to have ﬁnite growth does not depend on a choice of a base (Cartan matrix). Thus, by Theorem 2.2, any ﬁnite growth contragredient superalgebra is a Kac–Moody superalgebra; therefore, the hereditary principle (Corollary 5.2) is applicable to the Cartan matrices (the original one, and those obtained by odd reﬂections). That was used in [9] to obtain the following result. Theorem 7.2. (Hoyt, Serganova) A non-symmetrizable quasisimple superal(2) gebra of ﬁnite growth is isomorphic to q (n) or S (1, 2; b). In fact, [9] contains classiﬁcation of all ﬁnite-growth contragredient superalgebras (without assumption of quasisimplicity). We do not formulate it here, since it is rather technical. In [3] C. Hoyt classiﬁed all quasisimple regular Kac–Moody superalgebras. Theorem 7.3. (C. Hoyt) Any quasisimple regular Kac–Moody superalgebra with at least one isotropic simple root is isomorphic to a ﬁnite-dimensional (2) Kac–Moody superalgebra, its aﬃnization or twisted aﬃnization, q (n) , ± S (1, 2; b) with b ∈ / Z, or Q (k, l, m). Corollary 7.4. A quasisimple regular Kac–Moody superalgebra of type I is isomorphic to one of the following list: 1. 2. 3. 4. 5. 6. 7.

sl (m|n) with 1 ≤ m < n; gl (n|n) with n ≥ 2; osp (2|2n) with n ≥ 2; sl (m|n)(1) with 1 ≤ m < n; (1) psl (n|n) with n ≥ 2; (1) osp (2|2n) with n ≥ 2; S (1, 2; b) with b ∈ / Z.

8 Applications of classiﬁcation results In this section we prove some general results about regular quasisimple Kac– Moody superalgebras. We use the classiﬁcation (see Theorem 7.3). It would be very desirable to obtain proofs without use of classiﬁcation; unfortunately we were unable to do so. First, we can “improve” Lemma 3.12. Lemma 8.1. For any quasisimple regular Kac–Moody superalgebra g, the lattice Q has a ﬁnite index in Q0 .

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Proof. As follows from the proof of Lemma 3.12, for any odd roots α and β ∈ Π joined by a string, 2α + 2β ∈ Q . If Π contains an odd root α such that 2α ∈ Q (for example, if α is non-isotropic), then 2β ∈ Q for any root β, hence the rank of Q is n. If Γ contains a cycle with exactly three isotropic vertices α1 , α2 , α3 , then 2αi + 2αj ∈ Q implies 4αi ∈ Q , hence the rank of Q is also n. One can see from the classiﬁcation that the only regular quasisimple Kac– Moody superalgebras which have no base with non-isotropic odd roots are sl (m|n), gl (m|m), osp (2m|2n), D (2, 1; a), F (4), their aﬃnizations, S (1, 2; a), (1) q (m) , or Q± (k, l, m). The superalgebras of type II in this list are osp (2m|2n) (2) with m ≥ 2, D (2, 1; a), F (4), their aﬃnizations, q (m) , or Q± (k, l, m); they all have a graph Γ with a 3-cycle of isotropic vertices. Therefore if g has type II, the rank of Q is n (the same as the rank of Q0 ). Thus [Q0 : Q ] is ﬁnite. If g is of type I the rank of Q is n − 1 (again the same as the Q0 ). Hence again [Q0 : Q ] is ﬁnite. The lemma is proven.

Lemma 8.2. Let g (A) be a regular quasisimple Kac–Moody superalgebra. Then at least one of the conditions below is true: (a) A does not have zeros on the diagonal; (b) g (A) is ﬁnite dimensional; (c) There exists a ﬁnite set S ⊂ h∗ and δ ∈ ∆ such that ∆ ⊂ Zδ + S; (d) g (A) is isomorphic to Q± (k, l, m). Proof. Suppose (a) and (b) do not hold. Then, by Theorem 7.1 and (2) Theorem 7.3, g (A) is (twisted) aﬃne, S (1, 2; b), q (n) , or Q± (k, l, m). In the ﬁrst three cases (c) holds, as follows from the description of roots in Section 6. Theorem 8.3. Let Π and Π be two bases of a quasisimple regular Kac–Moody superalgebra g. Then Π or −Π can be obtained from Π by even and odd reﬂections. Proof. We already proved the statement for ﬁnite-dimensional g in Corollary 4.5. In fact, its proof implies that in general, it suﬃces to prove that either ∆+ (Π) ∩ ∆− (Π ) or ∆+ (Π) ∩ ∆+ (Π ) is ﬁnite. Assume ﬁrst that g satisﬁes (c) from Lemma 8.2. Choose h, h ∈ h such that α (h) = 1 for any α ∈ Π, and β (h ) = 1 for any β ∈ Π . Then γ (h) , γ (h ) = 0 for any root γ ∈ ∆, in particular, δ (h) , δ (h ) = 0. Suppose δ (h) > 0 and δ (h ) > 0. We claim that ∆+ (Π) ∩ ∆− (Π ) is ﬁnite. Indeed, let C be the maximum of {|s (h) |, |s (h ) | | s ∈ S}. Let α ∈ ∆+ (Π) ∩ ∆− (Π ). Since α = mδ + s for some s ∈ S, the conditions α (h) > 0, α (h ) < 0 imply |m| ≤ C. Thus there are only ﬁnitely many α in ∆+ (Π) ∩ ∆− (Π ). If δ (h) > 0, δ (h ) < 0, then ∆+ (Π) ∩ ∆− (Π ) is ﬁnite by a similar argument. Now let g satisfy (a) or (d). Then the matrix B is indecomposable, and the conditions of Lemma 3.10 hold (in the ﬁrst case B is obtained from A by dividing all odd rows by 2 and multiplying all odd columns by 2, in the second case it follows from the direct computation done in Section 6). Moreover,

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191

+ Lemma 3.10 implies that DΠ = w∈W w (C + (B)). It was shown in [2], Lemma 5.8, that the set of rays spanned by positive imaginary roots of g (B) is dense + in DΠ in any metric topology. We choose this metric so that the roots of Π form an orthonormal basis in Q. + + First, we claim that for any base Π , either DΠ , or −DΠ is contained in + C (Π ). Indeed, if this is not true, there exists h ∈ h such that α (h) ≥ 1 for + all α ∈ Π and γ (h) = 0 for some γ ∈ DΠ . Since the set of rays spanned by + positive imaginary roots of g (B) is dense in DΠ , one can choose a sequence γ . But each βi of positive imaginary roots of g (B) such that |ββii | approaches |γ| βi is a root of g, hence βi = α∈Π miα α where all miα are nonnegative (or i (h) i (h) nonpositive). Therefore β|β ≥ 1, or β|β ≤ −1, and we obtain contradiction i| i| with the assumption γ (h) = 0. + Without loss of generality, one may assume that DΠ ⊂ C + (Π ) (if necessary one can use −Π instead of Π ). Let us prove that in this case ∆+ (Π)∩∆− (Π ) is ﬁnite. Assume the opposite. Then one can ﬁnd an inﬁnite sequence of roots α1 , . . . , αn , · · · ∈ ∆+ (Π) ∩ ∆− (Π ). Let γ be a limit point αi of the sequence |α . By Lemma 4.4, after any chain of reﬂections r1 . . . rk ini| + . On the other ﬁnitely many of αi remain in ∆+ (r1 . . . rk (Π)). Hence γ ∈ DΠ + hand, γ ∈ −C (Π ). Contradiction. Remark 8.4. Theorem 8.3 implies Theorem 4.14.

9 Description of g (B) and g in examples Let us recall (see [1]) that if s is a simple classical Lie superalgebra with a nondegenerate even invariant symmetric form, then s 0 is either semisimple, or reductive with one-dimensional center. Every such s is contragredient. Now let us consider the case of (twisted) aﬃne superalgebras. Theorem 9.1. Let s be a simple classical Lie superalgebra with nondegenerate even invariant symmetric form (·, ·), g = s(1) , [s¯0 , s¯0 ] = s1 ⊕ · · · ⊕ sk be the sum of simple ideals s1 , . . . , sk , q : g → g (B)/c be as deﬁned in Lemma 3.7. Then (1) (1) (a) g (B) = s1 ⊕ · · · ⊕ sk ; (b) dim c = k − 1, Ker q = 0; (c) If s 0 is semisimple, then g¯0 = g ⊕ CD. If s¯0 has one-dimensional center Cc, then g¯0 = (g + i) ⊕ CD and g ∩ i = CK, where i is the subalgebra generated by c ⊗ tk . In particular, ∆0 (g) = ∆ (g (B)). (1) (1) (d) Let K1 , . . . , Kk denote the canonical central elements of s1 , . . . , sk , (·, ·)i denote the Killing form on si and (x, y) = bi (x, y)i for x, y ∈ si ; then c = a 1 K 1 + · · · + ak K k | ai b i = 0 . (8) (e) ∆0 (g) = ∆ (g ) = ∆ (g (B)), and imaginary roots are roots of the form mδ, where m ∈ Z {0}.

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Proof. (a) follows from the fact that even roots of g are the same as the roots (1) (1) of s1 ⊕ · · · ⊕ sk . This fact can be also used to prove (c). Indeed, (g¯0 )α = gα for any even root α of the form α◦ + mα with α◦ ∈ ∆0 (s), and (g¯0 )mδ = hs ⊗ tm ,

gmδ = h[s,s] ⊗ tm ,

where hs , h[s,s] are Cartan subalgebras of s and [s, s], respectively. If s¯0 is semisimple, then hs = h[s,s] and gmδ = (g¯0 )mδ . If s¯0 is not semisimple, then (g¯0 )mδ = gmδ ⊕ C (c ⊗ tm ) , and hence g¯0 = g + i. Next we will prove (d). Pick elements x1 ⊗ t, . . . , xk ⊗ t,

y1 ⊗ t−1 , . . . , yk ⊗ t−1

¯ i be such that xi , yi belong to the Cartan subalgebra of si , (xi , yi ) = 1. Let K the images of Ki under the natural projection g (B) → g (B)/c. Then q (K) = q

¯i = 1 K ¯ i. xi ⊗ t, yi ⊗ t−1 = (xi , yi )i K bi

Thus, c is generated by b1i Ki − b1j Kj , that implies (d). To prove (b) note that dim c = k − 1 by (d) and Ker q = 0 since g (B) is symmetizable (see Remark 4.7); this implies ∆ (g ) = ∆ (g (B)). Roots of i coincide with roots of h ⊗ tZ ; this implies ∆0 (g) = ∆ (g ). Since Weyl groups of g (B) and g coincide, even real roots of g coincide with real roots of g (B). Thus the required description of real even roots follows from the theory of aﬃne Lie algebras, see [2]. The absence of odd imaginary roots is not used in this paper; note only that it follows easily from the fact that the Weyl group of g (B) acts transitively on the subset of isotropic and nonisotropic odd roots in the set ∆◦1 + Zδ of odd roots of g. The latter fact can be checked case by case (see [11] for details); therefore, all odd roots are real. Remark 9.2. Observe that if g = s(1) , then k = 1 only for s = sl (1|n), osp (1|2n) or osp (2|2n). Example 9.3. Let s = D (2, 1; a). Then s¯0 ∼ = sl (2) ⊕ sl (2) ⊕ sl (2) ,

g (B) ∼ = sl (2)

(1)

(1)

⊕ sl (2)

(1)

⊕ sl (2)

,

and c is spanned by K2 − aK1 and (1 + a) K1 + K3 , where K1 , K2 and K3 are the standard central elements of the components in g (B) isomorphic to (1) sl (2) . In this case g¯0 = g ⊕ CD. Corollary 9.4. In the notations of Theorem 9.1 let k = 2. Then c is spanned by K1 + uK2 for some positive rational u.

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Proof. Follows from the fact that b1 , b2 are rational and have diﬀerent signs. The latter fact can be found in [1]. Below we list g (B) for all twisted aﬃne superalgebras. g sl (1|2n)(2) , n≥2 sl (2|2n)(2) , n≥2 (2) (p) sl (m|2n) , m ≥ 3, n ≥ 2 (2) sl (2m + 1|2) , m≥1 (4) sl (1|2n + 1) , n≥1 (p) sl (2m + 1|2n + 1)(4) , n ≥ m ≥ 1 (2) osp (2|2n) , n≥1 (2) osp (4|2n) , n≥1 (2) osp (2m|2n) , m ≥ 3, n ≥ 1

B) g (B sl (2n)(2) (1) sl (2) ⊕ sl (2n)(2) (2) (2) sl (m) ⊕ sl (2n) (2) (1) sl (2m + 1) ⊕ sl (2) (2) sl (2n + 1) sl (2m + 1)(2) ⊕ sl (2n + 1)(2) (1) sp (2n) (1) (1) sl (2) ⊕ sp (2n) (2) (1) o (2m) ⊕ sp (2n)

Remark 9.5. From this table one can see that g (B) is either a (twisted) aﬃne Lie algebra, or a direct sum of two (twisted) aﬃne Lie algebras. In the former case c = 0, and in the latter case c is one dimensional, and generated by K1 + uK2 ; here by K1 and K2 we denote the canonical central elements of (twisted) aﬃne superalgebras which appear as direct summands of g (B), and u is an appropriate positive rational number. It is not diﬃcult to see that g¯0 = g ⊕ CD for the twisted aﬃnization of osp (2m|2n), m ≥ 2, and of psl (n|n), n ≥ 3. If g is the twisted aﬃnization of sl (m|n) with m = n, or osp (2|2n), then g¯0 = (g + i) ⊕ CD, i ∩ g = CK; here i is generated by c ⊗ t2k+1 for k ∈ Z if the order of the twisting automorphism is 2, and by c ⊗ t4k+2 for k ∈ Z if the order of the twisting automorphism is 4 (as before, c denotes a central element of s¯0 ). (1) Strange twisted aﬃne superalgebra q (n)(2) . g (B) ∼ = sl (n) , c coincides (1) with the center of sl (n) , the homomorphism q : g → g (B)/c is injective, and g¯0 = g ⊕ CD ⊕ CK. (9)

Non-symmetrizable superalgebra S (1, 2; b), b ∈ / Z. To describe g¯0 , one has to use the realization Sb = S (1, 2; b)/CK; here K denotes a central element of S (1, 2; b). It is a straightforward calculation that Sb = Ct

∂ ⊕ [Sb , Sb ] , ∂t

and [Sb , Sb ]¯0 is a semidirect sum of the subalgebra L spanned by tm

∂ + (m + b) tm−1 E, ∂t

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and the ideal g /CK spanned by tm i,j=1,2 cij ξi ∂ξ∂ j with c11 + c22 = 0. Note −1 that L is isomorphic to the algebra of derivations of C t, t , and g /CK is −1 isomorphic to the loop algebra sl (2)⊗C t, t . Going to the central extension g = S (1, 2; b) of Sb , one obtains g¯0 = L + g ⊕ Ct

∂ , ∂t

g ⊕ Ct

∂ ∼ (1) = sl (2) , ∂t

L ∩ g = CK;

here L is isomorphic to the Virasoro algebra. The family Q± (l, m, n). Due to the lack of an explicit description, we can say only a few things about the roots and the structure of g and g (B) for g = Q± (l, m, n). For example, we would like to calculate the multiplicities of imaginary roots of these algebras; however, we have not yet succeeded. Let α1 , α2 , α3 be the base with three isotropic roots. Then there are three linearly independent principal roots β1 = α2 + α3 , β2 = α1 + α3 , β3 = α1 + α2 . Since [Q : Q ] = 2 and [Q : Q0 ] = 2, one has Q0 = Q . Note that the matrix B ⎞ ⎛ 2 −k −k ⎝ −l 2 −l ⎠ . −m −m 2 has a negative determinant (as easily follows from the condition k, l, m ≥ 1 and klm > 1). Therefore g (B) is a simple Kac–Moody Lie algebra of indeﬁnite type. In this case we do not know if q is injective.

10 Integrable modules and highest weight modules Let g = g (A) be a regular Kac–Moody superalgebra with a standard base Π. A g-module M is called a weight module if h acts semisimply on M ; in other words M has a weight decomposition M= Mµ , Mµ = {m ∈ M | hm = µ (h) m, ∀h ∈ h} , µ∈h∗

and dim Mµ < ∞ for all µ ∈ h∗ . The set P (M ) = {µ ∈ h∗ | Mµ = 0} is called the set of weights of M . The formal character ch M of M is deﬁned by the formula ch M = dim Mµ eµ . µ∈P (M)

A module M is integrable if M is a weight module, and Xβ and Yβ act locally nilpotently on M for every principal root β of g. Note that if α is isotropic, then Xα2 = 0, hence Xα acts locally nilpotently on any module. If

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β is a principal root, then exp Xβ exp (−Yβ ) exp Xβ is a well-deﬁned linear operator on an integrable module M ; Thus the Weyl group W of g acts on M ; therefore ch M is W -invariant. This implies also that Xα and Yα act locally nilpotently on M for any real root α. By deﬁnition of a Kac–Moody superalgebra, its adjoint module is integrable. Our next step is to deﬁne the category O of highest weight modules. Deﬁne a Verma module M (λ) with the highest weight λ as the induced module U (g) ⊗U (b) C (λ), where b = h ⊕ n+ , C (λ) is a one-dimensional bmodule with generator v such that hv = λ (h) v, Xα v = 0 for all simple roots α. A vector v is called a highest vector of M (λ). Any quotient V of M (λ) is an indecomposable module generated by the image of v under the natural projection M (λ) → V . The following lemma can be proven exactly as in the Lie algebra case. Lemma 10.1. Let m (α) = dim gα . (a) ch M (λ) = eλ

+ α∈∆1 + α∈∆0

m(α)

(1+e−α )

(1−e−α )m(α)

.

(b) M (λ) has a unique irreducible quotient which we denote by L (λ). The category O is a full subcategory of the category of g-modules, whose objects are weight modules M such that P (M ) ⊂ si=1 P (M (µi )) for some ﬁnite set {µ1 , . . . , µs } ∈ h∗ . In what follows we will often need to change the base Π to a base Σ by odd reﬂections. So we will write n+ Σ , bΣ , OΣ , MΣ (λ), and LΣ (λ) if we mean the corresponding object for a non-standard base. If the subindex is omitted, then Σ = Π. Lemma 10.2. Let Σ is obtained from Σ by an odd reﬂection rα . (a) MΣ (λ) and LΣ (λ) are objects of OΣ . (b) If λ (hα ) = 0, then MΣ (λ − α) ∼ = MΣ (λ), and LΣ (λ − α) ∼ = LΣ (λ). ∼ (c) Let λ (hα ) = 0, then LΣ (λ) = LΣ (λ). If p = CYα ⊕ b, then the b-module structure on C (λ) and C (λ − α) extends uniquely to a p-module structure, and the following exact sequences hold: 0 → U (g) ⊗U (p) C (λ − α) → MΣ (λ) → U (g) ⊗U(p) C (λ) → 0, 0 → U (g) ⊗U (p) C (λ) → MΣ (λ − α) → U (g) ⊗U (p) C (λ − α) → 0. Proof. The ﬁrst statement of the lemma follows from the identity ch MΣ (λ − α) = ch MΣ (λ) , which is a straightforward consequence of Lemma 10.1 (a). Now prove (b). Let v be a highest vector of MΣ (λ). Slightly abusing notations, we denote the image of v in LΣ (λ) by v. A simple calculation shows that Yα v is n+ Σ -invariant. If λ (hα ) = 0, then Xα Yα v is proportional to v with a nonzero coeﬃcient. Hence

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MΣ (λ) ∼ = U (g) ⊗U (bΣ ) (CYα v) ∼ = MΣ (λ − α) , and LΣ (λ − α) ∼ = LΣ (λ) . To show (c), assume that λ (hα ) = 0. Then Yα v generates a proper submodule in MΣ (λ). Therefore Yα v = 0 in LΣ (λ), thus v is a highest vector with respect to both Σ and Σ . Hence LΣ (λ) ∼ = LΣ (λ). Finally, note that the submodule of MΣ (λ) generated by Yα v is isomorphic to U (g) ⊗U(p) C (λ − α) (indeed, choose a subalgebra m ⊂ n− of codimension one, such that g = m ⊕ p. Then U (n− ) = U (m) ⊕ U (m) Yα , hence U (g) Yα v = U (m) Yα v). That implies the ﬁrst exact sequence. The second follows by symmetry. Corollary 10.3. If Σ is obtained from Π by odd reﬂections, then OΣ = OΠ . Remark 10.4. Corollary 10.3 guarantees that the category O does not change if we change a base by odd reﬂection. In what follows we omit the subindex in the notation for category O. A weight λ is called integrable if L (λ) is an integrable module. One can use Lemma 10.2 to describe integrable weights: it implies that for any Σ obtained from Π by odd reﬂections there is exactly one weight λΣ such that LΠ (λ) ∼ = LΣ (λΣ ). Theorem 10.5. Let g be a regular quasisimple Kac–Moody superalgebra. A weight λ is integrable iﬀ for any principal root β and any Σ obtained from Π by odd reﬂections such that β ∈ Σ, the condition λΣ (hβ ) ∈ Z≥0 holds. If it holds for one Σ such that β ∈ Σ, then it holds for any Σ such that β ∈ Σ. ∼ L (λ); Proof. If λ is integrable, then Yβ acts locally nilpotently on LΣ (λΣ ) = in particular, suﬃciently large power of Yβ annihilates a highest vector vΣ of LΣ (λΣ ). The standard sl (2)-calculation implies that λΣ (hβ ) ∈ Z≥0 . λ (h )+1 On the other hand, if λΣ (hβ ) ∈ Z≥0 , then Yβ Σ β vΣ = 0. Since LΣ (λ) = U (g) vΣ , and adYβ is locally nilpotent on U (g), Yβ acts locally nilpotently on LΣ (λ). If the condition λΣ (hβ ) ∈ Z≥0 holds for one Σ containing β, then Yβ is locally nilpotent; hence the condition holds for any other Σ containing β. Hence, in the absence of odd reﬂections, Corollary 10.6. Let g be a regular Kac–Moody superalgebra without isotropic simple roots. Then λ is integrable iﬀ λ (hα ) ∈ Z≥0 for any even simple root α, and λ (hα ) ∈ 2Z≥0 for any odd simple root α. Indeed, if α is odd, then hα /2 is hβ for β = 2α ∈ B. Since B is ﬁnite (see Remark 4.8), Theorem 10.5 is an explicit test for integrability. Using it, one can recover the description of integrable weights given in [1] for ﬁnite-dimensional g (see appendix in [13]). Let us illustrate the method with examples.

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Example 10.7. Let g = D (2, 1; a); choose the base Π = {α1 , α2 , α3 } such that all simple roots are isotropic. All principle roots are β1 = α2 + α3 , β2 = α1 + α3 and β3 = α1 + α2 . They are linearly independent. Therefore one can parameterize λ by setting λ = (c1 , c2 , c3 ), where c1 = λ (hβ1 ), c2 = λ (hβ2 ) and c3 = λ (hβ3 ). Using Remark 3.2 one gets hβ 1 =

h2 + ah3 , a

hβ 2 = −

h1 + ah3 , a+1

hβ 3 = h1 + h2 .

(10)

Let Σk = rαk (Π), k = 1, 2, 3. If λ (h1 ) = 0, then λΣ1 = λ − α1 = (c1 + 1, c2 − 1, c3 − 1) . Hence c2 , c3 ∈ Z>0 . If λ (h2 ) = 0, then λΣ2 = λ − α2 = (c1 − 1, c2 + 1, c3 − 1) and hence c1 ∈ Z>0 . We do not have to make the third odd reﬂection rα3 since Σ1 = {−α1 , β2 , β3 } and Σ2 = {β1 , −α2 , β3 } already contain all principal roots. By symmetry, if λ (hi ) , λ (hj ) = 0 for some i = j, then c1 , c2 , c3 ∈ Z>0 . On the other hand if λ (hi ) = λ (hj ) = 0 for some i = j, then two odd reﬂections do not change λ and hence c1 , c2 , c3 ∈ Z≥0 . Thus, λ = (c1 , c2 , c3 ) is integrable iﬀ c1 , c2 , c3 ∈ Z≥0 and, in addition, one of the following conditions holds: 1. c1 , c2 , c3 > 0; 2. c1 = (a + 1) c2 + c3 = 0; 3. c2 = −ac1 + c3 = 0; 4. c3 = −ac1 + (a + 1) c2 = 0. Note that for λ = 0, the conditions (2) − (4) imply a ∈ Q. Thus, there are more integrable weights for rational a than for irrational a. Now, do similar calculations for Q± (k, l, m), which also has a base with all isotropic roots. Example 10.8. Let g = Q± (m, k, l). Again Π = {α1 , α2 , α3 }, the principal roots are β1 = α2 +α3 , β2 = α1 +α3 and β3 = α1 +α2 . Since the principal roots are linearly independent, we again can use parameterization λ = (c1 , c2 , c3 ), where c1 = λ (hβ1 ), c2 = λ (hβ2 ) and c3 = λ (hβ3 ). It is easy to check that hβ 3 =

h1 + h2 , a

hβ 1 =

h2 + h3 , b

hβ 2 =

h3 + h1 . c

(11)

We notice that, as in the previous example, integrability implies ci ∈ Z≥0 for i = 1, 2, 3. Moreover, if at least for two hi , hj , λ (hi ) and λ (hj ) = 0, then c1 , c2 , c3 ∈ Z>0 . Assume that λ (h1 ) = λ (h2 ) = 0. Then by (11), c1 = λ (h3 ), c2 c = λ (h3 ). It was shown in [3] that a, b, c ∈ / Q. Hence c1 = c2 = c3 = 0. Similarly, λ (h1 ) = λ (h3 ) = 0 and λ (h3 ) = λ (h2 ) = 0 imply λ = 0. Thus, all integrable weights are (c1 , c2 , c3 ) ∈ Z3>0 and 0.

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The following theorem was proven in [11] for untwisted aﬃne superalgebras. Other cases were done in [3]. (2)

(1)

(2)

Theorem 10.9. Let g = q (n) , (p) sl (m|n) or (p) sl (m|n) with (1) (1) (1) (2) m, n ≥ 2, osp (m|2n) or osp (m|2n) with m ≥ 3, G3 or F4 ; then any integrable module L (λ) is one-dimensional and trivial over [g, g]. (2)

Proof. Let g = q (n)

. As follows from Section 9, (1) g ⊕ CD ∼ = sl (n) /c,

where c is the center of sl (n)(1) . Thus, λ is an integrable highest weight of (1) sl (n) with zero central charge. A nontrivial integrable highest weight module over an aﬃne Lie algebra must have a positive central charge (see [2]). Therefore λ (hβ ) = 0 for all principal β. Recall from (9) that Chβ ⊕ CK ⊕ CD, h ∩ [g, g] = Chβ ⊕ CK. h= β∈B

β∈B

We claim next that λ (K) = 0. Indeed, choose a base Π = {α1 , . . . , αn } such that α1 is odd (automatically isotropic), and αi is even for any i ≥ 2 (and therefore principal); the Dynkin diagram is a loop, so α1 is connected with α2 and αn . Let Σ = rα1 (Π). A direct calculation shows that the subspace of h spanned by hβ (for principal β) has codimension 1 in the space spanned by hα , α ∈ ∆, and K and hα1 are not in this subspace. (This does not contradict the algebra being of type II, since in the absence of duality this does not relate to the geometry of {hβ | β ∈ B} ⊂ h.) If λ (K) = 0, then λ (hα1 ) = 0, therefore λΣ = λ − α1 . Note that then λΣ (hαi ) = 0 for αi connected with α1 in ΓΠ . But λΣ (hβ ) = 0 for any principal β. Contradiction. Thus λ (h ∩ [g, g]) = 0, hence L (λ) is onedimensional. For other superalgebras in the list g (B) is a sum of two or three components. If there are two components, i.e., g = D (2, 1; a)(1) , one can use Corollary 9.4 or Remark 9.5, g ∼ = [g (B) , g (B)]/c, where c is spanned by K1 + uK2 for some positive u. Since L (λ) is integrable over g , the g submodule Lg (λ) ⊂ L (λ) generated by a highest vector is a highest weight g -integrable module. Therefore it is a g (B)-integrable module, and thus K1 and K2 have nonnegative eigenvalues (see [2]). But K1 + uK2 acts by zero on L (λ), therefore both K1 and K2 are zero. That implies λ (hβ ) = 0 for any principal β and, by the same argument as for q (n)(2) , λ (h ∩ [g, g]) = 0, and L (λ) is one-dimensional. (1) In the case D (2, 1; a) we have again g ∼ = [g (B) , g (B)]/c. The submodule Lg (λ) is the g (B)-module Lg(B) (µ); here µ is an integrable weight for g (B) such that µ (c) = 0. The description of c in Example 9.3 implies µ (K2 ) = aµ (K1 ) ;

µ (K3 ) = − (1 + a) µ (K1 ) ;

µ (Ki ) ∈ Z≥0 ;

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therefore µ (Ki ) = 0. Apply the representation theory of aﬃne Lie algebras again; hence Lg(B) (µ) is one-dimensional, and λ (hβ ) = 0 for all principal β. Since hβ generate h ∩ [g, g], as above, L (λ) is one-dimensional. The above theorem, Theorem 7.1, and Theorem 7.3 imply the following corollary. Corollary 10.10. Let g be an inﬁnite-dimensional quasisimple regular Kac–Moody superalgebra, and g has inﬁnite-dimensional integrable highest weight modules. Then g has no simple isotropic roots, or g is isomorphic (1) (1) to sl (1|n) , osp (2|2n) , S (1, 2; b) or Q± (k, l, m). (1)

(1)

We describe the set of integrable weights for sl (1|n) , osp (2|2n) S (1, 2; b) in Section 12 (Q± (m, k, l) is done in Example 10.8).

,

Remark 10.11. Theorem 10.9 shows that most aﬃne superalgebras do not have an interesting integrable highest weight representation. [11] and [12] propose a weaker condition of integrability by requiring integrability over one of the components of g (B) and a ﬁnite-dimensional part of another component. These papers contain a conjecture about character formulae of such representations; these formulae have interesting applications in number theory and combinatorics. Although here we do not discuss partially integrable modules, we will prove the Kac–Wakimoto conjecture for sl (1|n)(1) and osp (2|2n)(1) in Theorem 14.7 (in this case k = 1, hence their condition of integrability coincides with our condition).

11 General properties of category O Throughout this section g = g (A) is a regular quasisimple Kac–Moody superalgebra with the standard base Π. Here we investigate the properties of the category O which we need to calculate characters, and which are valid in the context of such g. Let ρ = ρΠ be any element of h∗ , satisfying the condition ρ (hα ) =

aαα 2

for all α ∈ Π; the choice of ρ is unique unless A is degenerate. When Σ is obtained from Π by odd reﬂections, deﬁne ρΣ = ρ + β. β∈∆+ (Π)∩∆− (Σ)

Lemma 11.1. Let Σ be a base obtained from Π by odd reﬂections. Then ρΣ (hα ) = aαα 2 for any α ∈ Σ.

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Proof. It is suﬃcient to check that if Σ = rβ (Σ) for some isotropic β ∈ Σ, rβ (α) hrβ (α) α(hα ) then ρΣ (hα ) = 2 for any α ∈ Σ implies ρΣ hrβ (α) = . Note 2 that, by Lemma 4.4, ρΣ = ρΣ + α. The statement can be checked by a direct calculation, which is not trivial only if rβ (α) = α + β. Then, by Remark 3.2, hα+β = α (hβ ) hα + β (hα ) hβ , so ρΣ (hα+β ) =

α (hβ ) α (hα ) α + β, hα+β + α (hβ ) β (hα ) = . 2 2

Remark 11.2. It is convenient to reformulate Lemma 10.2 in the following way. Let Σ = rα (Σ) for an odd isotropic α ∈ Σ. If λ + ρ, hα = 0, then λΣ + ρΣ = λΣ + ρΣ .

(12)

λΣ + ρΣ = λΣ + ρΣ + α.

(13)

If λ + ρ, hα = 0, then

A weight λ ∈ h∗ is called typical if λ + ρ, hα = 0 for any real isotropic root α. (12) implies that if λ is typical, then λΣ +ρΣ is invariant with respect to odd reﬂections. Hence the notion of typicality can be deﬁned for an irreducible highest weight module L (λ) independent of a choice of a base, i.e., if λ is typical, then λΣ is typical. If A is symmetrizable, then g admits an invariant symmetric nondegenerate even form (·, ·); this form induces an isomorphism η : h → h∗ such 2α that η (hα ) = (α,α) for every real even root α. If α is a real isotropic root, then η (hα ) is proportional to α. In this case the typicality condition can be rewritten as (λ + ρ, α) = 0 for any real isotropic root α. Note also that in this case (12) and (13) imply that for any Σ obtained from Σ by odd reﬂections (λΣ + ρΣ , λΣ + ρΣ ) = (λΣ + ρΣ , λΣ + ρΣ ) .

(14)

The following statement can be found in [11]; it is a straightforward generalization of the corresponding statement for Lie algebras. Lemma 11.3. Let g be a Kac–Moody superalgebra with an invariant nondegenerate symmetric even form. For each α ∈ ∆+ , let e1α , . . . , esαα be a basis in gα , fα1 , . . . , fαsα be the dual basis in g−α , u1 , . . . , ur and let u1 , . . . , ur be dual bases in h. The operator Ω = 2η −1 (ρ) +

r i=1

ui ui + 2

α∈∆+

(−1)p(α)

sα

fαj ejα

j=1

is well deﬁned on any module from the category O and commutes with the action of g. Ω acts on M (λ) as a scalar operator with eigenvalue (λ + 2ρ, λ).

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201

For any base Σ introduce a partial order on h∗ by putting iﬀ λ−µ= mα α µ ≤Σ λ α∈∆+ (Σ)

for some mα ∈ Z≥0 . For the standard base Π, we omit the subindex in ≤Π . The next three statements below are straightforward generalizations of the similar statements for Kac–Moody Lie algebras (which can be found in [2]). The proofs are omitted. Lemma 11.4. Let g be an arbitrary Kac–Moody superalgebra, M ∈ O and ν ∈ h∗ . There exists a ﬁltration 0 = F 0 (M ) ⊂ F 1 (M ) ⊂ F 2 (M ) ⊂ · · · ⊂ F s (M ) = M i (M ) = L (λ i ) for every and a subset of indices J such thatF i+1 (M )/F j i ∈ J and for every µ ≥ ν, µ ∈ / j ∈J P F (M )/F j−1 (M ) . For every / λ ≥ ν, deﬁne the multiplicity [M : L (λ)] as # {i | λi = λ}. The multiplicity [M : L (λ)] depends neither on a choice of ﬁltration, nor on ν ≤ λ. Hence [M : L (λ)] is well deﬁned for any λ.

Lemma 11.5. (a) [M (λ) : L (λ)] = 1; (b) [M (λ) : L (µ)] > 0 implies λ ≥ µ; (c) If g has an invariant form (·, ·), then [M (λ) : L (µ)] > 0 implies (λ + ρ, λ + ρ) = (µ + ρ, µ + ρ). Corollary 11.6. Let V = 0 be a quotient of M (λ). There is a unique way to write the character of V as an (inﬁnite) linear combination ch V = aµ ch M (µ) . µ≤λ

Furthermore, aλ = 1, and if T = {µ | aµ = 0}, then for any µ ∈ T there are ν1 , . . . , νk ∈ T such that [M (λ) : L (ν1 )] > 0,

[M (ν1 ) : L (ν2 )] > 0, . . . , [M (νk ) : L (µ)] > 0.

If g has an invariant form (·, ·), then all µ ∈ T satisfy the additional condition (λ + ρ, λ + ρ) = (µ + ρ, µ + ρ). Lemma 11.7. Let Σ be obtained from Π by odd reﬂections. Then [M (λ) : L (µ)] = [MΣ (λ + ρ − ρΣ ) : L (µ)] for any µ. Proof. Suﬃcient to check that [MΣ (λ) : L (µ)] = [MΣ (λ + ρΣ − ρΣ ) : L (µ)] for any µ if Σ = rα (Σ). In this case it follows directly from Lemma 10.2 (b) and (c).

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Corollary 11.8. Let V be a subquotient of M (λ). For any weight µ of V and any Σ obtained from Π by odd reﬂections, µ + ρΣ ≤Σ λ + ρ. Proof. If µ is a weight of M (λ), then µ is a weight of MΣ (λ + ρ − ρΣ ). Corollary 11.9. If [M (λ) : L (µ)] > 0, then µΣ + ρΣ ≤Σ λ + ρ for any Σ obtained from Π by odd reﬂections. Proof. If [M (λ) : L (µ)] > 0, then [MΣ (λ + ρ − ρΣ ) : L (µ)] > 0. But L (µ) = LΣ (µΣ ). Hence µΣ is a weight of MΣ (λ + ρ − ρΣ ). Now the statement follows from the previous corollary.

12 Lie superalgebras sl (1|n)(1) , osp (2|2n)(1) , and S (1, 2; b) (1)

(1)

In this section g is sl (1|n) , osp (2|2n) , or S (1, 2; b) (b ∈ / Z). The importance of these particular Lie superalgebras for our discussion is that that they are the only ﬁnite growth regular quasisimple Kac–Moody superalgebras with isotropic roots which have nontrivial integrable modules, and it is easy to describe all bases obtained by odd reﬂections. (1) If g = sl (1|n) or S (1, 2; b), a graph of any base is a cycle of length n + 1 with exactly two isotropic roots (which are neighbors); here n = 2 for the case of S (1, 2; b). If g = osp (2|2n), then we will see that all possible Dynkin graphs are ◦ ⇒ ◦ − · · · − ◦ − ⊗ − ⊗ − ◦ − · · · − ◦ ⇐ ◦,

⊗

\

| / ◦ − · · · − ◦ ⇐ ◦.

⊗

Here we write ⊗ instead of gray nodes. For some of our calculation we need an explicit description of roots of g. One can choose linearly independent ε, ε1 , . . . , εn , δ ∈ h∗ so that the even (1) roots of sl (1|n) are εi − εj + mδ, m ∈ Z, i = j, and mδ, m ∈ Z {0}; odd (1) (1) roots of sl (1|n) are ± (ε + εi ) + mδ, m ∈ Z. If g = osp (2|2n) , then the even roots of g are ±εi ± εj + mδ, ±2εi + mδ, m ∈ Z, and mδ, m ∈ Z {0}; odd roots of g are ±ε ± εi + mδ, m ∈ Z. Finally, the roots of S (2, 1; b) are the (1) same as the roots of sl (1|2) , as explained in Section 6. (1) For the standard base in case of sl (1|n) (or S (1, 2; b)) choose Π = {ε1 − ε2 , . . . , εn−1 − εn , εn + ε, −ε − ε1 + δ} ; for osp (2|2n)(1) choose Π = {−2ε1 , ε1 − ε2 , . . . , εn−1 − εn , εn + ε, εn − ε + δ} . Since g is of type I, it has a grading g−1 ⊕ g0 ⊕ g1 ; it is

Kac–Moody Superalgebras and Integrability

∆ (g1 ) = {ε + εi + mδ}, ∆ (g1 ) = {ε ± εi + mδ},

(1)

if g = sl (1|n)

203

or S (1, 2; b) , (1)

if g = osp (2|2n)

,

and ∆ (g−1 ) = −∆ (g1 ). This grading induces a Z-grading on the root lattice Qi . (15) Q= i∈Z

Set Q+ =

Qi ,

i>0

Q− =

Qi .

(16)

i<0

This grading induces the decomposition ± ± ∆± 1 = ∆ (g1 ) ∪ ∆ (g−1 ) .

The standard base Π has exactly two isotropic roots, one in ∆+ (g1 ), and one in ∆+ (g−1 ). The corresponding vertices in ΓΠ are connected, and each of them is connected with one even vertex (the latter might be the same for both isotropic vertices). These properties also hold for any base Σ; indeed, the set of diagrams described in the beginning of the section is invariant w.r.t. any odd reﬂection, and contains a diagram of Π. In particular, every base Σ has two isotropic roots; moreover |Σ ∩ ∆ (g−1 ) | = |Σ ∩ ∆ (g1 ) | = 1.

(17)

Let γ1 ∈ Σ ∩ ∆ (g−1 ) ,

α1 ∈ Σ ∩ ∆ (g1 ) .

The set B of principal roots coincides with the set of even simple roots of Σ taken together with α1 + γ1 ; indeed, the latter set is preserved by both odd reﬂections. In particular, B is linearly independent. Deﬁne an inﬁnite sequence {αk } ∈ ∆ (g1 ) by α2 ∈ rα1 (Σ) ∩ ∆ (g1 ) , . . . , αk+1 ∈ rαk . . . rα1 (Σ) ∩ ∆ (g1 ) , . . . . Similarly, deﬁne {γk } ∈ ∆ (g−1 ) by γ2 ∈ rγ1 (Σ) ∩ ∆ (g−1 ) , . . . , γk+1 ∈ rγk . . . rγ1 (Σ) ∩ ∆ (g−1 ) , . . . . (1)

(1)

Let N = n if g = sl (1|n) , N = 2n for g = osp (2|2n) . Using the explicit description of roots, one can see that αk+N − αk = γk+N − γk = δ; moreover, γk = −αN +1−k + δ if k ≤ N . If g = sl (1|n)(1) , and Σ = Π, α1 = εn + ε, (1)

If g = osp (2|2n)

α2 = εn−1 + ε, . . . , αn = ε1 + ε;

, Σ = Π, then

α1 = εn + ε, . . . ,αn = ε1 + ε, αn+1 = −ε1 + ε, . . . , α2n = −εn + ε.

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Unless stated otherwise αi and γi are for Σ = Π. It follows from (17) that every base Σ which can be obtained from Π by odd reﬂection is either rαk . . . rα1 (Π), or rγk . . . rγ1 (Π) for some k > 0. Moreover, every odd root is real and isotropic, and ∆+ (g1 ) = {α1 , . . . , αk , . . . } ,

∆+ (g−1 ) = {γ1 , . . . , γk , . . . } .

Imaginary even roots are (by Theorem 9.1) {mδ | m ∈ Z {0}}. Lemma 12.1. Let α1 and γ1 be isotropic roots of Σ, and γ ∈ Qi ∩ C + (Σ). Then γ = iα1 + mβ β, if i ≥ 0, γ = −iγ1 + mβ β, if i ≤ 0, β∈B

β∈B

for some mβ ∈ Z≥0 ; the representation is unique. Proof. Uniqueness is obvious. Express γ as a linear combination of simple roots γ = n1 γ1 + n2 α1 + mβ β. β∈B∩Σ

Note that i = n2 − n1 . If i ≥ 0, one can write it as mβ β. γ = iα1 + n1 (γ1 + α1 ) + β∈B∩Σ

If i < 0, one can write it as

γ = −iγ1 + n2 (γ1 + α1 ) +

mβ β.

β∈B∩Σ

Since γ1 + α1 ∈ B, this ﬁnishes the proof. Lemma 12.2. Let k ≥ 0. Then C + (rαs . . . rα1 (Π)) = ∅, Q+ ∩

Q− ∩

s≥k

C + (rγs . . . rγ1 (Π)) = ∅.

s≥k

Proof. Let γ ∈ Q+ ∩ Σ = rαs . . . rα1 (Π))

s≥k

C + (rαs . . . rα1 (Π)). By Lemma 12.1 (applied to γ = iαs +

msβ β,

β∈B

for every s ≥ k. Let βs = αs+1 − αs ; note that βs ∈ B. Then ms+1 = msβ β

if β = βs ,

ms+1 = msβ − i β

if β = βs .

Since msβ ≥ 0 for all β ∈ B and all s ≥ k, i ≤ 0. Contradiction. The proof of the second statement is similar.

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Lemma 12.3. Let [M (λ) : L (µ)] > 0. Then there exists a base Σ obtained from Π by odd reﬂections, and numbers mβ ∈ Z≥0 such that λ + ρ − µΣ − ρΣ = mβ β. β∈B

Proof. Let γΣ = λ + ρ − µΣ − ρΣ . By Corollary 11.9, 0 ≤Σ γΣ for any Σ obtained from Π by odd reﬂections. Let Σ = rα (Σ) for some odd α ∈ Σ. Then by (12) and (13) γΣ = γΣ

if µΣ + ρΣ , hα = 0,

γΣ = γΣ − α if µΣ + ρΣ , hα = 0. (18) Suppose that γΣ1 ∈ Q+ for some Σ1 , and γΣ2 ∈ Q− for some Σ2 . Since Σ1 and Σ2 are connected by odd reﬂections, and every odd root α is in Q±1 , there exists Σ such that γΣ ∈ Q0 . In this case the statement is true by Lemma 12.1. Now assume that γΣ ∈ Q+ for all Σ. Then one can ﬁnd i > 0 and k > 0 such that γrαs ...rα1 (Π) ∈ Qi for all s ≥ k. Hence, by (18), γrαs ...rα1 (Π) = γrα

s

...rα1 (Π)

=γ

for all s, s > k. Thus γ ∈ Q+ ∩ s≥k C + (rαs . . . rα1 (Π)), which is impossible by Lemma 12.2. Similarly, the case when γΣ ∈ Q− for all Σ is impossible. Lemma 12.4. Let λ ∈ h∗ . LΣ (λ) is integrable iﬀ λ + ρΣ , hβ ∈ Z>0 for any β ∈ B ∩ Σ, and one of the following two conditions holds: 1. λ + ρΣ , hα1 +γ1 ∈ Z>0 , 2. λ + ρΣ , hα1 = λ + ρΣ , hγ1 = 0. Proof. There is only one β ∈ B Π. To check the conditions of Theorem 10.5 make an odd reﬂection rα1 . The details are left to the reader. Remark 12.5. Call µ ∈ h∗ regular if µ, hβ = 0 for any even real root β. As follows from the lemma above, if LΣ (λ) is integrable, then λ + ρΣ is regular iﬀ the ﬁrst condition holds. Lemma 12.6. Let LΣ (λ) be integrable, λ + ρΣ be not regular, and λ (hβ ) = 0 at least for one β ∈ B. Then one can ﬁnd a base Σ obtained from Σ by odd reﬂections such that LΣ (λ) ∼ = LΣ (λ), and λ + ρΣ is regular. Moreover, one can choose this base Σ equal to rγk . . . rγ1 (Σ), and ﬁnd w ∈ W such that λ + ρΣ = λ + ρΣ + γ1 + · · · + γk = w (λ + ρ − kα1 ) , w (α1 ) = −γk . The choice of k is unique.

w

k

(−1) = (−1) ,

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Proof. If λ + ρΣ is not regular, then λ + ρΣ , hα1 = λ + ρΣ , hγ1 = 0. Let Σk = rγk . . . rγ1 (Σ). We claim that LΣk (λ) is not isomorphic to LΣ (λ) at least for one k. Indeed, otherwise (λ, γi ) = 0 for all i, but that would imply λ, hβ = 0 for all β ∈ B. Choose the last k such that LΣk (λ) and LΣ (λ) are isomorphic. Then λ + ρΣk = λ + ρΣ + γ1 + · · · + γk , λ + ρΣk , hγk+1 = 0. By Remark 12.5 applied to Σ = Σk , λ + ρΣk must be regular. Next consider even roots β1 = α1 + γ1 ,

β2 = γ2 − γ1 ,

. . . , βk = γk − γk−1 .

Note that for any i < k, βi+1 ∈ B Σi , and that λ + ρΣ + γ1 + · · · + γi , hβi+1 = 0. Furthermore, rβi+1 (γi+1 ) = γi (recall γ0 = −α1 ). Therefore λ + ρΣ + γ1 + · · · + γk = rβk (λ + ρΣ + γ1 + · · · + γk−1 + γk−1 ) = rβk (λ + ρΣ + γ1 + · · · + 2γk−1 ) = rβk−1 rβk (λ + ρΣ + γ1 + · · · + 3γk−2 ) = · · · = rβ1 . . . rβk (λ + ρΣ + kγ0 ) . Remark 12.7. Recall that g ⊕ CD is isomorphic to g (B) which is sl (n) for (1) (1) (1) (1) g = sl (1|n) , sl (2) for g = S (1, 2; b), and sp (2n) for g = osp (2|2n) . Principal roots B form a base of g . Let (1)

C = {µ ∈ h∗ | µ (hβ ) ∈ Z≥0 , β ∈ B} . It follows from Lemma 12.4, that if LΣ (λ) is integrable, then λ ∈ C. We need two followingfacts about aﬃne Weyl group action, see [2]. If λ ∈ C, then w (λ) = λ − β∈B mβ β for some mβ ∈ Z≥0 . If µ (hβ ) ∈ Z for all β ∈ B and µ (K) > 0, then W µ intersects C in exactly one point. Consider the ring R of all formal expressions µ∈P cµ eµ for all P satisfying the condition that there is a ﬁnite set L ⊂ h∗ such that P ⊂ L − Z≥0 α. (19) α∈Π

It is not diﬃcult to check that indeed such R is a commutative ring without zero divisors. Let R be a subring satisfying the additional condition that (19) holds for w (P ) for any w ∈ W . Then R enjoys the natural action of W ; this action preserves multiplication and addition. For an arbitrary µ ∈ h∗ and Σ obtained from Π by odd reﬂections, deﬁne m(α) e−ρΣ α∈∆+ (Σ) (1 + e−α ) w 1 UΣ (µ) = (−1) ew(µ) . m(α) −α ) w∈W α∈∆+ (Σ) (1 − e 0

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It is an immediate calculation that for any Σ obtained from Σ by odd reﬂections, and for any w ∈ W w

UΣ (µ) = UΣ (µ) ,

UΣ (w (µ)) = (−1) UΣ (µ) .

(20)

Hence one can drop the index Σ in UΣ (µ). Assume now that µ ∈ C. Then w (µ) ≤ µ and hence UΣ (µ) ∈ R. It easily follows from Lemma 10.1 (a) that w U (µ) = (−1) ch M (w (µ) − ρ) . (21) w∈W

If eν appears in ch M (κ) with nonzero coeﬃcient, then ν ≤ κ. Therefore, if eν appears in U (µ) with nonzero coeﬃcient, then ν ≤ w (µ) − ρ ≤ µ − ρ. Lemma 12.8. If µ is regular and integrable, then U (µ) ∈ R . Moreover, w (U (µ)) = U (µ) for any w ∈ W . Proof. Let ρ0 ∈ h∗ be such that ρ0 (hβ ) = Introduce the expressions

D0 = eρ0

1 − e−α

m(α)

aββ 2

for all β ∈ B, and let ρ1 = ρ0 −ρ.

D1 = eρ1

,

α∈∆+ 0

1 + e−α

m(α)

.

α∈∆+ 1

Then U (µ) =

D1 (−1)w ew(µ) . D0 w∈W

Note that ∆ (g ) = ∆0 , and the multiplicity of kδ in g is m (kδ) − 1 (see Theorem 9.1 (c), the case of s with one-dimensional center) D0 = D0

∞

1 − e−nδ ,

n=1

where D0 the corresponding expression for the Lie subalgebra g + h. The expression w (−1) ew(µ) S (µ) = w∈W D0 gives a character of the simple integrable module over Kac–Moody Lie algebra g +h with highest weight µ−ρ0 (see [2]). (Note that regularity and integrability of µ implies µ − ρ0 ∈ C.)Therefore S (µ) is W -invariant, and S (µ) ∈ R . ∞ On the other hand, n=1 1/ 1 − e−nδ is W -invariant, since W δ = δ. Therefore, it is suﬃcient to show that D1 is W -invariant. One has to check that rβ (D1 ) = D1 for all β ∈ B; assume ﬁrst that β ∈ Π; then ρ1 (hβ ) = 0, so rβ (ρ1 ) = ρ1 ; moreover, by Lemma 4.4, rβ permutes roots of ∆+ 1 . Hence rβ (D1 ) = D1 . If β ∈ / Π, then

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β = γ1 +α1 ,

ρ (hβ ) = 0,

ρ1 (hβ ) = 1,

rβ (ρ1 ) = ρ1 −β = ρ1 −α1 −γ1 .

Furthermore, since β = α1 + γ1 , rβ (α1 ) = −γ1 ,

rβ (γ1 ) = −α1 ,

therefore rβ rγ1 = r−α1 rβ by (5), which implies that rβ permutes the roots of ∆+ 1 {α1 , γ1 }. Hence rβ eρ1 1 + e−α1 1 + e−γ1 = eρ1 −α1 −γ1 (1 + eα1 ) (1 + eγ1 ) = eρ1 1 + e−α1 1 + e−γ1 , m(α) m(α) rβ 1 + e−α 1 + e−α = . α∈∆+ 1 {α1 ,γ1 }

α∈∆+ 1 {α1 ,γ1 }

This ﬁnishes the proof.

Remark 12.9. It is useful to note that D1 is not only W -invariant, but also independent of a choice of Σ. In other words, D1 = eρ0 −ρΣ 1 + e−α α∈∆+ 1 (Σ)

for any Σ obtained from Π by odd reﬂections.

13 Lie superalgebras sl (1|n)(1) , osp (2|2n)(1) (1)

From now on we assume that g is sl (1|n)

(1)

or osp (2|2n)

.

Lemma 13.1. There exists an invariant symmetric even form on g such that the corresponding form on Q × Q is semi-positive, takes integer values, and (α, α) > 0 for any even real root α. Proof. Just use the form in (7). The positivity conditions follow from those on sl (1|n) and osp (2|2n). Remark 13.2. One can normalize an invariant form on g so that the corresponding form on h∗ satisﬁes the relations (ε, ε) = −1,

(εi , εj ) = δij ,

(ε, εi ) = (ε, δ) = (εi , δ) = 0.

One can see that (δ, Q) = 0. On the other hand, (λ, δ) = λ (K) for any (1) λ ∈ h∗ . In the case of sl (1|n) , all real even roots have the same length. (1) If g = osp (2|2n) with n ≥ 2, then (β, β) = 2 for a short real even root, 2β and (β, β) = 4 for a long real even root. Since η (hβ ) = (β,β) , given an integrable weight λ, we have (λ + ρ, β) ∈ Z for any real even short root β, and (λ + ρ, β) ∈ 2Z for any real even long root β. Moreover, (ρ, δ) = N − 1 for sl (1|n)(1) , (ρ, δ) = N for osp (2|2n)(1) .

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For the convenience of the reader, let us recall the properties of weights we use in what follows (we reformulate them here using the invariant form). λ is regular if (λ, β) = 0 for any even real β; λ is typical if (λ + ρ, β) = 0 for any real isotropic β; λ is integrable if L (λ) is an integrable module. Lemma 13.3. Let µ ∈ h∗ be regular and (µ, δ) = 0. Then there is at most one α ∈ ∆+ 1 such that (µ, α) = 0. Proof. Let α, γ ∈ ∆+ 1 and α = γ. By a direct inspection of the list of roots, one can check that either α+γ, or α−γ is an even root β. So if (µ, α) = (µ, γ) = 0, then (µ, β) = 0. Contradiction, since every β is either real or mδ. Lemma 13.4. Let Π be now an arbitrary base, let ν ∈ h∗ be such that ν (hβ ) ∈ Z≥0 for all β ∈ B, Σ be obtained from Π by odd reﬂections. Then νΣ + ρΣ , hβ ∈ Z≥0 for any β ∈ B ∩ Π, and νΣ + ρΣ , hβ ∈ Z≥−1 for β ∈ B Π. Proof. First, observe that Yβ is locally nilpotent on L (ν) for any β ∈ B ∩ Π. Since LΣ (νΣ ) ∼ = LΠ (ν), then νΣ , hβ ∈ Z≥0 for any β ∈ B ∩ Π. Note also that ρΣ (hβ ) = 1 if β ∈ B ∩ Σ, and ρΣ (hβ ) = 0 if β ∈ B Σ. Hence it suﬃces to consider the case β = α1 + γ1 ∈ B Π. Let α0 = −γ1 , Π0 = Π, Πk = rαk . . . rα1 (Π). We will prove the statement k for Σ = Πk (for Σ = rγk . . . rγ1 (Π) the proof is similar). Let λ = νΠk + ρΠk , λki = λk , αi . Let k be such that λk−1 (hβ ) ≥ 0, λk (hβ ) < 0; let s > k be the minimal such that αs+1 − αs = αk+1 − αk (it exists since αt+1 − αt is N -periodic). Note that β = α1 − α0 . Lemma 13.5. One has λkk = 0, αk+1 − αk = β, λk = λk+1 , λk (hβ ) = λk+1 (hβ ) = · · · = λs−1 (hβ ) = −1. Proof. By inspection of the formulae for roots, and by (12) and (13), αi (hβ ) = −1 if β = αi+1 − αi ,

αi (hβ ) = 1 if β = αi − αi−1 ,

αi (hβ ) = 0 otherwise; λt = λt−1

if λt−1 = 0, t

λt = λt−1 + αt

if λt−1 = 0. t

Therefore, λkk = 0; αk+1 −αk = β, since αk (hβ ) < 0 (by λk (hβ ) < λk−1 (hβ )); this implies λkk+1 < 0, thus λk+1 = λk . Moreover, λk (hβ ) = λk+1 (hβ ) = · · · = λs−1 (hβ ) = −1, because αt (hβ ) = 0, k + 2 ≤ t < s, and αk (hβ ) = −1. Since αi+N = αi + δ, one has λki+N = λki + M , here M = (ν + ρ, δ). Moreover, (ν, δ) ≥ 0 (since δ is a positive root of g (B)), thus M ≥ (ρ, δ); (1) (1) therefore M ≥ N − 1 if g = sl (1|n) , M ≥ N if g = osp (2|2n) .

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Lemma 13.6. One has λks − λkk ≥ s − k − 2 + (β, β)/2. For k + 2 ≤ i ≤ s − 1, one has (αi , αs ) = −1. Proof. We prove it case-by-case. Let g = sl (1|n)(1) ; hence N = n. In this case αi+1 − αi = αj+1 − αj iﬀ i ≡ j mod N . Therefore, s = k + N . It is enough to note that (αi , αj ) = −1 if i ≡ j mod N , λks − λkk = M ≥ s − k − 1, and (β, β) = 2. (1) Let g = osp (2|2n) ; hence N = 2n. Then αi+1 − αi = αj+1 − αj iﬀ i ≡ j mod N or i + j ≡ 2p mod N ; here p is the smallest 0 ≤ p < n such that αp+1 − αp is a long root. Moreover, (αi , αj ) = −1 if i ≡ j, i + j ≡ 2p + 1 mod N . If p = 0, then (β, β) = 4, n | k, s = k + N , and M ≥ s − k imply what is needed. Now assume p > 0; then k ≡ 0 mod N,

s ≡ 2p mod N,

or k ≡ 2p

mod N,

s≡0

mod N.

Since (β, β) = 2, what remains to prove is λks − λkk ≥ s − k − 1. Note that for t between k and s, the only long root among αt+1 − αt is one with t = def r = (k + s)/2. We already know that λkk+1 − λkk = − (β, β)/2 = −1, and λkt+1 − λkt ≥ 1 for t ≡ k, s mod N . Note that λkr+1 − λkr ≥ 2; indeed, β = αr+1 − αr is a long root, β = β, thus ρΠr (hβ ) = 1, and (ρΠr , β ) = 2. Hence λks − λkk ≥ s − k − 1 indeed. Since λs = λk + αi1 + · · · + αij for some k + 1 < i1 < i2< · · · < ij ≤ s, and s k (αs , αs ) = 0, at most s−k −2 terms αit can ; hence contribute to λ − λ , α s λss −λks ≥ − (s − k − 2). Therefore λss = λss − λks + λks − λkk ≥ (β, β)/2 > 0. Either λs−1 = λs , or λs−1 = λs − αs ; since (αs , αs ) = 0, one can conclude s−1 λs > 0. Therefore λs = λs−1 , hence λs (hβ ) = −1. Lemma 13.7. If λss ≥ 1 + (β, β)/2, then λt = λs for any t > s. Proof. By induction in t, it is enough to show that λst ≥ 1 for any t > s. Since λki+N = λki + M > λki , it is enough to consider t < s + N . Since λt (hβ ) ≥ 0 for β = β, one has λtj+1 − λtj ≥ 0 unless αj+1 − αj = αk+1 − αk ; as shown at the beginning of the proof, the inequalities are strict for t = k, s. Hence, λsj may decrease only after j such that αj+1 − αj = β, and the decrease is (β, β)/2. (1) By inspection, unless g = osp (2|2n) and β is a short root, there are at most one such value of j per an interval of length N . Otherwise, there are two such values, they are not adjacent, and the decrease in each such value is by (β, β)/2 = 1. Now strict increase at other values implies λst ≥ 1. Therefore, if λss ≥ 1 + (β, β)/2, then λt , hβ = −1 for any t ≥ k. On the other hand, if λss = (β, β)/2, then λss+1 = λss +(λs , β) = (1 + λs (hβ )) (β, β)/2 = 0; hence λs+1 = λs + αs+1 . Since αs+1 (hβ ) = 1, this implies λs+1 (hβ ) = 0.

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In any case, this implies that if λt (hβ ) < 0 for t0 ≤ t ≤ t1 , then λt (hβ ) = −1 (indeed, taking t0 the minimal possible with the given t1 , we may assume t0 = k). This ﬁnishes the proof of Lemma 13.4. Lemma 13.8. Let λ, µ ∈ h∗ , λ (hβ ) > 0, µ (hβ ) ≥ 0 for any β ∈ B, and λ − µ = β∈B mβ β for some mβ ∈ Z≥0 . If (λ, λ) = (µ, µ), then λ = µ. Proof. We use the fact that all principal roots have positive square (see Remark 13.2). ⎞ ⎛ mβ β ⎠ = 0. (λ, λ) − (µ, µ) = (λ + µ, λ − µ) = ⎝λ + µ, β∈B

But (λ + µ, β) =

(β,β) 2

λ + µ, hβ is positive. Hence all mβ = 0.

14 On aﬃne character formulae Let g be a regular Kac–Moody superalgebra with a ﬁxed base Π and Cartan matrix A; let λ be an integrable weight. For a symmetrizable matrix A without zeros on diagonal, the character is described by the famous Weyl character formula ch L (λ) = U (λ + ρ) , (22) which was proven by Kac, see [10]. The proof is a straightforward generalization of his proof in Lie algebra case (see, for example, [2]); it is based on existence of the Casimir element corresponding to the invariant form. If g is a ﬁnite-dimensional Kac–Moody superalgebra, (22) holds for a typical weight. It was also proven by Kac [6], but the proof is more complicated. One has to use either the Shapovalov form, or a complete description of the center of the universal enveloping algebra U (g). The reason why a simpler proof from [2] does not work is the existence of real roots of nonpositive square of length. In this section we provide a generalization of these results to some inﬁnitedimensional algebras, as well as to the case of atypical weights: we calculate the characters of all simple integrable highest weight modules over sl (1|2n)(1) , and over osp (2|2n)(1) . We use the invariant form, odd reﬂections and the fact (1) (1) that the defect of sl (1|2n) and osp (2|2n) is 1. Recall that the defect is the maximal number of linearly independent pairwise orthogonal real isotropic roots; defect is 1 iﬀ for any real isotropic α and γ one of α ± γ is a root (compare 13.3). Our proof is an adaptation of the Bernstein–Leites proof of character formula for sl (1|n), see [14]. Since in our case the defect is 1, our formulae are easier than those for general ﬁnite-dimensional superalgebras (see [4, 15]).

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(1)

In what follows we assume g = sl (1|n) or osp (2|2n) , and we use the same notations as in the previous section. Let LΣ (λ) be an integrable simple g-module, and let λ + ρΣ be regular. Let v be a highest vector of MΣ (λ). For any principal root β put k (λ, β) = k(λ,β) λ + ρΣ , hβ . For β ∈ Σ, deﬁne vβ = Yβ v. If β ∈ / Σ, i.e., β = α1 + γ1 for isotropic α1 , γ1 ∈ Σ, then, by regularity of λ+ρΣ , one can choose α ∈ {α1 , γ1 } k(λ,β) such that (λ, α) = (λ + ρΣ , α) = 0; deﬁne vβ as Yβ Yα v. Let VΣ (λ) be the quotient of MΣ (λ) by the submodule generated by vβ , β ∈ B. Lemma 14.1. Let LΣ (λ) be an integrable simple g-module, and λ + ρΣ be regular. Then VΣ (λ) is an integrable g-module, and λ−α1 , λ−γ1 ∈ P (VΣ (λ)). Proof. Let v¯ be the image of v under the natural projection MΣ (λ) → VΣ (λ). k(λ,β) Then Yβ v¯ = 0 for any β ∈ Σ ∩ B. Since v¯ generates VΣ (λ), Yβ is locally k(λ,β)

Yα v¯ = 0. nilpotent on VΣ (λ) for any β ∈ Σ ∩ B. If β ∈ B Σ, we have Yβ But Xα Yα v¯ = λ (hα ) v¯, therefore Yα v¯ generates VΣ (λ), and Yβ is also locally nilpotent if β ∈ B Σ. Thus, VΣ (λ) is integrable. Denote by µβ the weight of vβ . To show that λ − α1 , λ − γ1 ∈ P (VΣ (λ)) it is suﬃcient to check that λ − α1 , λ − γ1 are not weights of a submodule + generated − by vβ for each β ∈ B. If β ∈ Σ, then vβ is nΣ -invariant, so U (g) vβ = U nΣ vβ , and obviously the inequalities −λ − α 1 ≤Σ µβ , λ − γ1 ≤Σ µβ do not / P U nΣ vβ . If β ∈ B Σ, consider hold. Therefore λ − α1 , λ − γ1 ∈ = −Σ + rα (Σ) and note that vβ is nΣ -invariant. In this case U (g) vβ = U nΣ vβ . Assume without loss of generality that α = α1 . Then the inequality λ−α1 ≤Σ µβ never holds, and λ − γ1 ≤Σ µβ holds only if k (λ, β) = 1. But then µβ = λ − α1 − β = λ − 2α1 − γ1 , and λ − γ1 = µβ + 2α1 ∈ v / P U n− Σ β v . / P U n− since −α1 ∈ Σ is isotropic. Therefore λ − α1 , λ − γ1 ∈ β Σ One can immediately see that VΣ (λ) is the maximal integrable quotient of MΣ (λ). The following lemma can be proven exactly as Lemma 10.2 (b). Therefore we omit the proof. Lemma 14.2. If λ + ρ is regular, Σ is obtained from Π by odd reﬂections, and λΣ + ρΣ = λ + ρ, then VΣ (λΣ ) ∼ = VΠ (λ). Recall that λΣ + ρΣ = λ + ρ holds when λΣ is obtained from λ by a chain of “typical” reﬂections. Lemma 14.3. Let LΣ (λ) be integrable, and let λ + ρΣ be regular. If λ is typical, then VΣ (λ) = LΣ (λ). If (λ + ρΣ , α) = 0 for a simple isotropic root α ∈ Σ, then there is a short exact sequence of g-modules 0 → LΣ (λ − α) → VΣ (λ) → LΣ (λ) → 0. Proof. Let [VΣ (λ) : LΣ (µ)] > 0. Since VΣ (λ) is integrable, LΣ (µ) is integrable. Hence µ is an integrable weight. By Lemma 12.3, λ + ρΣ =

Kac–Moody Superalgebras and Integrability

213

µ + ρΣ + β∈B mβ β for some mβ ∈ Z≥0 , some Σ obtained from Σ by odd reﬂections, and µ such that LΣ (µ) ∼ = LΣ (µ ). Since (µ + ρΣ , µ + ρΣ ) = (µ + ρΣ , µ + ρΣ ) = (λ + ρΣ , λ + ρΣ ) , Lemma 13.8 and regularity of λ + ρΣ imply λ + ρΣ = µ + ρΣ . Without loss of generality, one may assume that Σ = rαk . . . rα1 (Σ). Then (23) µ + ρΣ = µ + ρΣ + αi1 + αi2 + · · · + αir , for some 1 ≤ i1 < · · · < ir ≤ k such µ + ρΣ + αi1 + · · · + αij , αij+1 = 0 for all j < r. In particular, r = 0 or (λ + ρΣ , αir ) = 0. If λ is typical, the latter case is impossible; hence µ = λ, therefore VΣ (λ) = LΣ (λ). Assume that λ is atypical. Since λ + ρΣ is regular, by Lemma 13.3, there is only one α such that (λ + ρΣ , α) = 0; hence r = 0, or α = αir . Since α ∈ Σ, and Σ contains only two isotropic simple roots α1 and γ1 , either α = α1 or α = γ1 . Therefore, either λ = µ, or α = αir = α1 and λ = µ + α. Thus [VΣ (λ) : LΣ (µ)] > 0 implies µ ∈ {λ, λ − α}. It remains to show that [VΣ (λ) : LΣ (λ − α)] = 1. Since the multiplicity of λ − α in MΣ (λ) is 1, [VΣ (λ) : LΣ (λ − α)] ≤ 1. On the other hand, notice that Yα v¯ ∈ VΣ (λ) is not zero (by Lemma 14.1) and generates a submodule with the highest weight λ − α (see the proof of Lemma 10.2). Hence [VΣ (λ) : LΣ (λ − α)] = 1. The next step is to check that (22) holds if we replace LΣ (λ) by VΣ (λ). Theorem 14.4. Let λ be integrable, λ + ρΣ be regular. Then ch VΣ (λ) = U (λ + ρΣ ) . Proof. First, we note that, by regularity, there is only one α ∈ ∆+ 1 such that (λ + ρΣ , α) = 0. Without loss of generality, one may assume that α ∈ Σ (if necessary, one can change Σ to Σ by “typical” odd reﬂections so that VΣ (λ + ρΣ − ρΣ ) ∼ = VΣ (λ), α ∈ Σ , as in Lemma 14.2). By Corollary 11.6, ch VΣ (λ) = aκ ch MΣ (κ) , aκ = 0. κ∈T

Since VΣ (λ) is integrable, ch VΣ (λ) is W -invariant. Therefore bν eν ch VΣ (λ) − U (λ + ρΣ ) = =

κ∈T

ν∈F

aκ ch MΣ (κ) −

w

(−1) ch MΣ (w (λ + ρΣ ) − ρΣ )

w∈W

is W -invariant. We assume that bν = 0 for any ν ∈ F . Assume that the theorem does not hold. Then F is nonempty. Not that F is a W -invariant set. Pick up a maximal ν ∈ F . First, maximality of ν implies that

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ν ∈ T ∪ (W (λ + ρΣ ) − ρΣ ). Furthermore, rβ (ν) ≥ ν for any β ∈ B such that ν (hβ ) < 0; hence ν (hβ ) ∈ Z≥0 . In particular, ν = w (λ + ρΣ ) − ρΣ , because w (λ + ρΣ ) − ρΣ ∈ C only when w = 1 and, obviously, λ ∈ / F (see Remark 12.7). Recall that by Corollary 11.6, there exist ν1 , . . . , νk ∈ T , νk = ν, such that [MΣ (λ) : LΣ (ν1 )] > 0, [MΣ (ν1 ) : LΣ (ν2 )] > 0, . . . [MΣ (νk ) : LΣ (ν)] > 0. We claim that νk = w (λ + ρΣ ) − ρΣ for some w ∈ W . Indeed, ν < νk , therefore νk ∈ / F ; since νk ∈ T and ch VΣ (λ) − U (λ + ρΣ ) does not contain the term MΣ (νk ) with nonzero coeﬃcient, mβ β, νk + ρΣ = w (λ + ρΣ ) = λ + ρΣ − β∈B

for some mβ ∈ Z≥0 (see Remark 12.7). On the other hand, by Lemma 12.3, ν + ρΣ = νk + ρΣ −

mβ β

β∈B

for some base Σ obtained from Σ by odd reﬂections, ν such that LΣ (ν ) ∼ = LΣ (ν) and some mβ ∈ Z≥0 . As a result ν + ρΣ = λ + ρΣ −

mβ β,

mβ ∈ Z≥0 .

β∈B

By Lemma 13.4, ν + ρΣ , hβ ≥ 0 for all β ∈ B ∩ Σ and ν + ρΣ , hβ ≥ −1 for β ∈ B Σ. Since λ + ρΣ is regular and λ is integrable, (λ + ρΣ + ν + ρΣ , β) > 0 for all β ∈ B ∩Σ, and (λ + ρΣ + ν + ρΣ , β) ≥ 0 for β ∈ B Σ. The condition (λ + ρΣ , λ + ρΣ ) = (ν + ρΣ , ν + ρΣ ) implies λ + ρ Σ + ν + ρΣ , mβ β = 0. β∈B

If (λ + ρΣ + ν + ρΣ , β) > 0 for β ∈ B Σ, then ν + ρΣ = λ + ρΣ . Then, as in the proof of Lemma 14.3, one can show that ν = λ − α, but we already proved that the weight λ − α has multiplicity one in VΣ (λ), as well as in U (λ + ρΣ ). Hence ν = λ − α. Therefore, (λ + ρΣ + ν + ρΣ , β) = 0 for β ∈ B Σ; thus mβ = 0 for β ∈ B ∩ Σ, and λ + ρΣ , hβ = 1, ν + ρΣ , hβ = −1 (hence mβ = 1) for β ∈ B Σ. Therefore, ν + ρΣ = rβ (λ + ρΣ ) = λ + ρΣ − β. Without loss of generality, assume that Σ = rαk . . . rα1 (Σ). Then rβ (λ + ρΣ ) = ν + ρΣ = ν + ρΣ + αi1 + · · · + αir .

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Assume r = 0; then (rβ (λ + ρΣ ) , αir ) = 0. By our assumption in the beginning of the proof, there are only two possibilities: (λ + ρΣ , α1 ) = 0 or (λ + ρΣ , γ1 ) = 0. The ﬁrst case is impossible, because then we have rβ (α1 ) = −γ1 , (rβ (λ + ρΣ ) , γ1 ) = 0, and by regularity of rβ (λ + ρΣ ), αir = γ1 . In the second case, we have αir = α1 and ν = λ − β − α1 . Since (ρΣ , β) = 0, ν (hβ ) = λ (hβ ) − β (hβ ) − α1 (hβ ) = 1 − 2 − 2 (α1 , β)/(β, β) = −2, and that contradicts the condition rβ (ν) ≤ ν. Corollary 14.5. For any typical integrable λ the character of L (λ) is given by (22). Corollary 14.6. Let LΣ (λ) be integrable, and λ+ ρΣ be regular. Assume that (λ + ρΣ , α) = 0 for some isotropic α ∈ Σ. Then ch LΣ (λ) + ch LΣ (λ − α) = U (λ + ρΣ ) . Theorem 14.7. Let λ be an atypical integrable weight, Π be such that λ + ρ is regular, and there exists an isotropic α ∈ Π such that (λ + ρ, α) = 0. Then w λ+ρ D1 (−1) w e w∈W 1+e−α ch L (λ) = (24) D0 Proof. Without loss of generality, assume that α = α1 = −γ0 . We start with constructing a sequence (λi , Σi , k (i)), where Σi = rγk(i) . . . rγ1 Π, λi is a weight such that λi + ρΣi , γk(i) = 0, and λi + ρΣi is regular. This sequence is deﬁned uniquely by the following rule: 1. 2. 3. 4.

(λ0 , Σ0 , k (0)) = (λ, Π, 0); λi+1 = λi + γk(i) ; Σi+1 = Σi , k (i + 1) = k (i) if λi+1 + ρΣi is regular; if λi+1 + ρΣi is not regular, set, using Lemma 12.6, Σi+1 = rγk(i+1) . . . rγk(i)+1 (Σi )

such that LΣi (λi+1 ) ∼ = LΣi+1 (λi+1 ), and λi+1 + ρΣi+1 is regular. These rules ensure that λi + ρΣi , γk(i) = 0 and λi + ρΣi is regular. Now deﬁne D1 1 w λi +ρΣi . χi = ch LΣi (λi ) , ϕi = (−1) w e D0 1 + eγk(i) w∈W

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(The quotients are taken in R ; obviously, they make sense.) The series converges by the same reasons as for the series for U . We want to show that χi = ϕi ; they are both elements of R. Our next step is to prove the identity ϕi + ϕi+1 = U (λi + ρΣi ) . If Σi+1 = Σi , then the identity is straightforward. Otherwise, by Lemma 12.6 (with Σ = Σi , Σ = Σi+1 , λ = λi+1 , k = s − 1) λi+1 + ρΣi+1 = g λi+1 + ρΣi + (s − 1) γk(i) = g λi + ρΣi + sγk(i) , where s is the positive integer such that λi + ρΣi + tγk(i) is not regular for any g s+1 1 ≤ t < s; and g ∈ W is such that γk(i+1) = g γk(i) and (−1) = (−1) . Thus, we obtain ϕi + ϕi+1 D1 D1 1 w g(λi +ρΣi +sγk(i) ) = (−1) w eλi +ρΣi + e D0 1 + eγk(i) 1 + eg(γk(i) ) w∈W w s+1 λi +ρΣ +sγk(i) D1 λi +ρΣi i + (−1) e γk(i) w∈W (−1) w e 1+e = D0 " s−1 ! s−1 D1 w λi +ρΣi +tγk(i) = = (−1) w e U λi + ρΣi + tγk(i) . D0 t=0 t=0 w∈W

However, λi + ρΣi + tγk(i) is not regular for all 1 ≤ t < s. Hence U λi + ρΣi + tγk(i) = 0, t > 0; we obtain the desired identity. Now recall that by Corollary 14.6 the similar identity holds for χi : χi + χi+1 = U (λi + ρΣi ) . Therefore we can conclude that there exists Φ ∈ R such that i

χi = ϕi + (−1) Φ. We want to show now that Φ = 0. It suﬃces to prove that for every ν ∈ h∗ the monomial eν appears with nonzero coeﬃcient only in ﬁnitely many ϕi and in ﬁnitely many χi . We claim that if eν appears in χi (or ϕi ) with non-zero coeﬃcient, then ν ≤Σi λi . For χi it follows from the fact that ν must be a weight of LΣi (λi ). To check this in case of ϕi , note that every term which appears in ϕi appears in the expression for U (λi + ρΣi ), so the same argument as for U works. Recall the Z-grading on Q deﬁned in (15), (16). The condition ν ≤Σi λi can be rewritten in the form (see Lemma 12.1 for Π = Σi )

Kac–Moody Superalgebras and Integrability

λi − ν = −mi γk(i) +

miβ β,

if λi − ν ∈ Q+ ,

miβ β,

if λi − ν ∈ Q− ;

217

β∈B

λi − ν = mi γk(i)+1 +

β∈B

miβ

here mi , ∈ Z≥0 . Note that if λi ∈ Qj , then λi+1 ∈ Qj−1 . Hence for suﬃciently large i only the second case is possible. Then λi+1 − ν = mi γk(i)+1 + γk(i) + miβ β. β∈B

Now rewrite it in a suitable form λi+1 − ν = (mi + 1) γk(i+1)+1 + miβ β + mi γk(i)+1 − γk(i+1)+1 + γk(i) − γk(i+1)+1 . β∈B

However, γk(i)+1 − γk(i+1)+1 is either zero or a negative real root, γk(i) − γk(i+1)+1 is always a negative even root. Thus β∈B

mi+1 ≤ β

miβ − 1.

β∈B

Hence, for suﬃciently large i, β∈B miβ becomes negative, therefore ν ≤Σi λi does not hold. The theorem is proven. Remark 14.8. Under conditions of Theorem 14.7, it is not hard to show that the complex · · · → VΣi+1 (λi+1 ) → VΣi (λi ) → · · · → VΣ0 (λ0 ) → 0 with arrows deﬁned by Corollary 14.6 is a resolution of L (λ). Kac and Wakimoto call a representation with highest weight satisfying the condition of Theorem 14.7 tame. Character formula (24) coincides with (1) the Kac–Wakimoto conjectural formula in [12] in case g = sl (1|n) or osp (2|2n)(1) . On the other hand, using Lemma 12.6, one can construct a chain of odd reﬂections transforming any atypical integrable weight (except weights of one-dimensional representations) to a weight satisfying the conditions of the last theorem. Therefore, we found character formulae for all (1) (1) integrable simple highest weight modules for sl (1|n) and osp (2|2n) . It seems possible that using the Shapovalov form calculated in [16] as a substitute for a missing invariant symmetric form, one can obtain similar formulae for S (1, 2; b).

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References 1. V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977), 8–96. 2. Victor G. Kac, Inﬁnite-dimensional Lie Algebras, Third edition. Cambridge University Press, Cambridge, 1990. 3. C. Hoyt, Kac-Moody superalgebras of ﬁnite growth, Ph.D. thesis, UC Berkeley, Berkeley, 2007. 4. V. Serganova, Characters of simple Lie superalgebras, Proceedings of ICM, 1998, pp. 583–594. 5. V. V. Serganova, Automorphisms of simple Lie superalgebras, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 48 (1984), no. 3, 585–598 (Russian). 6. Victor G. Kac, Representations of classical Lie superalgebras, Diﬀerential geometrical methods in mathematical physics, II (Berlin), Lecture Notes in Math., vol. 676, Proc. Conf., Univ. Bonn, Bonn, 1977, Springer, 1978, pp. 597–626. 7. V. G. Kac and J. van de Leur, On classiﬁcation of superconformal algebras, Strings’88, World Sci. Publ. Teaneck, NJ, 1089, pp. 77–106. 8. J. van de Leur, A classiﬁcation of contragredient Lie superalgebras of ﬁnite growth, Comm. Algebra 17 (1989), 1815–1841. 9. C. Hoyt and V. Serganova, Classiﬁcation of ﬁnite-frowth general Kac-Moody superalgebras, Comm. Algebra 35 (2007), 851–874. 10. V. G. Kac, Inﬁnite-dimensional algebras, Dedekind’s η-function, classical Mobius function and the very strange formula, Advances in Mathematics 30 (1978), no. 2, 85–136. 11. V. G. Kac and M. Wakimoto, Integrable highest weight modules over aﬃne superalgebras and number theory, Progress in Math. 123 (1994), 415–456. 12. V.G. Kac and M. Wakimoto, Integrable highest weight modules over aﬃne superalgebras and Appel’s function, Commun. Math. Phys. 215 (2001), 631–682. 13. D. A. Leites, M. V. Savelev, and V. V. Serganova, Embeddings of osp(n/2) and the associated nonlinear supersymmetric equations, Group theoretical methods in physics (Utrecht), vol. I, Yurmala, 1985, VNU Sci. Press, 1986, pp. 255–297. 14. Joseph N. Bernstein and D. A. Leites, A formula for the characters of the irreducible ﬁnite-dimensional representations of Lie superalgebras of series Gl and sl, Doklady Bolgarskoi Akademii Nauk. Comptes Rendus de l’Academie Bulgare des Sciences 33 (1980), no. 8, 1049–1051 (Russian). 15. Jonathan Brundan, Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m, n), J. of AMS 16 (2003), no. 1, 185–231. 16. V.G. Kac and M. Wakimoto, Quantum reduction and representation theory of conformal superalgebras, Adv. Math. 185 (2004), 400–458.

Part B

Geometry of Infinite-Dimensional Lie (Transformation) Groups

Jordan Structures and Non-Associative Geometry Wolfgang Bertram Institut Elie Cartan, Universit´e Nancy I, Facult´e des Sciences, B.P. 239, 54506 Vandœuvre-l`es-Nancy, Cedex, France, [email protected] Summary. We give an overview of constructions of geometries associated to Jordan structures (algebras, triple systems and pairs), featuring analogues of these constructions with the Lie functor on one hand and with the approach of non-commutative geometry on the other. Key words: Jordan pair, Lie triple system, graded Lie algebra, ﬁltered Lie algebra, generalized projective geometry, ﬂag geometries, (non-) associative algebra and geometry. 2000 Mathematics Subject Classifications: 17C37, 17B70, 53C15.

Introduction Let us compare two aspects of the vast mathematical topic “links between geometry and algebra”: on the one hand, the Lie functor establishes a close relation between Lie groups (geometric side) and Lie algebras (algebraic side); this is generalized by a correspondence between symmetric spaces and Lie triple systems (see [Lo69]). On the other hand, the philosophy of noncommutative geometry generalizes the relation between usual, geometric point spaces M (e.g., manifolds) and the commutative and associative algebra Reg(M, K) of “regular” (e.g., bounded, smooth, algebraic, according to the context) K-valued functions on M , where K = R or C, by replacing the algebra Reg(M, K) by more general, possibly non-commutative algebras A. The interaction between these two aspects seems to be rather weak; indeed, in some sense they are “orthogonal” to each other. First, the classical setting of Lie’s third theorem is ﬁnite dimensional, whereas the algebras A of commutative or non commutative geometry are typically inﬁnite dimensional; second, and more importantly, taking a commutative and associative algebra as input, we obtain as a Lie bracket [x, y] = xy − yx = 0, and hence (if we forget the associative structure and retain only the Lie bracket) we are left with constructing a Lie group with a zero Lie bracket. But this is not very interesting: K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_7, © Springer Basel AG 2011

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it does not even capture the speciﬁc information encoded in “commutative geometries”.1 This remark suggests that, if one looks for a link between the two aspects just mentioned, it should be interesting to ask for an analogue of the Lie functor for a class of algebras that faithfully contains the class of commutative associative algebras, but rather sacriﬁces associativity instead of commutativity. At this point let us note that the algebras A of non-commutative geometry are of course always supposed to be associative, so that the term “associative geometry” might be more appropriate than “non-commutative geometry”. Indeed, behind the garden of associative algebras starts the realm of general algebras, generically neither associative nor commutative, and, for the time being, nobody has an idea of what their “geometric interpretation” might be. Fortunately, two oﬀsprings of associative algebras grow not too far behind the garden walls: Lie algebras right on the other side (the branches reach over the wall so abundantly that some people even consider them as still belonging to the garden), and Jordan algebras a bit further. Let us just recall that the former are typically obtained by skew-symmetrizing an associative algebra, [x, y] = xy − yx, whereas the latter are typically obtained by symmetrizing them, x • y = 12 (xy + yx) (the factor 12 being conventional, in order to obtain the same powers xk as in the associative algebra). So let us look at Jordan algebras—their advantage being that the class of commutative associative algebras is faithfully embedded. In this survey paper we will explain their geometric interpretation via certain generalized projective geometries, emphasizing that this interpretation really combines both aspects mentioned above. Of course, it is then important to include the case of inﬁnite-dimensional algebras, and to treat them in essentially the same way as ﬁnite-dimensional ones. This is best done in a purely algebraic framework, leaving aside all questions of topologizing our algebras and geometric spaces. Such questions form an interesting topic for the further development of the theory: in a very general setting (topological algebraic structures over general topological ﬁelds or rings) basic results are given in [BN05], and many results from the more speciﬁc setting of Jordan operator algebras (Banach– Jordan structures over K = R or C; see [HOS84]) admit interesting geometric interpretations in our framework. Beyond the Banach setting, it should be interesting to develop a theory of locally convex topological Jordan structures and their geometries, taking up the associative theory (cf. [Bil04]). The general construction for Jordan algebras (and for other members of the family of Jordan algebraic structures, namely Jordan pairs and Jordan triple systems, which in some sense are easier to understand than Jordan algebras; see Section 1) being explained in the main text (Section 2 and Section 3), here 1

Of course, this does not exclude other ways of associating Lie groups to function algebras from being interesting, for example by looking at their derivations and automorphisms; but our point is precisely that such constructions need more input than the trivial Lie bracket on the algebra of functions.

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let us just look at the special case of “commutative geometry”, i.e., the case of the commutative and associative algebra A = Reg(M, K) of “regular K-valued functions” on a geometric space M (here K is a commutative ﬁeld with unit 1; in the main text we will also allow commutative rings with 1). Looking at A as a Jordan algebra, we associate to it the space X := Reg(M, KP1 ) of “regular functions from M into the projective line KP1 ”. Of course, X is no longer an algebra or a vector space; however, we can recover all these structures if we want: we call two functions f, g ∈ X “transversal” and write f g, if f (p) = g(p) for all points p ∈ M . Now choose three functions f0 , f1 , f∞ ∈ X that are mutually transversal. For each point p ∈ M , the value f∞ (p) singles out an “aﬃne gauge” (by taking f∞ (p) as a point at inﬁnity in the projective line KP1 and looking at the aﬃne line KP1 \ {f∞ (p)}); then f0 (p) singles out a “linear gauge” (origin in the aﬃne line) and f1 (p) a “unit gauge” (unit element in the vector line). Obviously, any such choice of f0 , f1 , f∞ leads to an identiﬁcation of the set (f∞ ) of all functions that are transversal to f∞ with the usual algebra A of regular functions on M . In other words, A and (X ; f0 , f1 , f∞ ) carry the same information, and thus we see that the encoding via A depends on various choices, from which we are freed by looking instead at the “geometric space” X . It thus becomes evident that A really is a sort of “tangent algebra” of the geometric (“non-ﬂat”) space X at the point f0 , in a similar way as the Lie algebra of a Lie group reﬂects its tangent structure at the origin. This way of looking at ordinary “commutative geometry” leads to generalizations that are diﬀerent from non-commutative geometry [Co94] but still have much in common with it. For instance, both theories have concepts of “states”, i.e., a notion that, in the commutative case, amounts to recovering the point space M from the algebra of regular functions. However, in the case of non-commutative geometry, this concept relies heavily on positivity (and hence on the ordered structure of the base ﬁeld R), and the corresponding concept for generalized projective geometries is purely geometric and closely related to the notion of inner ideals in Jordan theory (Section 4). There are many interesting open problems related to these items, some of which are mentioned in Section 4. Summing up, it seems that the topic of geometrizing Jordan structures, incorporating both ideas from classical Lie theory and basic ideas from associative geometry, is well suited for opening the problem of “general nonassociative geometry”, i.e., the problem of ﬁnding geometries corresponding to more general non-associative algebraic structures. Remark (added in may 2010): While this paper was being processed, the joint work [BeKi09] with Michael Kinyon gave a new turn to the whole question. We cannot go into details here, and we strongly suggest the interested reader to have a look at that work. Acknowledgements: This paper is partly based on notes of a series of lectures given at the Universit´e Paul Verlaine (Metz university) in April 2007, and I would like to thank Said Benayadi for inviting me on this occasion.

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Notation: Throughout, K is a commutative base ring with unit 1 and such that 2 and 3 are invertible in K.

1 Jordan pairs and graded Lie algebras 1.1 Z/(2)-graded Lie algebras and Lie triple systems Let (Γ, +) be an abelian group. A Lie algebra g over K is called Γ -graded if it is of the form g= gn , with [gm , gn ] ⊂ gn+m . n∈Γ

Let us consider the example Γ = Z/(2). Here, g = g0 ⊕ g1 , with a subalgebra g0 and a subspace g1 which is stable under g0 and such that [g1 , g1 ] ⊂ g0 . It is then easily seen that the linear map σ : g → g which is 1 on g0 and −1 on g1 is an involution of g (automorphism of order 2). Conversely, every involution gives rise to a Z/(2)-grading of g, and hence such gradings correspond bijectively to symmetric Lie algebras (g, σ), i.e., to Lie algebras g together with an automorphism σ of order 2. Then g1 (the −1-eigenspace of σ) is stable upon taking triple Lie brackets [[x, y], z]. This leads to the notion of a Lie triple system. Definition 1.1. A Lie triple system (LTS) is a K-module q together with a trilinear map q × q × q → q, (X, Y, Z) → [X, Y, Z], or, equivalently, with a bilinear map R : q × q → End(q),

(X, Y ) → R(X, Y ) =: [X, Y, ·],

such that (LT1) R(X, X) = 0 (LT2) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0 (the Jacobi identity) (LT3) the endomorphism D := R(X, Y ) is a derivation of the trilinear product, i.e., D[U, V, W ] = [DU, V, W ] + [U, DV, W ] + [U, V, DW ]. If g is a Z/(2)-graded Lie algebra, then g1 with [X, Y, Z] := [[X, Y ], Z] becomes an LTS, and every LTS arises in this way since one may reconstruct a Lie algebra from an LTS via the standard imbedding (see [Lo69]): let h ⊂ Der(q) be the subalgebra of derivations of the LTS q spanned by all operators R(X, Y ), X, Y ∈ q, and deﬁne a bracket on g := q ⊕ h by [(X, D), (Y, E)] := DY − EX, [D, E] − R(X, Y ) . One readily checks that g is a Z/(2)-graded Lie algebra. As a side remark, one may note that this construction (the standard imbedding) is in general not functorial. See [Sm05] for a discussion of this topic.

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1.2 3-graded Lie algebras and Jordan pairs A Lie algebra is called 2n + 1-graded if it is Z-graded, with gj = 0 if j ∈ / {−n, −n + 1, . . . , n}. The linear map D : g → g with Dx = jx for x ∈ gj is then a derivation of g; if it is an inner derivation, D = ad(E), the element E ∈ g is called an Euler element. To any Z-graded algebra one may associate a Z/(2)-grading g = geven ⊕ godd , by putting together all homogeneous parts with even, resp. odd index. In the sequel we are mainly interested in 3-graded and 5-graded Lie algebras; in both cases, godd = g1 ⊕ g−1 . We then let V ± := g±1 and deﬁne trilinear maps by T ± : V ± × V ∓ × V ± → V ±,

(x, y, z) → [[x, y], z].

The maps T± satisfy the identity (LJP2) T ± (u, v, T ± (x, y, z)) = T ± (T ± (u, v, x), y, z) − T ± (x, T ∓ (v, u, y), z) + T ± (x, y, T ± (u, v, z)). Indeed, this is just another version of the identity (LT3), reﬂecting the fact that ad[u, v] is a derivation of g respecting the grading since [u, v] ∈ g0 . In the 3-graded case, we have moreover, for all x, z ∈ V ± , y ∈ V ∓ : (LJP1) T ± (x, y, z) = T ± (z, y, x). This follows from the fact that g1 and g−1 are abelian, and so by the Jacobi identity [[x, y], z] − [[z, y], x] = [[z, x], y] = 0. Definition 1.2. A pair of K-modules (V + , V − ) together with trilinear maps T ± : V ± × V ∓ × V ± → V ± is called a (linear) Jordan pair if (LJP1) and (LJP2) hold. (The term “linear” refers to the fact that the identities (LJP1) and (LJP2) are linear in each of their variables, whereas the deﬁnition given in [Lo75], which is also valid in the case where 2 or 3 are not invertible in K, is based on identities that are quadratic in some of their variables.) Every linear Jordan pair arises by the construction just described: if (V + , V − ) is a linear Jordan pair, let q := V + ⊕ V − and deﬁne a trilinear map T˜ : q3 → q via T˜ (x, x ), (y, y ), (z, z ) := T + (x, y , z), T −(x , y, z ) . Then the map T˜ satisﬁes the same identities as (T + , T − ), with T + and T − both replaced by T˜ (in other words, (q, T˜) is a Jordan triple system, see below). Now we deﬁne a trilinear bracket q3 → q by (x, x ), (y, y ), (z, z ) := T˜ ((x, x ), (y, y ), (z, z )) − T˜ ((y, y ), (x, x ), (z, z )).

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By a direct calculation (cf. Lemma 1.4 below), one checks that q with this bracket is an LTS, and its standard imbedding g = q ⊕ h with g±1 = V ± and g0 = h, is a 3-graded Lie algebra. Without loss of generality we may assume that g contains an Euler element E: in fact, the endomorphism E : q → q which is 1 on V + and −1 on V − is a derivation of q commuting with h, and hence we may replace h by h + KE in the construction of the standard imbedding. For 5-graded Lie algebras, the identity (LJP1) has to be replaced by another, more complicated identity, which leads to the notion of a Kantor pair, see [AF99]. As for Jordan pairs, Kantor pairs give rise to LTSs of the form q = V + ⊕ V − , where now the standard imbedding leads back to a 5-graded Lie algebra. 1.3 Involutive Z-graded Lie algebras An involution of a Z-graded Lie algebra is an automorphism θ of order 2 such that θ(gj ) = g−j for j ∈ Z. If g is 3- or 5-graded, we let V := g1 and deﬁne T : V × V × V → V,

(X, Y, Z) → [[X, θY ], Z].

Then T satisﬁes the identity (LJT2) obtained from (LJP2) by omitting the indices ±, and if g is 3-graded, we also have the analogue of (LJP1). Definition 1.3. A K-module V with a trilinear map T : V 3 → V satisfying the identities (JP1) and (JP2) obtained from (LJP1) and (LJP2) by omitting indices is called a (linear) Jordan triple system (JTS). Every linear JTS arises by the construction just described: just let V + := V := V and T ± := T ; then (V + , V − ) is a Jordan pair carrying an involution (= isomorphism τ from (V + , V − ) onto the Jordan pair (V − , V + ), namely τ (x, x ) = (x , x)). Let g be the 3-graded Lie algebra associated to this Jordan pair; then the involution τ of the Jordan pair induces an involution θ of g. By the way, these arguments show that JTSs are nothing but Jordan pairs with involution (cf. [Lo75]). For any JTS (V, T ), the Jordan pair (V + , V − ) = (V, V ) with T ± = T is called the underlying Jordan pair. −

The Lie algebra g now carries two automorphisms of order 2, namely θ and the automorphism σ corresponding to the Z/(2)-grading g = geven ⊕ godd . These two automorphisms commute. In particular, σ restricts to the θ-ﬁxed subalgebra gθ , with the −1-eigenspace being {X + θ(X)| X ∈ g1 }. The LTSstructure of this space is described by the following lemma. Lemma 1.4. (The Jordan–Lie functor.) If T is a JTS on V , then [X, Y, Z] = T (X, Y, Z) − T (Y, X, Z) deﬁnes an LTS on V .

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Proof. The proof of (LT1) is trivial, and (LT2) follows easily from the symmetry of T in the outer variables. In order to prove (LT3), we deﬁne the endomorphism R(X, Y ) := T (X, Y, ·) − T (Y, X, ·) of V . Then R(X, Y ) is a derivation of the trilinear map T , as follows from the second deﬁning identity (JT2). But every derivation of T is also a derivation of R, since R is simply deﬁned by skew-symmetrization of T in the ﬁrst two variables. The lemma deﬁnes a functor from the category of JTS to the category of LTS over K, which we call the Jordan–Lie functor. In general, it is neither injective nor surjective; all the more surprising is the fact that, in the real, simple and ﬁnite-dimensional case, the Jordan–Lie functor is not too far from setting up a one-to-one correspondence (classiﬁcation results due to E. Neher, cf. tables given in [Be00]). 1.4 The link with Jordan algebras Whereas all results from the preceding sections are naturally and easily understood from the point of view of Lie theory, the results presented next are “genuinely Jordan theoretic”—by this we mean that proofs by direct calculation in 3-graded Lie algebras are much less straightforward. The reader may try some of these proofs, in order to better appreciate the examples to be given in the next section. In the following, g is a 3-graded Lie algebra, with associated Jordan pair (V + , V − ) = (g1 , g−1 ). For x ∈ V − , we deﬁne a K-linear map Q− (x) : V + → V − ,

y → Q− (x)y :=

1 − 1 T (x, y, x) = − [x, [x, y]]. 2 2

In the same way Q+ (y) for y ∈ V + is deﬁned. Then the following hold. (1) The Fundamental Formula. For all x ∈ V − , y ∈ V + , Q− (Q− (x)y) = Q− (x)Q+ (y)Q− (x). (2) Meyberg’s theorem. Fix a ∈ V − . Then Va := V + with product x •a y :=

1 + T (x, a, y) 2

is a Jordan algebra (called the a-homotope algebra). Recall (see, e.g., [McC04]) that a (linear) Jordan algebra is a K-module J with a bilinear and commutative product x • y such that the identity (J2) x • (x2 • y) = x2 • (x • y) holds. (3) Invertible elements and unital Jordan algebras. An element a ∈ V − is called invertible in (V + , V − ) if the operator Q− (a) : V + → V − is invertible.

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The element a ∈ V − is invertible in (V + , V − ) if, and only if, the Jordan algebra Va described in the preceding point admits a unit element; this unit element is then a := (Q(a))−1 (a). Moreover, the Jordan pair (V + , V − ) can then be recovered from the Jordan algebra (V, •) = (Va , •a ) as follows: let V + := V − := V as K-modules and let T : V 3 → V , T (x, y, z) := (x • y) • z − y • (x • z) + x • (y • z). Then (V, T ) is a JTS, and its underlying Jordan pair (V + , V − ) is the one we started with. 1.5 Some examples (1) Jordan pairs of rectangular matrices. Let A and B be two K-modules and W := A ⊕ B. Let I : W → W be the linear map that is 1 on A and −1 on B. Then the Lie algebra g := glK (W ) is 3-graded, with, for i = 1, 0, −1, gj = {X ∈ g| [I, X] = 2jX}. In an obvious matrix notation, this corresponds to describing g by 2 × 2-matrices, g0 as diagonal and g1 as upper and g−1 as lower triangular matrices: g0 V + . V − g0 Therefore,

(V + , V − ) = (HomK (B, A), HomK (A, B))

carries the structure of a linear Jordan pair. The Jordan pair structure is given by T ± (X, Y, Z) = XY Z + ZY X. In fact, this is proved by the following calculation (using matrix notation for elements of g): XY 0 Z 0 XY Z + ZY X 0 0 0 0Z 0X . = , = , , 0 0 0 0 0 −Y X 0 0 Y 0 0 0 The proof of the Fundamental Formula for this Jordan pair is easy since we may calculate in the associative algebra HomK (W, W ), where Q(x) = x ◦ rx is simply composition of left and of right translation by x. Similarly, the proof of Meyberg’s theorem becomes an easy exercise: for any a ∈ HomK (W, W ), the product (x, y) → xay is again an associative product, and by symmetrizing and restricting to V + , if a ∈ V − , we get the Jordan algebra •a . In general, the Jordan pair (V + , V − ) does not contain invertible elements. For instance, if K is a ﬁeld and A and B are not isomorphic as vector spaces, then it is easily seen that Q− (x) is never bijective. Finally, it is clear that, if A and B are isomorphic as K-modules, and ﬁxing such an isomorphism in order to identify A and B, an element a ∈ V + = A = V −

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is invertible if it is invertible in the usual sense in Hom(A, A), and then •a has a−1 as a unit element. Note that, in particular, this happens for the Lie algebra gl(2, K), with V + ∼ =V− ∼ = K, where x •a y = xay = yax. (2) Function spaces. As for any algebraic structure deﬁned by identities, spaces of functions with values in such a structure, equipped with the “pointwise product”, again form a structure of the given type. In our case, if M is a set and (V + , V − ) a Jordan pair, then (Fun(M, V + ), Fun(M, V − )) is again a Jordan pair, and similarly for JTSs and Jordan algebras. Moreover, if (V + , V − ) corresponds to a 3-graded Lie algebra g, then the function space corresponds to the 3-graded Lie algebra Fun(M, g). In particular, pairs of scalar functions (Fun(M, K), Fun(M, K)) form a Jordan pair with pointwise product (T (f, g, h))(p) = 2f (p) g(p) h(p). Under suitable assumptions, “regular” functions (continuous, smooth, algebraic, . . .) will form subpairs. (3) The classical examples. Besides rectangular matrices, these include: • full associative algebras A. Here (V + , V − ) = (A, A), and g ⊂ gl(2, A) is the subalgebra generated by the strictly upper triangular and strictly lower triangular matrices; • Hermitian elements of an involutive associative algebra (A, ∗). Here, (V + , V − ) = (Ah , Ah ) is the ﬁxed space of ∗, and g is the symplectic algebra of (A, ∗) (cf. [BN04], Section 8.2). Note that symmetric and Hermitian matrices are a special case; • skew-Hermitian elements of an involutive associative algebra (A, ∗): similar to that above, replacing ∗ by its negative. As above, Jordan pairs of skew-symmetric or skew-Hermitian matrices are a special case; • conformal geometries (or “spin factors”). Here, V + = V − = V is a K-module with a non-degenerate symmetric bilinear form (·|·) : V × V → K and trilinear map T (x, y, z) = (x|z)y − (x|y)z − (z|y)x . Then g is the orthogonal Lie algebra of the quadratic form on K ⊕ V ⊕ K given by β((r, v, s), (r, v, s)) = rs + (v|v) . There are also exceptional Jordan systems, namely octonionic 1 × 2-matrices, resp. the Hermitian octonionic 3 × 3-matrices, corresponding to 3-gradings of the exceptional Lie algebras of type E6 , resp. E7 .

2 The generalized projective geometry of a Jordan pair 2.1 The construction The following construction of a pair (X + , X − ) of homogeneous spaces associated to a Jordan pair (V + , V − ) is due to J.R. Faulkner and O. Loos

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(see [Lo95]). Starting with a (linear) Jordan pair (V + , V − ), let g be the 3-graded Lie algebra g constructed in Section 1.2; as explained there, we may assume that g contains an Euler element E, i.e., an element such that [E, X] = iX for X ∈ gi . Next we deﬁne two abelian groups U ± = exp(g±1 ) by observing that, for X ∈ g±1 , the operator ad(X) : g → g is 3-step nilpotent, and that 1 exp(X) := ead(X) = id + ad(X) + ad(X)2 2 is an automorphism of g (for this we need that 2 and 3 are invertible in K). Since g±1 are abelian, U ± are abelian subgroups of Aut(g). The maps exp± : g± → Aut(g) are injective (here we use our Euler operator: exp± (X) = id implies exp± (X)E = E ± X = E, whence X = 0). Thus U ± is isomorphic to (V ± , +). The elementary projective group associated to the Jordan pair (V + , V − ) is the subgroup G := PE(V + , V − ) := U + , U − of Aut(g) generated by U + and U − . Let H be the subgroup of G stabilizing the grading (i.e., commuting with ad(E)); then U + and U − are normalized by H, and the groups P ± generated by H and U ± are semidirect products: P ± := H, U ± ∼ = H V ±. Finally, we deﬁne two homogeneous spaces X ± := G/P ∓ with base points o± = e/P ∓ (e being the unit element of G). The pair of “geometric spaces” (X + , X − ) is called the generalized projective geometry associated to (V + , V − ). It is indeed the geometric object associated to the Jordan pair (V + , V − ), in a similar way as a Lie group is the geometric object associated to a Lie algebra. Let us explain this brieﬂy. 2.2 Generalized projective geometries The spaces X ± being deﬁned as above, the direct product X + ×X − is equipped with a transversality relation: call (x, α) ∈ X + × X − transversal, and write xα, if they are conjugate under G to the base point (o+ , o− ). It is easy to describe the set α := {x ∈ X + | xα} : if α = o− is the base point (whose stabilizer is P + ), then this is the P + -orbit P + .o+ which is isomorphic to V+ ∼ = U + .o+ . Note here that the set α carries a canonical structure of an aﬃne space over K, since the vector group U + acts on it simply transitively. By homogeneity, the same statements hold for any α ∈ X − . This observation leads to the following deﬁnition. Definition 2.1. A pair geometry is a pair of sets X = (X + , X − ) together with a binary relation ⊂ X + × X − (called transversality, and we write xα for (x, α) ∈ ) such that X ± is covered by subsets of the form

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α = {x ∈ X ± | xα},

231

α ∈ X ∓.

A linear pair geometry (over K) is a pair geometry (X + , X − , ) such that, for any transversal pair xα, the set α is equipped with a structure of linear space over K (i.e., a K-module), with origin x, and the same property holds by exchanging the rˆ oles of X + and X − . An aﬃne pair geometry is a linear pair geometry such that the underlying aﬃne structure of the linear space (α , x) does not depend on x, for all transversal pairs (x, α), and dually. In other words, for all α ∈ X − , the set α carries the structure of an aﬃne space over K, and dually. The discussion above shows that, to any Jordan pair (V + , V − ) over K, we can associate an aﬃne pair geometry (X + , X − , ). The link with triple products is given by introducing the structure maps of a linear pair geometry: if xα, yα, zα and r ∈ K, then let rx,α (y) := ry denote the product r · y, and y +x,α z the sum of y and z in the K-module α with zero vector x. In other words, we deﬁne maps of three (resp. four) arguments by Pr : (X + × X − × X + ) → X + , (x, α, y) → Pr (x, α, y) := rx,α (y), + S : (X × X − × X + × X + ) → X + , (x, α, y, z) → S(x, α, y, z) := y +x,α z, where the domain of Pr is the “space of transversal triples”, (X + × X − × X + ) = {(x, α, y) ∈ X + × X − × X + | xα, yα}, and the domain of S is the similarly deﬁned space of generic quadruples. Of + course, we should write P+ instead of S; dually, the r instead of Pr and S − − maps Pr and S are then also deﬁned. The structure maps encode all the information of a linear pair geometry: by ﬁxing the pair (x, α), the structure maps describe the linear structure of (α , x), resp. of (x , α). In this way, linear pair geometries can be regarded as objects whose structure is deﬁned by (one or several) “multiplication maps”, just like groups, rings, modules, symmetric spaces, etc. This point of view naturally leads to the question of whether there are more speciﬁc “identities” satisﬁed by the structure maps. If (X + , X − ; ) is the geometry associated to a Jordan pair, then this is indeed the case. There are two such identities, denoted by (PG1) and (PG2) in [Be02]. The ﬁrst identity (PG1) can be seen as an “integrated version” of the deﬁning identity (LJP2) of a Jordan pair; it implies the existence of a “big” automorphism group, whereas (PG2) is rather a global version of the Fundamental Formula and implies the existence of “structural maps” exchanging the two partners X + and X − (we refer the reader to [Be02] for details). The important point about these identities is that we can also go the other way: by “deriving” them in a suitable way, one can show that any aﬃne pair geometry satisfying (PG1) and (PG2) in all scalar extensions gives rise to a Jordan pair (V + , V − ) as a “tangent geometry” with respect to a ﬁxed base point (o+ , o− ).

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This construction clearly parallels the correspondence between Lie groups and Lie algebras (and even more closely the one between symmetric spaces and Lie triple systems [Lo69]; see also [Be08]). However, in contrast to Lie theory, the constructions are algebraic in nature and therefore work much more generally in arbitrary dimension and over general base ﬁelds and rings. They deﬁne a bijection (even an equivalence of categories) between Jordan pairs and connected generalized projective geometries with base point and, in the same way, between Jordan triple systems (resp. unital Jordan algebras) and generalized polar (resp. null) geometries, i.e., generalized projective geometries with the additional structure of a certain kind of involution, cf. [Be02], [Be03]. Thus, in principle, all Jordan-theoretic notions can be translated into geometric ones; in general, it is not at all obvious what the correct translation should be, but once it has been found, it often sheds new light on the algebraic notion. In Section 4 we illustrate this with the example of the notion of inner ideals and their complementation relation. 2.3 The geometric Jordan–Lie functor Symmetric spaces are well-known examples of “non-associative geometries” (cf. [Lo69]): they are manifolds M together with a point reﬂection σx attached to each point x ∈ M such that the multiplication map µ : M × M → M , (x, y) → σx (y) satisﬁes the properties (M1) µ(x, x) = x, (M2) µ(x, µ(x, y)) = y, (M3) µ(x, µ(y, z)) = µ(µ(x, y), µ(x, z)), (M4) the tangent map Tx (σx ) is the negative of the identity of the tangent space Tx M . Recall from Lemma 1.4 the (algebraic) Jordan–Lie functor, associating an LTS to a JTS. The geometric analogue of this functor associates to each generalized polar geometry (X + , X − ; p) the symmetric space M (p) of its nonisotropic points. The geometric construction is very simple: essentially, the multiplication map µ is obtained from the structure maps Pr by taking the scalar r = −1. More precisely, given a polarity (i.e., a pair of bijections (p+ , p− ) : (X + , X − ) → (X − , X + ) that are inverses of each other, compatible with the structure maps and such that there exists a non-isotropic point x ∈ X + , which means that xp+ (x)), for each non-isotropic point x we deﬁne the point-reﬂection by σx (y) := P−1 x, p+ (x), y . The above-mentioned identity (PG1) then implies in a straightforward way that the set M (p) of non-isotropic points of p becomes a symmetric space (cf. [Be02] for the purely algebraic setting and [BN05] for the smooth setting). Moreover, the LTS of the symmetric space M (p) is precisely the one deﬁned

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by Lemma 1.4. The well-known construction of ﬁnite- and inﬁnite-dimensional bounded symmetric domains (cf. [Up85]) arises as a special case of the geometric Jordan–Lie functor. The remarks following Lemma 1.4 imply that in fact “most” symmetric spaces (in the ﬁnite-dimensional real case) are obtained by the preceding construction (essentially, all classical ﬁnite-dimensional real ones and about half of the exceptional ones; cf. the tables in [Be00]). 2.4 Examples revisited (1) Rectangular matrices correspond to Grassmannian geometries. Let W = A ⊕ B be as in Example (1) of Section 1.5 and let GrasB A (W ) be the set of all submodules V ⊂ W that are isomorphic to A (“type”) and admit a complement U ⊂ W isomorphic to B (“cotype”). Then A (X + , X − ) = (GrasB A (W ), GrasB (W )),

with transversality of (V, U ) ∈ X + × X − being the usual complementarity of subspaces (W = U ⊕ V ), is an aﬃne pair geometry over K. It is a standard exercise in linear algebra to show that the set U of complements of U carries a natural structure of aﬃne space. Note that by our deﬁnition of Grassmannians as complemented Grassmannians this aﬃne space is never empty; if K is a ﬁeld this condition is automatic. (However, if K is a topological ﬁeld or ring and W a topological K-module, then one will prefer modiﬁed deﬁnitions of “restricted Grassmannians” by imposing conditions closedness, boundedness, Fredholm type or other; cf., e.g., [PS86], Chapter 7. All such conditions lead to subgeometries of the algebraically deﬁned geometries considered here.) Fixing the decomposition W = A⊕B as base point (o+ , o− ) in (X + , X − ; ), by elementary linear algebra one may identify the pair of linear spaces ((o− ) , (o+ ) ) with the pair (V + , V − ) = (Hom(B, A), Hom(A, B)). By still elementary (though less standard) linear algebra, one can now give explicit expressions for the structure maps introduced in Section 2.2 and check their fundamental identities, thus describing a direct link between the geometry and its associated Jordan pair (see [Be04]). Namely, elements of X + are realized as images of injective maps f : A → W , modulo equivalence under the general linear group GlK (A) (f ∼ f iﬀ ∃g ∈ GlK (A): f = f ◦ g), and similarly elements of X − are realized as kernels of surjective maps φ : W → A, again modulo equivalence under GlK (A). Two such elements are transversal, [f ][φ], if and only if φ ◦ f : A → A is a bijection, and the structure map Pr is now given by Pr ([f ], [φ], [h]) = (1 − r)f ◦ (φ ◦ f )−1 + rh ◦ (φ ◦ h)−1 . As in ordinary projective geometry, an aﬃnization is given by writing f : A → W = B ⊕A as a “column vector” and normalizing the second component to be 1A , the identity map of A, and similarly for the “row vector” φ : B ⊕ A → A

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(see [Be04]). To get a feeling for the kind of non-linear formulas that appear in such contexts, the reader may rewrite the preceding formula for Pr by replacing f, φ and h by such column, resp. row vectors, and then renormalize the right-hand side, in order to get the formula for the multiplication map in the aﬃne picture. The special case r = 12 (“midpoint map”) is particularly important from a Jordan-theoretic point of view. One may as well also consider the total Grassmannian geometry of all complemented subspaces of W , Gras(W ) := {V ⊂ W | (V submodule), ∃V : W = V ⊕ V (V submodule)}. Then the pair (Gras(W ), Gras(W )) is still a generalized projective geometry, but it is not connected in general (there is a natural notion of connectedness for any aﬃne pair geometry; see [Be02]). In inﬁnite dimension, the geometries (X + , X − ) introduced above need not be connected either, but in ﬁnite dimension over a ﬁeld they are; in fact, they then reduce to the usual Grassmannians Grask (Kn ) ∼ = Gl(n, K)/(Gl(k, K) × Gl(n − k, K)) of k-spaces in Kn . The case n = 2k, or, more generally, A ∼ = B, deserves special attention: in this case, X + and X − really are the same sets; the identity map X + → X − is a canonical antiautomorphism of the geometry. Its properties are similar to the well-known null systems from classical projective geometry. In particular, in the case of the projective line (Gras1 (K2 ), Gras1 (K2 )), this really is the canonical null system coming from the (up to a scalar) unique symplectic form on K2 . (2) Geometries of maps and functions. We ﬁx some generalized projective geometry X = (X + , X − ), and let M be any set. Then the geometry of maps (Fun(M, X + ), Fun(M, X − )) with “pointwise transversality and structure maps” is again a generalized projective geometry. By arguments similar to those above, we see that it corresponds to the Jordan pair of functions introduced in Example (2) of Section 1.5. As explained there, the case of scalar-valued functions corresponds to V + = V − = Fun(M, K) with pointwise triple product (T (f, g, h))(p) = 2f (p) g(p) h(p). On the geometric side, then X + = X − = KP1 is the projective line, and Fun(M, X + ) = Fun(M, X − ) is the space of functions from M into the projective line. Note that these deﬁnitions parallel the usual ones for groups (cf. [PS86]); in particular, taking for M the unit circle, we would get “loop geometries”. (3) The classical examples. We brieﬂy characterize the geometries (X + , X − ) corresponding to the examples of item (3) of Section 1.5: • full associative algebras A correspond to the projective line (AP1 , AP1 ) over A (cf. [BN04]);

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• Hermitian elements of an involutive associative algebra (A, ∗) correspond to the ∗-Hermitian projective line over A (cf. [BN04], Section 8.3), and similarly for skew-Hermitian elements. Lagrangian geometries are a special case of this construction, corresponding to (skew-) Hermitian or (skew-) symmetric matrices or operators; • conformal geometries (or “spin factors”): here, X + ∼ = X − is the projective quadric of K ⊕ V ⊕ K with the same quadratic form as in Section 1.5. As to the exceptional Jordan systems, their geometries are among those constructed by Tits; to our knowledge, their structure of generalized projective geometry has not yet been fully investigated.

3 The universal model We have given the construction of the generalized projective geometry (X + , X − ) associated to a 3-graded Lie algebra g via homogeneous spaces (G/P − , G/P + ). For a more detailed study, a geometric realization of these spaces is useful, which in some sense generalizes the example of the Grassmannian geometries. This “universal model” has been introduced in [BN04] and used in [BN05] to deﬁne (under suitable topological assumptions) a manifold structure on (X + , X − ). 3.1 Ordinary flag geometries We ﬁx some K-module g and denote by Fk the set of all ﬂags f = 0 = f0 ⊂ f1 ⊂ · · · ⊂ fk = g of length k in E (i.e., the fi are linear subspaces of E, and all inclusions are supposed to be strict). We say that two ﬂags e, f ∈ F are transversal if they are “crosswise complementary” in the sense that ∀i = 1, . . . , k :

g = fi ⊕ ek−i .

It is a nice exercise in linear algebra to show that e and f are transversal if, and only if, they come from a grading of g of length k. Given a grading g = (g1 , . . . , gk ), i.e., a decomposition g = g1 ⊕ · · · ⊕ gk , we deﬁne two ﬂags f+ (g) and f− (g) via f+ i (g) := g1 ⊕ · · · ⊕ gi ,

f− k−i (g) := gi+1 ⊕ · · · ⊕ gk .

Then f+ (g) and f− (g) are obviously transversal, but the converse is also true: every pair of transversal ﬂags can be obtained in this way! From this exercise, one deduces easily that we get a linear pair geometry (X + , X − ; ) by taking X + := X − := X := {f ∈ Fk | ∃e ∈ Fk : ef},

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the space of all ﬂags that admit at least one transversal ﬂag (cf. [BL08]). For k = 2, this is the total Grassmannian geometry, and similarly as in that case, one may for general k single out connected components by taking ﬂags of ﬁxed type and cotype. In general, these geometries will be linear, but not aﬃne pair geometries, and we do not know what kind of “laws” (in a sense generalizing the laws of generalized projective geometries) can be used to describe them. 3.2 Filtrations and gradings of Lie algebras Let us assume now that g is a Lie algebra with Lie bracket denoted by [·, ·], and that all our gradings and ﬁltrations are compatible with the Lie bracket in the sense that [fi , fj ] ⊂ fi+j , [gi , gj ] ⊂ gi+j . We will assume that our Lie algebra is 2k + 1-graded, i.e., as an index set we take Ik = {−k, −k + 1, . . . , k}, and we say that a 2k + 1-grading is inner if it can be deﬁned by an Euler element, i.e., by an element E ∈ g such that [E, X] = iX for all X ∈ gi , i ∈ Ik . We denote by Gk the set of all inner 2k + 1-gradings of g (it is harmless to assume that the center of g is reduced to zero, and so Gk can be identiﬁed with the set of all Euler elements). An inner 2k + 1-ﬁltration of g is a ﬂag f = (0 = fk+1 ⊂ fk ⊂ · · · ⊂ f−k = g) such that there is some inner 2k+1-grading with fi = gi ⊕gi+1 ⊕· · ·⊕gk , for all i ∈ Ik ; we write Fk for the set of all such ﬁltrations. Then the following holds ([BN04] for the case of 3-gradings (k = 1), and [Ch09] for the general case). We have to impose some mild restrictions on the characteristic

∞ of K since we want to use, for X ∈ f1 , the operator exp(X) = ead(X) = j=0 j!1 ad(X)j , the sum being ﬁnite since [f1 , fi ] ⊂ fi+1 , so that ad(X) is nilpotent. Theorem. Assume that K is a commutative ring such that the integers 2, 3, . . . , 2k + 1 are invertible in K. (1) Two inner 2k + 1-ﬁltrations f and e are transversal if, and only if, there exists a 2k+1-grading g = (g−k , . . . , gk ) such that f = f+ (g) and e = f− (g). (2) The geometry of inner 2k + 1-ﬁltrations (Fk , Fk ; ) is a linear pair geometry. More precisely, for any f ∈ Fk , the nilpotent Lie algebra f1 is in bijection with f via X → exp(X).e, for an arbitrary choice of e ∈ f . Now assume that k = 1, i.e., we are in the 3-graded case. Then [f1 , f1 ] ⊂ f2 = 0 (i.e., f1 is an abelian subalgebra), whence part (2) of the theorem says that we have a simply transitive action of the vector group f1 ∼ = exp(f1 ) on f , and thus we see that the geometry of inner 3-ﬁltrations is an aﬃne pair geometry. In fact, ﬁxing some inner 3-grading as base point in the space of all inner 3-gradings, it now easily follows that the geometry (G/P − , G/P + ) constructed

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in Section 2 is imbedded into the geometry from the theorem simply by taking the G-orbit of the base point. As mentioned above, this “universal geometric realization” turns out to be useful for giving a precise description of the intersections of the “chart domains” α and β for α, β ∈ X − and for calculating the corresponding “transition functions” (see [BN04]); calculations become similar to the case of Grassmannian geometries (Example (1) in Section 2.4 above), the diﬀerence being that the usual fractional linear transformations have to be replaced by certain fractional quadratic transformations (this diﬀerence corresponds to the fact that we are now working with true ﬂags instead of single subspaces). Using these purely algebraic results, one can now give necessary and suﬃcient conditions for deﬁning on (X + , X − ) the structure of a smooth manifold ([BN05]). The theory works nicely over general topological base ﬁelds and rings (cf. [Be08] for basic diﬀerential geometry and Lie theory in this general framework); complex or real Banach, Fr´echet or locally convex manifolds are special cases of it. For higher gradings, similar results hold (see [Ch09] and [Ch10]).

4 The geometry of states Let us come back to the analogies, mentioned in the Introduction, of the preceding constructions with methods of commutative and non-commutative geometry. In the language of classical physics, one wants to recover the pure states of a system (the point space M ) from its observables (the function algebra A = Reg(M, R)). The usual procedure is to look at the (maximal) ideals Ip = {f ∈ A| f (p) = 0} corresponding to points p ∈ M . Left or right ideals in associative algebras are generalized by inner ideals in Jordan theory. Therefore, we shall interpret states in generalized projective geometries as the geometric objects corresponding to inner ideals. 4.1 Intrinsic subspaces Assume (X + , X − ; ) is a linear pair geometry over the commutative ring K. Definition 4.1. A pair (Y + , Y − ) of subsets Y ± ⊂ X ± is called a subspace of (X + , X − ) if for all x ∈ Y ± there exists an element α ∈ Y ∓ such that xα, and if for every such pair (x, α) the set Yα := Y ∩ α is a linear subspace of (α , x) and Yx := Y ∩ x is a linear subspace of (x , α). A subset I ⊂ X + is called a state or an intrinsic subspace (in X + ) if I “appears linearly to all possible observers”: for all α ∈ X − with α ∩ I = ∅ and for all x ∈ α ∩ I, the set (α ∩ I, x) is a linear subspace of (α , x). For instance, if (X + , X − ) = (Gras1 (Kn+1 ), Grasn (Kn+1 )) is an ordinary projective geometry, then all projective subspaces of X + are intrinsic

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subspaces, and every aﬃne subspace of an aﬃnization of X + is obtained in this way. In contrast, for Grassmannian geometries of higher rank the situation becomes more complicated: only rather speciﬁc linear subspaces of the aﬃnization M (p, q; K) of Grasp (Kp+q ) are obtained in this way, namely the inner ideals. This observation generalizes to all geometries (X + , X − ) associated to a Jordan pair (V + , V − ): subspaces containing the base point (o+ , o− ) correspond to subpairs of (V + , V − ), and intrinsic subspaces of X + containing o+ to inner ideals I ⊂ V + , i.e., submodules such that T + (I, V − , I) ⊂ I (see [BL08]). The notions of, e.g.s minimal, maximal, principal inner ideals may also be suitably translated into a geometric language. (Minimal intrinsic subspaces will also be called intrinsic lines and maximal ones intrinsic hyperplanes. Readers coming from axiomatic geometry or from Jordan theory will consider the former as more fundamental and call them accordingly pure states, whereas readers coming from algebraic geometry or C ∗ -algebra theory would instead reserve this term for the latter ones.) The collection of states associated to a linear pair geometry can again be turned into a pair geometry by introducing the following transversality relation. Definition 4.2. Let I be an intrinsic subspace in X + and J one in X − . We say that I and J are transversal, and we write again IJ if (1) the pair (I, J ) is a subspace, (2) the linear pair geometry (I, J ) is faithful in the following sense. A linear pair geometry (Y + , Y − ) is called faithful if Y − is faithfully represented by its eﬀect of linearizing Y + , and vice versa: whenever α = β as sets and as linear spaces (with respect to some origin o), then α = β, and the dual property holds. Let us give some motivation for this deﬁnition (which does not appear in [BL08]). First of all, in general, a linear pair geometry need not be faithful (take, for instance, a pair of K-modules with trivial structure maps), but ordinary projective geometries over a ﬁeld are. (In case of geometries corresponding to Jordan pairs, faithfulness corresponds to non-degeneracy in the Jordan-theoretic sense.) Next note that, even if the geometry (X + , X − ) was faithful, the geometry (I, X − ) will in general not be faithful: there is a kernel, i.e., the equivalence relation “α ∼I β iﬀ α and β induce the same linear structure on I” will be non-trivial on X − . The condition IJ means, then, that the space J ⊂ X − is transversal to the ﬁbers of this equivalence relation, in the usual sense. (This is the geometric translation of the notion of complementation of inner ideals introduced in [LoN95]; the ﬁber of the equivalence relation ∼I corresponds to the kernel of an inner ideal introduced in [LoN95].) For instance, if I is a projective line in an ordinary projective space X + = Gras1 (Kn+1 ), corresponding to a 2-dimensional subspace I of Kn+1 , then J should be a projective subspace of the dual projective space X − = Grasn (Kn+1 ) such that diﬀerent elements of J deﬁne diﬀerent aﬃne

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lines in I. If J is the set of hyperplanes lying over some k-dimensional subspace of Kn+1 , then this means that I should contain some complement of J. Vice versa, J then also should contain some complement of I, and hence I and J have to be complements of each other in Kn+1 . Thus (I, J ) is a projective line which is faithfully imbedded in the projective geometry. Moreover, in this example we clearly recover the Grassmannian geometry of 2-spaces as the geometry of intrinsic lines in the 1-Grassmannian. Parts of the observations from the example generalize. Let S ± be the set of all intrinsic subspaces I in X ± that admit a transversal intrinsic subspace J in X ∓ . It is a natural question to ask whether (S + , S − ) is again a linear pair geometry, or under which conditions on (X + , X − ) and on the states we obtain one. For the time being, we have no general answers, but results from [BL08] point into the direction that, indeed, for the most interesting cases we get geometries (S + , S − ) that are again linear pair geometries. More precisely, if (X + , X − ) corresponds to a Jordan pair (V + , V − ) and we consider only states that are associated, via a Peirce-decomposition, to idempotents, we obtain geometries coming from 5-graded Lie algebras ([BL08], Theorem 5.8), and these are indeed linear pair geometries, according to Theorem 3.2. 4.2 Examples (1) Classical states revisited. Let M be a set and A = Fun(M, K) the Jordan algebra of all functions from M to K. It corresponds to the geometry (X , X ) with X = Fun(M, KP1 ). We claim that (if K is a ﬁeld) the set M can be recovered from the geometry X as the geometry of all pure states (intrinsic lines) running through the “zero function” f0 ≡ o, by associating to a point p ∈ M the intrinsic line Lp = {f : M → KP1 | f (x) = o if x = p} ⊂ Fun(M, KP1 ). Indeed, note ﬁrst that Lp is indeed a minimal intrinsic subspace since Lp ∼ = KP1 and the geometry (KP1 , KP1 ) itself does not admit any proper intrinsic subspaces that are not points. Now assume f : M → KP1 is a function taking non-zero values at two diﬀerent points p, q ∈ M . Then a look at the direct product geometry (KP1 , KP1 ) × (KP1 , KP1 ) shows that the intrinsic subspace generated by f contains at least all functions obtained from f by altering the values at p and q in an arbitrary way, and hence contains a homomorphic image of KP1 × KP1 and thus is not minimal, proving our claim. Of course, we may as well work with the maximal intrinsic subspaces Hp = {f : M → KP1 | f (p) = o } ⊂ Fun(M, KP1 ). This is certainly closer to the philosophy of algebraic geometry and to the usual imbedding of “classical” into “non-commative” geometry, when M is a smooth or algebraic manifold over R or C and we look at geometries of

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smooth or algebraic functions f : M → KP1 . Both points of view are dual to each other in the sense that one really should look at the geometry (S + , S − ), where one of the S ± is the collection of minimal, and the other of maximal intrinsic subspaces, each having a complement (in the sense of Deﬁnition 4.2) of the other type. (2) Grassmannian geometries X + = GrasB A (W ). Let us ﬁx a ﬂag in W of length two, f : 0 ⊂ f1 ⊂ f2 ⊂ W . Then the set of all subspaces of W that are “squeezed” by this ﬂag, If := {E ∈ X + | f1 ⊂ E ⊂ f2 }, is an intrinsic subspace of X , and if K is a ﬁeld and W is ﬁnite dimensional over K, then all intrinsc subspaces are of this form for a suitable ﬂag f (see [BL08], Theorem 3.11). Moreover, one can show that the intrinsic subspaces If and Ie are transversal in the sense deﬁned above if, and only if, the ﬂags e and f are transversal in the sense deﬁned in Section 3.1 above. Therefore, the geometry of intrinsic subspaces of the Grassmannian geometry is a geometry of ﬂags of length two and hence is a linear pair geometry. The case of ordinary projective geometry (i.e., A ∼ = K) is somewhat degenerate. In this case there is not much choice for the ﬁrst component f1 of the ﬂag f, and therefore the intrinsic subspace If is already determined by f2 alone; we get back the usual projective subspaces of a projective space. In all other cases, the geometry of intrinsic subspaces is a true ﬂag geometry corresponding to 5-gradings and hence is a linear, but no longer an aﬃne, pair geometry. (3) Quantum states. In the language of quantum mechanics (also widely used in non-commutative geometry), states are deﬁned as positive normalized linear functionals φ : A → C on a C ∗ -algebra (A, ∗). They form a convex set, whose extremal points are interpreted as pure states. These, in turn, admit a Jordan-theoretic interpretation via (primitive) idempotents. See [FK94] for the theory of ﬁnite-dimensional symmetric cones and their geometry; there is a vast literature on inﬁnite-dimensional generalizations; cf., e.g., [HOS84], [ER92], [Up85]. Whereas the notion of state just mentioned depends highly on the ordered structure of the base ﬁeld R (via positivity and convexity), this is not the case for the notion of inner ideal and intrinsic subspace, which hence seems to be more general and more geometric. Accordingly, it has been advocated to use inner ideals as a basic ingredient for an approach to quantum mechanics (see [F80]).

References [AF99] [Be00] [Be02]

Allison, B.N. and J.R. Faulkner, Elementary groups and invertibility for Kantor pairs, Comm. Algebra 27 (1999), no. 2, 519–556. Bertram, W., The geometry of Jordan- and Lie structures, Springer Lecture Notes in Mathematics 1754, Berlin 2000. Bertram, W., Generalized projective geometries: general theory and equivalence with Jordan structures, Adv. Geom. 2 (2002), 329–369.

Jordan Structures and Non-Associative Geometry [Be03] [Be04] [Be08] [BeKi09]

[BL08] [BN04] [BN05] [Bil04] [Ch09] [Ch10] [Co94] [ER92] [FK94] [F80] [HOS84] [Lo69] [Lo75] [Lo95] [LoN95] [McC04] [PS86] [Sm05] [Up85]

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Bertram, W., The geometry of null systems, Jordan algebras and von Staudt’s Theorem, Ann. Inst. Fourier 53 (2003) fasc. 1, 193–225. Bertram, W., From linear algebra via aﬃne algebra to projective algebra, Linear Algebra and its Applications 378 (2004), 109–134. Bertram, W., Diﬀerential geometry, Lie groups and symmetric spaces over general base ﬁelds and rings. Mem. AMS 192, 900 (2008), math.DG/ 0502168. Bertram, W. and M. Kinyon, Associative Geometries. I: Torsors, Linear Relations and Grassmannians. II: Involutions, the Classical Torsors, and their Homotopes. To appear in Journal of Lie Theory, arxiv: math.Ra/0903.5441 and math.Ra/0909.4438 Bertram, W. and H. L¨ owe, Inner ideals and intrinsic subspaces, Adv. in Geom. 8 (2008), 53–85. Bertram, W. and K.-H. Neeb, Projective completions of Jordan pairs. I: The generalized projective geometry of a Lie algebra, J. Algebra 277(2) (2004), 193–225; math.RA/0306272. Bertram, W. and K.-H. Neeb, Projective completions of Jordan pairs. II: Manifold structures and symmetric spaces, Geom. Dedicata 112 (2005), 73–113; math.GR/0401236. Biller, H., Continuous Inverse Algebras and Inﬁnite-Dimensional Linear Lie Groups, Habilitationsschrift, Technische Universit˝at Darmstadt, 2004. Chenal, J., Generalized ﬂag geometries and manifolds associated to short Z-graded Lie algebras, C.R.Acad. Sci. Paris, Ser. I 347 (2009) 21–25. Chenal, J.: G´eom´etries li´ees aux alg`ebres de Lie gradu´ees, thesis, Nancy 2010. Connes, A.: Non-commutative Geometry. Academic Press, San Diego 1994. Edwards, C.M. and G.T. R¨ uttimann, A characterization of inner ideals in JB ∗ -triples, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1049–1057. Faraut, J. and A. Koranyi, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994. Faulkner, J.R., Incidence Geometry of Lie Groups in Quantum Theory, Group theoretical methods in physics (Proc. Eigth Internat. Colloq., Kiryat Anavim, 1979) Ann. Israel Phys. Soc., 3, 73–79, Hilger, Bristol, 1980. Hanche-Olsen, H. and E. Størmer, Jordan Operator Algebras, Pitman, Boston, 1984. Loos, O., Symmetric Spaces I, Benjamin, New York, 1969. Loos, O., Jordan Pairs, Springer LNM 460, New York, 1975. Loos, O., Elementary groups and stability for Jordan pairs, K-Theory 9 (1995), no. 1, 77–116. Loos, O. and E. Neher, Complementation of inner ideals in Jordan pairs, J. Alg. 166 (1994), 255–295. Mc. Crimmon, K. A Taste of Jordan Algebras. Springer, New York, 2004. Pressley, A. and G. Segal, Loop Groups, Clarendon Press, Oxford 1986. Smirnov, O.N., Imbedding of Lie triple systems into Lie algebras, preprint, 2005. arxiv: math/0906.1170 Upmeier, H., Symmetric Banach Manifolds and Jordan C ∗ -algebras, North-Holland Mathematics Studies, Amsterdam, 1985.

For electronic versions of some of the above-mentioned papers that are not available on the arXiv, see the Jordan preprint server at the address http://homepage.uibk.ac.at/∼c70202/jordan/index.html

Direct Limits of Infinite-Dimensional Lie Groups Helge Gl¨ ockner Universit¨ at Paderborn, Institut f¨ ur Mathematik, Warburger Str. 100, 33098 Paderborn, Germany, [email protected]

Ë

Summary. Many inﬁnite-dimensional Lie groups G of interest can be expressed as the union G = n∈N Gn of an ascending sequence G1 ⊆ G2 ⊆ · · · of (ﬁnite- or inﬁnite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples and explain what the general theory tells us about them. Key words: Lie group, direct limit, inductive limit, ascending sequence, ascending union, directed union, Silva space, regularity, initial Lie subgroup, homotopy group, small subgroup. 2000 Mathematics Subject Classiﬁcations: Primary 22E65. Secondary 26E15, 46A13, 46G20, 46T05, 46T20, 46T25, 54B35, 54D50, 55Q10, 58B05, 58D05.

1 Introduction Many inﬁnite-dimensional Lie groups G can be expressed as the union G = n∈N Gn of a sequence G1 ⊆ G2 ⊆ · · · of (ﬁnite- or inﬁnite-dimensional) Lie groups, such that the inclusion maps jn : Gn → G and jm,n : Gn → Gm (for n ≤ m) are smooth homomorphisms. Typically, the steps Gn are Lie groups of a simpler type, and one hopes to (and often does) deduce results concerning G from information available for the Lie groups Gn . The goals of this article are twofold: • •

To survey general results on ascending unions of Lie groups and their properties; To collect concrete classes of examples and explain how the general theory specializes in these cases.

One typical class of examples is given by the groups Diﬀ c (M ) of smooth diﬀeomorphisms φ : M → M of σ-compact, ﬁnite-dimensional smooth manifolds M K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_8, © Springer Basel AG 2011

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which are compactly supported in the sense that the set {x ∈ M : φ(x) = x} has compact closure. The group operation is composition of diﬀeomorphisms. It is known that Diﬀ c (M ) is a Lie group (see [Mc80] or [Gl02d]). Furthermore, Diﬀ Kn (M ) Diﬀ c (M ) = n∈N

for each exhaustion K1 ⊆ K2 ⊆ · · · of M by compact sets (with Kn in the interior of Kn+1 ), where Diﬀ Kn (M ) is the Lie group of smooth diﬀeomorphisms of M supported in Kn . The manifold structure of Diﬀ c (M ) is modelled on the space Vc (M ) of compactly supported smooth vector ﬁelds, which is an LF-space with a complicated topology. In contrast, Diﬀ Kn (M ) is modelled on the space VKn (M ) of smooth vector ﬁelds supported in Kn , which is a Fr´echet space. Many speciﬁc tools of inﬁnite-dimensional calculus can be applied to VKn (M ), e.g., to clarify diﬀerentiability questions for functions on this space. In other typical cases, each Gn is ﬁnite dimensional (a particularly well-understood situation) or modelled on a Banach space, whence again special tools are available to deal with the Lie groups Gn (but not a priori for G). Besides diﬀeomorphism groups, we shall also discuss the following major classes of examples (described in more detail in Section 6): ∞ • The “test function groups” Cc∞ (M, H) = n∈N CK (M, H) of compactly n supported Lie group-valued smooth mappings on a σ-compact smooth manifold M = n∈N Kn ; ∗ n • Weak direct products n∈N Hn := n∈N k=1 Hk of Lie groups Hn ; • Unions A× = n∈N A× n of unit groups of Banach algebras A1 ⊆ A2 ⊆ · · · ; • The groups Germ(K, H) of germs of analytic mappings on open neighbourhoods of a compact subset K of a metrizable complex locally convex space, with values in a complex Banach–Lie group H; 1 • The group H ↓s (K, F ) = t>s H t (K, F ) = n∈N H s+ n (K, F ), where K is a compact smooth manifold, s ≥ dim(K)/2, F a ﬁnite-dimensional Lie group and H t (K, F ) ⊆ C(K, F ) the integral subgroup whose Lie algebra is the Sobolev space H t (K, L(F )) of functions with values in the Lie algebra L(F ) of F . For s = dim(K)/2, the Lie group H ↓s (K, F ) is particularly interesting, because a Hilbert–Lie group H s (K, F ) is not available in this case. In some situations, H ↓s (K, F ) may serve as a substitute for the missing group. We shall also discuss the group GermDiﬀ(K, X) of germs of analytic diﬀeomorphisms γ around a compact set K in a ﬁnite-dimensional complex vector space X, such that γ|K = idK . This group is not considered as a union of groups, but as a union of Banach manifolds M1 ⊆ M2 ⊆ · · · . Among others, we shall discuss the following topics in our general setting (and for the preceding examples): •

Direct limit properties of ascending unions;

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• • •

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Homotopy groups of ascending unions; When ascending unions are regular Lie groups in Milnor’s sense; Questions concerning subgroups of ascending unions.

We now describe the main problems and questions in more detail, together with some essential concepts. As a rule, references to the literature, answers (and partial answers) will only be given in later sections. 1.1 Direct limit properties of ascending unions Consider a Lie group G which is an ascending union G = groups, and a map f : G → X. It is natural to ask:

n∈N

Gn of Lie

(a) If X is a smooth manifold (modelled on a locally convex space) and f |Gn is smooth for each n ∈ N, does it follow that f is smooth? (b) If X is a topological space and f |Gn is continuous for each n ∈ N, does it follow that f is continuous? (c) If X is a Lie group, f is a homomorphism of groups and f |Gn is smooth for each n ∈ N, does it follow that f is smooth? (d) If X is a topological group, f is a homomorphism and f |Gn is continuous for each n ∈ N, does it follow that f is continuous? As we shall see, (a) and (b) are frequently not true (unless compactness can be brought into play), while (c) and (d) hold for our typical examples. The preceding questions can be recast in category-theoretic terms: They amount to asking if G is the direct limit lim Gn in the categories of smooth −→ manifolds, topological spaces and Lie groups, resp., topological groups (see 2.8). The relevant concepts from category theory will be recalled in Section 2. Questions (b) and (d) can be asked just as well if G and each Gn are merely a topological group, and each inclusion map is a continuous homomorphism. Essential progress concerning direct limits of topological groups and their relations to direct limits of topological spaces was achieved in the last ten years, notably by N. Tatsuuma, E. Hirai, T. Hirai and N. Shimomura (see [TSH98] and [HST01]) as well as A. Yamasaki [Ya98]. In Section 3, we recall the most relevant results. 1.2 Existence of direct limit charts: an essential hypothesis Meaningful results concerning the preceding topics can only be expected under additional hypotheses. For instance, our general setting includes the situation where each Gn is discrete but G is not (as we only assume that the inclusion maps Gn → G are smooth). In this situation, algebraic properties of the groups Gn (like simplicity or perfectness) pass to G, but we cannot expect to gain information concerning the topological or diﬀerentiable structure of G from information on the groups Gn .

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A very mild additional hypothesis is the existence of a direct limit chart. Roughly speaking, this is a chart of G juxtaposed from charts of the Lie groups Gn . The formal deﬁnition reads as follows (cf. [Gl07b, Deﬁnition 2.1]). Deﬁnition. A Lie group G = n∈N Gn is said to admit a weak direct limit chart if there exists n0 ∈ N, charts φn : Un → Vn from open identity neighbourhoods Un ⊆ Gn onto open 0-neighbourhoods Vn ⊆ L(Gn ) in the tangent space L(Gn ) := T1 (Gn ) at 1 for n ≥ n0 and a chart φ : U → V from an open identity neighbourhood U ⊆ G onto an open 0-neighbourhood V ⊆ L(G), such that (a) U = n≥n0 Un and Un ⊆ Un+1 for each integer n ≥ n0 ; and (b) φn+1 |Un = L(jn+1,n ) ◦ φn and φ|Un = L(jn ) ◦ φn for each n ≥ n0 . If, furthermore, L(G) = lim L(Gn ) as a locally convex space,1 then −→ G = n∈N Gn is said to admit a direct limit chart. Note that (b) implies that the linear maps L(jn ) and L(jn+1,n ) are injective on some 0-neighbourhood and thus injective. Hence, identifying L(Gn ) with its image under L(jn ) in L(G), we can rewrite (b) as (b) φ|Un = φn and φn+1 |Un = φn , for each n ≥ n0 . Furthermore, we now simply have V = n≥n0 Vn . To assume the existence of a direct limit chart is a natural requirement, which is satisﬁed by all of our main examples. It provides a link between the topologies (resp., manifold structures) on G and the Lie groups Gn , and will be encountered in connection with most of the topics from above. 1.3 Homotopy groups of ascending unions of Lie groups Given a Lie group G = n∈N Gn , it is natural to ask if its k-th homotopy group can be calculated in terms of the homotopy groups πk (Gn ) in the form πk (G) = lim πk (Gn ) , (1) −→ for each k ∈ N0 . This is quite obvious if G = n∈N Gn is compactly regular in the sense that each compact subset K of G is a compact subset of some Gn (see [Gl10, Proposition 3.3]; cf. [Gl05b, Remark 3.9] and [Ne04c, Lemma A.7] for special cases, as well as works on stable homotopy theory and K-theory). There is another, non-trivial condition: If G = n∈N Gn admits a weak direct limit chart, then(1) holds [Gl10, Theorem 1.2]. A variant of this condition even applies if n∈N Gn is merely dense in G (see Theorem 1.13 in [Gl10]). Moreover, ascending unions can be replaced with directed unions over uncountable families, and Lie groups with manifolds (see Section 9). These results are based on approximation arguments. Analogous results for open subsets of locally convex spaces are classical [Pa66]. 1

Here, we use the bonding maps L(jm,n ) : L(Gn ) → L(Gm ) and the limit maps L(jn ) : L(Gn ) → L(G).

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We mention that knowledge of π0 (G) = G/G0 , the fundamental group π1 (G) and π2 (G) is essential for the extension theory of G. It is needed to → G → 1 of G with abelian understand the Lie group extensions 1 → A → G kernels, by recent results of K.-H. Neeb (see [Ne02b], [Ne04b] and [Ne07]). 1.4 Regularity in Milnor’s sense Roughly speaking, a Lie group G (modelled on a locally convex space) is called a regular Lie group if all diﬀerential equations on G which are of relevance for Lie theory can be solved, and their solutions depend smoothly on parameters. To make this more precise, given g, h ∈ G and v ∈ Th (G), let us write g · v := (Th λg )(v) ∈ Tgh (G), where λg : G → G, x → gx denotes left translation by g. Deﬁnition. A Lie group G modelled on a locally convex space is called a regular Lie group (in Milnor’s sense) if for each smooth curve γ : [0, 1] → L(G), there exists a (necessarily unique) smooth curve η = ηγ : [0, 1] → G (a “product integral”) which solves the initial value problem η(0) = 1

(2)

η (t) = η(t) · γ(t) for all t ∈ [0, 1]

(3)

(with 1 ∈ G the identity element), and the “evolution map” evol: C ∞ ([0, 1], L(G)) → G ,

evol(γ) := ηγ (1)

(4)

is smooth (see [Mr84], [GN10] and [Ne06]). Regularity is a useful property which provides a link between G and its Lie algebra. In particular, regularity ensures the existence of a smooth exponential map expG : L(G) → G, i.e., a smooth map such that, for each v ∈ L(G), γv : R → G ,

t → expG (tv)

is a homomorphism of groups with initial velocity γv (0) = v (cf. [Mr84]). The modelling space E of a regular Lie group is necessarily Mackey 1 complete in the sense that the Riemann integral 0 γ(t) dt exists in E for each smooth curve γ : R → E (cf. Lemma A.5 (1) and p. 21 in [NW08]). Lie groups modelled on non-Mackey complete locally convex spaces need not even have an exponential map. For example, this pathology occurs for group G = A× of invertible elements in the normed algebra A ⊆ C[0, 1] of all restrictions to [0, 1] of rational functions without poles in [0, 1] (with the supremum norm). Because A× is an open subset of A, it is a Lie group, and it is not hard to see that a smooth homomorphism γv : R → A× with γ (0) = v exists for v ∈ A = L(A× ) only if v is a constant function [Gl02c, Proposition 6.1]. Further information concerning Mackey completeness can be found in [KM97]. At the time of writing, it is unknown whether non-regular Lie groups modelled on Mackey complete locally convex spaces exist. However, there

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is no general method of proof; for each individual class of Lie groups, very speciﬁc arguments are required to verify regularity. It is natural to look for conditions ensuring that a union G = n∈N Gn is a regular Lie group if each Gn is one. Already the case of ﬁnite-dimensional Lie groups Gn is not easy [Gl05b]. In Section 8, we preview work in progress concerning the general case. We also describe a construction which might lead to non-regular Lie groups (Proposition 8.7). The potential counterexamples are weak direct products of suitable regular Lie groups. 1.5 Subgroups of direct limit groups It is natural to try to use information concerning the subgroups of Lie groups Gn to deduce results concerning the subgroups of a Lie group G = n∈N Gn . Aiming at a typical example, let us recall that a topological group G is said to have no small subgroups if there exists an identity neighbourhood U ⊆ G containing no subgroup of G except for the trivial subgroup. Although ﬁnite-dimensional (and Banach–) Lie groups do not have small subgroups, already for Fr´echet–Lie groups the situation changes: The additive group of the Fr´echet space RN has small subgroups. In fact, every 0-neighbourhood contains ]−r, r[n ×R{n+1,n+2,...} for some n ∈ N and r > 0. It therefore contains the non-trivial subgroup {0} × R{n+1,n+2,...} . It is natural to ask whether a Lie group G = n∈N Gn does not have small subgroups if none of the Lie groups Gn has small subgroups. In Section 10, we describe the available answers to this question and various other results concerning subgroups of direct limit groups. 1.6 Constructions of Lie group structures on ascending unions So far, we assumed that G is already equipped with a Lie group structure. Sometimes, only an ascending sequence G1 ⊆ G2 ⊆ · · · of Lie groups is given such that all inclusion maps Gn → Gn+1 are smooth homomorphisms. It is then natural to ask whether the union G = n∈N Gn can be given a Lie group structure making each inclusion map Gn → G a smooth homomorphism.2 We shall also discuss this complementary problem (in Section 5). If each Gn is ﬁnite dimensional, then a Lie group structure on G is always available. 1.7 Properties of locally convex direct limits To enable an understanding of direct limits of Lie groups, an understanding of various properties of locally convex direct limits is essential, i.e., of direct limits in the category of locally convex spaces. For instance, we shall see that if a Lie group G = n∈N Gn admits a direct limit chart, then G = lim Gn as −→

2

Or even making G the direct limit lim Gn in the category of Lie groups. −→

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a topological space if and only if L(G) = lim L(Gn ) as a topological space. −→ The latter property is frequently easier to prove (or refute) than the ﬁrst. Also compact regularity of G (as in 1.3 above) can be checked on the level of the modelling spaces (see Lemma 6.1). Another property is useful: Consider a locally convex space E which is a union n∈N En of locally convex spaces, such that all inclusion maps are continuous linear maps. We say that E is regular (or boundedly regular, for added clarity) if every bounded subset of E is a bounded subset of some En . If one wants to prove that a Lie group n Gn is regular in Milnor’s sense, then it helps a lot if one knows that L(G) = n L(Gn ) is compactly or boundedly regular (see Section 8). 1.8 Further comments, and some historical remarks The most typical examples of direct limits of ﬁnite-dimensional Lie groups are unions of classical groups like GL (C) = GL (C) and its sub∞ n n∈N groups GL (R) = GL (R), O (R) = O (R) and U (C) = ∞ n ∞ n ∞ n∈N n∈N n∈N Un (C), where A ∈ GLn (C) is identiﬁed with the block matrix A 0 0 1 in GLn+1 (C). Thus GL∞ (C) is the group of invertible matrices of countable size which diﬀer from the identity matrix only at ﬁnitely many entries. Groups of this form (and related ascending unions of homogeneous spaces) have been considered for a long time in (stable) homotopy theory and K-theory. Furthermore, results concerning their representation theory can be traced back to the 1970s (more details are given below). However, only the group structure or topology was relevant for these studies. Initially, no attempt was made to consider them as Lie groups. The Lie group structure on GL∞ (C) was ﬁrst described in [Mr82] and that on U∞ (C) and O∞ (R) was mentioned (cf. also p. 1053 in the survey article [Mr84]). The ﬁrst systematic discussion of direct limits of ﬁnite-dimensional Lie groups was given in [NRW91] and [NRW93]. Notably, a Lie group structure on G = lim Gn was constructed there under technical conditions which ensure, −→ in particular, that lim expGn : lim L(Gn ) → lim Gn , x → expGn (x) for all n ∈ N, x ∈ L(Gn ) −→

−→

−→

is a local homeomorphism at 0 (see Section 5 for the sketch of a more general construction from [Gl05b]). Moreover, situations were described in [NRW93] where ascending unions of Lie groups (or the corresponding Lie algebras) can be completed with∞respect to some coarser topology. Also, the Lie group Cc∞ (M, H) = n∈N CK (M, H) is brieﬂy discussed in [NRW93] (for ﬁniten dimensional H), and in [AHK93]. Test function groups with values in possibly inﬁnite-dimensional Lie groups were treated in [Gl02b]. Compare [ACM89]

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for an early discussion of gauge groups and automorphism groups of principal bundles over non-compact manifolds in the inequivalent “convenient setting” of inﬁnite-dimensional calculus (cf. also [KM97]). The ﬁrst construction of the Lie group structure on Diﬀ c (M ) was given in [Mc80], as part of a discussion of manifold structures on spaces of mappings between non-compact manifolds. Groups of germs of complex analytic diﬀeomorphisms of Cn around 0 were studied in [Pi77]. The real analytic analogue was discussed in [Le94], and groups of germs of more general diﬀeomorphisms in [KR01]. Further recent works will be described later. It should be stressed that the current article focusses on direct limit groups as such, i.e., on their structure and properties. Representation theory and harmonic analysis on such groups are outside its scope. For completeness, we mention that the study of (irreducible) unitary representations of ascending unions of ﬁnite groups started in the 1960s (see [Th64a] and [Th64b]), notably for the symmetric group S∞ := lim Sn . Representations −→ of direct limits of ﬁnite-dimensional Lie groups were ﬁrst investigated in the 1970s (see [Vo76] for representations of U∞ (C), and [KS77] for representations of U∞ (C) and SO∞ (R)). Ol’shanski˘ı [Ol83] studied representations for inﬁnite-dimensional spherical pairs like (GL∞ (R), SO∞ (R)),(GL∞ (C),U∞ (C)) and (U∞ (C), SO∞ (R)). The representation theory of direct limits of both ﬁnite groups and ﬁnite-dimensional Lie groups remains an active area of research. Representations of O(∞, ∞), U(∞, ∞) and Sp(∞, ∞) were studied in [Dv02] using inﬁnite-dimensional adaptations of Howe duality. Novel results concerning the representation theory of U∞ (C) and S∞ were obtained in [Ol03] and [KOV04], respectively. Versions of the Bott–Borel–Weil theorem for direct limits of ﬁnite-dimensional Lie groups were established in [NRW01] and [DPW02] (in a more algebraic setting). J. A. Wolf also investigated principal series representations of suitable direct limit groups [Wo05], as well as the regular representation on some direct limits of compact symmetric spaces [Wo08]. The paper [AK06] discusses representations of an inﬁnite-dimensional subgroup of unipotent matrices in GL∞ (R). Finally, a version of Bochner’s theorem for inﬁnite-dimensional spherical pairs was obtained in [Ra08]. There also is a body of literature devoted to irreducible representations of diﬀeomorphism groups of (compact or) non-compact manifolds, as well as quasi-invariant measures and harmonic analysis thereon (see, e.g., [VGG75], [Ki81], [Hi93], [Sh01] and [Sh05]). Representations of Cc∞ (M, H) were studied by R. S. Ismagilov [Is76] and in [AHK93]. Such groups, Diﬀ c (M ) and semidirect products thereof, arise naturally in mathematical physics [Go03]. Direct limits of ﬁnite-dimensional Lie groups are also encountered as dense subgroups of some interesting Banach–Lie groups (like the group U2 (H) of unitary operators on a complex Hilbert space H which diﬀer from idH by a Hilbert–Schmidt operator) and other groups of operators. This frequently enables the calculation of the homotopy groups of such groups (see [Pa65], also [Ne02a]), exploiting that the homotopy groups of many direct limits of

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classical groups (like U∞ (C)) can be determined using Bott periodicity. Dense unions of ﬁnite-dimensional Lie groups are also useful in representation theory (see [Ne98] and [Ne04c]). We mention a more specialized result: For very particular classes of direct limits G of ﬁnite-dimensional Lie groups, a classiﬁcation is possible which uses the homotopy groups of G (notably π1 (G) and π3 (G)); see [Ku06]. In contrast to direct (or inductive) limits, the dual notion of an inverse (or projective) limit of Lie groups was used much earlier in inﬁnite-dimensional Lie theory. Omori’s theory of ILB-Lie groups (which are inverse l imits of B anach manifolds) gave a strong impetus to the development of the area in the late 1960s and early 1970s (see [Om97] and the references therein). Many important examples of inﬁnite-dimensional Lie groups could be discussed in this approach, e.g., the group C ∞ (K, H) = k∈N0 C k (K, H) of smooth maps on a compact manifold K with values in a ﬁnite-dimensional Lie group H, and the group Diﬀ(K) = k∈N Diﬀ k (K) of C ∞ -diﬀeomorphisms of a compact manifold. The passage from compact to non-compact manifolds naturally leads to the consideration of direct limits of compactly supported objects.

2 Preliminaries, terminology and basic facts General conventions. We write N := {1, 2, . . .}, and N0 := N∪{0}. As usual, R and C denote the ﬁelds of real and complex numbers, respectively. If (E, .) is a normed space, x ∈ E and r > 0, we write BrE (x) := {y ∈ E : y − x < r}. Topological spaces, topological groups and locally convex topological vector spaces are not assumed Hausdorﬀ. However, manifolds are assumed Hausdorﬀ, and whenever a locally convex space serves as the domain or range of a diﬀerentiable map, or as the modelling space of a Lie group or manifold, it is tacitly assumed Hausdorﬀ. Moreover, all compact and all locally compact topological spaces are assumed Hausdorﬀ. We allow non-Hausdorﬀ topologies because direct limits are much easier to describe if the Hausdorﬀ property is omitted (further explanations will be given at the end of this section). Inﬁnite-dimensional calculus. We are working in the setting of Keller’s Cck theory [Ke74], in a topological formulation that avoids the use of convergence structures (as in [Mc80], [Mr84], [Gl02a], [Ne06] and [GN10]). For more information on analytic maps, see, e.g., [Gl02a], [GN10] and (for K = C) [BS71]. 2.1 Let K ∈ {R, C}, r ∈ N ∪ {∞}, E and F be locally convex K-vector spaces and f : U → F be a map on an open set U ⊆ E. If f is continuous, we say that f is C 0 . We call f a CKr -map if f is continuous, the iterated real (resp., complex) directional derivatives dk f (x, y1 , . . . , yk ) := (Dyk · · · Dy1 f )(x) exist for all k ∈ N such that k ≤ r, x ∈ U and y1 , . . . , yk ∈ E, and the maps dk f : U × E k → F so obtained are continuous. If K is understood, we write

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C r instead of CKr . If f is C ∞ , we also say that f is smooth. If K = R, we say → FC on an that f is real analytic (or CRω ) if f extends to a CC∞ -map f : U open neighbourhood U of U in the complexiﬁcation EC of E. 2.2 We mention that a map f : E ⊇ U → F is CC∞ if and only if it is complex analytic, i.e., f is continuous and for each x ∈ U , there exists a 0-neighbourhood Y ⊆ E with x + Y ⊆ U and continuous homogeneous polynomials pn : E → F of degree n such that ∞

pn (y) . (∀y ∈ Y ) f (x + y) = n=0

Complex analytic maps are also called C-analytic or CCω . 2.3 It is known that compositions of composable CKr -maps are CKr , for each r ∈ N0 ∪ {∞, ω}. Thus a CKr -manifold M modelled on a locally convex K-vector space E can be deﬁned in the usual way, as a Hausdorﬀ topological space, together with a maximal set of homeomorphisms from open subsets of M to open subsets of E, such that the domains cover M and the transition maps are CKr . Given r ∈ {∞, ω}, a CKr -Lie group is a group G, equipped with a structure of CKr -manifold modelled on a locally convex space, such that the group multiplication and group inversion are CKr -maps. Unless the contrary is stated, we consider CK∞ -Lie groups. Throughout the following, the words “manifold” and “Lie group” will refer to manifolds and Lie groups modelled on locally convex spaces. We shall write Tx M for the tangent space of a manifold M at x ∈ M and L(G) := T1 (G) for the (topological) Lie algebra of a Lie group G. Given a CK1 -map f : M → N between CK1 -manifolds, we write Tx f : Tx M → Tf (x) N for the tangent map at x ∈ M . Given a smooth homomorphism f : G → H, we let L(f ) := T1 (f ) : L(G) → L(H). Direct limits. We recall terminology and basic facts concerning direct limits. 2.4 (General deﬁnitions). Let (I, ≤) be a directed set, i.e., I is a non-empty set and ≤ a partial order on I such that any two elements have an upper bound. Recall that a direct system (indexed by (I, ≤)) in a category A is a pair S := ((Xi )i∈I , (φji )j≥i ), where each Xi is an object of A and φji : Xi → Xj a morphism such that φii = idXi and φkj ◦ φji = φki , for all elements k ≥ j ≥ i in I. A cone over S is a pair (X, (φi )i∈I ), where X is an object of A and each φi : Xi → X a morphism such that φj ◦ φji = φi whenever j ≥ i. A cone (X, (φi )i∈I ) is a direct limit of S (and we write (X, (φi )i∈I ) = lim S or X = −→ lim Xi ), if for every cone (Y, (ψi )i∈I ) over S, there exists a unique morphism −→ ψ : X → Y such that ψ ◦ φi = ψi for all i ∈ I. If T = ((Yi )i∈I , (ψji )j≥i ) is another direct system over the same index set, (Y, (ψi )i∈I ) a cone over T and (ηi )i∈I a family of morphisms ηi : Xi → Yi which is compatible with the direct systems in the sense that ψji ◦ηi = ηj ◦φji for all j ≥ i, then (Y, (ψi ◦ηi )i∈I ) is a cone over S. We write lim ηi for the induced morphism ψ : X → Y , determined −→ by ψ ◦ φi = ψi ◦ ηi .

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Direct limits in the categories of sets, groups and topological spaces are particularly easy to understand, and we discuss them now. Direct limits of topological groups (which are a more diﬃcult topic) and direct limits of locally convex spaces will be discussed in Sections 3 and 4, respectively. We concentrate on direct sequences (viz., the case I = N) and actually on ascending sequences, to avoid technical complications. This is the more justiﬁed because (except for some counterexamples) hardly anything is known about direct limits of direct systems of Lie groups which do not admit a coﬁnal subsequence. 2.5 (Ascending unions of sets). If X1 ⊆ X2 ⊆ · · · is an ascending sequence of sets, let φm,n : Xn → Xm be the inclusion map for m, n ∈ N with m ≥ n. Then S := ((Xn )n∈N , (φm,n )m≥n ) is a direct system in the category SET of sets and maps. Deﬁne X := n∈N Xn and let φn : Xn → X be the inclusion map. Then (X, (φn )n∈N ) is a cone over S in SET. A sequence of maps ψn : Xn → Y to a set Y gives rise to a cone (Y, (ψn )n∈N ) if and only if ψm |Xn = ψn

for all m, n ∈ N with m ≥ n.

Then ψ : X → Y , ψ(x) := ψn (x) if n ∈ N and x ∈ Xn is a well-deﬁned map, and is uniquely determined by the requirement that ψ ◦ φn = ψ|Xn = ψn for each n ∈ N. Thus (X, (φn )n∈N ) = lim S in SET. −→

2.6 (Direct limits of groups). If each Xn is a group in the situation of 2.5 and each φm,n a homomorphism, then S is a direct system in the category G of groups and homomorphisms. If x, y ∈ X, there exists n ∈ N such that x, y ∈ Xn . We deﬁne the product of x and y in X as their product in Xn , i.e., x · y = φn (x) · φn (y) := φn (x · y). Since each φm,n is a homomorphism, x · y is independent of the choice of n, and it is clear that the product so deﬁned makes X a group and each φn : Xn → X a homomorphism. If (Y, (ψn )n∈N ) is a cone over S in G, let ψ : X → Y be the unique map such that ψ ◦ φn = ψn for each n ∈ N, as in 2.5. Given x, y ∈ X, say x, y ∈ Xn , we then have ψ(xy) = ψ(φn (xy)) = ψn (xy) = ψn (x)ψn (y) = ψ(x)ψ(y), whence ψ is a homomorphism. Thus (X, (φn )n∈N ) = lim S in G. −→

If S = ((Xn )n∈N , (φm,n )) is a direct sequence of groups (with φm,n : Xn → Xm not necessarily injective), then Kn := m≥n ker(φm,n ) is a normal subgroup of Xn . Consider the quotient groups Gn := Xn /Kn , the canonical quotient maps qn : Xn → Gn and the homomorphisms ψm,n : Gn → Gm determined by ψm,n ◦ qn = qm ◦ φm,n . Then each ψm,n is injective, and it is clear that the direct limit (G, (ψn )n∈N ) of the “injective quotient system” ((Gn )n∈N , (ψm,n )) yields a direct limit (G, (ψn ◦ qn )n∈N ) of S (cf. [NRW91, §3]). 2.7 (Direct limits of topological spaces). If each Xn is a topological space in the situation of 2.5 and each φm,n : Xn → Xm a continuous map, we equip X = n∈N Xn with the ﬁnest topology ODL making each inclusion

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map φn : Xn → X continuous (called the direct limit topology). Thus U ⊆ X is open (resp., closed) if and only if φ−1 n (U ) = U ∩ Xn is open (resp., closed) in Xn for each n ∈ N. Then (X, (φn )n∈N ) = lim S in the category TOP of −→ topological spaces and continuous maps. To see this, let (Y, (ψn )n∈N ) be a cone over S in TOP. Let ψ : X → Y be the unique map with ψ ◦ φn = ψn for each n. If U ⊆ Y is open, then ψ −1 (U ) ∩ Xn = (ψ|Xn )−1 (U ) = (ψn )−1 (U ) is open in Xn , for each n. Hence ψ −1 (U ) is open in X and thus ψ is continuous. The direct system S is called strict if each φn is a topological embedding (i.e., Xn+1 induces the topology of Xn ). Then also the inclusion map φn : Xn → X is a topological embedding for each n [NRW93, Lemma A.5]. It is also known that X has the separation property T1 if each Xn is T1 (see, e.g., [Gl05b, Lemma 1.7 (a)]). And in the case of a direct sequence X1 ⊆ X2 ⊆ · · · of locally compact spaces Xn , the direct limit topology on n∈N Xn is Hausdorﬀ (as observed in [Gl05b, Lemma 1.7 (c)], the strictness hypotheses in [Ha71, Proposition 4.1 (ii)] and [Gl03a, Lemma 3.1] is unnecessary). First remarks on ascending unions of Lie groups and direct limits. Consider an ascending sequence G1 ⊆ G2 ⊆ · · · of CK∞ -Lie groups, such that the inclusion maps jm,n : Gn → Gm are C ∞ -homomorphisms for all m, n ∈ N with m ≥ n. Then S := ((Gn )n∈N , (jm,n )m≥n ) is a direct system in the category LIEK of CK∞ -Lie groups and CK∞ -homomorphisms. One would not expect that S always has a direct limit in the category of CK∞ -Lie groups (although no counterexamples are known at the time of writing). Also, there is no general construction principle for a Lie group structure on n∈N Gn such that all inclusion maps jn : Gn → G are CK∞ -homomorphisms (unless restrictive conditions are imposed, as in Section 5). 2.8 However, in many concrete cases we are given such a Lie group structure on G := n∈N Gn . Then (G, (jn )n∈N ) is a cone over S in LIEK , and it is natural to ask if G = lim Gn as a Lie group. A sequence of CK∞ -homomorphisms −→ fn : Gn → H to a CK∞ -Lie group H is a cone over S if and only if fm |Gn = fn

for all m, n ∈ N with m ≥ n.

Then f : G → H, f (x) := fn (x) if n ∈ N and x ∈ Gn is a well-deﬁned homomorphism. This map is uniquely determined by the requirement that f ◦ jn = f |Gn = fn for each n ∈ N. Therefore, (G, (jn )n∈N ) = lim S holds in −→ LIEK if and only if each f of the preceding form is CK∞ . A similar argument applies if H is a topological group, smooth manifold or topological space. Thus questions (a)–(d) posed in 1.1 amount to asking: Does G = lim Gn −→ hold : (a) in the category of smooth manifolds (modelled on locally convex spaces) and smooth maps between them? (b) in the category of topological spaces and continuous maps?

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(c) in the category LIEK of Lie groups? (d) in the category of topological groups and continuous homomorphisms? The Hausdorﬀ property. We allow non-Hausdorﬀ topologies because direct limits are much easier to describe if the Hausdorﬀ property is omitted. In fact, we have already seen that it is always possible to topologize a union X = n∈N Xn of topological spaces in such a way that it becomes the direct limit lim Xn in the category of topological spaces (see 2.7), and likewise a −→ union of topological groups (resp., locally convex spaces) can always be made the direct limit in the category of topological groups, resp., locally convex spaces (see Sections 3 and 4). A mere union X = n∈N Xn is a very concrete object, and easy to work with. In contrast, if each Xn is Hausdorﬀ, then the direct limit lim Xn in the −→ category of Hausdorﬀ topological spaces (resp., Hausdorﬀ topological groups, resp.,Hausdorﬀ locally convex spaces) can only be realized as a quotient of X = n∈N Xn in general, and is a much more elusive object in this case. Luckily, in all situations of relevance for this article, X from above injects continuously into a Lie group and thus X is Hausdorﬀ. Then automatically X also is the direct limit in the category of Hausdorﬀ topological spaces (resp., Hausdorﬀ topological groups, resp., Hausdorﬀ locally convex spaces).

3 Direct limits of topological groups As an intermediate step towards the study of Lie groups, let us consider a sequence G1 ⊆ G2 ⊆ · · · of topological groups, such that all inclusion maps Gn → Gn+1 are continuous homomorphisms. We make G = n∈N Gn the direct limit group (as in 2.6) and give it the ﬁnest group topology ODLG making each inclusion map Gn → G continuous. Then G = lim Gn in the −→ category of (not necessarily Hausdorﬀ) topological groups. Moreover, if each Gn is Hausdorﬀ, then the factor group of G modulo the closure {1} ⊆ G is the direct limit in the category of Hausdorﬀ topological groups.3 Unfortunately, the preceding description of the topology ODLG on the direct limit topological group is not at all concrete. Various questions are natural (and also relevant for our studies of Lie groups): Does ODLG coincide with the direct limit topology ODL (as in 2.7)? Can ODLG be described more explicitly? Given a group topology on G = n∈N Gn , how can we prove that it agrees with ODLG ? We now give some answers to the ﬁrst and last questions. An answer to the second question, namely the description of ODLG as a “bamboo-shoot” topology, can be found in [TSH98] and [HST01] (under suitable hypotheses).

3

If G is Hausdorﬀ, then no passage to the quotient is necessary.

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Comparison of ODL and ODLG . It is clear from the deﬁnition that the direct limit topology ODL is ﬁner than ODLG . Moreover, ODL may be properly ﬁner than ODLG , as emphasized by Tatsuuma et al. [TSH98].4 To understand this diﬃculty, let ηn : Gn → Gn , x → x−1 and η : G → G be the inversion maps and µn : Gn × Gn → Gn , (x, y) → xy as well as µ : G × G → G be the respective group multiplication. Then η = lim ηn : lim Gn , ODL → lim Gn , ODL −→

−→

−→

is always continuous. However, it may happen that µ is discontinuous (with respect to the product topology on G × G), in which case (G, ODL ) is not a topological group and hence ODL = ODLG . We recall a simple example for this pathology from [TSH98]. Example 3.1. Let Gn := Q × Rn−1 with the addition and topology induced by Rn . Identifying Rn−1 with the vector subspace Rn−1× {0} of Rn , we obtain a strict direct sequence G1 ⊆ G2 ⊆ · · · of metrizable topological groups. It can be shown by direct calculation that the direct limit topology ODL does not make the group multiplication on G := n∈N Gn continuous (see [TSH98, Example 1.2]). To understand the diﬃculties concerning the group multiplication (in contrast to the group inversion) on G = n∈N Gn , note that we always have a continuous map lim µn : lim (Gn × Gn ), ODL → lim Gn , ODL . −→

−→

−→

Thus µ is continuous as a map from (G×G, ODL ) to (G, ODL ); i.e., it becomes continuous if, instead of the product topology, the topology ODL is used on G × G, which makes it the direct limit topological space lim (Gn × Gn ). This −→ topology is ﬁner than the product topology and, in general, properly ﬁner. If the direct limit topology on G × G happens to coincide with the product topology, then (G, ODL ) is a topological group and thus ODL = ODLG (cf. [HST01] and [Gl03a, §3]). The following proposition describes a situation where the two topologies coincide. We recall that a topological space X is said to be a kω -space if it is the direct limit topological space of an ascending sequence K1 ⊆ K2 ⊆ · · · of compact topological spaces (see, e.g., [GGH10] and the references therein).5 Such spaces are always Hausdorﬀ (see 2.7). For example, every σ-compact, locally compact space is a kω -space. A topological space X is called locally kω if every point x ∈ X has an open neighbourhood in X which is a kω -space in the induced topology [GGH10, Deﬁnition 4.1]. For example, every locally compact topological space is locally kω . The topological space underlying a topological group G is locally kω if and only if G has an open subgroup which is a kω -space [GGH10, Proposition 5.3]. See [GGH10, 4 5

In part of the older literature, there was some confusion concerning this point. These spaces can also be characterized as the hemicompact k-spaces.

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Proposition 4.7] for the following fact. The special case where each Xn and Yn is locally compact was ﬁrst proved in [HST01, Theorem 4.1] (cf. also [Gl03a, Proposition 3.3] for the strict case). Proposition 3.2. Let X1 ⊆ X2 ⊆ · · · and Y1 ⊆ X2 ⊆ · · · be topological spaces with continuous inclusion maps Xn → Xn+1 and Yn → Yn+1 . If each Xn and each Yn is locally kω , then lim (Xn × Yn ) = lim Xn × lim Yn −→

−→

−→

as a topological space. Using the fact that direct limits of ascending sequences of locally kω -spaces are locally kω by [GGH10, Proposition 4.5] (and thus Hausdorﬀ), the preceding discussion immediately entails the following conclusion from [GGH10] (cf. [TSH98, Theorem 2.7] for locally compact Gn , as well as [Gl03a, Corollary 3.4] (in the case of a strict direct system)). Corollary 3.3. Consider a sequence G1 ⊆ G2 ⊆ · · · of topological groups such that each inclusion map Gn → Gn+1 is a continuous homomorphism. If the topological space underlying Gn is locally kω for each n ∈ N (for example, if each Gn islocally compact), then the direct limit topology is Hausdorﬀ and makes G = n∈N Gn the direct limit topological group. 3.4 Given topological groups G1 ⊆ G2 ⊆ · · · such that all inclusion maps Gn → Gn+1 are continuous homomorphisms, consider the conditions: (a) Gn is an open subgroup of Gn+1 (with the induced topology) for all suﬃciently large n. (b) For each suﬃciently large n, the topological group Gn has an identity neighbourhood U whose closure in Gm is compact for some m ≥ n. Then ODL = ODLG holds if (a) or (b) is satisﬁed [Ya98, Theorems 2 and 3]. By a most remarkable theorem of Yamasaki [Ya98, Theorem 4], the validity of (a) or (b) is also necessary in order that ODL = ODLG , provided that each Gn is metrizable and the inclusion maps Gn → Gn+1 are topological embeddings. Criteria ensuring that a given group topology coincides with ODLG . Frequently, a given topological group G is a union G = n∈N Gn of topological groups, such that all inclusion maps Gn → Gn+1 and Gn → G are continuous homomorphisms. In many cases, a criterion from [Gl07b] helps to show that the given topology on G coincides with ODLG (cf. [Gl07b, Proposition 11.8]). The criterion uses the weak direct product ∗n∈N Gn as a tool. The latter can be formed for any sequence (Gn )n∈N of topological groups. It is deﬁned as the subgroup of all (gn )n∈N ∈ n∈N Gn such that gn = 1 for all but ﬁnitely many n. The weak direct product is a topological group; a basis for ∗its topology (the “box topology”) is given by sets of the form n∈N Un ∩ n∈N Gn (the “boxes”), where Un ⊆ Gn is open for each n and 1 ∈ Un for almost all n.

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Returning to the case where G1 ⊆ G2 ⊆ · · · and G = n∈N Gn , we can consider the “product map” ∗ π : n∈N Gn → G , (gn )n∈N → g1 g2 · · · gN , where N ∈ N is so large that gn = 1 for all n > N . ∗ Proposition 3.5. If the product map π : n∈N Gn → G is open at 1, then the given topology on G coincides with ODLG and thus G = lim Gn as a −→ topological ∗group. The openness of π at 1 is guaranteed if there exists a map σ : Ω → n∈N Gn on an identity neighbourhood Ω ⊆ G such that π ◦ σ = idΩ , σ(1) = 1 and σ is continuous at 1. Remark 3.6. Such a section σ to π might be called a fragmentation map, in analogy to concepts in the theory of diﬀeomorphism groups (cf. [Ba97, §2.1]). Example 3.7. It canbe shown that the Lie groups Diﬀ c (M ) = n∈N Diﬀ Kn (M ) r and Ccr (M, H) = n∈N CK (M, H) (as deﬁned in the Introduction) always n admit fragmentation maps (even smooth ones); cf. [Gl07b, Lemmas 5.5 and r 7.7]. Hence Diﬀ c (M ) = lim Diﬀ Kn (M ) and Ccr (M, H) = lim CK (M, H) as n −→ −→ topological groups.

4 Non-linear mappings on locally convex direct limits Consider a sequence E1 ⊆ E2 ⊆ · · · of locally convex spaces, such that each inclusion map En → En+1 is continuous and linear. Then there is a ﬁnest locally convex vector topology Olcx on E := n∈N En , making each inclusion map En → E continuous, called the locally convex direct limit topology.6 4.1 Some basic properties of locally convex direct limits are frequently used. (a) If the direct sequence E1 ⊆ E2 ⊆ · · · is strict, then (E, Olcx ) induces the given topology on En , for each n ∈ N (see Proposition 9 (i) in [Bo87, Chapter II, §4, no. 6]). (b) If the direct sequence E1 ⊆ E2 ⊆ · · · is strict and each En is Hausdorﬀ, then (E, Olcx ) is also Hausdorﬀ (see Proposition 9 (i) in [Bo87, Chapter II, §4, no. 6]). (c) If the direct sequence E1 ⊆ E2 ⊆ · · · isstrict and each En complete, then the locally convex direct limit E = n∈N En is boundedly regular (cf. Proposition 6 in [Bo87, Chapter III, §1, no. 4]) and hence also compactly regular, in view of (a).

6

This topology can be described in various ways. We mention: (1) A convex set U ⊆ E is open if and only if U ∩En is open in En , for each n ∈ N. (2) A seminorm q : E → [0, ∞[ is continuous if and only if q|En is continuous, for each n ∈ N.

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(d) If the direct sequence E1 ⊆ E2 ⊆ · · · is strict and each En complete, then also the locally convex direct limit E is complete (see Proposition 9 (iii) in [Bo87, Chapter II, §4, no. 6]). (e) If also F1 ⊆ F2 ⊆ · · · is an ascending sequence of locally convex spaces, with locally convex directlimit F = n∈N Fn , then the locally convex direct limit topology on n∈N (En × Fn ) and the product topology on E × F coincide [HST01, Theorem 3.4] (because ﬁnite direct products coincide with ﬁnite direct sums in the category of locally convex spaces). The reader may ﬁnd [Fl80] and [Bi88] convenient points of entry to the research literature on locally convex direct limits. We mention that few general results ensuring the Hausdorﬀ property for locally convex direct limits E = lim En are known (besides 4.1 (b) just encoun−→ tered and Proposition 4.4 below). Some Hausdorﬀ criteria for direct limits of Banach spaces (and normed spaces) can be found in [Fl79] (see also [Fl80, p. 214]). In concrete examples, a very simple argument frequently works: If one can ﬁnd an injective continuous linear map from E to some Hausdorﬀ locally convex space, then E is Hausdorﬀ. However, the fact remains that non-Hausdorﬀ locally convex direct limits do exist: See [Mk63] for examples where each En is a Banach space, [Fl80, p. 207] for a simple example (due to L. Waelbroeck) with each En a normed space and [Fl80, p. 227, Corollary 2] for an example where each En is a nuclear Fr´echet space (cf. also [Sv69]). It is well known that O and O coincide on E = lcx DLG n∈N En , because

n on E = E , both O and O coincide with the box n k lcx DLG n∈N n∈N k=1 topology and n∈N En (with either topology) can be considered as a quotient of the direct sum (see [Gl07b, Lemma 2.7]; cf. [HST01, Proposition 3.1] for a diﬀerent argument. See also [Ko69] and [Bo87, Chapter II, Exercise 14 to §4]). It is also known that Olcx need not coincide with ODL (see [Sr59] or Exercise 16 (a) to §4 in [Bo87, Chapter II]; cf. also [Du64, p. 506]). For example, Olcx is properly coarser than ODL if each En is an inﬁnite-dimensional Fr´echet space and En is a proper vector subspace of En+1 with the induced topology, for each n ∈ N ([KM97, Proposition 4.26 (ii)]; cf. Yamasaki’s theorem recalled in Section 3). The following concrete example shows that not even smoothness or analyticity of f |En ensures that a map f : E → F on a locally convex direct limit E = n∈N En is continuous (let alone smooth or analytic). Example 4.2. Consider the map g : Cc∞ (R, C) → Cc∞ (R × R, C) ,

g(γ) := γ ⊗ γ

between spaces of compactly supported smooth functions, where (γ ⊗ γ)(x, y) := γ(x)γ(y) for x, y ∈ R. It can be shown that g is discontinuous, although ∞ ∞ ∞ g|C[−n,n] (R,C) : C[−n,n] (R, C) → Cc (R × R, C) is a continuous homogeneous polynomial (and hence complex analytic), for each n ∈ N (see Remark 7.9 in [Gl07b], based on [HST01, Theorem 2.4]).

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Remark 4.3. Consider the locally convex direct limit E = n∈N En of Hausdorﬀ locally convex spaces E1 ⊆ E2 ⊆ · · · over K ∈ {R, C}. Let U1 ⊆ U2 ⊆ · · · be an ascending sequence of open sets Un ⊆ En , and U := n∈N Un . Let r ∈ N ∪ {∞}, F be a Hausdorﬀ locally convex space and f : U → F be a map such that f |Un : En ⊇ Un → F is CKr for each n ∈ N. Assume that E is Hausdorﬀ and U ⊆ E is open.7 Then the iterated directional derivatives dk f (x, y1 , . . . , yk ) = (Dyk · · · Dy1 f )(x) exist for all k ∈ N with k ≤ r and all x ∈ U and y1 , . . . , yk ∈ E, because x ∈ Un and y1 , . . . , yk ∈ En for some n ∈ N and then (Dyk · · · Dy1 f )(x) = dk (f |Un )(x, y1 , . . . , yk ). Hence only continuity of the maps dk f , which satisfy dk f |Un ×(En )k = dk (f |Un ) for all n ∈ N,

(5)

may be missing for some k, and may prevent f from being a CKr -map. We mention that locally convex direct limits of ascending sequences of Banach spaces (resp., Fr´echet spaces) are called (LB)-spaces (resp., (LF)-spaces). If the sequence is strict, we speak of LB-spaces (resp., LF-spaces).8 A locally convex space E is called a Silva space if it is the locally convex direct limit of an ascending sequence E1 ⊆ E2 ⊆ · · · of Banach spaces, such that all inclusion maps En → En+1 are compact operators (cf. [Se55] and [Fl71]).9 Silva spaces are very well-behaved direct limits. We recall the following from [Fl71]. Proposition 4.4. If E = n∈N En is a Silva space, then the following hold: (a) E is Hausdorﬀ and complete; (b) E = n∈N En is boundedly regular and hence also compactly regular; 10 (c) The locally convex direct limit topology on E coincides with the direct limit topology ODL ; (d) If F = F n∈N n is also a Silva space, with Fn → Fn+1 compact, then E × F = n∈N (En × Fn ) is a Silva space.11 Some interesting inﬁnite-dimensional Lie groups are modelled on Silva spaces, e.g., the group Diﬀ ω (K) of real analytic diﬀeomorphisms of a compact real analytic manifold K (see [Ls82]; cf. [KM97, Theorem 43.4]). More examples will be encountered below.

7 8 9

10 11

For example, we might start with an open set U ⊆ E and set Un := U ∩ En . These conventions are local. The meanings of LF and (LF) vary in the literature. A locally convex space is a Silva space if and only if it is isomorphic to the dual of a Fr´echet–Schwartz space [Fl71]; therefore, Silva spaces are also called (DFS)-spaces. Using the fact that the inclusion maps En → En+1 are compact operators. The inclusions En × Fn → En+1 × Fn+1 are compact operators, and 4.1 (e) holds.

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Mappings on Silva spaces or unions of kω -spaces. In good cases, the pathology described in Remark 4.3 cannot occur (see [Gl07b, Lemma 9.7] and [GGH10, Proposition 8.13]). Proposition 4.5. Consider the locally convex direct limit E = n∈N En of Hausdorﬀ locally convex spaces E1 ⊆ E2 ⊆ · · · over K ∈ {R, C}. Let U1 ⊆ U2 ⊆ · · · be an ascending sequence of open sets Un ⊆ En , and U := n∈N Un . Let r ∈ N0 ∪ {∞}, F be a Hausdorﬀ locally convex space and f : U → F be a map such that f |Un is CKr for each n ∈ N. Assume that (a) Each En is a kω -space, or (b) En is a Banach space and the inclusion map En → En+1 is a compact operator, for each n ∈ N (in which case E is a Silva space). Then E is Hausdorﬀ and the locally convex direct limit topology on E coincides with ODL . Moreover, U is open in E and f : U → F is CKr . In the Silva case, the hypotheses of Proposition 4.5 can be relaxed (cf. [Ls85, Proposition 2.8]). Real analyticity is more elusive. For example, there exists a real-valued map f on the Silva space R(N) := lim Rn which is not real analytic −→ although f |Rn is real analytic for each n ∈ N (cf. [KM97, Example 10.8]). Complex analytic maps on (LB)-spaces. A very useful result from [Da10] frequently helps us to check complex analyticity beyond Silva spaces. Theorem 4.6 (Dahmen’s theorem). Let E1 ⊆ E2 ⊆ · · · be an ascending sequence of normed spaces (En , .n ) over C such that, for each n ∈ N, the inclusion map En → En+1 is continuous and complex linear, of operator norm at most 1. Let r ∈ ]0, ∞[, Un := {x ∈ En : xn < r} for n ∈ N and F be a complex locally convex space. Assume that the locally convex direct limit E = n∈N En is Hausdorﬀ. Then U := n∈N Un is open in E, and if f: U → F is a map such that f |Un : En ⊇ Un → F is complex analytic and bounded for each n ∈ N, then f is complex analytic. Mappings between direct sums. If (En )n∈N is a sequence of locally

convex spaces, we equip n∈N En with the box topology (as introduced before Proposition 3.5). See [Gl03b, Proposition 7.1] for the following result. Proposition 4.7. Let (En )n∈N and (Fn )n∈N be sequences of Hausdorﬀ locally convex spaces, r ∈ N0 ∪ {∞}, Un ⊆ En be open and fn : Un → Fn be C r . Assume that 0 ∈ Un and fn (0) = 0 for all but ﬁnitely many n ∈ N. Then U := ( E ) ∩ n∈N n∈N n∈N Un is open in n∈N En and the map

n

n r ⊕n∈N fn : U → F , (x ) → (f (x )) n n∈N n n n∈N is C . n∈N n n∈N n Non-linear maps between spaces of test functions. Let r, s ∈ N0 ∪ {∞}, M be a σ-compact, ﬁnite-dimensional C r -manifold, N be a σ-compact,

262

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ﬁnite-dimensional C s -manifold, E, F be Hausdorﬀ locally convex spaces, Ω ⊆ Ccr (M, E) be open and f : Ω → Ccs (N, F ) be a map. We say that f is almost local if there exist locally ﬁnite covers (Un )n∈N and (Vn )n∈N of M (resp., N ) by relatively compact, open sets Un ⊆ M (resp., Vn ⊆ N ) such that f (γ)|Vn only depends on γ|Un , i.e., (∀n ∈ N) (∀γ, η ∈ Ω)

γ|Un = η|Un ⇒ f (γ)|Vn = f (η)|Vn .

For example, f is almost local if M = N and f is local in the sense that f (γ)(x) only depends on the germ of γ at x ∈ M . As shown in [Gl02d] (see also [Gl04b, Theorem 10.4]), almost locality prevents pathologies as in Example 4.2. Proposition 4.8. Let r, s, t ∈ N0 ∪ {∞} and f : Ccr (M, E) ⊇ Ω → Ccs (N, F ) r be an almost local map. Assume that the restriction of f to Ω ∩ CK (M, E) is t t C , for each compact set K ⊆ M . Then f is C . An analogous result is available for mappings between open subsets of spaces of compactly supported sections in vector bundles. Almost local maps between subsets of the space of compactly supported smooth vector ﬁelds occur in the construction of the Lie group structure on Diﬀ c (M ) (see [Gl02d]; cf. [Gl05a] and [Gl04b]). Proof of Proposition 4.8. The proof exploits the fact that the map σ : Ccs (N, F ) → C s (Vn , F ) , γ → (γ|Vn )n∈N n∈N

is a linear topological embedding with closed image [Gl04b, Proposition 8.13], for each locally ﬁnite cover (Vn )n∈N of N by relatively compact, open sets Vn . It hence suﬃces to show that σ◦f is C t . Let us assume that Ω = Ccr (M, E) for n )n∈N of M by relatively compact, simplicity. There is a locally ﬁnite cover (U n → R be a open sets such that Un contains the closure of Un . Let hn : U compactly supported smooth map such that hn |Un = 1. Then the following map is C t : n , E) → C s (Vn , F ) , fn (γ) := f (hn · γ)|Vn . fn : C r (U

n , E), γ → (γ| )n . Then σ◦f = (⊕n∈N fn )◦ρ, Set ρ : Ccr (M, E) → n∈N C r (U Un t where ⊕n∈N fn is C by Proposition 4.7. Hence σ ◦ f and thus f is C t .

5 Lie group structures on directed unions of Lie groups In some situations, it is possible to construct Lie group structures on ascending unions of Lie groups. Unions of ﬁnite-dimensional Lie groups. In the case of ﬁnite-dimensional manifolds, an extension lemma for charts is available [Gl05b, Lemma 2.1].

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263

If M and N are ﬁnite-dimensional C ∞ -manifolds such that M ⊆ N and the inclusion map M → N is an immersion, then each chart φ : U → V of M which is deﬁned on a relatively compact, (smoothly) contractible subset U ⊆ M extends to a chart of N on a domain with analogous properties. Now consider a sequence G1 ⊆ G2 ⊆ · · · of ﬁnite-dimensional Lie groupssuch that the inclusion maps are smooth homomorphisms. Let x ∈ G := n∈N Gn , say x ∈ Gn0 . We then pick a chart φn0 of Gn0 around x whose domain is relatively compact and contractible, and use the extension lemma to obtain charts φn of Gn for n > n0 which are deﬁned on relatively compact, contractible open sets, and such that φn extends φn−1 . One then easily veriﬁes (using Proposition 4.5) that the homeomorphisms φ := lim φn so obtained deﬁne a C ∞ -atlas on G (equipped with the −→ direct limit topology), which makes the lattera Lie group modelled on lim L(Gn ) (see [Gl05b]).12 By construction, G = n∈N Gn admits direct limit −→ charts. Moreover, it is clear from the construction that G = lim Gn as a topo−→ logical space and as a topological group. Using Proposition 4.5, one easily infers that G = lim Gn also as a smooth manifold and as a Lie group (see −→ [Gl05b, Theorem 4.3]). Remark 5.1. The preceding construction applies just as well to ascending unions of ﬁnite-dimensional smooth manifolds Mn , such that all inclusion maps are immersions.13 This enables G/H to be turned into the direct limit C ∞ -manifold lim Gn /(H ∩ Gn ), for each closed subgroup H ⊆ G (see [Gl05b, −→ Proposition 7.5]). Then the quotient map G → G/H makes G a principal H-bundle over G/H, using a suitable extension lemma for sections in nested principal bundles [Gl05b, Lemma 6.1]. We mention that an equivariant version of the above extension lemma (namely [Wk07, Lemma 1.13]) can be used to turn the gauge group Gau(P ) into a Lie group, for each smooth principal bundle P → K over a compact smooth manifold K whose structure group is a direct limit G = lim Gn of −→ ﬁnite-dimensional Lie groups (see [Wk07, Lemma 1.14 (e) and Theorem 1.11]). Remark 5.2. Direct limits G = lim Gn of ﬁnite-dimensional Lie groups are −→ regular Lie groups in Milnor’s sense [Gl05b, Theorem 8.1], but they can be quite pathological in other ways. For example, the exponential map expG = lim expGn need not be injective on any 0-neighbourhood, and the exponential −→ image need not be an identity neighbourhood in G. Both pathologies occur for G := C(N) × R = lim Cn × R , −→

via t.(zk )k∈N := (eikt zk )k∈N . This can be checked where t ∈ R acts on C quite easily, using the fact that the exponential map of G is given explicitly ikt by expG ((zk )k∈N , t) = e ikt−1 zk k∈N , t (see [Gl03a, Example 5.5]). (N)

12 13

Ë

See [NRW91], [KM97, Theorem 47.9] and [Gl03a] for earlier, less general results. Compare [Ha71] for n∈N Mn as a topological manifold.

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The preceding general construction implies thatevery countably dimensional locally ﬁnite Lie algebra g (i.e., each union g = n∈N gn of ﬁnite-dimensional Lie algebras g1 ⊆ g2 ⊆ · · · ), when endowed with the ﬁnest locally convex vector topology, arises as the Lie algebra of some regular Lie group.14 Such locally ﬁnite Lie algebras have been much studied in recent years, e.g., by Yu. Bahturin, A. A. Baranov, G. Benkart, I. Dimitrov, K.-H. Neeb, I. Penkov, H. Strade, N. Stumme, A. E. Zalesski˘ı and others (see [BBZ04], [BB04], [DP04], [Ne00], [NS01], [PS03], [St99] and the references therein). Unions of Banach–Lie groups. These are Lie groups under additional hypotheses (which, e.g., exclude the pathologies described in Remark 5.2). Theorem 5.3. Let G1 ⊆ G2 ⊆ · · · be Banach–Lie groups over K ∈ {R, C}, ∞ such that all inclusion maps λn : Gn → Gn+1 are CK -homomorphisms. Set G := n∈N Gn . Assume that (a)–(c) are satisﬁed. (a) For each n ∈ N, there exists a norm .n on L(Gn ) deﬁning its topology, such that [x, y]n ≤ xn yn for all x, y ∈ L(Gn ) and the continuous linear map L(λn ) : L(Gn ) → L(Gn+1 ) has operator norm at most 1. (b) The locally convex direct limit topology on g := n∈N L(Gn ) is Hausdorﬀ. (c) expG := lim expGn : g → G is injective on some 0-neighbourhood. −→

Then there exists a K-analytic Lie group structure on G which makes expG a K-analytic local diﬀeomorphism at 0. If, furthermore, g = n∈N L(Gn ) is compactly regular, then G is a regular Lie group in Milnor’s sense. Sketch of proof. The Lie group structure is constructed in [Da10], along the following lines: Applying Dahmen’s theorem 4.6 (to the complexiﬁcation gC , if K = R), one ﬁnds that the Baker–Campbell–Hausdorﬀ (BCH-) series con L(G ) L(G ) verges to a K-analytic map n∈N Br n (0) × Br n (0) → g for some r > 0. Because expG is locally injective, it induces an isomorphism φ of local groups L(G ) from some 0-neighbourhood U ⊆ n∈N Br n (0) onto some subset V of G. We give V the CKω -manifold structure making φ a CKω -diﬀeomorphism. Now standard arguments can be used to make G a Lie group with V as an open submanifold. The proof of regularity will be sketched in Section 8. The author does not know whether the Lie groups G in Theorem 5.3 are always the direct limit lim Gn in the category of Lie groups (unless additional −→ hypotheses are satisﬁed). Another construction principle. There is another construction principle for a Lie group structureon a union G = n∈N Gn of Lie groups (or a group which is a union G = n∈N Mn of manifolds), which produces Lie groups modelled on Silva spaces or ascending unions of kω -spaces. A direct limit Lie group structure can be constructed on G if (1) there are compatible charts φn 14

One chooses a simply connected Lie group Gn with Lie algebra gn and forms the direct limit group G = lim Gn (see [Gl05b, Theorem 5.1] for the details). −→

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265

of the Lie groups Gn (resp., the manifolds Mn ) around each point in G; and (2) suitable hypotheses are satisﬁed which ensure that the transition maps between charts of the form lim φn are CK∞ , because they are mappings of the −→ form discussed in Proposition 4.5 (see [Gl07b, Lemma 14.5]).

6 Examples of directed unions of Lie groups The main examples of ascending unions of inﬁnite-dimensional Lie groups were already brieﬂy described in the Introduction. We now provide more details. Notably, we discuss the existence of direct limit charts and compact regularity. As already mentioned, the latter gives information on the homotopy groups (see (1)) and can help to verify regularity in Milnor’s sense (see Theorem 5.3 and Section 8). A special case of [Gl10, Corollary 3.6] is useful. Lemma 6.1. If the Lie group G = n∈N Gn admits a weak direct limit chart, then G = n∈N Gn is compactly regular if and only if L(G) = n∈N L(Gn ) is compactly regular. In the case of an (LF)-space E = n∈N En , there is a quite concrete characterization of compact regularity only in terms of properties of the steps En (see [We03, Theorem 6.4 and its corollary]). Theorem 6.2. Let E1 ⊆ E2⊆ · · · be Fr´echet spaces, with continuous linear inclusion maps. Give E = n∈N En the locally convex direct limit topology. Then E = n∈N En is compactly regular if and only if for each n ∈ N, there exists m ≥ n such that for all k ≥ m, there is a 0-neighbourhood U in En on which Ek and Em induce the same topology. In this case, E is also boundedly regular and complete. We mention that a Hausdorﬀ (LF)-space is boundedly regular if and only if it is Mackey complete [Fl80, 1.4 (f), p. 209]. Groups ofcompactly supported diﬀeomorphisms. The Lie group Diﬀ c (M ) = n∈N Diﬀ Kn (M ) (discussed in the Introduction) admits a direct limit chart (cf. [Gl07b, §5.1]). Moreover, the LF-space Vc (M ) = n∈N VKn (M ) is compactly regular (see 4.1 (c)) and hence also Diﬀ c (M ) (by Lemma 6.1). To avoid exceptional cases in our later discussions of direct limit properties, we assume henceforth that M is non-compact and of positive dimension. Test function groups. Let M and an exhaustion K1 ⊆ K2 ⊆ · · · of M be as in the deﬁnition of Diﬀ c (M ), H be a Lie group modelled on a locally convex space and r ∈ N0 ∪ {∞}. We consider the “test function group” Ccr (M, H) of C r -maps γ : M → H such that the closure of {x ∈ M : γ(x) = 1} (the r support of γ) is compact. Let CK (M, H) be the subgroup of functions supn r r ported in Kn . Then CKn (M, H) is a Lie group modelled on CK (M, L(H)), n r and Cc (M, H) is a Lie group modelled on the locally convex direct limit

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r Ccr (M, L(H)) = lim CK (M, L(H))([Gl02b]; cf. 1.8 for special cases). Also, n −→

Ccr (M, H) =

n∈N

r CK (M, H) n

admits a direct limit chart (cf. [Gl07b, §7.1]). Furthermore, Ccr (M, L(H)) = r n CKn (M, L(H)) is compactly regular as a consequence of 4.1 (c). We now assume that H is non-discrete and M non-compact, of positive dimension. Weak direct products of Lie groups. Given a sequence (Hn )n∈N of Lie groups, its weak direct product G := ∗n∈N Hn (as introduced before Proposition 3.5) has a natural §7], modelled on the locally

Lie group structure [Gl03b, convex direct sum L(H ). Then G = n n∈N Gn , identifying the n n∈N partial product Gn := k=1 Hk with a subgroup of G. By construction, G = n∈N Gn

has a direct limit chart. Furthermore, L(G) = n∈N L(Hn ) = lim L(Gn ) is −→ compactly regular, as locally convex direct sums are boundedly regular [Bo87, Ch. 3, §1, no. 4, Proposition 5] and induce the given topology on each ﬁnite partial product (cf. Propositions 7 or 8 (i) in [Bo87, Ch. 2, §4, no. 5]). To avoid exceptional cases, we assume henceforth that each Hn is non-discrete. Unit groups of unions of Banach algebras. Let A1 ⊆ A2 ⊆ · · · be unital complex Banach algebras (such that all inclusion maps are continuous homo morphisms of unital algebras). Give A := n∈N An the locally convex direct limit topology. Then A× is open in A and if A is Hausdorﬀ (which we assume now), then A× is a complex Lie group [Gl07b, Proposition 12.1]. Moreover, A× = n∈N A× n , and the identity map idA× is a direct limit chart (cf. [DW97] and [Ed99] for related results). If each inclusion map An → An+1 is a topological embedding or each inclusion map a compact operator, then A = A and hence also A× = n∈N n × An are compactly regular. However, for particular choices of the steps, n∈N A = n∈N An is not compactly regular (see [Gl10, Example 7.8], based on [BMS82, Remark 1.5]). Lie groups of germs of analytic mappings. Let H be a complex Banach– Lie group, . be a norm on L(H) deﬁning its topology, X be a complex metrizable locally convex space and K ⊆ X be a non-empty compact set. Let W1 ⊇ W2 ⊇ · · · be a fundamental sequence of open neighbourhoods of K in X such that each connected component of Wn meets K. Then the set Germ(K, H) of germs around K of H-valued complex analytic functions on open neighbourhoods of K can be made a Lie group modelled on the locally convex direct limit Germ(K, L(H)) = lim Holb (Wn , L(H)) −→

of the Banach spaces gn := Holb (Wn , L(H)) of bounded L(H)-valued complex analytic functions on Wn , equipped with the supremum norm (see [Gl04a]). The group operation arises from pointwise multiplication of representatives of germs. The identity component Germ(K, H)0 is the union

Direct Limits of Inﬁnite-Dimensional Lie Groups

Germ(K, H)0 =

267

Gn

n∈N

of the Banach–Lie groups Gn := [expH ◦ γ] : γ ∈ gn , and Germ(K, H)0 = n∈N Gn admits a direct limit chart [Gl07b, §10.4]. Theorem 6.2 implies that Germ(K, L(H)) = n∈N gn is compactly regular (see [DG10]), and thus Germ(K, H)0 = n∈N Gn is compactly regular (see [Ch85, Theorems 21.15 and 21.23] for the bounded regularity and completeness of Germ(K, L(H)) if X is a normed space; cf. [Mj79]). In the most relevant case where X and H are ﬁnite-dimensional, we can choose Wn+1 relatively compact in Wn . Then the restriction maps Holb (Wn , L(H)) → Holb (Wn+1 , L(H)) are compact operators [Gl07b, §10.5] and thus Germ(K, L(H)) is a Silva space. Lie groups of germs of analytic diﬀeomorphisms. If X is a complex Banach space and K ⊆ X a non-empty compact set, let GermDiﬀ(K, X) be the set of germs around K of C-analytic diﬀeomorphisms γ : U → V between open neighbourhoods U and V of K (which may depend on γ), such that γ|K = idK . Then GermDiﬀ(K, X) is a Lie group modelled on the locally convex direct limit Germ(K, X)K := lim Holb (Wn , X)K , −→

where Wn and Holb (Wn , X) are as in the last example and Holb (Wn , X)K := {ζ ∈ Holb (Wn , X) : ζ|K = 0} (see [Gl07b, §15] for the case dim(X) < ∞, and [Da10] for the general result). The group operation arises from composition of representatives of germs. Now the set Mn of all elements of GermDiﬀ(K, X) having a representative in Holb (Wn , X)K is a Banach manifold, and Mn GermDiﬀ(K, X) = n∈N

has a direct limitchart (see [Da10]; cf. [Gl07b, Lemma 14.5 and §15]). GermDiﬀ(K, X) = n Mn is compactly regular by Theorem 6.2 and Lemma 6.1 (see [Da10]); if X is ﬁnite dimensional, then Germ(K, X)K is a Silva space. Unions of Lie groups1 modelled on Sobolev spaces. The Lie groups H ↓s (K, F ) = n∈N H s+ n (K, F ) (as in the Introduction) are studied in the work in progress [DG10]. By construction, they admit a direct limit chart, and 1 they are modelled on the Silva space H ↓s (K, L(F )) = n∈N H s+ n (K, L(F )) (and hence compactly regular). We mention that the Lie group structure on H ↓s (K, F ) can be obtained via Theorem 5.3; therefore, H ↓s (K, F ) is a regular Lie group in Milnor’s sense. Compare [Pk08] (in this volume) for analysis and probability theory on variants of the Lie groups H s (K, F ) (with s > dim(K)/2), and limit processes as s ↓ dim(K)/2.

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7 Direct limit properties of ascending unions We now discuss the direct limit properties of ascending unions of inﬁnitedimensional Lie groups in the categories of Lie groups, topological groups, smooth manifolds and topological spaces. Tools to prove or disprove direct limit properties. Such tools were provided in [Gl07b]. Recall that a real locally convex space E is said to be smoothly regular (or to admit smooth bump functions) if the topology on E is initial with respect to C ∞ (E, R). Remark 7.1. If U ⊆ E is a 0-neighbourhood and the topology is initial n with respect to C ∞ (E, R), then j=1 fj−1 (]−ε, ε[) ⊆ U for suitable ε > 0 and f1 , . . . , fn ∈ C ∞ (E, R) such that f1 (0) = · · · = fn (0) = 0. Then f −1 (]−δ, δ[) ⊆ U with f := f12 + · · · + fn2 and δ := ε2 . Let g : R → R be a smooth function such that g(R) ⊆ [0, 1], g(0) = 1 and g(x) = 0 if |x| ≥ δ/2. Then h := g ◦ f : E → R is a smooth function such that h(0) = 1 and supp(h) ⊆ U (a “smooth bump function” supported in U ). This explains the terminology. Example 7.2. Every Hilbert space H admits smooth bump functions (because H → R, x → x2 is smooth). As a consequence, every locally convex space which admits a linear topological embedding into a direct product of Hilbert spaces (for example, every nuclear locally convex space) admits smooth bump functions (cf. also [KM97, Chapter III]). Proposition 7.3. Consider a Lie group G = n∈N Gn , where G1 ⊆ G2 ⊆ · · · are Lie groups and all inclusion maps Gn → Gn+1 and Gn → G are smooth homomorphisms. Assume that G = n∈N Gn admits a direct limit chart. Then the following hold. (a) If G = lim Gn as a topological group, then G = lim Gn as a Lie group. −→ −→ (b) G = lim Gn as a topological space if and only if L(G) = lim L(Gn ) as a −→ −→ topological space. (c) If L(G) admits smooth bump functions, then G = lim Gn as a −→ CR∞ -manifold if and only if L(G) = lim L(Gn ) as a CR∞ -manifold. −→

Direct limit properties of the main examples. Using Proposition 7.3, Proposition 3.5 (to recognize direct limits of topological groups) and a counterpart of Proposition 4.5 for analogous ascending unions of manifolds [Gl07b, Proposition 9.8], one obtains the following information concerning the direct limit properties of the examples from Section 6 (see [Gl07b]; the properties of H ↓s (K, F ) follow from [Gl07b, Proposition 9.8]). The entries in Table 7.1 indicate whether G = lim Gn holds in the category −→ shown on the left, for the Lie group described at the top. The abbreviation “dep” is used if the answer depends on special properties of the group(s) involved. We abbreviate “category” by “cat,” “group” by “gp,” “space” by “sp,” “topological” by “top” and “smooth manifold” by “mfd.”

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Table 7.1: Direct limit properties in various categories cat\gp Diﬀc (M ) Cc∞ (M,H) Lie gps top gps mfds top sps

yes yes no no

yes yes no no

ÉH ∗ n

Germ(K,H)0 GermDiﬀ H ↓s (K,F ) (K,X) yes yes yes — yes yes yes yes — yes dep* dep** yes† yes†† yes dep* dep** yes† yes†† yes n

A×

* “yes” if each Hn is ﬁnite dimensional or modelled on a kω -space; “no” if each Hn is modelled on an inﬁnite dimensional Fr´echet space (which we assume nuclear when dealing with the category of smooth manifolds). Other cases unclear. ** “yes” if each An is ﬁnite dimensional or each inclusion map λn : An → An+1 a compact operator; “no” (when dealing with the category of topological spaces), if An is inﬁnite dimensional, An ⊂ An+1 and λn a topological embedding for each n. Other cases unclear. † “yes” if X and H are ﬁnite dimensional; general case unknown. †† “yes” if X is ﬁnite dimensional; general case unknown.

8 Regularity in Milnor’s sense Experience tellsus that if one tries to prove regularity in Milnor’s sense for a Lie group G = n∈N Gn , then regularity of the Lie groups Gn does not suﬃce to carry out the desired arguments. But strengthened regularity properties increase the chances for success. Deﬁnition 8.1. Given k ∈ N0 , we say that a Lie group G is C k -regular if it is a regular Lie group in Milnor’s sense and evolG : C ∞ ([0, 1], L(G)) → G is smooth with respect to the C k -topology on C ∞ ([0, 1], L(G)) (induced by C k ([0, 1], L(G))). If each γ ∈ C k ([0, 1], L(G)) has a product integral ηγ and the map evolG : C k ([0, 1], L(G)) → G , γ → ηγ (1) is smooth, then we say that the Lie group G is strongly C k -regular. For example, every Banach–Lie group is strongly C 0 -regular [GN10]. Although much of the following remains valid for C k -regular Lie groups, we shall presume strong C k -regularity, as this simpliﬁes the presentation. We also suppress possible variants involving bounded regularity instead of compact regularity. All results presented in this section are taken from [DG10]. In the regularity proofs for our main classes of direct limit groups, we always use an isomorphism C k ([0, 1], lim En ) ∼ = lim C k ([0, 1], En ) at a pivotal −→ −→ point. Let us begin with the elementary case of locally convex direct sums.

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Lemma 8.2.

If (En )n∈N

is a sequence of Hausdorﬀ locally convex spaces, then C k ([0, 1], n∈N En ) = n∈N C k ([0, 1], En ), for all k ∈ N0 .

Sketch of proof. The locally convex direct sum n∈N En = n∈N (E1 ×· · ·×En ) is compactly regular, because it is boundedly regular by [Bo87, Chapter 3, §1, no. 4, Proposition 5] and induces the given topology on each ﬁnite partial product (cf. or 8 (i) in [Bo87, Chapter 2, §4, no. 5]). Therefore,

Propositions 7 k C k ([0, 1], n∈N En ) and n∈N C ([0, 1], En ) coincide as sets. Comparing 0-neighbourhoods, we see that both vector topologies coincide (using the fact that boxes are typical 0-neighbourhoods in a countable direct sum). ∞ (N) ∞ n Remark 8.3. Although C ([0, 1], R ) = n∈N C ([0, 1], R ) as a set, the topology on the left-hand side is properly coarser than the locally convex direct limit topology Olcx on the right-hand side, because n γ = (γn )n∈N ∈ C ∞ ([0, 1], R(N) ) : (∀n ∈ N) ddxγnn (0) ∈ ]−1, 1[ ∈ Olcx is not a 0-neighbourhood in C ∞ ([0, 1], R(N) ) = lim

←− m∈N0

C m ([0, 1], R(N) ). Thus

Lemma 8.2 becomes false for k = ∞, explaining the need for C k -regularity with ﬁnite k. Weak direct products of Lie groups.If k ∈ N0 and (Hn )n∈N is a sequence ∗ of strongly C k -regular Lie groups, then n∈N Hn is strongly C k -regular (and hence regular) since its evolution map can be obtained as the composition ∼

⊕n∈N evolHn ∗ = k C k [0, 1], n L(Hn ) −→ → n Hn n C ([0, 1], L(Hn )) −−−−−−−− (cf. Proposition 4.7 for the deﬁnition and smoothness of ⊕n evolHn ). Test function groups. Given a σ-compact, ﬁnite-dimensional smooth manifold M and a C k -regular Lie group H, pick a locally ﬁnite family (Mn )n∈N of compact submanifolds with boundary of M , the interiors of which cover M . Then standard arguments (based on suitable exponential laws for function r k spaces) show that H n := C (Mn , H) is C -regular, for each n ∈ N. The map r r σ : Cc (M, L(H)) → n C (Mn , L(H)), γ → (γ|Mn )n∈N is continuous linear k and hence also the map τ := C ([0, 1], σ) from C k ([0, 1], Ccr (M, L(H))) to

k r C ([0, 1], n C (Mn , L(H))) ∼ = n C k ([0, 1], C r (Mn , L(H))). Furthermore, ρ : G := Ccr (M, H) → ∗n∈N C r (Mn , H) , γ → (γ|Mn )n∈N is an isomorphism of Lie groups onto a closed Lie subgroup (and embedded submanifold) of the weak direct product P . Using point evaluations, one ﬁnds that the composition evolP ◦ τ = ⊕n evolHn ◦ τ : Ccr (M, H) → ∗n∈N C r (Mn , H) (which is smooth by the preceding example) takes its image in the image of ρ. Then f := ρ−1 ◦ evolP ◦ τ : C k ([0, 1], Ccr (M, L(H))) → Ccr (M, H) is a smooth

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271

map, and one veriﬁes using point evaluations that f = evolG . A similar (but more complicated) argument shows that Diﬀ c (M ) is regular. Ascending unions of Banach–Lie groups. As a preliminary, observe that regularity in Milnor’s sense (and strong C k -regularity) can be deﬁned just as well for local Lie groups G; in this case, one requires that a smooth evolution evolG exists on some open 0-neighbourhood in C ∞ ([0, 1], L(G)) (resp., C k ([0, 1], L(G))). In the case of global Lie groups, the local notions of regularity are equivalent to the corresponding global ones (see [DG10] and [GN10]; cf. [KM97, lemma on p. 409]). See [Sm78] for the next theorem (and [Mj83] or [Sm83, Theorem I.7.2] for a variant beyond compact regularity). Theorem 8.4. Consider a Hausdorﬀ locally convex space E which is the locally convex direct limit of Hausdorﬀ locally convex spaces E1 ⊆ E2 ⊆ · · · . If E = n∈N En is compactly regular, then the natural continuous linear map lim C([0, 1], En ) → C([0, 1], E) −→

is an isomorphism of topological vector spaces. Remark 8.5. If E is a locally convex space which is integral complete,15 then C k ([0, 1], E) ∼ = E k × C([0, 1], E)

(6)

naturally via γ → (γ(0), . . . , γ (k−1) (0), γ (k) ), for each k ∈ N. If E = lim En is a −→ locally convex direct limit of integral complete locally convex spaces and E = n∈N En is compactly regular, then E is also integral complete. Moreover, (6), 4.1 (e) and Theorem 8.4 imply that lim C k ([0, 1], En ) ∼ = C k ([0, 1], lim En ) . −→

−→

(7)

Alternatively, (7) follows from Theorem 8.4 because C k ([0, 1], E) ∼ = C([0, 1], E) naturally if E is integral complete [DG10], as follows from (6) and the fact that C([0, 1], E) ∼ = E m × C([0, 1], E) naturally for each m ∈ N, by a suitable (elementary) variant of Miljutin’s theorem [Mi66] provided in [DG10]. 8.6 Assume that g = n∈N L(Gn ) is compactly regular in Theorem 5.3. Following [DG10], we now explain in the essential case where K = C that G is regular in Milnor’s sense.16 Let r > 0 be as in the earlier parts of the L(G ) proof and Un := Br n (0). Because the BCH-series has the same shape for each n ∈ N, one ﬁnds s > 0 such that an evolution evolUn exists as a map from L(G ) C([0, 1], Bs n (0)) to Un , for each n ∈ N. Since Un is bounded, Theorem 4.6 L(G ) shows that evolU := lim evolUn : n∈N C([0, 1], Bs n (0)) → U := n∈N Un −→ is C-analytic and hence also expG ◦ evolU , which is a local group version of the evolution map for G. Hence G is regular and in fact strongly C 0 -regular. 15 16

That is, every continuous curve γ : [0, 1] → E has a Riemann integral in E [LS00]. The real case follows easily via complexiﬁcation on the level of local groups.

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Using Theorem 5.3, one readily deduces H) and H ↓s (K, F ) are that ×Germ(K, 0 × strongly C -regular, and also A = n∈N An if A = n∈N An is compactly regular (see [DG10]). The proof of regularity for GermDiﬀ(K, X) is more involved, but eventually also reduces to Theorem 4.6 (see [Da10]). An idea which might lead to non-regular Lie groups. An observation from [DG10] might be a source of Lie groups which are not regular in Milnor’s sense, although they are modelled on Mackey complete locally convex spaces. Proposition 8.7. Suppose that, for each n ∈ N, there exists a Lie group Hn modelled on a Mackey complete locally convex space which is regular but not C n -regular because evolHn : C ∞ ([0, 1], L(H ∗ n )) → Hn is discontinuous with respect to the C n -topology. Then G := n∈N Hn is a Lie group modelled on the Mackey complete locally convex space L(G) = n∈N L(Hn ). It has an evolution evolG : C ∞ ([0, 1], L(G)) → G, but evolG fails to be continuous and thus G is not a regular Lie group in Milnor’s sense.

9 Homotopy groups of ascending unions of Lie groups We have seen that all main examples of ascending unions G = Lie groups admit a direct limit chart, and thus πk (G) = lim πk (Gn ) −→

for all k ∈ N0

n∈N

Gn of (8)

(see 1.3). Alternatively, many (but not all) of them are compactly regular. In this case, (8) holds by an elementary argument, but one has to pay the price that the proof of compact regularity may require specialized functionalanalytic tools (like Wengenroth’s theorem recalled above). It is an interesting feature that the approach via (weak)direct limit charts even extends to Lie groups G in which an ascending union n∈N Gn is merely dense (and to similar, more general situations). Weak direct limit charts (as deﬁned in 1.2) then have to be replaced by certain “well-ﬁlled charts.” The precise setting will be described now. Besides smooth manifolds, it applies to topological manifolds and more general topological spaces (like manifolds with boundary or corners). Given a subset A of a real vector space V , let us write conv2 (A) := {tx + (1 − t)y : x, y ∈ A, t ∈ [0, 1]}. Deﬁnition 9.1. Let M be a topological space and (Mα )α∈A be a directed family of topological spaces such that M∞ := α∈A Mα is dense in M and all inclusion maps Mα → M and Mα → Mβ (for α ≤ β) are continuous. We say that a homeomorphism φ : U → V ⊆ E from an open subset U ⊆ M onto an arbitrary subset V of a topological vector space E is a well-ﬁlled chart of M if there exist α0 ∈ A and homeomorphisms φα : Uα → Vα ⊆ Eα from open subsets Uα ⊆ Mα onto subsets Vα of certain topological vector spaces Eα for α ≥ α0 such that the following conditions are satisﬁed.

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(a) Eα ⊆ E, Eα ⊆ Eβ if α ≤ β and the inclusion maps Eα → E and Eα → Eβ are continuous and linear. (b) For all α ≥ α0 , we have Uα ⊆ U and φ|Uα = φα . (c) For all β ≥ α ≥ α0 , we have Uα ⊆ Uβ and φβ |Uα = φα . (d) U∞ := α≥α0 Uα = U ∩ M∞ . (e) There exists a non-empty (relatively) open set V (2) ⊆ V such that (2) conv2 (V (2) ) ⊆ V and conv2 (V∞ ) ⊆ V∞ , where V∞ := α≥α0 Vα and (2)

V∞ := V (2) ∩ V∞ . (2) (f) For each α ≥ α0 and compact set K ⊆ Vα := V (2) ∩ Vα , there exists β ≥ α such that conv2 (K) ⊆ Vβ . Then U (2) := φ−1 (V (2) ) is an open subset of U , called a core of φ. If cores of well-ﬁlled charts cover M , then M is said to admit well-ﬁlled charts. On a ﬁrst reading, the reader may ﬁnd the notion of a well-ﬁlled chart somewhat elusive. Special cases of particular interest (which are more concrete and easier to understand) are described in [Gl10, Examples 1.11 and 1.12]. See [Gl10, Theorem 1.13] for the following result. Theorem 9.2.Let M be a Hausdorﬀ topological space containing a directed union M∞ := α∈A Mα of Hausdorﬀ topological spaces Mα as a dense subset, such that all inclusion maps Mα → Mβ (for α ≤ β) and Mα → M are continuous. If M admits well-ﬁlled charts, then πk (M, p) = lim πk (Mα , p) −→

for all k ∈ N0 and p ∈ M∞ .

For a typical application, let H be a Lie group, m ∈ N, S(Rm , L(H)) be the Schwartz space of rapidly decreasing L(H)-valued smooth functions on Rm and S(Rm , H) be the correspondingLie group, as in [BCR81] (for spe∞ m cial H) and [Wa10]. Then Cc∞ (Rm , H) = n∈N C[−n,n] , H) is dense in m (R m m S(R , H), and S(R , H) admits well-ﬁlled charts [Gl10, Example 8.4]. Using Theorem 9.2 and approximation results from [Ne04a], it is then easy to see that ∞ m πk (S(Rm , H)) ∼ , H)) ∼ = lim πk (C[−n,n] = πk (Cc∞ (Rm , H)) m (R −→

∼ = πk (C(Sm , H)∗ ) ∼ = πk+m (H) = πk (C0 (Rm , H)) ∼ (see [Gl10, Remark 8.6]). This had been conjectured in [BCR81] and was open since 1981.

10 Subgroups of ascending unions and related topics We now discuss various results concerning subgroups of ascending unions of Lie groups (notably for direct limits of ﬁnite-dimensional Lie groups).

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Non-existence of small subgroups. It is an open problem whether inﬁnitedimensional Lie groups may contain small torsion subgroups [Ne06, p. 293]. For direct limits of ﬁnite-dimensional Lie groups, the pathology could be ruled out by proving that they do not contain small subgroups [Gl07a, Theorem A]. Theorem 10.1. If G1 ⊆ G2 ⊆ · · · is a direct sequence of ﬁnite-dimensional Lie groups, then the Lie group G = lim Gn does not have small subgroups. −→

Idea of proof. Given a compact identity neighbourhood C1 ⊆ G1 which does not contain non-trivial subgroups of G1 , there exists a compact identity neighbourhood C2 ⊆ G2 with C1 in its interior relative G2 , which does not contain non-trivial subgroups of G2 (see [Gl07a, Lemma 2.1]). Proceeding in this way, we ﬁnd a sequence (Cn )n∈N of compact identity neighbourhoods Cn ⊆ Gn 0 not containing non-trivial subgroups, such that Cn ⊆ Cn+1 for each n. Then C := n∈N Cn is an identity neighbourhood in G and we may hope that C does not contain non-trivial subgroups this is not true of G. Unfortunately, in general, as the example R(N) = n∈N Rn = n∈N Cn with Cn := [−n, n]n shows. However, if the sets Cn are chosen carefully (which requires much work), then indeed C will not contain non-trivial subgroups [Gl07a]. We mention that an analogous result is available for certain ascending unions of inﬁnite-dimensional Lie groups G1 ⊆ G2 ⊆ · · · (see [Gl07a, Theorem B]). To enable compactness arguments, each Gn has to be locally kω or each Gn a Banach–Lie group and the tangent map L(λn ) : L(Gn ) → L(Gn+1 ) of the inclusion map λn : Gn → Gn+1 a compact operator.17 Initial Lie subgroups. If G is a Lie group and H ⊆ G a subgroup, then H is called an initial Lie subgroup 18 if it admits a Lie group structure making the inclusion map ι : H → G a smooth map, such that L(ι) is injective and mappings from C k -manifolds M to H are C k if and only if they are C k as mappings to G, for each k ∈ N ∪ {∞}. Answering an open problem from [Ne06] in the negative, it was shown in [Gl08] that subgroups of inﬁnite-dimensional Lie groups need not be initial Lie subgroups. In fact, one can take G = RN (with the product topology) and H = ∞ (see [Gl08, Theorem 1.3]). For direct limits of ﬁnite-dimensional Lie groups, G = n∈N Gn , it was already shown in [Gl05b] that every subgroup H ⊆ G admits a natural Lie group structure. By [Gl08, Theorem 2.1], this Lie group structure makes H an initial Lie subgroup of G, and thus the preceding pathology does not occur for such direct limit Lie groups G. Continuous one-parameter groups and the topology on L(G). If G = n∈N Gn is a direct limit of ﬁnite-dimensional Lie groups, then every 17 18

Further technical hypotheses need to be imposed, which we suppress here. Some readers may prefer to omit the second condition, or allow M to be a manifold with C k -boundary, with corners or (more generally) a C k -manifold with rough boundary (as introduced in [GN10]). The following results carry over to these varied situations (see [Gl08]).

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continuous homomorphism (R, +) → G (i.e., each continuous one-parameter subgroup) is a continuous homomorphism to some Gn (by compact regularity) and hence smooth. It easily follows from this that the natural map θ : L(G) → Homcts (R, G) ,

x → (t → expG (tx))

is a bijection onto the set Homcts (R, G) of continuous one-parameter subgroups of G. It was asked in [Ne06, Problem VII.2] whether G is a topological group with Lie algebra in the sense of [HM07, Deﬁnition 2.11]. This holds if θ is a homeomorphism onto Homcts (R, G), equipped with the compact-open topology (which is not obvious because expG need not be a local homeomorphism at 0). As shown in [Gl08, Theorem 3.4], the latter property is always satisﬁed. Thus L(G) is determined by the topological group structure of G. For example, this implies that every continuous homomorphism from a locally exponential Lie group to G is smooth [Gl08, Proposition 3.7] (where a Lie group is called locally exponential if it has an exponential function and the latter is a local diﬀeomorphism at 0). It is an open problem whether continuous homomorphisms between arbitrary Lie groups are automatically smooth. Acknowledgements: The author is grateful to K.-D. Bierstedt (†) and S. A. Wegner (Paderborn) for discussions related to regularity properties of (LF)spaces, and thanks J. Bonet (Valencia) for comments which entered into Remark 8.5.

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Lie Groups of Bundle Automorphisms and Their Extensions Karl-Hermann Neeb Department of Mathematics, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Germany, [email protected] Summary. We describe natural abelian extensions of the Lie algebra aut(P ) of inﬁnitesimal automorphisms of a principal bundle over a compact manifold M and discuss their integrability to corresponding Lie group extensions. The case of a trivial bundle P = M × K is already quite interesting. In this case, we show that essentially all central extensions of the gauge algebra C ∞ (M, k) can be obtained from three fundamental types of cocycles with values in one of the spaces z := C ∞ (M, V ), Ω 1 (M, V ) and Ω 1 (M, V )/dC ∞ (M, V ). These cocycles extend to aut(P ), and, under the assumption that T M is trivial, we also describe the space H 2 (V(M ), z) classifying the twists of these extensions. We then show that all fundamental types have natural generalizations to non-trivial bundles and explain under which conditions they extend to aut(P ) and integrate to global Lie group extensions. Key words: gauge group, automorphism group, inﬁnite-dimensional Lie group, central extension, abelian extension, aﬃne connection. 2000 Mathematics Subject Classiﬁcations: Primary 22E65. Secondary 22E67, 17B66.

Introduction Two of the most important classes of inﬁnite-dimensional Lie groups are groups of smooth maps, such as the Lie group C ∞ (M, K) of smooth maps of a compact smooth manifold M with values in a Lie group K, and groups of diﬀeomorphisms, such as the group Diﬀ(M ) of diﬀeomorphisms of a compact smooth manifold M . A case of particular importance arises for the circle M = S1 , where LK := C ∞ (S1 , K) is called the loop group of K. If K is a compact sim by the circle ple Lie group, then LK has a universal central extension LK 1 group T. Furthermore, the group Tr ⊆ Diﬀ(S ) of rigid rotations of S1 acts smoothly by automorphisms on LK, this action lifts to the central extension

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and we thus obtain the aﬃne Kac–Moody Lie groups LK Tr .1 The LK twisted aﬃne Kac–Moody Lie groups can be realized in this picture as the ﬁxed point groups for an automorphism σ acting trivially on Tr , and inducing on LK an automorphism of the form σ(f )(t) = ϕ(f (ζt)), where ϕ ∈ Aut(K) is an automorphism of ﬁnite order m and ζ ∈ Tr satisﬁes ζ m = 1. On each aﬃne Kac–Moody group, the group Diﬀ + (S1 ) of orientation-preserving diﬀeomor Tr embeds into LK phisms of S1 acts by automorphism, so that LK 1 Diﬀ + (S ). Actually, the latter group permits an interesting twist, where the subgroup T×Diﬀ + (S1 ) is replaced by the Virasoro group, a non-trivial central extension of Diﬀ + (S1 ) by T. The purpose of this article is to discuss several inﬁnite-dimensional Lie groups that are constructed in a similar fashion from higher dimensional compact manifolds M . Since the one-dimensional manifold M = S1 is a rather simple object, the general theory leaves much more room for diﬀerent Lie group constructions, extensions and twistings thereof. Here we shall focus on recent progress in several branches of this area, in particular relating the Lie algebra picture to global objects. We shall also explain how this relates to other structures, such as multiloop algebras, which are currently under active investigation from the algebraic point of view ([ABP06], [ABFP08]). We also present open problems and describe several areas where the present knowledge is far from satisfactory. The analogue of the loop algebra C ∞ (S1 , k) of a Lie algebra k is the Lie algebra of smooth maps g = C ∞ (M, k) on the compact manifold M . Here the compactness assumption is convenient if we want to deal with Lie groups, because it implies that g is the Lie algebra of any group G = C ∞ (M, K), where K is a Lie group with Lie algebra k. This group can also be identiﬁed with the gauge group of the trivial K-principal bundle P = M × K, so that gauge groups of principal bundles are natural generalizations of mapping groups. Twisted loop groups are gauge groups of certain non-trivial principal bundles over S1 (see Section 4). For a K-principal bundle q : P → M over the compact manifold M , we write Aut(P ) := Diﬀ(P )K for the group of all diﬀeomorphisms of P commuting with the K-action, i.e., the group of bundle automorphisms. The gauge group Gau(P ) is the normal subgroup of all bundle isomorphisms inducing the identity on M . Writing Diﬀ(M )P for the set of all diﬀeomorphisms of M that can be lifted to bundle automorphisms, which is an open subgroup of the Lie group Diﬀ(M ), we obtain a short exact sequence of Lie groups 1 → Gau(P ) → Aut(P ) → Diﬀ(M )P → 1

(1)

(cf. [ACM89], [Wo07]). On the Lie algebra level, we have a corresponding short exact sequence of Lie algebras of vector ﬁelds 0 → gau(P ) → aut(P ) := V(P )K → V(M ) → 1. 1

(2)

Strictly speaking, these are the “unitary forms” of the aﬃne Kac–Moody groups. Starting with a complex simple Lie group K instead, we obtain complex versions.

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It is an important problem to understand the central extensions of gauge groups by an abelian Lie group Z on which Diﬀ(M )P acts naturally and the extent to which they can be enlarged to abelian extensions of the full group Aut(P ), or at least its identity component. Whenever such an enlargement exists, one has to understand the set of all enlargements, the twistings, which leads to the problem of classifying all abelian extensions of Diﬀ(M )P by Z. Below we shall discuss various partial results concerning these questions. We also describe several tools to address them and to explain what still remains to be done. The content of the paper is as follows. After discussing old and new results on Lie group structures on gauge groups, mapping groups and diffeomorphism groups in Section 1, we turn in Section 2 to central extensions of mapping groups G := C ∞ (M, K). Here we ﬁrst exhibit three fundamental classes of cocycles over which all other non-trivial cocycles can be factored, up to cocycles vanishing on the commutator algebra. The fundamental cocycles are deﬁned by an invariant symmetric bilinear form κ : k × k → V and an alternating map η : k × k → V as follows: 1

ωκ (ξ1 , ξ2 ) := [κ(ξ1 , dξ2 )] ∈ Ω (M, V ) := Ω 1 (M, V )/dC ∞ (M, V ), ωη (ξ1 , ξ2 ) := η(ξ1 , ξ2 ) ∈ C ∞ (M, V ) for dk η = 0, and if dk η(x, y, z) = κ([x, y], z), we also have ωκ,η (ξ1 , ξ2 ) := κ(ξ1 , dξ2 ) − κ(ξ2 , dξ1 ) − d(η(ξ1 , ξ2 )) ∈ Ω 1 (M, V ). We then discuss the action of G0 Diﬀ(M ) on the corresponding central Lie algebra extension and explain under which conditions it integrates to an extension of Lie groups. Since we are also interested in the corresponding abelian extensions of the full automorphism group Aut(M ×K) ∼ = GDiﬀ(M ) and its Lie algebra, we turn in Section 3 to abelian extensions of V(M ), resp., Diﬀ(M ) by the three types of target spaces z from above. If (the tangent bundle) T M is trivial and V = R, then a full description of H 2 (V(M ), z) in all cases has been obtained recently in [BiNe08] (Theorem 3.8). As explained in some detail in Appendix B (Corollary 8.6), the space H 2 (V(M ), z) classiﬁes the twists of the abelian extensions of the semidirect sum g V(M ) which are of the form g V(M ), where g is a central extension of g by z. See [FRST90] for a discussion of central extensions of Lie algebras of the form C ∞ (M, k) V(M, µ), where µ is a volume form on M . In Section 4 we then turn to gauge and automorphism groups of general principal bundles, where the ﬁrst step consists in the construction of analogs of the three fundamental types of cocycles. A crucial diﬃculty arises from the fact that one has to consider principal bundles with non-connected structure groups to realize natural classes of Lie algebras such as twisted aﬃne algebras as gauge algebras. We therefore have to consider target spaces V on which the quotient group π0 (K) = K/K0 acts non-trivially and consider K-invariant forms κ : k×k → V . A typical example is the target space V (k) of the universal

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invariant symmetric bilinear form of a semisimple Lie algebra k on which K = Aut(k) acts non-trivially. Replacing the exterior diﬀerential by a covariant derivative, we thus obtain cocycles 1

ωκ (ξ1 , ξ2 ) := [κ(ξ1 , dξ2 )] ∈ Ω (M, V) := Ω 1 (M, V)/dΓ V, where V → M is the ﬂat vector bundle associated to P via the K-module structure on V . For the cocycles ωκ there are natural criteria for integrability whenever π0 (K) is ﬁnite ([NeWo09]), but it is not clear how to extend ωκ to aut(P ). However, the situation simpliﬁes considerably for the cocycle d ◦ ωκ with values in Ω 2 (M, V), which extends naturally to a cocycle on aut(P ) that integrates to a smooth group cocycle on Aut(P ). The analogs of C ∞ (M, V )-valued cocycles are determined by a central extension k of k by V to which the adjoint action of K lifts. Here the central Lie algebra extension g au(P ) is a space of sections of an associated Lie algebra bundle with ﬁber k. We then show that for any central extension K ∼ of K by V there exists a K-bundle P with P /V = P , so that Aut(P ) is an abelian extension of Aut(P ) by C ∞ (M, V ) containing the central extension Gau(P ) of Gau(P ) with Lie algebra g au(P ). There are also analogs of the Ω 1 (M, V )-valued cocycles which exist whenever the 3-cocycle κ([x, y], z) is a coboundary. For non-trivial bundles the fact that the extension aut(P ) of V(M ) generally does not split makes the analysis of the abelian extensions of aut(P ) considerably more diﬃcult than in the trivial case, where we have a semidirect product Lie algebra. As a consequence, the results on group actions on the Lie algebra extensions and on integrability are much less complete than for trivial bundles. To connect our geometric setting with the algebraic setup, we describe in Section 5 the connection between multiloop algebras and gauge algebras of ﬂat bundles over tori which can be trivialized by ﬁnite coverings. The paper concludes with some remarks on bundles with inﬁnite-dimensional structure groups and appendices on integrability of abelian Lie algebra extensions, extensions of semidirect products (such as C ∞ (M, k) V(M )) and the triviality of the action of a connected Lie group on the corresponding continuous Lie algebra cohomology. Throughout we only consider Lie algebras and groups of smooth maps endowed with the compact open C ∞ -topology. To simplify matters, we focus on compact manifolds to work in a convenient Lie-theoretic setup for the corresponding groups. For non-compact manifolds, the natural setting is provided by spaces of compactly supported smooth maps with the direct limit LF -topology (cf. [KM97], [ACM89], [Mi80], [Gl02]). However, there are some classes of non-compact manifolds on which we still have (a) Lie group structure on the full group of smooth maps (Theorem 1.4). For groups of sections of group bundles one can also develop a theory for C k - and Sobolev sections, but this has the disadvantage that smooth vector ﬁelds do not act as derivations ([Sch04]).

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In [Pi93] D. Pickrell studies central extensions of loop groups C ∞ (S1 , K), K a compact connected Lie group, which are invariant under the group Diﬀ(S1 )0 of orientation-preserving diﬀeomorphisms of the circle. Such extensions turn out to deﬁne modular functors, in the sense of three-dimensional topological quantum ﬁeld theory, if they have a certain reciprocity property. Some generalizations of these results to non-connected compact groups K can be found in [Pi00]. Unitary representations of central extensions of loop groups and the existence of intertwining actions of Diﬀ(S1 ) are discussed in [TL99]. Smooth mapping groups on surfaces and realizations of their central extensions via complex geometry have been studied in [EF94] and [FK96]. Connections between central extensions of mapping groups and algebraic geometry and rational conformal ﬁeld theory are exhibited in [LMNS96]. The automorphism group Aut(P ) also shows up naturally as a symmetry group in the symplectic approach to classical ﬁeld theory ([GIMM04]). Notation and basic concepts A Lie group G is a group equipped with a smooth manifold structure modeled on a locally convex space for which the group multiplication and the inversion are smooth maps. We write 1 ∈ G for the identity element and λg (x) = gx, resp., ρg (x) = xg for the left, resp., right multiplication on G. Then each x ∈ T1 (G) corresponds to a unique left invariant vector ﬁeld xl with xl (g) := dλg (1).x, g ∈ G. The space of left invariant vector ﬁelds is closed under the Lie bracket of vector ﬁelds, and hence it inherits a Lie algebra structure. In this sense we obtain on T1 (G) a continuous Lie bracket which is uniquely determined by [x, y]l = [xl , yl ] for x, y ∈ T1 (G). We write L(G) = g for the so-obtained locally convex Lie algebra and note that for morphisms ϕ : G → H of Lie groups we obtain with L(ϕ) := T1 (ϕ) a functor from the category of Lie 0 → G0 groups to the category of locally convex Lie algebras. We write qG : G for the universal covering map of the identity component G0 of G and identify 0 with π1 (G) ∼ the discrete central subgroup ker qG of G = π1 (G0 ). In the following we always write I = [0, 1] for the unit interval in R. A Lie group G is called regular if for each ξ ∈ C ∞ (I, g), the initial value problem γ(0) = 1,

γ (t) = γ(t).ξ(t) = T (λγ(t) )ξ(t)

has a solution γξ ∈ C ∞ (I, G), and the evolution map evolG : C ∞ (I, g) → G,

ξ → γξ (1)

is smooth (cf. [Mil84]). For a locally convex space E, the regularity of the Lie group (E, +) is equivalent to the Mackey completeness of E, i.e., to the existence of integrals of smooth curves γ : I → E. We also recall that for each regular Lie group G, its Lie algebra g is Mackey complete and that all Banach–Lie groups are regular ([GN10]). For a smooth map f : M → G we deﬁne the left, resp., right logarithmic derivative in Ω 1 (M, g) by

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δ l (f )v := f (m)−1 ·Tm (f )v and δ r (f )v := Tm (f )v·f (m)−1 , where · refers to the two-sided action of G on its tangent bundle T G. A smooth map expG : L(G) → G is said to be an exponential function if for each x ∈ L(G), the curve γx (t) := expG (tx) is a homomorphism R → G with γx (0) = x. Currently, all known Lie groups modeled on complete locally convex spaces are regular, and hence possess an exponential function. For Banach–Lie groups, the existence of an exponential function follows from the theory of ordinary diﬀerential equations in Banach spaces. A Lie group G is called locally exponential if it has an exponential function mapping an open 0-neighborhood in L(G) diﬀeomorphically onto an open neighborhood of 1 in G. For more details, we refer to Milnor’s lecture notes [Mil84], the survey [Ne06a] or the monograph [GN10]. If q : E → B is a smooth ﬁber bundle, then we write Γ E := {s ∈ C ∞ (B, E) : q ◦ s = idB } for its space of smooth sections. If g is a topological Lie algebra and V a topological g-module, we write (C • (g, V ), dg ) for the corresponding Lie algebra complex ([ChE48]).

1 Lie group structures on mapping groups and automorphism groups of bundles Before we turn to extensions, in this short section we collect some results on Lie group structures on gauge groups and automorphism groups of bundles and introduce the notation and conventions used below. Some of the material is classical, but there are also some new interesting results. 1.1 Automorphism groups of bundles Theorem 1.1. Let M be a compact manifold and K a Lie group with Lie algebra k. Then the following assertions hold. (1) Diﬀ(M ) carries the structure of a Fr´echet–Lie group whose Lie algebra is the space V(M ) of smooth vector ﬁelds on M . (2) G := C ∞ (M, K) carries a natural Lie group structure with Lie algebra g := C ∞ (M, k), endowed with the pointwise bracket. If (ϕ, U ) is a k-chart of K, then (ϕG , UG ) with UG := {f ∈ G : f (M ) ⊆ U } and ϕG (f ) = ϕK ◦f is a g-chart of G. For (1) we refer to [Les67], [Omo70] and [Ha82], and for (2) we refer to [Mil84] and [Mi80] (see also [GN10] for both assertions). Theorem 1.2. If q : P → M is a smooth principal bundle over a compact manifold M with locally exponential structure group K, then its gauge group Gau(P ) and its automorphism group Aut(P ) carry Lie group structures turning (1) into a Lie group extension, where the group Diﬀ(M )P is open in Diﬀ(M ). In particular, the conjugation action of Aut(P ) on Gau(P ) is smooth.

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For the case where K is ﬁnite dimensional, this can be found in [KYMO85], [ACM89] and [KM97]. For general locally exponential Lie groups K, this has recently been proved by Wockel [Wo07]. Remark 1.3. (a) If P = M ×K is a trivial bundle, the group extension Aut(P ) splits and we have Aut(P ) ∼ = Gau(P ) Diﬀ(M )

where

Gau(P ) ∼ = C ∞ (M, K).

Here Diﬀ(M ) acts on P by ϕ.(m, k) := (ϕ(m), k) and C ∞ (M, K) acts by f.(m, k) = (m, f (m)k). (b) For any principal K-bundle P , each gauge transformation ϕ ∈ Gau(P ) determines a unique smooth function f : P → K by ϕ(p) = p.f (p) for p ∈ P . Then f satisﬁes f (p.k) = k −1 f (p)k, and any such function corresponds to a gauge transformation ϕf , so that Gau(P ) ∼ = {f ∈ C ∞ (P, K) : (∀p ∈ P )(∀k ∈ K) f (p.k) = k −1 f (p)k}. Accordingly, we have on the level of vector ﬁelds gau(P ) ∼ = {ξ ∈ C ∞ (P, k) : (∀p ∈ P )(∀k ∈ K) ξ(p.k) = Ad(k)−1 ξ(p)}. (c) For any connected manifold M , the universal covering map → M deﬁnes on M the structure of a π1 (M )-principal bundle, where qM : M π1 (M ) denotes the group of deck transformations of this bundle, considered as a discrete group. Then ) = Z(π1 (M )) Gau(M

and

) = C Aut(M (π (M )). Diﬀ(M) 1

) carries a natural Lie group structure, turning If M is compact, then Aut(M it into a covering of the open subgroup Diﬀ(M )[M] of Diﬀ(M ) with kernel ) carries a Lie group Z(π1 (M )). The normalizer Diﬀ(M ) of π1 (M ) in Diﬀ(M structure that leads to a short exact sequence 1 → π1 (M ) → Diﬀ(M ) → Diﬀ(M ) → 1. This is due to the fact that each diﬀeomorphism ϕ of M lifts to some dif normalizing π1 (M ), but in general it does not centralize feomorphism of M π1 (M ). (d) Let ρ : π1 (M ) → K be a homomorphism and ×ρ K := (M × K)/π1 (M ) Pρ := M via ρ. be the corresponding ﬂat K-bundle associated to the π1 (M )-bundle M × K for the right action (m, Then Pρ is the set of π1 (M )-orbits in M k).γ := (m.γ, ρ(γ)−1 k). We write [(m, k)] for the orbit of the pair (m, k). Then → P, s(m) s: M := [(m, 1)] is a smooth function with q ◦ s = qM , satisfying

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s(m.γ) = s(m).ρ(γ)

for

, γ ∈ π1 (M ). m ∈M

).K, any bundle automorphism ϕ In view of P = s(M is determined by its values on the image of s. If ϕ ∈ Diﬀ(M ) is the corresponding diﬀeomorphism , then ϕ of M and ϕ is a lift of ϕ to a diﬀeomorphism of M can be written ϕ(s( m)) = s(ϕ( m)).f (m)

(3)

→ K, satisfying for some smooth function f : M f (m)ρ(γ) = ρ(ϕγ ϕ −1 )f (m.γ)

for

, γ ∈ π1 (M ). m ∈M

(4)

Conversely, if (4) is satisﬁed, then (3) deﬁnes an automorphism of P . We thus obtain a gauge transformation if and only if ϕ = idM , which leads to , K) : (∀γ ∈ π1 (M )) f ◦ γ = c−1 ◦ f }, Gau(P ) ∼ = {f ∈ C ∞ (M ρ(γ) ) deﬁned under (c) acts naturally where ck (h) := khk −1 . The group Aut(M ), via ϕ(s( on P , preserving s(M m)) := s(ϕ( m)). An element of this group induces the identity on M if and only if it comes from some γ ∈ Z(π1 (M )), → K with the value ρ(γ). and this corresponds to the constant function M We thus obtain an open subgroup of Aut(P ) as the quotient ))/Z(π1 (M )). (Gau(P ) Aut(M of vector ﬁelds on M to vector Accordingly, the horizontal lift X → X ﬁelds on P is a homomorphism because the bundle is ﬂat. This yields an isomorphism aut(P ) ∼ = gau(P ) V(M ), , k) : (∀γ ∈ π1 (M )) f ◦ γ = Ad(ρ(γ))−1 ◦ f }. where gau(P ) ∼ = {f ∈ C ∞ (M 1.2 Mapping groups on non-compact manifolds For a non-compact smooth manifold M , the preceding construction of the atlas on C ∞ (M, K) no longer works because the sets UG are not open. However, it turns out that in some interesting cases there are other ways to obtain charts. To formulate the results, we say that a Lie group structure on G = C ∞ (M, K) is compatible with evaluations if g := C ∞ (M, k) is its Lie algebra and all K k evaluation maps evK m : G → K, m ∈ M , are smooth with L(evm ) = evm . We then have the following (cf. [NeWa08b]). Theorem 1.4. Let K be a connected regular real Lie group and M a real ﬁnite-dimensional connected manifold (with boundary). Then the group C ∞ (M, K) carries a Lie group structure compatible with evaluations if one of the following conditions holds.

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of K is diﬀeomorphic to a locally convex (1) The universal covering group K space, which is the case if K is ﬁnite-dimensional solvable. If, in addition, π1 (M ) is ﬁnitely generated, the Lie group structure is compatible with the smooth compact open topology. (2) dim M = 1. (3) M ∼ = Rk × C, where C is compact. For complex groups we have the following ([NeWa08b], Thms. III.12, IV.3). Theorem 1.5. Let K be a connected regular complex Lie group and M a ﬁnite-dimensional connected complex manifold without boundary. Then the group O(M, K), endowed with the compact open topology, carries a Lie group structure with Lie algebra O(M, k) compatible with evaluations if is diﬀeomorphic to a locally convex space. If, in addition, π1 (M ) is (1) K ﬁnitely generated, the Lie group structure is compatible with the compact open topology. (2) dimC M = 1, π1 (M ) is ﬁnitely generated and K is a Banach–Lie group Problem 1.6. (1) Do the results of Section 3 on central extensions of mapping group C ∞ (M, K) extend to corresponding groups of smooth and holomorphic maps on non-compact manifolds whenever Theorem 1.4 or Theorem 1.5 provides a suitable Lie group structure? For corresponding results on groups of compactly supported maps we refer to [Ne04b]. (2) Find an example of a connected non-compact smooth manifold M and a ﬁnite-dimensional Lie group K for which C ∞ (M, K) does not carry a Lie group structure compatible with evaluations. The ﬁrst candidate causing problems is the compact group K = SU2 (C), and M should be a manifold that is not a product of some Rn with a compact manifold (see Theorem 1.4(3)), which excludes all Lie groups and all Riemannian symmetric spaces. (3) Do the preceding theorems generalize in a natural way to gauge groups? Which restrictions does one have to impose on the bundles?

2 Central extensions of mapping groups In this section we discuss several issues concerning central extensions of mapping groups. We start with a description of the fundamental types of 2-cocycles on the Lie algebra g := C ∞ (M, k), where M is a ﬁnite-dimensional smooth manifold and k a ﬁnite-dimensional Lie algebra over K ∈ {R, C}. These 1 cocycles have values in spaces like C ∞ (M, V ), Ω 1 (M, V ) and Ω (M, V ) = Ω 1 (M, V )/dC ∞ (M, V ). In particular, we describe what is known about their integrability to corresponding central Lie group extensions. This is best understood for the identity component C ∞ (M, K)0 , but the extendibility to all of C ∞ (M, K) is not clear (see Problem 7.5 in Appendix A). In Section 4 we outline some generalizations of the results in the present section to gauge groups of non-trivial bundles.

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2.1 Central extensions of C ∞ (M, k) 1

We write [α] ∈ Ω (M, V ) for the image of a 1-form α ∈ Ω 1 (M, V ) in this space. The subspace dC ∞ (M, V ) of exact V -valued 1-forms is characterized by the vanishing of all integrals over loops in M , hence a closed subspace, and 1 therefore Ω (M, V ) carries a natural Hausdorﬀ locally convex topology. For a trivial k-module V , we write Sym2 (k, V )k for the space of V -valued symmetric invariant bilinear forms, and recall the Cartan map Γ : Sym2 (k, V )k → Z 3 (k, V ),

Γ (κ)(x, y, z) := κ([x, y], z).

We call κ exact if Γ (κ) is a coboundary. Theorem 2.1. Each continuous 2-cocycle ω with values in the trivial g-module K can be written as ω = β1 ◦ ω1 + β2 ◦ ω2 + β3 ◦ ω3 + ω4 , 1

where β1 , β2 , resp., β3 is a continuous linear map on a space Ω (M, V1 ), C ∞ (M, V2 ), resp., Ω 1 (M, V3 ), and ωi , i = 1, . . . , 4, is a cocycle of type i described as follows: 1

(I) Cocycles ωκ ∈ Z 2 (g, Ω (M, V )) of the form ωκ (ξ1 , ξ2 ) = [κ(ξ1 , dξ2 )],

κ ∈ Sym2 (k, V )k ,

where κ(ξ1 , dξ2 ) is considered as a V-valued 1-form on M ; (II) Cocycles ωη ∈ Z 2 (g, C ∞ (M, V )) of the form ωη (ξ1 , ξ2 ) = η(ξ1 , ξ2 ) := η ◦ (ξ1 , ξ2 ),

η ∈ Z 2 (k, V );

(III) Cocycles ωκ,η ∈ Z 2 (g, Ω 1 (M, V )) of the form ωκ,η (ξ1 , ξ2 ) = κ(ξ1 , dξ2 ) − κ(ξ2 , dξ1 ) − d(η(ξ1 , ξ2 )), where η ∈ C 2 (k, V ) and κ ∈ Sym2 (k, V )k are related by Γ (κ) = dk η; (IV) Cocycles vanishing on g × g, where g = C ∞ (M, k ) is the commutator algebra, i.e., pull-backs of cocycles of an abelian quotient of g. Proof. First we apply [NeWa08a] (Lemma 1,1, Thm. 3.1, Cor. 3.5 and Section 6) to g = C ∞ (M, K) ⊗ k to show that, after subtracting a cocycle of type (IV), each continuous cocycle ω ∈ Z 2 (g, K) can be written as ω(f ⊗ x, g ⊗ y) = βa (f dg − gdf )(x, y) − βs (f g)(x, y), where βa : Ω 1 (M, K) → Sym2 (k, K)k

and

βs : C ∞ (M, K) → C 2 (k, K)

(5)

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291

are continuous linear maps coupled by the condition Γ (βa (df )) = dk (βs (f ))

for

f ∈ C ∞ (M, K).

(6)

Conversely, any pair (βa , βs ) satisfying (6) deﬁnes a cocycle via (5): ω(f f ⊗ [x, x ], f ⊗ x ) (dg ω)(f ⊗ x, f ⊗ x , f ⊗ x ) = − =−

cyc.

βa (f f df − f d(f f ))([x, x ], x ) +

cyc.

βs (f f f )([x, x ], x )

cyc.

= Γ (βa (d(f f f )))(x, x , x ) − dk (βs (f f f ))(x, x , x ). Let V (k) := S 2 (k)/k.S 2 (k) be the target space of the universal invariant symmetric bilinear form κu (x, y) := [x ∨ y] and observe that Sym2 (k, K)k ∼ = V (k)∗ . We write Sym2 (k, K)kex for the subspace of Sym2 (k, K)k consisting of exact forms, and V (k)0 ⊆ V (k) for its annihilator. Then there exists a linear map χ : Sym2 (k, K)kex → C 2 (k, K) with dk (χ(β)) = Γ (β). As βa (dC ∞ (M, K)) ⊆ Sym2 (k, K)kex , we may use the Hahn–Banach extension theorem to extend βa from dC ∞ (M, K) to a continuous linear map βa : Ω 1 (M, K) → Sym2 (k, K)kex and put βs (f ) := χ(βa (df )), so that ω (f ⊗ x, f ⊗ x ) := βa (f df − f df )(x, x ) − βs (f f )(x, x ) is a 2-cocycle. Further, βa := βa − βa vanishes on dC ∞ (M, K), so that the values of βs := βs − βs are cocycles. Hence both pairs (βa , 0), resp., (0, −βs ) deﬁne cocycles, ω2 , resp., ω3 , satisfying ω=ω + ω2 − ω3 . We now show that each of these summands factors through a cocycle of the form (I)–(III). For ω2 we put V := V (k), so that we may write ω2 (f ⊗ x, f ⊗ x ) = βa (f df − f df )(x, x ) = β a (f df − f df ⊗ κu (x, x )), where β a : Ω 1 (M, V ) ∼ = Ω 1 (M, K) ⊗ V → K is a continuous linear map van∞ ishing on dC (M, V ). This shows that ω2 factors through a cocycle of type 1 (I) with values in Ω (M, V ). For ω3 we put V := Z 2 (k, K)∗ , and write ηu ∈ Z 2 (k, V ) for the 2-cocycle deﬁned by ηu (x, x)(f ) = f (x, x ) for f ∈ Z 2 (k, K). Then ω3 (f ⊗ x, f ⊗ x ) = βs (f f )(x, x ) = ηu (x, x )(βs (f f )) = β s (f f ⊗ ηu (x, x )), where β s : C ∞ (M, V ) ∼ = C ∞ (M, K) ⊗ V → K is a continuous linear map. Hence ω3 factors through a cocycle of type (II) with η = ηu .

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Finally, we turn to ω . Now we put V := V (k)/V (k)0 , so that V ∗ ∼ = 2 k Sym (k, K)ex and write κ : k × k → V for the symmetric bilinear map obtained from κu . We note that χ : V ∗ → C 2 (k, K) deﬁnes a map χ ∈ C 2 (k, V ) by χ(λ)(x, y) = λ(χ (x, y))

for

x, y ∈ k, λ ∈ V ∗ .

For each λ ∈ V ∗ we then identify λ with the corresponding bilinear form λ ◦ κ and obtain λ ◦ dk (χ ) = dk (λ ◦ χ ) = dk (χ(λ)) = Γ (λ) = Γ (λ ◦ κ) = λ ◦ Γ (κ), showing that dk (χ ) = Γ (κ). Let β a : Ω 1 (M, V ) ∼ = Ω 1 (M, K) ⊗ V → K be the continuous linear functional deﬁned by β a (α ⊗ v) = βa (α)(v). Then β a κ(f ⊗ x, df ⊗ x ) − κ(f ⊗ x , df ⊗ x) − d(χ (f ⊗ x, f ⊗ x )) = β a (f df − f df ) ⊗ κ(x, x ) − d(f f ⊗ χ (x, x )) = βa (f df − f df )(x, x ) − βa (d(f f ))(χ (x, x )) = βa (f df − f df )(x, x ) − χ(βa (d(f f )))(x, x ) = βa (f df − f df )(x, x ) − βs (f f )(x, x ) = ω (f ⊗ x, f ⊗ x ). This completes the proof.

Remark 2.2. (a) The central extension g := C ∞ (M, V ) ⊕ωη g deﬁned by a cocycle ωη of type (II) can also be described more directly: If k = V ⊕η k is the central extension deﬁned by η, then g∼ = C ∞ (M, k) ∼ = C ∞ (M, K) ⊗ k, i.e., ∞ the C (M, K)-Lie algebra g is obtained from k by extension of scalars from K to the ring C ∞ (M, K). (b) For any η ∈ C ∞ (M, Z 2 (k, V )), ωη(ξ1 , ξ2 )(m) := η(m)(ξ1 (m), ξ2 (m)) also deﬁnes a continuous C ∞ (M, V )-valued 2-cocycle on g. To see how these cocycles ﬁt into the scheme of the preceding theorem, we observe that Z 2 (k, V ) is ﬁnite dimensional (if k and V are ﬁnite dimensional), so that n η = i=1 αi ·ηi for some αi ∈ C ∞ (M, R) and ηi ∈ Z 2 (k, V ). This leads to ωη = factors through the cocycle ω(η1,...,ηn ) with values i αi · ωηi , showing that ωη in C ∞ (M, V )n ∼ = C ∞ (M, V n ). All the cocycles ωη are C ∞ (M, R)-bilinear and if such a cocycle is a coboundary, there exists a continuous linear map β : g → C ∞ (M, V ) with ωη (ξ1 , ξ2 ) = β([ξ2 , ξ1 ]) for ξ1 , ξ2 ∈ g. This relation easily implies that the restriction of β to the commutator algebra g = C ∞ (M, k ) is C ∞ (M, R)bilinear so that we may w.l.o.g. assume that β itself has this property. Then there exists a smooth map β : M → Hom(k, V ) with β(ξ)(m) = β(m)(ξ(m)) and we obtain

Lie Groups of Bundle Automorphisms and Their Extensions

= η(m) dk β(m)

for

293

m ∈ M.

Thus ωη is a coboundary if and only if η(M ) consists of coboundaries. In particular, [ωη ] = 0 is equivalent to [η] = 0 if η = η is constant, which corresponds to cocycles of type (II). (c) Each cocycle ωκ,η of type (III) may be composed with the quotient 1 map q : Ω 1 (M, V ) → Ω (M, V ), which leads to the cocycle q ◦ ωκ,η = 2ωκ of type (I). Here the exactness of κ ensures the existence of a lift of ωκ to an Ω 1 (M, V )-valued cocycle. If η ∈ C 2 (k, V ) also satisﬁes dk η = Γ (κ), then η − η ∈ Z 2 (k, V ) and ωκ,η = ωκ,η + d ◦ ωη−η (7) is another lift of 2ωκ . (d) If k is a semisimple Lie algebra and κ : k × k → V (k) is the universal invariant symmetric bilinear form, then the corresponding cocycle ωκ with 1 values in Ω (M, V (k)) is universal; i.e., up to coboundaries each 2-cocycle with values in a trivial module can be written as f ◦ ωκ for a continuous linear 1 map f : Ω (M, V (k)) → V (cf. [Ma02], [PS86]). (e) Cocycles of type (III) exist if and only if k carries an exact invariant symmetric bilinear form κ with Γ (κ) = 0. Typical examples of Lie algebras with this property are cotangent bundles k = T ∗ (h) := h∗ h with κ((α, x), (α , x )) := α(x ) + α (x) and η((α, x), (α , x )) := α (x) − α (x) (cf. [NeWa08a], Example 5.3). (f) Cocycles of the types (I)–(III) can also be deﬁned if k is an inﬁnite-dimensional locally convex Lie algebra. In this case we require κ and η to be continuous. The only point where we have used the ﬁnite dimension of k in the proof of Theorem 2.1 is to show that every K-valued cocycle is a sum of cocycles factoring through a cocycle of type (I)–(IV); the corresponding arguments do not carry over to the inﬁnite-dimensional case. Remark 2.3. Many results concerning cocycles of type (I)–(III) can be reduced from general manifolds to simpler ones as follows.

1 For cocycles of type (I), the integration maps γ : Ω (M, V ) → V, γ ∈ C ∞ (S1 , M ), separate the points. Accordingly, we have pull-back homomorphisms of Lie groups γ ∗ : C ∞ (M, K) → C ∞ (S1 , K), f → f ◦ γ satisfying 1 ◦ωκM = ◦ωκS ◦ (L(γ ∗ ) × L(γ ∗ )), (8) γ

S1

which can be used to reduce many things to M = S1 (cf. [MN03]). For cocycles of type (II), the evaluation maps evVm : C ∞ (M, V ) → V,

f → f (m)

separate the points, and the corresponding evaluation homomorphism of Lie ∞ groups evK m : C (M, K) → K, f → f (m), satisﬁes

294

Karl-Hermann Neeb K k k evVm ◦ ωη = η ◦ (L(evK m ) × L(evm )) = η ◦ (evm × evm ).

(9)

Finally, we observe that for cocycles of type (III), the integration maps 1 ∞ : γ Ω (M, V ) → V, γ ∈ C (I, M ), separate the points and that the ∗ pull-back homomorphism γ : C ∞ (M, K) → C ∞ (I, K) satisﬁes M I ◦ωκ,η = ◦ωκ,η ◦ (L(γ ∗ ) × L(γ ∗ )). (10)

γ

I

2.2 Covariance of the Lie algebra cocycles In this subsection we discuss the covariance of the cocycles of types (I)–(III) under Diﬀ(M ) and C ∞ (M, K), i.e., the existence of actions of these groups on the centrally extended Lie algebras g = z ⊕ω g, compatible with the actions on g and z. For Diﬀ(M ), the situation is simple because the cocycles are invariant, but for C ∞ (M, K) several problems arise. For the identity component C ∞ (M, K)0 , the existence of a corresponding action on g is equivalent to the vanishing of the ﬂux homomorphism (see Proposition 7.1 of Appendix A), but for the existence of an extension to the full group there is no such simple criterion. Remark 2.4. (Covariance under Diﬀ(M )) (a) The Lie algebra V(M ) acts in a p natural way on all spaces Ω p (M, V ) and Ω (M, V ) := Ω p (M, V )/dΩ p−1 (M, V ) and it acts on the Lie algebra g = C ∞ (M, k) by derivations. With respect to this action, all cocycles ωκ , ωη and ωκ,η are V(M )-invariant, and hence extend by ω ((f, X), (f , X )) := ω(f, f ) to cocycles of the semidirect sum g V(M ) (cf. [Ne06a], Def. I.4.2). (b) These cocycles are invariant under the full diﬀeomorphism group Diﬀ(M ), which implies that the diagonal action of this group on the corresponding central extension deﬁned by ϕ.(z, f ) := ((ϕ−1 )∗ z, ϕ.f ) is a smooth action by Lie algebra automorphisms ([MN03], Thm. VI.3). 1

Remark 2.5. (Covariance under C ∞ (M, K)) (a) For ωκ ∈ Z 2 (g, Ω (M, V )) of type (I) we also have a natural action of the full group C ∞ (M, K) on the 1 extended Lie algebra g = Ω (M, V ) ⊕ωκ g ([MN03], Thm. VI.3). If δ(f ) := l δ (f ) denotes the left logarithmic derivative of f , then θ(f )(x) := [κ(δ(f ), x)] 1 deﬁnes a linear map g → Ω (M, V ), satisfying Ad(f )∗ ωκ − ωκ = dg (θ(f )), so that

Adg (f ).(z, ξ) := (z − θ(f )(ξ), Ad(f )ξ)

deﬁnes an automorphism of g (Lemma 7.6), and we thus obtain a smooth action of the Lie group C ∞ (M, K) on g.

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(b) Let K be a connected Lie group with Lie algebra k, V a trivial k-module and η ∈ Z 2 (k, V ) a 2-cocycle. For the cocycle ωη ∈ Z 2 (g, C ∞ (M, V )) of type (II), the situation is slightly more complicated. We recall from Proposition 7.1 of Appendix A that the action of K on k lifts to an action on k = V ⊕η k if and only if the ﬂux Fη : π1 (K) → HomLie (k, V ) = H 1 (k, V ) vanishes. Let m0 ∈ M be a base point and C∗∞ (M, K) C ∞ (M, K) be the normal subgroup of all maps vanishing in m0 . Then this also is a Lie group, and C ∞ (M, K) ∼ = C∗∞ (M, K) K → K denotes the universal covering group, then a as Lie groups. If qK : K if and only smooth map f : M → K vanishing in m0 lifts to a map M → K if the induced homomorphism π1 (f ) : π1 (M ) → π1 (K) vanishes. We therefore have an exact sequence of groups → C ∞ (M, K) → Hom(π1 (M ), π1 (K)), 1 → C∗∞ (M, K) ∗

(11)

and since π1 (M ) is ﬁnitely generated because M is compact, we may consider as an open subgroup of C ∞ (M, K). From the canonical action Ad C∗∞ (M, K) k on of K on k we immediately get a smooth action of C ∞ (M, K) g∼ = C ∞ (M, k) by (Adg (f ).ξ)(m) := Adk (f (m)).ξ(m). This shows that the ﬂux Fωη : π1 (C ∞ (M, K)) ∼ = π1 (C∗∞ (M, K)) × π1 (K) → H 1 (g, C ∞ (M, V )) vanishes on the subgroup π1 (C∗∞ (M, K)), and hence factors through the ﬂux Fη of η. For any f ∈ C ∞ (M, K) and m ∈ M , the connectedness of K implies that Ad(f (m))∗ η − η is a 2-coboundary (Theorem 9.1 of Appendix C), and from that we derive the existence of a smooth map θ: M → C 1 (k, V ) with Ad(f (m)−1 )∗ η − η = dk (θ(m)) for m ∈ M . Now the automorphism Ad(f ) lifts to an automorphism of g (Lemma 7.6). (c) To determine the covariance of the cocycles ωκ,η of type (III) under the group C ∞ (M, K), we ﬁrst observe that for f ∈ C ∞ (M, K) and ξ1 , ξ2 ∈ g we obtain similarly as in [MN03], Prop. III.3: Ad(f )∗ ωκ,η (ξ1 , ξ2 ) = κ(Ad(f ).ξ1 , d(Ad(f ).ξ2 )) − κ(Ad(f ).ξ2 , d(Ad(f ).ξ1 )) − d (Ad(f )∗ η)(ξ1 , ξ2 ) = κ(ξ1 , dξ2 ) − κ(ξ2 , dξ1 ) + 2κ(δ l (f ), [ξ2 , ξ1 ]) − d (Ad(f )∗ η)(ξ1 , ξ2 ) = ωκ,η (ξ1 , ξ2 ) + 2κ(δ l (f ), [ξ2 , ξ1 ]) − d (Ad(f )∗ η − η)(ξ1 , ξ2 ) .

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It is clear that the term κ(δ l (f ), [ξ2 , ξ1 ]) is a Lie algebra coboundary. To see that the second term also has this property, we note that Γ (κ) = dk η implies that for each x ∈ k the cochain ix (dk η)(y, z) = κ([x, y], z) = κ(x, [y, z]) is a coboundary. Hence Lx η = ix dk η + dk (ix η) also is a coboundary and since K is connected, for each m ∈ M the cocycle Ad(f (m))∗ η −η is a coboundary. With Lemma 7.6 we now see that, for each smooth function f : M → K, the adjoint action on g lifts to an automorphism of the central extension g, deﬁned by ωκ,η . If f is not in C ∞ (M, K)0 , i.e., homotopic to a constant map, this does not follow from Theorem 9.1. Problem 2.6. For which groups K and which connected smooth manifolds M does the short exact sequence 1 → C ∞ (M, K)0 → C ∞ (M, K) → π0 (C ∞ (M, K)) ∼ = [M, K] → 1 split? If K ∼ = k/ΓK is abelian, then C ∞ (M, K)0 is a quotient of C ∞ (M, k), hence divisible, and therefore the sequence splits. If K is non-abelian, the situation is more involved and even the case M = S1 is non-trivial. Then [M, K] ∼ = π1 (K) and if K is abelian, then we obtain a splitting from the isomorphism π1 (K) ∼ = Hom(T, K). If K is solvable and T ⊆ K is a maximal torus, then K ∼ = Rm × T as a manifold ([Ho65]), so that [M, K] ∼ = [M, T ] and we obtain a splitting from the splitting [M, T ] → C ∞ (M, T ) ⊆ C ∞ (M, K). The case of compact (semisimple) groups is the crucial one. If M is a product of d spheres (for instance a d-dimensional torus), then [M, K] is nilpotent of length ≤ d which is not always abelian ([Whi78], Thm. X.3.6). Problem 2.7. Can the automorphisms of the central extensions g = z ⊕ω g of types (I)–(III) corresponding to elements of C ∞ (M, K) be chosen in such a way as to deﬁne a group action? If k and hence g is perfect, this follows trivially from the uniqueness of lifts to g. In general, lifts are only unique up to an element of the group H 1 (g, z), so that only an abelian extension acts. Are the corresponding cohomology classes in H 2 (C ∞ (M, K), z) that arise this way always trivial? Since the action of g on g is determined by the cocycle, we obtain an action of the identity component C ∞ (M, K)0 whenever the ﬂux vanishes, and then the restriction of the cocycle to the identity component is trivial. Here even the case M = S1 of loop groups is of particular interest. Then π0 (C ∞ (S1 , K)) ∼ = π1 (K), so that we have a natural inﬂation map H 2 (π1 (K), z) → H 2 (C ∞ (S1 , K), z), and it would be nice if all classes under consideration could be determined this way. If we have a semidirect decomposition C ∞ (M, K) ∼ = C ∞ (M, K)0 [M, K], then the present problem simpliﬁes signiﬁcantly because then semidirect product techniques similar to those discussed in Appendix B for Lie algebras apply.

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Problem 2.8. Generalize the description of H 2 (A ⊗ k, K) obtained in [NeWa08a] under the assumption that A is unital to non-unital commutative associative algebras. This would be of particular interest for algebras of the form C∗∞ (M, R) (functions vanishing in one point) or the algebra Cc∞ (M, R) of compactly supported functions. If A+ := A ⊕ K is the algebra with unit 1 = (0, 1), then we have A+ ⊗ k ∼ = (A ⊗ k) k, so that we may use the semidirect product techniques in Appendix B. 2.3 Corresponding Lie group extensions A natural question is: To which extent do the cocycles of the form ωκ , ωη and ωκ,η on g = C ∞ (M, k) actually deﬁne central extensions of the corresponding group C ∞ (M, K)? Cocycles of the form ωκ for a vector-valued κ : k × k → V are treated in [MN03], where it is shown that the corresponding period homomorphism 1

perωκ : π2 (C ∞ (M, K)) → Ω (M, V ) 1 (M, V ), and the image con(see Appendix A) has values in the subspace HdR sists of cohomology classes whose integrals over circles take values in the image of the homomorphism perκ : π3 (K) → V, [σ] → Γ (κ)eq , σ

where Γ (κ)eq ∈ Ω 3 (K, V ) is the left invariant closed 3-form whose value in 1 is Γ (κ). The ﬂux homomorphism 1

Fω : π1 (C ∞ (M, K)) → H 1 (g, Ω (M, V )) vanishes because the action of g on the central extension g deﬁned by ωκ integrates to a smooth action of the group C ∞ (M, K) on g (Remark 2.5(a), [MN03], Prop. III.3). This leads to the following theorem ([MN03], Thms. I.6, II.9, Cor. III.7). Theorem 2.9. For the cocycles ωκ and the connected group G := C ∞ (M, K)0 , the following are equivalent. (1) ωκ integrates for each compact manifold M to a Lie group extension of G. (2) ωκ integrates for M = S1 to a Lie group extension of G. 1 1 (3) The image of perωκ in HdR (M, V ) ⊆ Ω (M, V ) is discrete. (4) The image of perκ in V is discrete. These conditions are satisﬁed if κ is the universal invariant bilinear form with values in V (k).

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Remark 2.10. Composing a cocycle ωκ with the exterior derivative, we obtain the Ω 2 (M, V )-valued cocycle (d ◦ ωκ )(ξ1 , ξ2 ) = κ(dξ1 , dξ2 ). In view of [MN03], Thm. III.9, we have a smooth 2-cocycle c(f1 , f2 ) := δ l (f1 ) ∧κ δ r (f2 ) = δ r (f1 ) ∧κ Ad(f1 ).δ r (f2 ) on the full group C ∞ (M, K) which deﬁnes a central extension by Ω 2 (M, V ) whose corresponding Lie algebra cocycle is 2d ◦ ωκ . Here ∧κ denotes the natural product Ω 1 (M, k) × Ω 1 (M, k) → Ω 2 (M, V ) deﬁned by κ : k × k → V . In Subsection 4.3, this construction is generalized to gauge groups of non-trivial bundles. For the cocycles of type (II), the situation is much simpler (see Theorem 7.2). Theorem 2.11. Let K be a connected ﬁnite-dimensional Lie group with Lie algebra k, η ∈ Z 2 (k, V ), and k = V ⊕η k. Then the following are equivalent. (a) The central Lie algebra extension g = C ∞ (M, V ) ⊕ωη g of g = C ∞ (M, k) integrates to a central Lie group extension of C ∞ (M, K)0 . → K. (b) k → k integrates to a Lie group extension V → K 1 (c) The ﬂux homomorphism Fη : π1 (K) → H (k, V ) vanishes. If this is the case, then → C ∞ (M, K) → 1 1 → C ∞ (M, Z) → C ∞ (M, K) g. deﬁnes a central Lie group extension of the full group C ∞ (M, K) integrating ∗ k ∗ Proof. For m ∈ M , the relation evVm ◦ ωη = L(evK m ) η = (evm ) η (see (9) in V K Remark 2.3) leads to evm ◦ perωη = perη ◦π2 (evm ) (see (17) in Remark 7.4). Since π2 (K) vanishes ([CaE36]), the period map perη is trivial; therefore, all maps evVm ◦ perωη vanish, which implies that perωη = 0. A direct calculation further shows that the ﬂux homomorphisms Fη and Fωη : π1 (G) → H 1 (g, C ∞ (M, V )) satisfy

Fωη ([γ])(ξ)(m) = Fη ([evK m ◦ γ])(ξ(m)),

ξ ∈ g, [γ] ∈ π1 (G)

(see (18) in Remark 7.4). Therefore, Fη vanishes if and only if Fωη does. Now Theorem 7.2 of Appendix A shows that (a) is equivalent to (b), which in turn is equivalent to (c) because all period maps vanish. If these conditions are satisﬁed, we apply Theorem 7.2 with V = A to → K. This extension is a principal V obtain a central extension V → K bundle, hence trivial as such, so that there exists a smooth section σ : K → K. is a central Lie group extension of C ∞ (M, K) with a Therefore, C ∞ (M, K) g (cf. [GN10]).

smooth global section. Its Lie algebra is C ∞ (M, k) ∼ =

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→ Remark 2.12. If, in the context of Theorem 2.11, qK : K → K is a connected Lie group extension of K by Z = V /ΓZ , ΓZ ⊆ V a discrete subgroup and generally does not have a smooth global section, = k = V ⊕η k, then K L(K) → C ∞ (M, K) only is an open and the image of the canonical map C ∞ (M, K) subgroup. In our context, where K is ﬁnite dimensional, the obstruction for a map can be made quite explicit. The existence of a smooth f : M → K to lift to K of f is equivalent to the triviality of the smooth Z-bundle lift f: M → K → M , and the equivalence classes of these bundles are parametrized by f ∗K ˇ H 2 (M, ΓZ ) (cf. [Bry93]). Describing bundles in terms of Cech cocycles, it is easy to see that we thus obtain a group homomorphism C ∞ (M, K) → H 2 (M, ΓZ ),

f → [f ∗ K]

which factors through an injective homomorphism π0 (C ∞ (M, K)) ∼ = [M, K] → H 2 (M, ΓZ ),

[f ] → [f ∗ K]

of discrete groups. Since π2 (K) and Fη vanish, Remark 6.12 in [Ne04a] implies that the ∈ H 2 (K, ΓZ ) vanishes on H2 (K). Therefore, the cohomology class [K] universal coeﬃcient theorem shows that it is deﬁned by an element of as Ext(H1 (K), ΓZ ) = Ext(π1 (K), ΓZ ). Accordingly, we may interpret [f ∗ K] an element of the group Ext(H1 (M ), ΓZ ). The long exact homotopy sequence contains the short exact sequence of the Z-bundle K → π1 (K) → 1, 1 → ΓZ = π1 (Z) → π1 (K) as a central extension of π1 (K) corresponding to the which exhibits π1 (K) class in Ext(π1 (K), ΓZ ). We now see that the homomorphism from above yields an exact sequence of groups → C ∞ (M, K) → Ext(H1 (M ), ΓZ ). C ∞ (M, K) Here the rightmost map can be calculated by ﬁrst assigning to f ∈ C ∞ (M, K) ∈ the homomorphism H1 (f ) : H1 (M ) → H1 (K) ∼ = π1 (K) and then H1 (f )∗ [K] Ext(H1 (M ), ΓZ ), which can be evaluated easily in concrete cases. Now we turn to cocycles of type (III). We start with a key example. Example 2.13. Let k be a locally convex real Lie algebra and consider the locally convex Lie algebra g := C ∞ (I, k), where I := [0, 1] is the unit interval. Let κ : k × k → V be a continuous invariant symmetric bilinear form with values in the Mackey complete space V . Consider V as a trivial g-module and let η ∈ C 2 (k, V ) with Γ (κ) = dk η. Then the cocycle ωκ,η ∈ Z 2 (g, Ω 1 (I, V )) vanishes on the subalgebra k ⊆ g of constant maps.

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Let K be a connected Lie group with Lie algebra k and G := C ∞ (I, K). The map H : I×G → G, H(t, f )(s) := f (ts) is smooth because the correspond : I × G × I → K, H(t, f, s) := f (ts) is smooth (cf. [NeWa08b], ing map H Lemma A.2). Since H1 = idG and H0 (f ) = f (0), H is a smooth retraction of G to the subgroup K of constant maps. Therefore, the inclusion j : K → G induces an isomorphism π2 (j) : π2 (K) → π2 (G). The period map perωκ,η : π2 (G) → V satisﬁes perωκ,η ◦π2 (j) = perj ∗ ωκ,η = 0 because the cocycle j ∗ ωκ,η = ωκ,η |k×k vanishes, so that the period group of ωκ,η is trivial. Theorem 2.14. Let M be a smooth compact manifold, k a locally convex Lie algebra, κ : k × k → V a continuous invariant symmetric bilinear form and let η ∈ C 2 (k, V ) with Γ (κ) = dk η. For a connected Lie group K with Lie algebra k and G := C ∞ (M, K), the following assertions hold: (a) The period map perωκ,η : π2 (G) → Ω 1 (M, V ) vanishes. (b) If K is simply connected, then the ﬂux Fωκ,η vanishes. (c) If K is ﬁnite dimensional, then there exists η ∈ C 2 (k, V ) with dk η = Γ (κ) for which the ﬂux Fωκ,η vanishes. (d) If Fωκ,η = 0, then ωκ,η integrates to a central extension of G0 by Ω 1 (M, V ). Proof. (a) Combining (10) in Remark 2.3 with (17) in Remark 7.4, we obtain for each γ ∈ C ∞ (I, M ): ◦ωκ,η ◦ωκ,η ◦ perωκ,η ◦π2 (γ ∗ ) = ◦ perωκ,η ◦π2 (γ ∗ ), M = per M = per I I γ

γ

I

I

and Example 2.13 implies that perωκ,η vanishes. Since γ was arbitrary, all I M periods of ωκ,η vanish. (b) Next we assume that K is 1-connected. In Example 2.13 we have seen I that the group C ∞ (I, K) is also 1-connected, so that the ﬂux of ωκ,η vanishes. Now we apply (18) in Remark 7.4 with (10) in Remark 2.3 to see that the ﬂux Fωκ,η also vanishes. (c) Assume that K is ﬁnite dimensional and pick a maximal compact subgroup C ⊆ K. Since κ is K-invariant, K acts on the aﬃne space {η ∈ C 2 (k, V ) : dk η = Γ (κ)}, and the restriction of this action to the compact subgroup C has a ﬁxed point η. Then ωκ,η is C-invariant, and therefore C acts diagonally on g = Ω 1 (M, V ) ⊕ωκ,η g. It follows, in particular, that the ﬂux vanishes on the image of π1 (C) ∼ = π1 (K) in π1 (G) (Proposition 7.1). For M = I = [0, 1] we immediately derive that the ﬂux vanishes, and in the general case we argue as in (b) by reduction via pull-back maps γ ∗ : C ∞ (M, K) → C ∞ (I, K). (d) This assertion follows from Theorem 7.2.

Remark 2.15. We have seen in the preceding theorem that if K is simply connected, the period homomorphism and the ﬂux of ω := ωκ,η vanish. Since G := C ∞ (I, K) is also 1-connected, one may therefore expect an explicit formula for a corresponding group cocycle integrating ωκ,η because

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eq 2 the cohomology class [ωκ,η ] ∈ HdR (C ∞ (I, K), C ∞ (I, R)) vanishes (cf. [Ne02], Thm. 8.8). Such a formula can be obtained by a method due to Cartan, combined with ﬁber integration and the homotopy H : I × G → G to K. Since ω vanishes on k, the ﬁber integral θ :==I H ∗ ω eq ∈ Ω 1 (G, V ) satisﬁes dθ = ω eq , so that we may use Prop. 8.2 in [Ne04a] to construct an explicit cocycle. For a more general method, based on path groups, we refer to [Vi08].

Remark 2.16. In view of Remarks 2.4 and 2.5, for cocycles ω of types (I)–(III), the diﬀeomorphism group Diﬀ(M ) acts diagonally on g. If, in addition, K is 1-connected, then the ﬂux Fω vanishes, so that the identity component G0 := C ∞ (M, K)0 also acts naturally on g (Proposition 7.1), and this action is uniquely determined by the adjoint action of g on g (cf. [GN10]). From that it easily follows that these two actions combine to a smooth action of C ∞ (M, K)0 Diﬀ(M ) on g and the Lifting Theorem 7.7 provides a smooth action of this group by automorphisms on any corresponding central of the simply connected covering group G 0 . extension G Problem 2.17. Do the central extensions of the connected group C ∞ (M, K)0 constructed above extend to the full group? The answer is positive for M = S1 , K compact simple and ω = ωκ if κ is universal ([PS86], Prop. 4.6.9). See Problem 7.5 for the general framework in which this problem can be investigated.

3 Twists and the cohomology of vector fields We have seen above how to obtain central extensions of the Lie algebra C ∞ (M, k) and that the three types of cocycles ωκ , ωη and ωκ,η are V(M )-invariant, which leads to central extensions of the semidirect sum C ∞ (M, k) V(M ) by spaces of the type z := C ∞ (M, V ), Ω 1 (M, V ) and 1 Ω (M, V ). The bracket on such a central extension has the form [(z1 , (ξ1 , X1 )), (z2 , (ξ2 , X2 ))] = X1 .z2 − X2 .z1 + ω(ξ1 , ξ2 ), (X1 .ξ2 − X2 .ξ1 + [ξ1 , ξ2 ], [X1 , X2 ]) , and there are natural twists of these central extensions that correspond to replacing the cocycle ω ((ξ1 , X1 ), (ξ2 , X2 )) := ω(ξ1 , ξ2 ) by ω + η, where η((ξ1 , X1 ), (ξ2 , X2 )) := η(X1 , X2 )

for some

η ∈ Z 2 (V(M ), z)

(see Corollary 8.6 in Appendix B). To understand the diﬀerent types of twists, one has to determine the cohomology groups Hc2 (V(M ), R),

Hc2 (V(M ), Ω 1 (M, R))

and

1

Hc2 (V(M ), Ω (M, R)).

For a parallelizable manifold M , these spaces have been determined in [BiNe08], and in the following subsections we describe the diﬀerent types of cocycles that appear.

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We also discuss the integrability of these twists to abelian extensions of Diﬀ(M ), or at least of the simply connected covering Diﬀ(M )0 of its identity component Diﬀ(M )0 . 3.1 Some cohomology of the Lie algebra of vector ﬁelds The most obvious source of 2-cocycles of V(M ) with values in diﬀerential forms is described in the following lemma, applied to p = 2 ([Ne06c], Prop. 6). Lemma 3.1. For each closed (p + q)-form ω ∈ Ω p+q (M, V ), the prescription q

ω [p] (X1 , . . . , Xp ) := [iXp . . . iX1 ω] ∈ Ω (M, V ) q

deﬁnes a continuous p-cocycle in Z p (V(M ), Ω (M, V )). The preceding lemma is of particular interest for p = 0. In this case it associates Lie algebra cohomology classes to closed diﬀerential forms. For the • following theorem, we recall that, for a subset E ⊆ HdR (M, R), the ideal of • all classes subordinated to E (cf. [Th54]) is the minimal ideal of HdR (M, R) ∼ = • ∗ Hsing (M, R) containing E, which is the kernel of a pull-back map ϕ for some continuous map ϕ : N → M , N a topological space. Theorem 3.2. Each smooth p-form ω ∈ Ω p (M, R) deﬁnes a p-linear alternating continuous map V(M )p → C ∞ (M, R), which leads to an inclusion of chain complexes (Ω • (M, R), d) → (C • (V(M ), C ∞ (M, R)), dV(M) ) inducing an algebra homomorphism • Ψ : HdR (M, R) → H • (V(M ), C ∞ (M, R))

whose kernel is the ideal of all classes suborbinated to the Pontrjagin classes of M , i.e., the characteristic classes of the tangent bundle. Proof. The main point is the determination of the kernel of Ψ , for which we refer to [ST77, p. 224]. That ker Ψ contains the Pontrjagin classes follows from Cor. 3.2 in [Lec85], so that the representability of H • (V(M ), C ∞ (M, R)) as the cohomology of a bundle space over M ([Tsu81]) further implies that ker Ψ contains all classes subordinated to the Pontrjagin classes.

Remark 3.3. There are several classes of compact manifolds for which the Pontrjagin classes all vanish but the tangent bundle is non-trivial. According to [GHV72], II, Proposition 9.8.VI, this is in particular the case for all Riemannian manifolds with constant curvature, and hence in particular for all spheres. A second source of cocycles is the following ([Kosz74]).

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Lemma 3.4. Any aﬃne connection ∇ on M deﬁnes a 1-cocycle ζ : V(M ) → Ω 1 (M, End(T M )),

X → LX ∇,

where (LX ∇)(Y )(Z) = [X, ∇Y Z] − ∇[X,Y ] Z − ∇Y [X, Z]. For any other aﬃne connection ∇ , the corresponding cocycle ζ has the same cohomology class. Deﬁnition 3.5. We use the cocycle ζ, associated to an aﬃne connection ∇ to deﬁne k-cocycles Ψk ∈ Zck (V(M ), Ω k (M, R)). Note that A → Tr(Ak ) deﬁnes a homogeneous polynomial of degree k on gld (R), invariant under conjugation. The corresponding invariant symmetric k-linear map is given by β(A1 , . . . , Ak ) = Tr(Aσ(1) · · · Aσ(k) ), σ∈Sk

and we consider it as a linear GLd (R)-equivariant map gld (R)⊗k → R, where GLd (R) acts trivially on R. This map leads to a vector bundle map βM : End(T M )⊗k → M × R, where M × R denotes the trivial vector bundle with ﬁber R. On the level of bundle-valued diﬀerential forms, this in turn yields an alternating k-linear Diﬀ(M )-equivariant map 1 βM : Ω 1 (M, End(T M ))k → Ω k (M, R).

The Diﬀ(M )-equivariance implies the invariance of this map under the natural 1 action of V(M ), so that we can use βM to multiply Lie algebra cocycles (cf. [Fu86] or App. F in [Ne04a]). In particular, we obtain for each k ∈ N a Lie algebra cocycle Ψk ∈ Zck (V(M ), Ω k (M, R)), deﬁned by Ψk (X1 , . . . , Xk ) := (−1)k

1 sgn(σ)βM (ζ(Xσ(1) ), . . . , ζ(Xσ(k) )).

σ∈Sk

For any other aﬃne connection ∇ , the corresponding cocycle ζ and the associated cocycles Ψk , the diﬀerence ζ −ζ is a coboundary, and since products of cocycles and coboundaries are coboundaries, Ψk −Ψk is a coboundary. Hence its cohomology class in Hck (V(M ), Ω k (M, R)) does not depend on ∇. Remark 3.6. The Lie algebra 1-cocycle ζ is the diﬀerential of the group cocycle ζD : Diﬀ(M ) → Ω 1 (M, End(T M )),

ϕ → ϕ.∇ − ∇,

where (ϕ.∇)X Y := ϕ.(∇ϕ−1 .X ϕ−1 .Y ) for ϕ.X = T (ϕ) ◦ X ◦ ϕ−1 . Now the k-fold cup product of ζD with itself deﬁnes an Ω k (M, R)-valued smooth k-cocycle on Diﬀ(M ) whose corresponding Lie algebra cocycle is Ψk (cf. [Ne04a], Lemma F.3).

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Remark 3.7. Assume that M is a parallelizable d-dimensional manifold and that the 1-form τ ∈ Ω 1 (M, Rd ) implements this trivialization in the sense that each map τm is a linear isomorphism Tm (M ) → Rd . Then, for each X ∈ V(M ), LX τ can be written as LX τ = −θ(X)·τ for some smooth function θ(X) ∈ C ∞ (M, gld (R)) and θ is a crossed homomorphism, i.e., θ([X, Y ]) = X.θ(Y ) − Y.θ(X) + [θ(X), θ(Y )] ([BiNe08], Example II.3). Moreover, ∇X Y := τ −1 (X.τ (Y )) deﬁnes an aﬃne connection on M for which the parallel vector ﬁelds correspond to constant functions, and for the corresponding cocycle ζ, the map −1 τ : Γ (End(T M )) → C ∞ (M, gld (R)), τ(ϕ)(m) = τm ◦ ϕm ◦ τm satisﬁes τ ◦ ζ(X) = −d θ(X) ∈ Ω 1 (M, gld (R)). This leads to sgn(σ) Tr dθ(Xσ(1) ) ∧ . . . ∧ dθ(Xσ(k) ) . Ψk (X1 , . . . , Xk ) =

(12)

σ∈Sk

If T M is trivial, we further obtain the cocycles Φk ∈ Zc2k−1 (V(M ), C ∞ (M, R)), Φk (X1 , . . . , X2k−1 ) = sgn(σ) Tr θ(Xσ(1) ) · · · θ(Xσ(2k−1) ) σ∈S2k−1 k−1

and Ψ k ∈ Zck (V(M ), Ω (M, R)), deﬁned by Ψ k (X1 , . . . , Xk ) = sgn(σ)[Tr θ(Xσ(1) )dθ(Xσ(2) ) ∧ . . . ∧ dθ(Xσ(k) ) ] σ∈Sk

([BiNe08], Def. II.7). For each k ≥ 1 we have d ◦ Ψ k = Ψk . For k = 1 we have in particular 0

Ψ 1 (X) = Tr(θ(X)) ∈ C ∞ (M, R) = Ω (M, R),

Ψ1 (X) = Tr(dθ(X)),

and 1 Ψ 2 (X1 , X2 ) = [Tr θ(X1 )dθ(X2 ) − θ(X2 )dθ(X1 ) ] ∈ Ω (M, R). Collecting the results from [BiNe08], Section IV, we now obtain the following classiﬁcation result which, for a parallelizable manifold M , provides in particular all the information on the twists of the fundamental central extensions of the mapping Lie algebras deﬁned by cocycles of types (I)–(III). Theorem 3.8. Let M be a compact d-dimensional manifold with trivial tangent bundle. Then the following assertions hold. (1) For p ≥ 2 and d ≥ 2 the map p

p+2 HdR (M, R) → H 2 (V(M ), Ω (M, R)),

is a linear isomorphism.

[ω] → [ω [2] ]

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(2) For d ≥ 2 we have 3 H 2 (V(M ), Ω (M, R)) ∼ (M, R) ⊕ R[Ψ 1 ∧ Ψ1 ] ⊕ R[Ψ 2 ], = HdR 1

3 where HdR (M, R) embeds via [ω] → [ω [2] ]. (3) For d ≥ 2 the map 2 1 HdR (M, R)⊕HdR (M, R) → H 2 (V(M ), C ∞ (M, R)), ([α], [β]) → [α+β∧Ψ 1 ]

is a linear isomorphism. (4) For M = S1 , dim H 2 (V(M ), C ∞ (M, R)) = 2 and dim H 2 (V(M ), 1 Ω (M, R)) = dim H 2 (V(M ), R) = 1. In addition, for any compact connected manifold M , H 2 (V(M ), Ω 2 (M, R)) = R[Ψ2 ] ⊕ R[Ψ1 ∧ Ψ1 ] is 2-dimensional and 1 H 2 (V(M ), Ω 1 (M, R)) = R[Ψ 1 ∧ Ψ1 ] ⊕ {[α ∧ Ψ1 ] : [α] ∈ HdR (M, R)},

is of dimension 1 + b1 (M ). Remark 3.9. The main point of the preceding theorem is the precise description of the cohomology spaces, which is obtained under the assumption that the tangent bundle is trivial. The cocycles Ψk exist for all manifolds, whereas the construction of the cocycles Φk and Ψ k requires at least the existence of a ﬂat aﬃne connection on M : If ∇ is a ﬂat aﬃne connection on the d-dimensional manifold M , then we have a holonomy homomorphism ρ : π1 (M ) → GLd (R), for which the tangent bundle T M is equivalent to the ×ρ Rd . We then obtain a trivializing 1-form τ ∈ associated bundle M , Rd ) satisfying γ ∗ τ = ρ(γ)−1 ◦ τ for γ ∈ π1 (M ) and a corresponding Ω 1 (M crossed homomorphism ) → C ∞ (M , gld (R)). θ: V(M ∈ V(M )π1 (M) denote the canonical lift of X to M . Then For X ∈ V(M ), let X X) X) ◦ ρ(γ) = ρ(γ)−1 ◦ θ( γ ∗ θ(

for

γ ∈ π1 (M ),

which means that the cocycles Ψk , Φk and Ψ k lead on vector ﬁelds of the form to π1 (M )-invariant diﬀerential forms, and hence that they actually deﬁne X k−1 (M, R). cocycles on V(M ) with values in Ω k (M, R), C ∞ (M, R) and Ω 1

Problem 3.10. Compute the second cohomology of V(M ) in Ω (M, R) for all (compact) connected smooth manifolds. The cohomology with values in Ω p (M, R) can be reduced with results of Tsujishita ([Tsu81], Thm. 5.1.6; [BiNe08], Thm. III.1) to the description of H • (V(M ), C ∞ (M, R)), which depends very much on the topology of M ([Hae76]).

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3.2 Abelian extensions of diﬀeomorphism groups We now address the integrability of the cocycles described in the preceding subsection. Here a key method consists in a systematic use of crossed homomorphisms which are used to pull back central extensions of mapping groups (cf. [Bi03]). Deﬁnition 3.11. Let G and N be Lie groups and S : G → Aut(N ) a smooth action of G on N . A smooth map θ : G → N is called a crossed homomorphism if θ(g1 g2 ) = θ(g1 ) · g1 .θ(g2 ), i.e., if θ := (θ, idG ) : G → N S G is a morphism of Lie groups. Crossed homomorphisms are non-abelian generalizations of 1-cocycles. They have the interesting application that, for any smooth G-module V (considered as a module of N S G on which N acts trivially), we have a natural pull-back map θ∗ = (θ, idG )∗ : H 2 (N S G, V ) → H 2 (G, V ).

(13)

Two crossed homomorphisms θ1 , θ2 : G → N are said to be equivalent if there exists an n ∈ N with θ2 = cn ◦ θ1 , cn (x) = nxn−1 . Since N acts trivially by conjugation on the cohomology group H 2 (N S G, V ) (cf. [Ne04a], Prop. D.6), equivalent crossed homomorphisms deﬁne the same pull-back map on cohomology. Remark 3.12. (a) If θ : G → N is a crossed homomorphism, then θ = (θ, idG ) yields a new splitting on the semidirect product group N S G, which leads to an isomorphism N S G ∼ = N S G, where S (g) = cθ(g) ◦ S(g). As we shall see, it may very well happen that a Lie group extension of N S G is trivial on {1} × G but not on θ(G), which leads to a non-trivial extension of G (Example 3.16). (b) If θ : G → N is a crossed homomorphism of Lie groups, L(θ) := T1 (θ) : L(G) → L(N ) is a crossed homomorphism of Lie algebras because : L(G) → L(N ) L(G) is a morphism of Lie algebras. (L(θ), idL(G) ) = L(θ) (c) If the central extension n = V ⊕ω n is deﬁned by a g-invariant cocycle ω and ω is the corresponding cocycle of the abelian extension ng, corresponding to the diagonal action of g on n, then the pull-back of this cocycle with respect to a crossed homomorphism θ : g → n has the simple form (θ∗ ω )(x, y) = ω ((θ(x), x), (θ(y), y)) = ω(θ(x), θ(y)) = (θ∗ ω)(x, y). As we shall see here, crossed homomorphisms can be used to prove the integrability of many interesting cocycles of the Lie algebra of vector ﬁelds to the corresponding group of diﬀeomorphisms.

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Example 3.13. (a) Let q : V → M be a natural vector bundle over M with typical ﬁber V . Then the action of Diﬀ(M ) on M lifts to an action on V by bundle automorphisms. Any trivialization of V yields an isomorphism Aut(V) ∼ = Gau(V) Diﬀ(M ) → Aut(M × V ) ∼ = C ∞ (M, GL(V )) Diﬀ(M ), and the lift ρ : Diﬀ(M ) → Aut(V) can now be written as ρ = (θ, idDiﬀ(M) ) for some crossed homomorphism θ : Diﬀ(M ) → C ∞ (M, GL(V )). (b) Let τ ∈ Ω 1 (M, Rd ) be such that each τm is invertible in each m ∈ M , so that τ deﬁnes a trivialization of the tangent bundle of M . In [BiNe08], Ex. II.3, it is shown that LX τ = −θ(X) · τ deﬁnes a crossed homomorphism θ : V(M ) → C ∞ (M, gld (R)) (cf. Remark 3.7). A corresponding crossed homomorphism on the group level is deﬁned by Θ : Diﬀ(M ) → C ∞ (M, GLd (R)),

ϕ.τ = (ϕ−1 )∗ τ = Θ(ϕ)−1 · τ.

It satisﬁes L(Θ) = θ, taking into account that V(M ) is the Lie algebra of the opposite group Diﬀ(M )op , which causes a minus sign. We shall use this crossed homomorphism to write the cocycles Ψ 2 and Ψ 1 ∧ Ψ1 as pull-backs (cf. [Bi03] for the case where M is a torus). For the Lie algebra k := gld (R), the space Sym2 (k, R)k is two dimensional, spanned by the two forms κ1 (x, y) := tr(x) tr(y)

and

κ2 (x, y) := tr(xy).

Then the relations [tr(θ(X1 )) tr(dθ(X2 )) − tr(θ(X2 )) tr(dθ(X1 ))] = (Ψ 1 ∧ Ψ1 )(X1 , X2 ), [tr(θ(X1 )dθ(X2 ) − θ(X2 )dθ(X1 ))] = Ψ 2 (X1 , X2 ) imply that

2θ∗ ωκ1 = Ψ 1 ∧ Ψ1

and

2θ∗ ωκ2 = Ψ 2 .

With these preparations, we are now ready to prove the following. Proposition 3.14. If the tangent bundle T M is trivial, then the cocycles Ψ 1 ∧ Ψ1

and

Ψ2

in

Z 2 (V(M ), Ω 1 (M, R))

integrate to abelian Lie group extensions of some covering group of Diﬀ(M )0 . Proof. Fix i ∈ {1, 2}. Since the invariant form κi is one of the two components of the universal invariant symmetric bilinear form of gld (R), Theorem 2.9 implies that the cocycles 2ωκi of C ∞ (M, gld (R)) are integrable to a central ex of the simply connected covering group G of G := C ∞ (M, GLd (R))0 tension G

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by Z := Ω (M, V )/ im(perωκi ) on which Diﬀ(M ) acts smoothly by automorphisms (see Remark 2.4 and the Lifting Theorem 7.7). We thus obtain a := G Diﬀ(M semidirect product group H )0 which is an abelian extension Diﬀ(M )0 by Z. of H := G The crossed homomorphism Θ : Diﬀ(M )0 → G has a unique lift to a and the pull-back extension Ξ ∗H crossed homomorphism Ξ : Diﬀ(M )0 → G ∗ integrates 2θ ωκi . In view of the preceding example, the assertion follows.

Problem 3.15. It would be nice to know if we really need to pass to a covering group in the preceding proposition. As the argument in the proof shows, this is not necessary if the crossed homomorphism Θ lifts to a crossed homomorphism The obstruction for that is the homomorphism into G. π1 (Θ) : π1 (Diﬀ(M )) → π1 (G), but this homomorphism does not seem to be very accessible. In all cases where we have a compact subgroup C ⊆ Diﬀ(M ) for which π1 (C) → π1 (Diﬀ(M )) is surjective (this is in particular the case for dim M ≤ 2; [EE69]), we may choose the trivialization τ in a C-invariant fashion, so that Θ(C) is trivial. This clearly implies that π1 (Θ) vanishes. We now take a closer look at some examples of extensions of Diﬀ(M )0 deﬁned by the two cocycles discussed above. Example 3.16. (cf. [Bi03]) (The Virasoro group) For M = S1 , the preceding constructions can be used in particular to obtain the Virasoro group, resp., a corresponding global smooth 2-cocycle. The simply connected cover of C ∞ (S1 , GL1 (R))0 is the group N := ∞ 1 C (S , R) on which we have a smooth action of G := Diﬀ(S1 )op 0 by ϕ.f := f ◦ ϕ. In the notation of the preceding example, we have κ := κ1 = κ2 , and, identifying V(S1 ) with C ∞ (S1 , R), we obtain the 1-cocycle θ(X) = X . The corresponding 1-cocycle on G with values in N is given by Θ(ϕ) = log(ϕ ). 1 The cocycle ω := 2ωκ on V(S1 ) with values in Ω (S1 , R) ∼ = R is 1 ω(ξ1 , ξ2 ) = ξ1 ξ2 − ξ2 ξ1 dt = 2 ξ1 dξ2 , 0

S1

and 12 ω = ωκ is a corresponding group cocycle, deﬁning the central extension := R ×ω N on which G acts by ϕ.(z, f ) := (z, ϕ.f ), so that the semidirect N G is a central extension of N G (cf. [Ne02]). product N : G → N G. Since we have in We now pull this extension back with Θ G the relation N

(0, Θ(ϕ), ϕ)(0, Θ(ψ), ψ) = ωκ (Θ(ϕ), ϕ.Θ(ψ)), Θ(ϕ)ϕ.Θ(ψ), ϕψ , this leads to the 2-cocycle

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Ω(ϕ, ψ) = ωκ (Θ(ϕ), ϕ.Θ(ψ)) = ωκ (Θ(ϕ), Θ(ϕψ) − Θ(ϕ)) log(ψ ◦ ϕ) d(log ϕ ). = ωκ (Θ(ϕ), Θ(ϕψ)) = −ωκ (log(ψ ◦ ϕ) , log ϕ ) = − S1

This is the famous Bott–Thurston cocycle for Diﬀ(S1 )op 0 , and the corresponding central extension is the Virasoro group. On the Lie algebra level, the pull-back cocycle is (θ∗ ω)(ξ1 , ξ2 ) = ω(ξ1 , ξ2 ) =

1

ξ1 ξ2 − ξ2 ξ1 dt

0

(cf. [FF01]). Example 3.17. ([Bi03]) The preceding example easily generalizes to tori M = Td . Then we can trivialize T M with the Maurer–Cartan form τ of the group Td , which leads after identiﬁcation of V(M ) with C ∞ (M, Rd ) to θ(X) = −X (the Jacobian matrix of X). A corresponding crossed homomorphism is Θ : Diﬀ(Td ) → C ∞ (Td , GLd (R)),

Θ(ϕ) := (ϕ−1 ) ,

where the derivative ϕ of an element of Diﬀ(Td ) ⊆ C ∞ (Td , Td ) is considered as a GLd (R)-valued smooth function on Td . The main diﬀerence from the one-dimensional case is that here we do not have an explicit formula for the action of Diﬀ(Td ) on the central extension of the mapping group C ∞ (M, GLd (R))0 . As a consequence, we do not get explicit cocycles integrating Ψ 2 and Ψ 1 ∧ Ψ1 . On the integration of cocycles of the form ω [2] Now we turn to the other types of cocycles listed in Theorem 3.8(2), namely those of the form ω [2] , where ω is a closed V -valued 3-form on M . Let G be a Lie group and σ : M × G → M be a smooth right action on the (possibly inﬁnite-dimensional) smooth manifold M . Let ω ∈ Ω p+q (M, V ) be a closed (p + q)-form with values in a Mackey complete space V . If L(σ) : g → V(M ) is the corresponding homomorphism of Lie algebras, then q

ωg := L(σ)∗ ω [p] ∈ Z p (g, Ω (M, V )) is a Lie algebra cocycle, and we have a well-deﬁned period homomorphism q

perωg : πp (G) → Ω (M, V )G . If M is smoothly paracompact, which is in particular the case if M is ﬁnite q dimensional, then any class [α] ∈ Ω (M, V ) is determined by its integrals over smooth singular q-cycles S in M , which permits us to determine perωg geometrically.

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In the following we write for an oriented compact manifold F : = : Ω p+q (M × F, V ) → Ω q (M, V ) F

for the ﬁber integral (cf. [GHV72], Ch. VII). Theorem 3.18. (Period Formula) For any smooth singular q-cycle S on M and any smooth map γ : Sp → G, we have the following integral formula: perωg ([γ]) = (−1)pq ω, (14) S

S•γ

where S • γ ∈ Hp+q (M ) denotes the singular cycle obtained from the natural map Hq (M ) ⊗ Hp (G) → Hp+q (M ), [α] ⊗ [β] → [σ ◦ (α × β)] induced by the action map M × G → M . Proof. First, let γ : Sp → G be a smooth map and α : ∆q → M be a smooth singular simplex. We then obtain a smooth map α • γ := σ ◦ (α × γ) : ∆q × Sp → M,

(x, y) → α(x).γ(y).

Integration of ω yields with 7.12/14 in [GHV72] (on pull-backs and Fubini’s theorem): ω= (α • γ)∗ ω = (α × γ)∗ σ ∗ ω α•γ

= ∆q ×Sp

∆q ×Sp

∆q ×Sp

(α × idSp )∗ (idM ×γ)∗ σ ∗ ω =

∆q

= (α × idSp )∗ (idM ×γ)∗ σ ∗ ω Sp

α∗ = (idM ×γ)∗ σ ∗ ω = = (idM ×γ)∗ σ ∗ ω . = p ∆q S α Sp

∈Ω q (M,V )

It therefore remains to show that

perωg ([γ]) = (−1)pq = ((idM ×γ)∗ σ ∗ ω) .

(15)

Sp

So let Y1 , . . . , Yq ∈ V(M ) be smooth vector ﬁelds on M and write F jm : F → M × F, x → (m, x) for the inclusion map. We then have Sp ∗ = ((idM ×γ)∗ σ ∗ ω)(Y1 , . . . , Yq )m = (jm ) (iYq · · · iY1 ((idM ×γ)∗ σ ∗ ω)) Sp Sp Sp ∗ ∗ ∗ G ∗ = (jm ) (idM ×γ) (iYq · · · iY1 (σ ω)) = (jm ) (iYq · · · iY1 (σ ∗ ω)). Sp

γ

We further observe that for x1 , . . . , xp ∈ g = T1 (G) we have

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(iYq · · · iY1 (σ ∗ ω))(m,g) (g.x1 , . . . , g.xp ) = (σ ∗ ω)(m,g) (Y1 (m), . . . , Yq (m), g.x1 , . . . , g.xp ) = ω(m.g) (Y1 (m).g, . . . , Yq (m).g, m.(g.x1 ), . . . , m.(g.xp )) = ω(m.g) (Y1 (m).g, . . . , Yq (m).g, L(σ)(x1 )(m.g), · · · , L(σ)(xp )(m.g)) = (−1)pq (σg∗ (iL(σ)(xp ) · · · iL(σ)(x1 ) ω))(Y1 (m), . . . , Yq (m)) = (−1)pq (g.ωg (x1 , . . . , xp ))(Y1 , . . . , Yq )m = (−1)pq ωgeq (g.x1 , . . . , g.xp )(Y1 , . . . , Yq )m . We thus obtain in Ω q (M, V ) the identity ∗ ∗ pq = (idM ×γ) σ ω = (−1) ωgeq = (−1)pq perωg ([γ]), Sp

γ

which in turn implies (15) and hence completes the proof.

p

Example 3.19. For q = 0 and a closed p-form ω ∈ Ω (M, V ), the preceding theorem implies in particular that perωg : πp (G) → C ∞ (M, V ) satisﬁes perωg ([γ])(m) := ω with (m • γ)(x) = m.γ(x). m•γ

As an important consequence of the period formula in the preceding theorem, we derive that the 2-cocycles ω [2] of integral (p + 2)-forms integrate to abelian extensions of a covering group of Diﬀ(M )0 . This applies in particular to p = 1, which leads to the integrability of the cocycles occurring in Theorem 3.8(2). Corollary 3.20. Let ω ∈ Ω p+2 (M, V ) be closed with discrete period group Γω . Then there exists an abelian extension of the simply connected covering p group Diﬀ(M )0 by the group Z := Ω (M, V )/ΓZ , where p ΓZ := [α] ∈ HdR (M, V ) : α ⊆ Γω ∼ = Hom(Hp (M ), Γω ), Zp (M)

p

whose Lie algebra cocycle is ω [2] ∈ Z 2 (V(M ), Ω (M, V )). Proof. Applying the period formula in Theorem 3.18 to ωg = ω [2] and G = Diﬀ(M )op , which acts from the right on M , we see that the image of the period map is contained in the group ΓZ , which is discrete. Now Theorem 7.2 of Appendix A applies.

It is a clear disadvantage of the preceding theorem that it only says something about the simply connected covering group, and since π1 (Diﬀ(M )) is not very accessible, it would be nicer if we could say more about the ﬂux homomorphism of ω [2] . Unfortunately, it does not vanish in general, so that we cannot expect an extension of the group Diﬀ(M )0 itself. However, the following result is quite useful to evaluate the ﬂux.

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Theorem 3.21. (Flux Formula) Let ω ∈ Ω p+2 (M, V ) be a closed (p+2)-form

1 and γ : S1 → Diﬀ(M ) a smooth loop. Then ηγ := 0 γ(t)∗ iδr (γ)t ω dt is a closed (p + 1)-form and the ﬂux of ω [2] can be calculated as p

Fω[2] ([γ]) = [ηγ[1] ] ∈ H 1 (V(M ), Ω (M, V )). Proof. We recall from Appendix A that Fω[2] ([γ]) = [Iγ ] = −[Iγ −1 ] for 1 Iγ (X) = γ(t).ω [2] (Ad(γ(t))−1 .X, δ l (γ)t ) dt. 0 p

Hence Iγ −1 (X) ∈ Ω (M, V ) is represented by the p-form Iγ −1 (X), deﬁned on vector ﬁelds Y1 , . . . , Yp by Iγ −1 (X)(Y1 , . . . , Yp ) 1 γ(t)∗ ω(Ad(γ(t)).X, δ l (γ −1 )t , Ad(γ(t)).Y1 , . . . , Ad(γ(t)).Yp ) dt = 0

1

γ(t)∗

= 0

=− 0

1

iδr (γ)t ω (Ad(γ(t)).X, Ad(γ(t)).Y1 , . . . , Ad(γ(t)).Yp ) dt

γ(t)∗ iδr (γ)t ω (X, Y1 , . . . , Yp ) dt

= −ηγ (X, Y1 , . . . , Yp ) = −ηγ[1] (X)(Y1 , . . . , Yp .) That ηγ is closed follows directly from the fact that, for any smooth path γ : I → Diﬀ(M ), we have 1 1 ∗ ∗ γ(1) ω − ω = γ(t) Lδr (γ)t ω dt = γ(t)∗ diδr (γ)t ω dt 0

=d 0

0

1

γ(t)∗ iδr (γ)t ω dt = dηγ .

In general it is not that easy to get good hold of the ﬂux homomorphism, but there are cases where it factors through an evaluation homomorphism π1 (evm ) : π1 (Diﬀ(M )) → π1 (M, m). According to Lemma 11.1 in [Ne04a], this is the case if ω is a volume form on M . At this point we have discussed the integrability of all the 2-cocycles of V(M ) that are relevant for understanding the twists of the fundamental central 1 extensions of the mapping algebras with values in Ω (M, V ). The twists relevant for cocycles with values in Ω 1 (M, V ) and C ∞ (M, V ) are easier to handle. First we recall that a product of two integrable Lie algebra 1-cocycles is integrable to the cup product of the corresponding group cocycles ([Ne04a], Lemma F.3), so that, combined with the integrability of the Ψk (Remark 3.6), the relevant information is contained in the following remark.

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Remark 3.22. (a) If α ∈ Ω 1 (M, R) is a closed 1-form, then Example 3.19, applied to p = 1, provides a condition for the integrability of the 1-cocycle → M αg on V(M ) to a 1-cocycle on Diﬀ(M ). Passing to a covering q : M ∗ of M for which q α is exact shows that there exists a covering of the whole diﬀeomorphism group to which α integrates as a 1-cocycle with values in C ∞ (M, R) (cf. Remark 1.3(c)). (b) If ω ∈ Ω 2 (M, R) is a closed 2-form, then Example 3.19, applied with p = 2, shows that if ω has a discrete period group Γω , then ωg has discrete periods, and hence integrates to a group cocycle on some covering group of Diﬀ(M )0 . This can be made more explicit by the observation that in this case Z := R/Γω is a Lie group and there exists a Z-bundle q : P → M with a connection 1-form θ satisfying q ∗ ω = dθ. Then the group Aut(P ) is an abelian extension of the open subgroup Diﬀ(M )P of all diﬀeomorphisms lifting to bundle automorphisms of P by C ∞ (M, Z) and ω is a corresponding Lie algebra cocycle (cf. [Ko70], [NV03]). For these cocycles the ﬂux can also be made quite explicit (cf. [Ne04a], Prop. 9.11): For α ∈ C ∞ (S1 , M ) and γ ∈ C ∞ (S1 , Diﬀ(M )0 ), we have Fω ([γ]) = ω, where (γ −1 • α)(t, s) := γ(t)−1 (α(s)). α

γ −1 •α

(c) If µ is a volume form on M , then the cocycle Ψ 1 satisﬁes LX µ = −Ψ 1 (X)µ ([BiNe08], Lemma III.3), so that Ψ 1 (X) = − div X. Therefore, ζ(ϕ), deﬁned by ϕ.µ = ζ(ϕ)−1 µ, deﬁnes a C ∞ (M, R)-valued group cocycle integrating Ψ 1 . → M such If M is not orientable, then there is a 2-sheeted covering q : M that M is orientable. Accordingly, we have a 2-fold central extension Diﬀ(M ) of Diﬀ(M ) by Z/2 acting on M and the preceding construction provides a , R). If σ : M →M is the non-trivial deck transcocycle ζ : Diﬀ(M ) → C ∞ (M and there exists a volume formation, then σ reverses the orientation of M ∗ ), all functions form µ on M with σ µ = −µ. Since σ commutes with Diﬀ(M ∞ σ ∼ ∞ ζ(ϕ) are σ-invariant, and hence are in C (M , R) = C (M, R). We also note that σ ∗ µ = −µ implies that σ is not contained in the identity component of Diﬀ(M ), so that Diﬀ(M )0 ∼ = Diﬀ(M )0 and Ψ 1 also integrates to a cocycle on Diﬀ(M )0 . Toroidal groups and their generalizations 1

For the spaces z = Ω (M, V ), C ∞ (M, V ) and Ω 1 (M, V ), the cocycles ω of types (I)–(III) on g = C ∞ (M, k) are V(M )-invariant, and hence extend trivially to cocycles on h := g V(M ), and we have also seen how to classify the z-cocycles τ on V(M ) that occur as twists (cf. Theorems 3.8 and 8.5). We write ωτ for the corresponding twisted cocycle on g V(M ). Let G := C ∞ (M, K)0 . If ω is integrable to a Lie group extension, then for a corresponding extension of G by Z ∼ we write G = z/ΓZ . This extension

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is uniquely determined by [ω], which has the consequence that the natural which leads to a action of Diﬀ(M ) on g integrates to a smooth action on G, Diﬀ(M ). semidirect product G If p : Diﬀ(M ) → Diﬀ(M )0 is an abelian extension of Diﬀ(M )0 by Z cor Diﬀ(M ) is an extension of H := G Diﬀ(M )0 by responding to τ , then G Z × Z, and the anti-diagonal ∆Z in Z is a normal split Lie subgroup, so that := (G Diﬀ(M H ))/∆Z ∼ carries a natural Lie group structure, and it is now easy to verify that L(H) = z ⊕ωτ h. For the special case where M = Td is a torus, k is a simple complex algebra, κ is the Cartan–Killing form, V = C, ω = ωκ and τ is a linear combination whose Lie of the two cocycles Ψ 2 and Ψ 1 ∧ Ψ1 , we thus obtain Lie groups H algebras are Fr´echet completions of the toroidal Lie algebras. These algebras (with the twists) have been introduced by Rao and Moody in [ERMo94], and since then their representation theory has been an active research area (cf. [Bi98], [Lar99], [Lar00], [ERC04], [FJ07]).

4 Central extensions of gauge groups In this section we shall see how, and under which circumstances, the fundamental types of cocycles on C ∞ (M, k) generalize to gauge algebras gau(P ). Therefore, it becomes a natural issue to understand the corresponding central extensions of the gauge group Gau(P ), resp., its identity component. Depending on the complexity of the bundle P , this leads to much deeper questions, most of which are still open. 4.1 Central extensions of gau(P ) In this section we describe natural analogs of the cocycles of types (I)–(III) for gauge Lie algebras of non-trivial bundles. Let q : P → M be a K-principal bundle and gau(P ) its gauge algebra, which we realize as gau(P ) ∼ = {f ∈ C ∞ (P, k) : (∀p ∈ P )(∀k ∈ K) f (p.k) = Ad(k)−1 .f (p)}, which is the space of smooth sections of the associated Lie algebra bundle := P/K0 denote the Ad(P ) = P ×Ad k (see Remark 1.3(b)). Further, let M corresponding squeezed bundle, which is a, not necessarily connected, covering of M and a π0 (K)-principal bundle. The squeezed bundle is associated to the → M by the homomorphism δ1 : π1 (M ) → π0 (K) from π1 (M )-bundle qM : M the long exact homotopy sequence of P . Sometimes it is convenient to reduce

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matters to connected structure groups, which amounts to considering P as a of M . K0 -principal bundle over the covering space M Let ρ : K → GL(V ) be a homomorphism with K0 ⊆ ker ρ, so that V is a π0 (K)-module. Then the associated bundle V := P ×ρ V is a ﬂat vector bundle. It is associated to the squeezed bundle P/K0 via the representation ρ : π0 (K) → GL(V ). We have a natural exterior derivative on 1 the space Ω • (M, V) of V-valued diﬀerential forms and we deﬁne Ω (M, V) := 1 Ω (M, V)/d(Γ V). If V is ﬁnite dimensional, then d(Γ V) is a closed subspace of the Fr´echet space Ω 1 (M, V), so that the quotient inherits a natural Hausdorﬀ locally convex topology (cf. [NeWo09]). The Lie algebra aut(P ) ⊆ V(P ) acts in a natural way on all spaces of sections of vector bundles associated to P , via their realization as smooth functions on P , and hence in particular on the spaces Ω p (M, V), which can be realized as V -valued diﬀerential forms on P . Since V is ﬂat, the action on Ω p (M, V) factors through an action of the Lie algebra V(M ) ∼ = aut(P )/gau(P ). := P/K0 , the squeezed Remark 4.1. The passage from M to the covering M bundle associated to P , provides a simpliﬁcation of our setting to the case where the structure group under consideration is connected if we consider P . One has to pay for this reduction by changing M . In as a K0 -bundle over M is non-compact. particular, if π0 (K) is inﬁnite, the manifold M We now take a closer look at analogs of the cocycles of types (I)–(III) for gauge algebras g := gau(P ) of non-trivial bundles. Here the natural target 1 spaces are Γ V, Ω 1 (M, V) and Ω (M, V), replacing C ∞ (M, V ), Ω 1 (M, V ) and 1 Ω (M, V ). (I) First we choose a connection 1-form θ ∈ Ω 1 (P, k) and write ∇ for the corresponding covariant derivative, which induces covariant derivatives on all associated vector bundles such as Ad(P ). For the ﬂat bundle V the corresponding covariant derivative coincides with the Lie derivative. Let κ : k × k → V be a K-invariant symmetric bilinear map. For ξ ∈ gau(P ), the 1-form ∇ξ ∈ Ω 1 (P, k) is K-equivariant and basic; hence it deﬁnes a bundle-valued 1-form in Ω 1 (M, Ad(P )). Now κ deﬁnes a 2-cocycle 1

ωκ ∈ Z 2 (gau(P ), Ω (M, V)), 1

ωκ (ξ1 , ξ2 ) := [κ(ξ1 , ∇ξ2 )]

with values in the trivial module Ω (M, V) (cf. [NeWo09]; and [LMNS98] for bundles with connected K). (II) Let K be a connected Lie group with Lie algebra k and η ∈ Z 2 (k, V ) be a 2-cocycle, deﬁning a central Lie algebra extension qk : k → k of k by V of K by some quotient which integrates to a central Lie group extension K Z := V /ΓZ of V by a discrete subgroup. Then K acts trivially on V , so that the bundle V is trivial.

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) with The K action on k deﬁnes an associated Lie algebra bundle Ad(P ﬁber k which is a central extension of the Lie algebra bundle Ad(P ) by the ) of trivial bundle M × V (cf. [Ma05]). Now the Lie algebra g au(P ) := Γ Ad(P smooth sections of this bundle is a central extension of gau(P ) by the space C ∞ (M, V ) ∼ = Γ V. Any splitting of the short exact sequence of vector bundles ) → Ad(P ) → 0 0 → M × V → Ad(P leads to a description of the central extension g au(P ) by a 2-cocycle ω ∈ Z 2 (gau(P ), C ∞ (M, V )) which is C ∞ (M, R)-bilinear, and hence can be identiﬁed with a section of the associated bundle with ﬁber Z 2 (k, V ). (III) To see the natural analogs of cocycles of type (III), we ﬁrst note that the natural analogs of the cocycles ω0,η = d ◦ ωη are cocycles of the form d◦ω ∈ Z 2 (gau(P ), Ω 1 (M, V )), where ω ∈ Z 2 (gau(P ), C ∞ (M, V )) corresponds ) of the Lie algebra bundle Ad(P ) by the trivial to a central extension Ad(P bundle M × V . Let κ be an invariant V -valued symmetric bilinear form κ and θ a principal connection 1-form on P with associated covariant derivative ∇. We are looking for a C ∞ (M, R)-bilinear alternating map η ∈ C 2 (gau(P ), C ∞ (M, V )) for which ωκ,∇,η(ξ1 , ξ2 ) := κ(ξ1 , ∇ξ2 ) − κ(ξ2 , ∇ξ1 ) − d( η (ξ1 , ξ2 )) deﬁnes a Lie algebra cocycle with values in Ω 1 (M, V ). For ω κ (ξ1 , ξ2 ) := κ(ξ1 , ∇ξ2 ) − κ(ξ2 , ∇ξ1 ) we obtain as in Section 3: dg ( ωκ )(ξ1 , ξ2 , ξ3 ) = d(Γ (κ)(ξ1 , ξ2 , ξ3 )). Here we use the fact that, in the realization of g = gau(P ) in C ∞ (P, k), we have ∇ξ = dξ + [θ, ξ], so that ω κ (ξ1 , ξ2 ) = κ(ξ1 , dξ2 ) − κ(ξ2 , dξ1 ) + 2κ(θ, [ξ2 , ξ1 ]). Therefore, dg ωκ,∇,η = 0 is equivalent to Γ (κ) − dg η having values in constant functions, and since this map is C ∞ (M, R)-trilinear, we ﬁnd the condition Γ (κ) = dg η

in

Z 3 (gau(P ), C ∞ (M, V )).

This means that, for each m ∈ M , we have ﬁberwise in local trivializations Γ (κ) = dk η(m). In particular, κ has to be exact. Lemma 4.2. If Γ (κ) is a coboundary, then there exists a C ∞ (M, R)-bilinear alternating map η ∈ C 2 (gau(P ), C ∞ (M, V )) for which ωκ,∇,η is a 2-cocycle.

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Proof. The set C 2 (k, V )κ := {η ∈ C 2 (k, V ) : dk η = Γ (κ)} is an aﬃne space on which K acts by aﬃne map. Now the associated bundle A := P ×K C 2 (k, V )η has a smooth section η, and any such section deﬁnes a C ∞ (M, R)-bilinear element of C 2 (g, C ∞ (M, V )) with dg (d ◦ η) = d ◦ Γ (κ) = dg ( ωκ ).

In view of the preceding lemma, analogs of the type (III) cocycles on the Lie algebra gau(P ) with values in Ω 1 (M, V ) always exist if κ is exact. Problem 4.3. (Universal central extensions) Suppose that k is a semisimple Lie algebra and q : P → M a principal K-bundle. Find a universal central extension of gau(P ). If P is trivial, then we have a universal central extension 1 by Ω (M, V (k)), given by the universal invariant symmetric bilinear form κu of k (see Remark 2.2(d)). The construction described above yields a central 1 extension of gau(P ) by Ω (M, V) for the associated bundle with ﬁber V (k), but it is not clear that this extension is universal. A class of gauge algebras closest to those of trivial bundles are those of ﬂat bundles which can be trivialized by a ﬁnite covering of M . It seems quite probable that the above central extension is universal. The analogous result for multiloop algebras (see Section 5) has recently been obtained by E. Neher ([Neh07], Thm. 2.13; cf. also [PPS07], p. 147). 4.2 Covariance of the Lie algebra cocycles Remark 4.4. (Covariance for type (I)) (a) Realizing g = gau(P ) in C ∞ (P, k), we have ∇ξ = dξ + [θ, ξ],so that ωκ (ξ1 , ξ2 ) = [κ(ξ1 , ∇ξ2 )] = [κ(ξ1 , dξ2 ) + κ(θ, [ξ2 , ξ1 ])]. If X = Xξ ∈ aut(P ) corresponds to ξ ∈ gau(P ) in the sense that θ(Xξ ) = −ξ, then LXξ ξ = −[θ(Xξ ), ξ ] = [ξ, ξ ], and the fact that the curvature dθ + 12 [θ, θ] is a basic 2-form leads to 1 LXξ θ = d(iXξ θ) + iXξ dθ = −dξ − iXξ [θ, θ] = −dξ − [θ(Xξ ), θ] 2 = −dξ + [ξ, θ] = −∇ξ, so that ωκ can also be written as ωκ (ξ1 , ξ2 ) = −[κ(ξ1 , LXξ2 θ)]. The group Aut(P ) acts on the aﬃne space A(P ) ⊆ Ω 1 (P, k) of principal connection 1-forms by ϕ.θ := (ϕ−1 )∗ θ and on g = gau(P ) by ϕ.ξ = ξ ◦ ϕ−1 . We then have ϕ.∇ξ = ϕ.(dξ + [θ, ξ]) = d(ϕ.ξ) + [ϕ.θ, ϕ.ξ]) = ∇(ϕ.ξ) + [ϕ.θ − θ, ϕ.ξ].

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This leads to (ϕ.ωκ )(ξ1 , ξ2 ) = ϕ.ωκ (ϕ−1 .ξ1 , ϕ−1 .ξ2 ) = ωκ (ξ1 , ξ2 ) + [κ(ϕ.θ − θ, [ξ2 , ξ1 ])]. Note that ζ : Aut(P ) → Ω 1 (M, Ad(P )),

ϕ → ϕ.θ − θ

is a smooth 1-cocycle, so that 1

Ψ : Aut(P ) → Hom(g, Ω (M, V)),

Ψ (ϕ)(ξ) := [κ(ϕ.θ − θ, ξ)]

is a 1-cocycle with dg (Ψ (ϕ)) = ϕ.ωκ − ωκ . Therefore, Lemma 7.6 implies that ϕ.([α], ξ) := ([ϕ.α] + [κ(ϕ.θ − θ, ϕ.ξ)], ϕ.ξ) deﬁnes an automorphism of the central extension g, and it is easy to see that we thus obtain a smooth action of Aut(P ) on g (cf. [NeWo09]). If ϕf ∈ Gau(P ) is a gauge transformation corresponding to the smooth function f : P → K, then ϕ∗f θ = δ l (f ) + Ad(f )−1 .θ implies ϕf .θ = δ l (f −1 ) + Ad(f ).θ = −δ r (f −1 ) + Ad(f ).θ. (b) If P = M × K is the trivial bundle and pM : P → M and pK : P → K are the two projection maps, then the identiﬁcation of gauge and mapping groups can be written as C ∞ (M, K) → Gau(P ) ⊆ C ∞ (P, K),

∗ f → f := p−1 K · pM (f ) · pK ,

and the canonical connection 1-form is θ = δ l (pK ) ∈ Ω 1 (P, k). We further write α := Ad(pK )−1 .p∗M α for the realization of α ∈ Ω 1 (M, k) as an element 1 of Ω (M, Ad(P )) ⊆ Ω 1 (P, k). If ϕ ∈ Aut(P ) is the lift of ϕ ∈ Diﬀ(M ) deﬁned by ϕ(m, k) := (ϕ(m), k), then pK ◦ ϕ = pK implies that ϕ.θ = θ, so that ζ(ϕ) = 0. To determine ζ on the gauge group, we calculate ϕf.θ = −δ r (f) + Ad(f).θ = −δ r (f) − Ad(f).δ r (p−1 K )

r −1 ∗ r −1 −1 r ∗ .δ (pM (f )) = −δ r (f · p−1 K ) = −δ (pK · pM (f )) = −δ (pK ) − Ad(pK )

= δ l (pK ) − Ad(pK )−1 .(p∗M δ r (f )) = θ − δ r (f ). In this sense, the restriction of the cocycle ζ to Gau(P ) is ζ(ϕf) = −δ r (f ), so that it corresponds to the 1-cocycles deﬁned by the right logarithmic derivative δ r : C ∞ (M, K) → Ω 1 (M, k). We therefore recover the formulas for the actions of C ∞ (M, K) Diﬀ(M ) on g from Remarks 2.4 and 2.5. For later reference, we also note that the action of the gauge group on 1-forms is given in these terms by

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∗ ϕf. α = (ϕ−1 K ).(pM α) )∗ α = (ϕ−1 )∗ Ad(p−1 f

f

∗ ∗ )∗ α = Ad(p−1 = Ad(pK · f−1 )−1 .(pM ◦ ϕ−1 K · pM f )pM α f ∗ α. = Ad(p−1 K ).pM (Ad(f ).α) = Ad(f ).α = Ad(f ).

Remark 4.5. (Covariance for type (II)) In the type (II) situation we have con ), and since Ad(P ) structed the Lie algebra extension g = g au(P ) as Γ Ad(P is a Lie algebra bundle associated to P via an action of K on k by automorphisms, the group Aut(P ) and its Lie algebra aut(P ) act naturally on g au(P ) by automorphisms, resp., derivations, lifting the action on gau(P ). Problem 4.6. (Covariance for type (III)) It would be interesting to see if the type (III) cocycles on gau(P ) with values in Ω 1 (M, V ) are aut(P )-covariant in the sense that aut(P ) acts on g au(P ). Problem 4.7. (Corresponding crossed modules) For cocycles of type (I) and (II) we have seen that the action of aut(P ) on gau(P ) lifts to an action by derivations on the central extension g au(P ), which deﬁnes a crossed module µ : g au(P ) → aut(P ) of Lie algebras. For type (I), the characteristic class of this crossed module is an element 1 of H 3 (V(M ), Ω (M, V )), and for type (II), in H 3 (V(M ), C ∞ (M, V )). The vanishing of these characteristic classes is equivalent to the embeddability of the central extension g au(P ) to an abelian extension of aut(P ) (cf. [Ne06b], Thm. III.5). For type (II) it has been shown in [Ne06b], Lemma 6.2, that the characteristic class can be represented by a closed 3-form, so that it comes from an ele3 ment of HdR (M, V ). It seems that a proper understanding of the corresponding situation for type (I) requires a description of the third cohomology space 1 H 3 (V(M ), Ω (M, R)), so that we need a generalization of Theorem 3.8 for cohomology in degree 3. For type (III) cocycles, one similarly needs the cohomology spaces H 3 (V(M ), Ω 1 (M, R)), but they can be described with Tsujishita’s work ([Tsu81], Thm. 5.1.6; [BiNe08], Thm. III.1), which asserts that the space H • (V(M ), Ω • (M, R)) is a free module of the algebra H • (V(M ), C ∞ (M, R)), generated by the non-zero products of the cohomology classes [Ψk ]. This leads to H 3 (V(M ), Ω 1 (M, R)) ∼ = H 2 (V(M ), C ∞ (M, R)) · [Ψ1 ]. If T M is trivial, we obtain with Theorem 3.8 1 2 (M, R) · [Ψ1 ] ⊕ HdR (M, R) · [Ψ 1 ∧ Ψ1 ]. H 3 (V(M ), Ω 1 (M, R)) = HdR

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4.3 Corresponding Lie group extensions 1

(I) For cocycles of the form ωκ ∈ Z 2 (gau(P ), Ω (M, V)), it seems quite difﬁcult to decide their integrability. However, if π0 (K) is ﬁnite, we have the following generalization of Theorem 2.9 ([NeWo09]). Theorem 4.8. If π0 (K) is ﬁnite and G := Gau(P )0 , then the following are equivalent. (1) ωκ integrates for each principal K-bundle P over a compact manifold M to a Lie group extension of G. (2) ωκ integrates for the trivial K-bundle P = S1 × K over M = S1 to a Lie group extension of G. (3) The image of perκ : π3 (K) → V is discrete. These conditions are satisﬁed if κ is the universal invariant bilinear form with values in V (k). As we have already seen for trivial bundles, the cocycle d ◦ ωκ with values in Ω 2 (M, V) can be integrated quite explicitly. We now explain the geometric background for that in the general case. We start with a general group-theoretic remark. Remark 4.9. Let E and F be vector spaces and ω : E × E → F an alternating bilinear form. (a) We write H := H(E, F, ω) for the corresponding Heisenberg group, which is the product set F × E, endowed with the product (f1 , e1 )(f2 , e2 ) = (f1 + f2 + ω(e1 , e2 ), e1 + e2 ). The invariance group of ω is Sp(E, F, ω) := {(ϕ, ψ) ∈ GL(E) × GL(F ) : ϕ∗ ω = ψ ◦ ω}, and the natural diagonal action of this group on F × E preserves ω, so that we may form the semidirect product group HSp(E, F, ω) := H(E, F, ω) Sp(E, F, ω) with the product (f1 , e1 , ϕ1 , ψ1 )(f2 , e2 , ϕ2 , ψ2 ) = (f1 +ψ1 .f2 +ω(e1 , ϕ1 .e2 ), e1 +ϕ1 .e2 ,ϕ1 ϕ2 ,ψ1 ψ2 ). This group is an abelian extension of the group E Sp(E, F, ω) ⊆ Aﬀ(E) × GL(F ) by F , and the corresponding cocycle is

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Ω((e1 , ϕ1 , ψ1 ), (e2 , ϕ2 , ψ2 )) = ω(e1 , ϕ1 .e2 ). (b) Now let G be a group, and let ρE : G → Aﬀ(E) and ρF : G → GL(F ) be group homomorphisms such that ρ := (ρE , ρF ) maps G into E Sp(E, F, ω). := ρ∗ HSp(E, F, ω) is an abelian extension of G by V . To Then the pull-back G make the cocycle ωG of this extension explicit, we write ρE (g) = (µ(g), ϕ(g)) and obtain ωG (g1 , g2 ) = ω(µ(g1 ), ρ(g1 ).µ(g2 )). This 2-cocycle ωG ∈ Z 2 (G, F ) is the cup product ωG = µ ∪ω µ of the 1-cocycle µ ∈ Z 1 (G, E) with itself with respect to the equivariant bilinear map ω (cf. [Ne04a], App. F). If we think of E as an abstract aﬃne space without specifying an origin, then any point o ∈ E leads to a 1-cocycle µ(g) := g.o − o with values in the translation group (E, +) and, accordingly, we get ωG (g1 , g2 ) = ω(g1 .o − o, ρ(g1 )(g2 .o − o)). Remark 4.10. On the translation space Ω 1 (M, Ad(P )) of the aﬃne space A(P ) of principal connection 1-forms, any V -valued K-invariant symmetric bilinear map κ on k deﬁnes an alternating C ∞ (M, R)-bilinear map Ω 1 (M, Ad(P )) × Ω 1 (M, Ad(P )) → Ω 2 (M, V),

(α, β) → α ∧κ β,

so that A(P ) is an aﬃne space with a translation-invariant Ω 2 (M, V)-valued 2-form that is invariant under the aﬃne action of Aut(P ) by pull-backs. Proposition 4.11. The smooth 2-cocycle ζκ ∈ Z 2 (Aut(P ), Ω 2 (M, V)),

ζκ (ϕ1 , ϕ2 ) = (ϕ1 .θ − θ) ∧κ ϕ1 .(ϕ2 .θ − θ)

deﬁnes an abelian extension of Aut(P ) by Ω 2 (M, V) whose corresponding Lie algebra cocycle is given by (X1 , X2 ) → (LX1 θ) ∧κ (LX2 θ) − (LX2 θ) ∧κ (LX1 θ) = 2(LX1 θ) ∧κ (LX2 θ). Remark 4.12. If P = M × K is a trivial bundle, then we obtain on Gau(P ) ∼ = C ∞ (M, K) with Remark 4.4(b): g ) = δ l (f) ∧κ δ r ( g ) = δ l (f ) ∧κ δ r (g) . ζκ (ϕf, ϕg ) = δ r (f) ∧κ Ad(f).δ r ( Problem 4.13. From the existence of the cocycle ζκ , it follows that d ◦ c = 0 holds for the obstruction class c of the corresponding crossed module (see Problem 4.7). Therefore, the long exact cohomology sequence associated to the short exact sequence 1

1 2 0 → HdR (M, V) → Ω (M, V) → BdR (M, V) → 0

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1 implies that c lies in the image of the space H 3 (V(M ), HdR (M, V)). As the 1 action of V(M ) on HdR (M, V) is trivial, we have to consider H 3 (V(M ), R). This space vanishes for dim M ≥ 3 (cf. [BiNe08], Cor. D.6, resp., [Hae76], Thm. 3/4), so that in this case the central extension g au(P ) corresponding to ωκ embeds into an abelian extension of aut(P ). It is an interesting problem to describe these extensions more explicitly. Are there any two-dimensional manifolds for which the obstruction class in c is non-zero?

(II) Now we turn to cocycles of type (II). As before, we ﬁx a central of K by Z ∼ Lie group extension K = V /ΓZ . Then there is an obstruction 2 ˇ class χK ∈ H (M, Z) with values in the sheaf of germs of smooth Z-valued functions that vanishes if and only if there exists a K-principal bundle P with ∼ P /Z = P , as a K-bundle ([GP78]). Using the isomorphisms 3 ˇ 2 (M, Z) ∼ ˇ 3 (M, ΓZ ) ∼ H (M, ΓZ ), =H = Hsing 3 (M, V ), and it has been shown we see that χ maps to some class χ ∈ HdR in [LW08] that χ yields (up to sign) the characteristic class χ(µ) of the Lie algebra crossed module µ : g au(P ) → aut(P ). (see Theorem 3.2). On the level of de Rham classes, the following theorem can also be found in [LW08]; the present version is a reﬁnement.

→ K a central Theorem 4.14. Let K be a connected Lie group and qK : K Lie group extension by Z with Z0 ∼ and q : P → M a principal KV /Γ = Z bundle. Then the following assertions hold. (a) If Z is connected and π2 (K) vanishes, which is the case if dim K < ∞, then there is a K-bundle P with P /Z ∼ = P and we obtain an abelian Lie group extension 1 → C ∞ (M, Z) → Aut(P) → Aut(P ) → 1 containing the central extension 1 → C ∞ (M, Z) → Gau(P) → Gau(P ) → 1 integrating the Lie algebra g au(P ). 3 (b) If dim K < ∞, then the obstruction class in HdR (M, V ) vanishes. (c) If K is 1-connected and K is connected, then Z is connected. ´ Cartan Proof. (a) That π2 (K) vanishes if K is ﬁnite dimensional is due to E. ([CaE36]). If π2 (K) vanishes and Z is connected, then Theorem 7.12 of [Ne02] of K by Z ∼ implies that the central extension K = V /ΓZ can be lifted to a ∼ K /Γ central extension K of K by V with K = Z . This implies that the which obstruction class to lifting the structure group of the bundle P to K, 2 ˇ is an element of the sheaf cohomology group H (M, Z), is the image of an ˇ 2 (M, V ) = {0}, and hence trivial. Here we have used the fact that element of H

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ˇ p (M, V ), p > 0, vanish because the sheaf of the sheaf cohomology groups H germs of smooth maps with values in the vector space V is soft (cf. [God73]). (b) Since Z0 is divisible, we have a direct product decomposition Z ∼ = is a Baer sum of an Z0 × π0 (Z) and, accordingly, the central extension K extension by Z0 and an extension by π0 (Z). According to (a), the extension by Z0 does not contribute to the obstruction class, so that it is an element of H 2 (M, π0 (Z)), and therefore the corresponding de Rham class in H 3 (M, V ) vanishes. 0 is a connected covering group of K, and hence trivial, (c) The group K/Z and therefore Z = Z0 .

is a central T-extension of some inﬁnite-dimensional group Remark 4.15. If K 3 for which the corresponding de Rham obstruction class in HdR (M, R) is non-zero, Theorem 3.2 implies that the corresponding class in H 3 (V(M ), C ∞ (M, R)) is also non-zero because the degrees on the Pontrjagin classes are multiples of 4, so that the ideal they generate vanishes in degrees ≤ 3. This means that the class of the corresponding crossed module µ : g au(P ) → aut(P ) is non-trivial, and hence the central extension g au(P ) does not embed equivariantly into an abelian extension of aut(P ). In a similar fashion as the group H 2 (M, Z) classiﬁes T-bundles over M , the group H 3 (M, Z) classiﬁes certain geometric objects called bundle gerbes ([Bry93], [Mu96]). As the preceding discussion has shown, non-trivial bundle gerbes can only be associated to the central T-extensions of inﬁnitedimensional Lie groups. For more on bundle gerbes and compact groups, we refer to [SW07] in this volume.

5 Multiloop algebras In this section we brieﬂy discuss the connection between gauge algebras of ﬂat bundles and the algebraic concept of a (multi-)loop algebra. These are inﬁnite-dimensional Lie algebras which are presently under active investigation from the algebraic point of view ([ABP06], [ABFP08]). To simplify matters, we discuss only the case where k is a complex Lie algebra. 5.1 The algebraic picture Let k be a complex Lie algebra, m1 , . . . , mr ∈ N, ζi ∈ T with ζimi = 1 and σ1 , . . . , σr ∈ Aut(k) with σimi = idk . Then we obtain an action of the group ∆ := Z/m1 × · · · × Z/mr on the Lie algebra −1 k ⊗ C[t1 , t−1 1 , . . . , tr , tr ]

by letting the i-th generator σ i of ∆ act by σ i (x ⊗ tα ) := σi (x) ⊗ ζiαi tα .

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The algebra −1 ∆ M (k, σ1 , . . . , σr ) := (k ⊗ C[t1 , t−1 1 , . . . , tr , tr ])

of ﬁxed points of this action is called the corresponding multiloop algebra. For r = 1 we also write L(k, σ) := M (k, σ) and call it a loop algebra. The loop algebra construction can be iterated: If we start with σ1 ∈ Aut(k) with σ1m1 = id and pick σ2 ∈ Aut(L(k, σ1 )) with σ2m2 = id, we put L(k, σ1 , σ2 ) := L(L(k, σ1 ), σ2 ). Repeating this process, we obtain the iterated loop algebras L(k, σ1 , . . . , σr ) := L(L(k, σ1 , . . . , σr−1 ), σr ), m

where σj ∈ Aut(L(k, σ1 , . . . , σj−1 )) is assumed to satisfy σj j = id. Clearly, every multiloop algebra can also be described as an iterated loop algebra, but the converse is not true (see Example 5.3). 5.2 Geometric realization of multiloop algebras There is a natural analytic variant of the loop algebra construction. Here the main idea is that the analytic counterpart of the algebra C[t, t−1 ] of Laurent polynomials is the Fr´echet algebra C ∞ (S1 , C) of smooth complex-valued functions on the circle. Its dense subalgebra of trigonometric polynomials is isomorphic to the algebra of Laurent polynomials, so that it is a completion of this algebra. A drawback of this completion process is that the algebra of smooth functions has zero divisors, but this in turn facilitates localization arguments. −1 The analytic analogue of the algebra k ⊗ C[t1 , t−1 1 , . . . , tr , tr ] is the ∞ r Fr´echet–Lie algebra C (T , k) of smooth k-valued functions on the r-dimensional torus Tr , endowed with the pointwise Lie bracket. To describe the analytic analogs of the multiloop algebras, let ∆ := Z/m1 × · · · × Z/mr and ﬁx a homomorphism ρ : ∆ → Aut(k), which is determined by a choice of automorphisms σi of k with σimi = idk . Let m := (m1 , . . . , mr ) and consider the corresponding torus Trm := (R/m1 Z) × · · · × (R/mr Z), on which the group ∆ ∼ = Zr /(m1 Z ⊕ . . . ⊕ mr Z) acts in the natural fashion from the right by factorization of the translation action of Zr on Rr . Now we obtain an action of ∆ on the Lie algebra C ∞ (Trm , k) by (γ.f )(x) := γ.f (x.γ) Then we have a dense embedding M (k, σ1 , . . . , σr ) → C ∞ (Trm , k)∆ . To realize this Lie algebra geometrically, we note that the quotient map q : Trm → Tr is a regular covering for which ∆ acts as the group of deck transformations on

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Trm . Then the homomorphism ρ : ∆ → Aut(k) from above deﬁnes a ﬂat Lie algebra bundle K := (Trm × k)/∆, [t, x] → q(t), over Tr , where ∆ acts by γ.(t, x) := (t.γ −1 , γ.x), and the space Γ K of smooth sections of this Lie algebra bundle can be realized as {f ∈ C ∞ (Trm , k) : (∀γ ∈ ∆)(∀x ∈ Trm ) f (x.γ) = γ −1 .f (x)} = C ∞ (Trm , k)∆ . In this sense, the Lie algebra Γ K is a natural geometric analogue of a multiloop algebra. 5.3 A generalization of multiloop algebras Our geometric realization of multiloop algebras suggests the following geo → M be a regular metric generalization. Let ∆ be a discrete group and q : M as the group of covering, where ∆ acts from the right via (t, γ) → t.γ on M deck transformations. Let k be a locally convex Lie algebra and ρ : ∆ → Aut(k) an action of ∆ by topological automorphisms of k. Let K0 be a 1-connected (regular or locally exponential) Lie group with Lie algebra k. Then the action of ∆ integrates to an action ρK : ∆ → Aut(K0 ) by Lie group automorphisms ([GN10]), so that we may form the semidirect product Lie group K := K0 ∆ with π0 (K) = ∆. Next we form the ﬂat Lie algebra bundle × k)/∆, K := (M

qK : K → M, [t, x] := ∆.(t, x) → q(t),

over M , where ∆ acts on M × K by γ.(t, x) := (t.γ −1 , γ.x). Each section of → k, K can be written as s(q(t)) = [t, fs (t)] with a smooth function f : M −1 satisfying f (t.γ) = γ .f (t) for t ∈ M , γ ∈ ∆. We thus obtain a realization of the space Γ K of smooth sections of this Lie algebra bundle as , k) : (∀γ ∈ ∆) f (t.γ) = γ −1 .f (t)} = C ∞ (M , k)∆ , {f ∈ C ∞ (M

(16)

, k) is deﬁned by (γ.f )(t) := γ.f (t.γ). where the Γ -action on C ∞ (M × K)/∆ is the ﬂat K-bundle associated to the Proposition 5.1. If P := (M inclusion ∆ → K, then gau(P ) ∼ = Γ K. Hence the Lie algebra of sections of a ﬂat Lie algebra bundle can always be realized as a gauge algebra of a ﬂat principal bundle; the converse is trivial. Proof. Our assumptions on K0 imply that Aut(K0 ) ∼ = Aut(k) ([GN10]), so that the action of ∆ on k integrates to an action ρK on K0 by automorphisms. × K0 is a trivial K0 -bundle, but it is not trivial as a Clearly, P ∼ = M × K0 by K-bundle, because the group ∆ acts on M (m, k).γ = (m.γ, ρK (γ)−1 (k)).

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Let Ad(P ) = (P × k)/K denote the corresponding gauge bundle with Γ Ad(P ) ∼ = gau(P ). Then gau(P ) ∼ = {f ∈ C ∞ (P, k) : (∀p ∈ P )(∀k ∈ K) f (p.k) = Ad(k)−1 .f (p)} ∼ , k) : (∀m ∈ M )(∀γ ∈ ∆) f (m.γ) = Ad(γ)−1 .f (m)} = {f ∈ C ∞ (M , k) : (∀m ∈ M )(∀γ ∈ ∆) f (m.γ) = ρ(γ)−1 .f (m)}. = {f ∈ C ∞ (M

5.4 Connections to forms of Lie algebras over rings In this subsection we assume that ∆ is ﬁnite and brieﬂy describe the connection to forms of Lie algebras over rings. For more details on this topic we refer to [PPS07]. The Lie algebra Γ K is a module of the algebra R := C ∞ (M, C), and the Lie bracket is R-bilinear. The pull-back bundle × K : q(t) = qK (x)} q ∗ K = {(t, x) ∈ M × k → q ∗ K, (t, y) → (t, [t, y]) is an isomorphism is trivial, because the map M of Lie algebra bundles. Accordingly, the space Γ (q ∗ K), which is a module of , C), is isomorphic to Γ (M × k) ∼ , k). On the level of S := C ∞ (M = C ∞ (M sections, we have an R-linear embedding q ∗ : Γ K → Γ (q ∗ K),

(q ∗ s)(t) := (t, s(q(t))),

which corresponds to the realization of Γ K as the subspace (16) of ∆-ﬁxed , k). Using the local triviality of the bundle and the ﬁniteness points in C ∞ (M of ∆, one easily veriﬁes that the canonical map ∼ , k), Γ K ⊗R S → Γ K = C ∞ (M

s ⊗ f → (q ∗ s) · f

is a linear isomorphism (cf. Thm 7.1 in [ABP06]). In this sense, the R-Lie , k). algebra Γ K is an R-form of the S-Lie algebra C ∞ (M denote the completion of the algebraic tensor product Problem 5.2. Let ⊗ as a locally convex space. Determine to which extent the map , K) → Γ K ∼ , k), C ∞ (M,K) C ∞ (M Γ K⊗ = C ∞ (M

s ⊗ f → f · (q ∗ s)

is an isomorphism of Lie algebras if ∆ is not necessarily ﬁnite. In the proof of Theorem 7.1 in [ABP06], the ﬁniteness of ∆ is used to conclude that its cohomology with values in any rational representation space is trivial. Example 5.3. (cf. Example 9.7 in [ABP06]) We consider k = sln (C) for n ≥ 2 and the iterated loop algebra L(k, σ1 , σ2 ) deﬁned as follows. As σ1 ∈ Aut(k) we choose an involution not contained in the identity component of Aut(k),

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so that L(k, σ1 ) is a non-trivial loop algebra corresponding to the aﬃne (2) Kac–Moody algebra of type An−1 (cf. [Ka90], Ch. 8). As σ2 , we take the involution on L(k, σ1 ) = {f ∈ C ∞ (T, k) : f (−t) = σ1 (f (t))}, where T is realized as the unit circle in C× , deﬁned by σ2 (f )(t1 ) := f (t−1 1 ) and put L := L(k, σ1 , σ2 ) = {f ∈ C ∞ (T, L(k, σ1 )) : : f (−t2 ) = σ2 (f (t2 ))}. We may thus identify elements of L with smooth functions f : T2 → k satisfying f (t1 , −t2 ) = f (t−1 1 , t2 )

and

f (−t1 , t2 ) = σ1 (f (t1 , t2 )).

For τ1 (t1 , t2 ) := (−t1 , t2 ) and τ2 (t1 , t2 ) := (t−1 1 , −t2 ) we thus obtain a ﬁxed point free action of ∆ := τ1 , τ2 on T2 and a representation ρ : ∆ → Aut(k), deﬁned by ρ(τ1 ) := σ1 and ρ(τ2 ) := id. Then L∼ = {f ∈ C ∞ (T2 , k) : (∀γ ∈ ∆)(∀t ∈ T2 ) f (t.γ) = γ −1 .f (t)}. This means that L ∼ = Γ K for the Lie algebra bundle K over M := T2 /∆, where M is the Klein bottle (τ2 is orientation reversing). This implies that the centroid of L is C ∞ (M, C) ([Lec80]), and hence not isomorphic to C ∞ (T2 , C), so that L cannot be a multiloop algebra.

6 Concluding remarks From the structure-theoretic perspective on inﬁnite-dimensional Lie groups, it is natural to ask for a classiﬁcation of gauge groups of principal bundles and associated structural data, such as their central extensions, invariant forms, etc. One readily observes that two equivalent K-principal bundles Pi → M have isomorphic gauge groups, so that a classiﬁcation of gauge groups or automorphism groups of K-principal bundles can be achieved, in principle, by a classiﬁcation of all K-principal bundles and then an identiﬁcation of those whose gauge groups are isomorphic. A key observation is that if a bundle P is twisted by a Z(K)-bundle PZ , then the corresponding gauge groups are isomorphic Lie groups, where the isomorphism on the Lie algebra level is C ∞ (M, R)-linear. Taking this reﬁned structure into account leads to the concept of (transitive) Lie algebroids and the perspective on Gau(P ) as the group of sections of a Lie group bundle (cf. [Ma89] for results in this direction

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and [Ma05] for more on Lie groupoids). Currently the theory of Lie groupoids is only available for ﬁnite-dimensional structure groups K. It would be natural to develop it also for well-behaved inﬁnite-dimensional groups over ﬁnitedimensional manifolds. If K is ﬁnite dimensional with ﬁnitely many connected components, then the classiﬁcation of K-principal bundles can be reduced to the corresponding problem for a maximal compact subgroup, so that the well-developed theory of bundles with compact structure group applies ([Hu94]). If K is an inﬁnite-dimensional Lie group, it might not possess maximal compact subgroups,2 so that a similar reduction does not work. However, recent results of M¨ uller and Wockel show that each topological K-principal bundle over a ﬁnite-dimensional paracompact smooth manifold M is equivalent to a smooth one, and that two smooth K-principal bundles which are topologically equivalent are also smoothly equivalent ([MW09]). Therefore, the set Bun(M, K) of equivalence classes of smooth K-principal bundles over M can be identiﬁed with the set Bun(M, K)top of topological K-bundles over M , and the latter set can be identiﬁed with the set [M, BK] of homotopy classes of continuous maps of M into a classifying space BK of the topological group K ([Hu94], Thms. 9.9, 12.2, 12.4). On the topological level, each continuous map σ : M → BK deﬁnes an algebra homomorphism • H • (σ, R) : H • (BK, R) → H • (M, R) ∼ (M, R), = HdR

whose range is the characteristic classes of the corresponding bundle P . The advantage of these characteristic classes is that they can be represented by closed diﬀerential forms on M ; hence, they can be evaluated by integrals over compact submanifolds or other piecewise smooth singular cycles. These characteristic classes are quite well understood for ﬁnite-dimensional connected Lie groups K, because in this case essentially everything can be reduced to compact Lie groups, for which a powerful theory exists. However, for inﬁnite-dimensional structure groups, the corresponding theory of characteristic classes has only been explored in very special cases. We refer to Morita’s excellent textbook for more details and references (cf. [Mo01]). Central extensions of gauge groups of bundles with inﬁnite-dimensional ﬁber over the circle have recently been studied in the context of integrable systems in [OR07].

7 Appendix A. Abelian extensions of Lie groups In this appendix we result some facts on the integration of Lie algebra 2-cocycles from [Ne04a]. They provide a general set of tools to integrate abelian extensions of Lie algebras to extensions of connected Lie groups. 2

The representation theory of compact groups implies that the unitary group of an inﬁnite-dimensional Hilbert space is such an example.

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Let G be a connected Lie group and V a Mackey complete G-module. Further, let ω ∈ Z k (g, V ) be a k-cocycle and ω eq ∈ Ω k (G, V ) be the corresponding left equivariant V -valued k-form with ω1eq = ω. Then each continuous map Sk → G is homotopic to a smooth map, and ω eq = σ ∗ ω eq perω : πk (G) → V G , [σ] → σ

Sk

deﬁnes the period homomorphism whose values lie in the G-ﬁxed part of V ([Ne02], Lemma 5.7). For k = 2 we deﬁne the ﬂux homomorphism Fω : π1 (G) → H 1 (g, V ),

[γ] → [Iγ ],

where we deﬁne for each piecewise smooth loop γ : S1 → G the 1-cocycle 1 eq Iγ : g → V, Iγ (x) := ixr ω = γ(t).ω(Ad(γ(t))−1 .x, δ l (γ)t ) dt, γ

0

where xr is the right invariant vector ﬁeld on x with xr (1) = x. If V is a trivial module, then dg V = {0}, so that H 1 (g, V ) consists of linear maps g → V , and we may think of the ﬂux as a map Fω : π1 (G) × g → V. Proposition 7.1. ([Ne02], Prop. 7.6) If V is a trivial G-module, then the adjoint action of g on the central extension g := V ⊕ω g integrates to a smooth action Adg of G if and only if Fω = 0. Theorem 7.2. Let G be a connected Lie group, A a smooth G-module of the form A ∼ = a/ΓA , where ΓA ⊆ a is a discrete subgroup of the Mackey complete space a and qA : a → A the quotient map. Then the Lie algebra extension g := a ⊕ω g of g deﬁned by the cocycle ω integrates to an abelian Lie group extension of G by A if and only if qA ◦ perω : π2 (G) → A

and

Fω : π1 (G) → H 1 (g, a)

vanish. Remark 7.3. If only qA ◦ perω vanishes, then the preceding theorem applies of G, so that we only obtain an to the simply connected covering group G extension of G by a non-connected group A , which itself is a central extension of the discrete group π1 (G) by A. For a more detailed discussion of these aspects, we refer to Section 7 of [Ne04a]. Remark 7.4. To calculate period and ﬂux homomorphisms, it is often convenient to use related cocycles on diﬀerent groups. So, let us consider a morphism ϕ : G1 → G2 of Lie groups and ωi ∈ Z 2 (gi , V ), V a trivial Gi -module, satisfying L(ϕ)∗ ω2 = ω1 . Then a straightforward argument shows that perω2 ◦π2 (ϕ) = perω1 : π2 (G1 ) → V.

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For the ﬂux we likewise obtain Fω2 ◦ (π1 (ϕ) × L(ϕ)) = Fω1 : π1 (G1 ) × g1 → V,

(18)

if we consider the ﬂux as a bihomomorphism π1 (G) × g → V . Problem 7.5. It is a crucial assumption in the preceding theorem that the group G is connected, and it would be very desirable to have a suitable generalization to non-connected groups G. The generalization to non-connected Lie groups G means deriving accessible criteria for the extendibility of a 2-cocycle, resp., abelian extensions, from the identity component G0 to the whole group G. From the short exact sequence G0 → G → → π0 (G), we obtain maps I

R

H 2 (π0 (G), A)−−→H 2 (G, A)−−→H 2 (G0 , A)G = H 2 (G0 , A)π0 (G) , but it seems to be diﬃcult to describe the image of the restriction map R. If the identity component G0 of G splits, i.e., if G ∼ = G0 π0 (G), then it is easy to see that R is surjective, because the invariance of the cohomology 0 permits us to lift the conjugation action of π0 (G) to an action on class of G 0 π0 (G) is an abelian extension of G G0 , and then the semidirect product G 0 as a subgroup. by A, containing G If A is a trivial module, one possible approach is to introduce additional 0 of G0 by A, so that the map q : G 0 → G structures on a central extension G describes a crossed module of groups (cf. [Ne07]), which requires an exten 0 , resp., its Lie algebra sion of the natural G0 -action of G on G g, to an action of G. If we have such an action lifting the conjugation action of G 0 ) of this crossed module is an eleon G0 , then the characteristic class χ(G 3 ment of the cohomology group H (π0 (G), A), which vanishes if there exists an of G into which G 0 embeds in a G-equivariant fashion A-extension G (cf. [Ne07], Thm. III.8). The following lemma is easy to verify ([MN03], Lemma V.1; see also [Ne06b], Lemma II.5 for a generalization to general Lie algebra extensions). Lemma 7.6. Let V ⊕ω g be a central extension of the Lie algebra g and γ = (γV , γg ) ∈ GL(V ) × Aut(g). For θ ∈ C 1 (g, V ) the formula γ (z, x) := (γV (z) + θ(γg (x)), γg (x)),

x ∈ g, z ∈ V,

deﬁnes a continuous Lie algebra automorphism of g if and only if γ.ω − ω = dg θ,

resp.,

dg θ = ω − γ −1 .ω

for

θ := θ ◦ γg .

The following theorem can be found in [MN03], Thm. V.9. → G be a central Lie group Theorem 7.7. (Lifting Theorem) Let q : G extension of the 1-connected Lie group G by the Lie group Z ∼ = z/ΓZ . Let

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σG : H × G → G, resp., σZ : H × Z → Z be smooth automorphic actions of the Lie group H on G, resp., Z and σg a smooth action of H on g compatible with →G the actions on z and g. Then there is a unique smooth action σG : H × G by automorphisms compatible with the actions on Z and G, for which the corresponding action on the Lie algebra g is σg .

8 Appendix B. Abelian extensions of semidirect sums We consider a semidirect product of topological Lie algebras h = n S g and a topological h-module V . We are interested in a description of the cohomology space H 2 (h, V ) in terms of H 2 (n, V ) and H 2 (g, V ). To this end, we have to study the inﬂation, resp., the restriction map I : H 2 (g, V n ) → H 2 (h, V ),

resp., Rn : H 2 (h, V ) → H 2 (n, V )g ,

satisfying Rn I = 0. Here we use the fact that h acts trivially on H 2 (h, V ) to see that the image of Rn is contained in H 2 (n, V )h = H 2 (n, V )g . We further have a restriction map Rg : H 2 (h, V ) → H 2 (g, V ). The composition Rg ◦I : H 2 (g, V n ) → H 2 (g, V ) is the natural map induced by the inclusion V n → V of g-modules. If we interpret the elements of H 2 (h, V ) as abelian extensions of h by V , then the inﬂation map leads to twistings of these extensions by extensions of g by V n . We now construct an exact sequence describing the kernel and cokernel of the map (Rn , Rg ) which provides a quite accessible description of H 2 (h, V ). Deﬁnition 8.1. (a) We write C p (n, V ) for the space of continuous Lie algebra p-cochains with values in V and C p (g, C q (n, V ))c ⊆ C p (g, C q (n, V )) for the subspace consisting of those cochains deﬁning a continuous (p + q)-linear map gp × nq → V . Accordingly, we deﬁne Z p (g, C q (n, V ))c , H p (g, C q (n, V ))c , etc. (b) We further write H 2 (n, V )[g] ⊆ H 2 (n, V )g for the subspace of those cohomology classes [f ] for which there exists a θ ∈ C 1 (g, C 1 (n, V ))c with dn (θ(x)) = x.f

for x ∈ g.

(19)

Because of the continuity requirement for θ, this is stronger than the g-invariance of the cohomology class [f ] ∈ H 2 (n, V ).

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Remark 8.2. Any continuous g-module action by derivations on a central extension n = V ⊕f n, compatible with the actions on V and n is of the form x.(v, n) = (x.v + θ(x)(n), x.n), (20) where θ ∈ Z 1 (g, C 1 (n, V ))c satisﬁes (19). In particular, the existence of such an action implies that [f ] ∈ H 2 (n, V )[g] . Lemma 8.3. For [fn ] ∈ H 2 (n, V )[g] we choose θ ∈ C 1 (g, C 1 (n, V ))c as in (19). We thus obtain a well-deﬁned linear map, γ : H 2 (n, V )[g] → H 2 (g, Z 1 (n, V ))c ,

[fn ] → [dg θ].

Proof. Using θ, we obtain a linear map ψ : g → der( n),

ψ(x)(v, n) = (x.v + θ(x)(n), x.n)

which is a linear lift of the given Lie algebra homomorphism ψ = (S, ρV ) : g → (der(n) × gl(V )). In view of [Ne06b], Prop. A.7, the corresponding Lie algebra cocycle on g is given by fg = dg θ ∈ Z 2 (g, Z 1 (n, V ))c . The corresponding extension g = Z 1 (n, V ) ×fg g → g → 0 0 → Z 1 (n, V ) → acts naturally on n by (α, x).(v, n) = (α(n) + x.v + θ(x)(n), x.n). Next we observe that γ is well deﬁned. Indeed, if θ ∈ C 1 (g, C 1 (n, V ))c also satisﬁes (19), then θ − θ ∈ C 1 (g, Z 1 (n, V ))c , so that [dg (θ − θ)] = 0 in H 2 (g, Z 1 (n, V ))c . Moreover, if fn = fn + dn β for some β ∈ C 1 (n, V ), then x.fn = x.fn + dn (x.β) = dn (θ(x) + x.β), so that θ(x) := θ(x) + x.β leads to dn (θ(x)) = x.fn . Then dg θ = dg θ + d2g β = dg θ. This shows that γ is well deﬁned. The linearity is clear.

Lemma 8.4. For each θ ∈ Z 1 (g, Z 1 (n, V ))c , x.(v, n) := (x.v + θ(x)(n), x.n) deﬁnes a continuous action of g on the semidirect sum n := V n, so that hθ := n g is an extension of h by V . We thus obtain a well-deﬁned map, ϕ

ϕ : H 1 (g, Z 1 (n, V ))−−→H 2 (h, V ),

[θ] → [ hθ ].

Proof. A 2-cocycle deﬁning the extension hθ is given by ωθ ((n1 , x1 ), (n2 , x2 )) = [(0, (n1 , x1 )), (0, (n2 , x2 ))] − (0, [n1 , n2 ] + x1 .n2 − x2 .n1 , [x1 , x2 ]) = (0, θ(x1 )(n2 ) − θ(x2 )(n1 )).

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333

If θ is a coboundary, i.e., there exists a β ∈ Z 1 (n, V ), with θ(x) = x.β, x ∈ g, then θ(x)(n) = x.β(n) − β(x.n) leads to ωθ ((n1 , x1 ), (n2 , x2 )) = θ(x1 )(n2 ) − θ(x2 )(n1 ) = x1 .β(n2 ) − x2 .β(n1 ) + β(x1 .n2 − x2 .n1 ). If β : h → V denotes the continuous linear map extending β and vanishing on g, then ωθ is a coboundary: (dh β)((n 1 , x1 ), (n2 , x2 )) = x1 .β(n2 ) − x2 .β(n1 ) + dn (β)(n1 , n2 ) − β(x1 .n2 − x2 .n1 ) = x1 .β(n2 ) − x2 .β(n1 ) − β(x1 .n2 − x2 .n1 ) = ωθ ((n1 , x1 ), (n2 , x2 )).

Theorem 8.5. With the linear map η = (dn )∗ : H 2 (g, V ) → H 2 (g, Z 1 (n, V ))c ,

[fg ] → [−dn ◦ fg ],

the following sequence is exact: (Rn ,Rg )

ϕ

H 1 (g, Z 1 (n, V ))−−→H 2 (h, V )−−−−−−→H 2 (n, V )[g] ⊕ H 2 (g, V ) γ−η

−−−−→H 2 (g, Z 1 (n, V ))c . Proof. To see that the sequence is exact in H 2 (h, V ), we simply note that a V -extension corresponds to a class [f ] ∈ ker(Rn , Rg ) if and only if its restrictions to both subalgebras n and g are trivial. This is equivalent to h being of the form h = (V n)g, and hence equivalent to hθ for some θ ∈ Z 1 (g, Z 1 (n, V ))c . It remains to verify that im(Rn , Rg ) = ker(γ − η). If [fn ] ∈ im(Rn ), then there exists a topologically split extension h of h by the topological h-module V . Then h contains n as an ideal, so that the adjoint action deﬁnes an action of h on n. As V acts non-trivially on n, this action generally does not factor through an action of h. Let p : h → h denote the quotient map. Then n = p−1 (n), and we put g := p−1 (g). As we have seen in Remark 8.2, the action of g is described by some θ ∈ Z 1 ( g, C 1 (n, V ))c satisfying = x.f dn (θ(x))

for x ∈ g.

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x) satisﬁes (19), so that Writing g as V ⊕fg g, it follows that θ(x) := θ(0, 2 [g] [fn ] ∈ H (n, V ) . For x, y ∈ g we then obtain the relation y) − y.θ(0, x) = θ([(0, x.θ(y) − y.θ(x) = x.θ(0, x), (0, y)]) = −dn (fg (x, y)) + θ([x, y]), i.e., dg θ = −dn ◦ fg . Hence, any element ([fn ], [fg ]) ∈ im(Rn , Rg ) is contained in the kernel of γ − η : H 2 (n, V )[g] ⊕ H 2 (g, V ) → H 2 (g, Z 1 (n, V ))c .

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If, conversely, [fn ] ∈ ker(γ − η), then there exists θ ∈ C 1 (g, C 1 (n, V ))c x) := θ(x) − dn v deﬁnes a representation with dg θ = −dn ◦ fg , and then θ(v, g := V ⊕fg g → der n, compatible with the actions of g on n and V . Then the semidirect product n g is an extension of n g = h by the space V × V . Since both V -factors act in the same way on n, the anti-diagonal ∆V acts trivially, and hence commutes with n. Since g, resp., g, acts diagonally on V × V , it also preserves the anti-diagonal. We thus obtain the V -extension h := ( n g)/∆V of h. This extension contains n and g as subalgebras. This completes the proof.

Corollary 8.6. If n is topologically perfect and V = V n , then (Rn , Rg ) : H 2 (h, V ) → H 2 (n, V )[g] ⊕ H 2 (g, V ) is a linear bijection. Proof. In this case Z 1 (n, V ) vanishes, and the assertion follows from Theorem 8.5.

9 Appendix C. Triviality of the group action on Lie algebra cohomology Theorem 9.1. Let G be a connected Lie group with Lie algebra g and V a Mackey complete smooth G-module. Then the natural action of G on the Lie algebra cohomology H • (g, V ) is trivial. Proof. Since the space H • (g, V ) carries no natural locally convex topology for which the action of G on this space is smooth, we cannot simply argue that the triviality of the G-action follows from the triviality of the g-action. The assertion is clear for p = 0, because H 0 (g, V ) = V g = V G follows from the triviality of the g-action on the closed G-invariant subspace V g of V ([Ne06a], Rem. II.3.7). We may therefore assume p > 0. Let ω ∈ Z p (g, V ) be a continuous p-cocycle and g ∈ G. We have to ﬁnd an η ∈ C p−1 (g, V ) with g.ω − ω = dg η. Let γ : [0, 1] → G be a smooth curve with γ(0) = 1 and γ(1) = g. We write δ l (γ)t := γ(t)−1 .γ (t) for its left logarithmic derivative and recall the Cartan formula Lx = ix ◦ dg + dg ◦ ix for the action of g on C • (g, V ). We thus obtain in the pointwise sense on each p-tuple of elements of g: 1 1 d g.ω − ω = γ(t).ω dt = γ(t). Lδl (γ)t ω dt 0 dt 0 1 1 γ(t). iδl (γ)t dg ω + dg iδl (γ)t ω dt = dg γ(t).iδl (γ)t ω dt = 0

= dg 0

0

1

γ(t).iδl (γ)t ω dt.

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For p = 2 we obtain in particular g.ω − ω = dg ηg for 1 ηg (x) := γ(t).ω δ l (γ)t , Ad(γ(t))−1 .x dt.

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0

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Gerbes and Lie Groups Christoph Schweigert and Konrad Waldorf Organisationseinheit Mathematik, Universit¨ at Hamburg, Bundesstraße 55, D-20146, Germany, [email protected], [email protected]

Summary. Compact Lie groups do not only carry the structure of a Riemannian manifold, but also canonical families of bundle gerbes. We discuss the construction of these bundle gerbes and their relation to loop groups. We present several algebraic structures for bundle gerbes with connection, such as Jandl structures, gerbe modules and gerbe bimodules, and indicate their applications to Wess–Zumino terms in two-dimensional ﬁeld theories. Key words: bundle gerbe, loop group, Jandl structure, gerbe module and bimodule. 2000 Mathematics Subject Classifications: 22E67, 55R65, 81T40.

Introduction Compact Lie groups come not only with a canonical metric (the Killing form), but also with a canonical family of bundle gerbes. These bundle gerbes are geometric objects made of ﬁnite-dimensional manifolds and maps between them, and they provide a way of understanding structure over the inﬁnitedimensional loop group. As a motivation, consider a central extension of the loop group of a compact connected and simply connected Lie group G, 0 −→ C× −→ LG −→ LG −→ 0.

(1)

Such extensions are classiﬁed by H2 (LG, Z). By transgression, this cohomology group is in turn isomorphic to the cohomology of G, H3 (G, Z) ∼ = H2 (LG, Z).

(2)

While the cohomology group H2 (LG, Z) classiﬁes line bundles over LG by their Chern class, H3 (G, Z) classiﬁes bundle gerbes over G. In this way, every

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_10, © Springer Basel AG 2011

339

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bundle gerbe over the ﬁnite-dimensional manifold G gives rise to a line bundle over the inﬁnite-dimensional manifold LG. This contribution is organized as follows. In Section 1 we describe bundle gerbes on general manifolds and their classiﬁcation. In Section 2 we explain how bundle gerbes can be equipped with a connection which allows to deﬁne surface holonomies: from this point of view, bundle gerbes generalize the holonomy of principal bundles around curves. In case that the base manifold is a compact Lie group G, we construct examples of bundle gerbes over G in Section 3. Then we explain in Section 4 how a bundle gerbe gives rise to a line bundle over the loop space. In Section 5 we discuss additional structures for bundle gerbes like bundle gerbe modules, bimodules and Jandl structures. Finally, in Section 6 we outline the applications of bundle gerbes on Lie groups to two-dimensional conformal ﬁeld theory and string theory, which are closely related to the surface holonomy from Section 2. In these theories one also recovers the loop space as the space of conﬁgurations.

1 Bundle gerbes Let M be a smooth manifold. We shall brieﬂy review the classiﬁcation of complex line bundles over M . For this purpose, let us choose a good open cover V = {Vi }i∈I of M ; i.e., every ﬁnite intersection of open sets Vi is contractible. In particular, every line bundle L admits local non-zero sections which determine smooth transition functions gij : Vi ∩ Vj → C× .

(3)

On three-fold intersections Vi ∩ Vj ∩ Vk , these transition functions satisfy the cocycle condition gik = gij · gjk . (4) It is fair to call this equality a cocycle condition, since it means that δg = 0 ˇ for the element g := (gij ) ∈ Cˇ 1 (V, C× M ) in the Cech cohomology of the sheaf × of smooth C -valued functions on M with respect to the cover V. Since we have chosen the cover V to be good, there is a canonical isomorphism 2 ˇ 1 (V, C× ) ∼ H M = H (M, Z) using the exponential sequence. The image of the 2 class [g] in H (M, Z) is independent of the choice of the sections, and is called the (ﬁrst) Chern class c1 (L) of the line bundle L. This deﬁnes an isomorphism c1 : Pic(M ) → H2 (M, Z)

(5)

from the group Pic(M ) of isomorphism classes of line bundles to the cohomology group H2 (M, Z), providing a geometric realization of this group. A bundle gerbe is a geometric object which realizes the cohomology group H3 (M, Z) in a similar way. To prepare its deﬁnition, we ﬁx the

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following notation. For a surjective submersion π : Y → M we denote the k-fold ﬁbre product of Y over M by Y [k] = {(y1 , . . . , yk ) ∈ Y k |π(y1 ) = · · · = π(yk )}.

(6)

This is again a smooth manifold, having canonical projections πi1 ,...,i : Y [k] → Y [] on the respective factors. Definition 1.1 ([Mur96]). A bundle gerbe G over a manifold M is a triple (π, L, µ) consisting of a surjective submersion π : Y → M , a line bundle L over Y [2] and an isomorphism ∗ ∗ ∗ µ : π12 L ⊗ π23 L → π13 L

(7)

of line bundles over Y [3] , such that µ is associative in the sense that the diagram ∗ ∗ ∗ π12 L ⊗ π23 L ⊗ π34 L

∗ π123 µ⊗id

∗ ∗ / π13 L ⊗ π34 L

∗ id⊗π234 µ

(8)

∗ π134 µ

∗ ∗ π12 L ⊗ π24 L

∗ π124 µ

∗ / π14 L

of isomorphisms of line bundles over Y [4] is commutative. Let us now describe how bundle gerbes realize the cohomology group H3 (M, Z). Let us again choose a good open cover V = {Vi }i∈I of M which admits sections si : Vi → Y into the manifold Y of the surjective submersion of a bundle gerbe G = (π, L, µ). If we denote by MV the disjoint union of all the open sets Vi , the sections si patch together to a smooth map s : MV → Y sending a point x ∈ Vi to si (x) ∈ Y . Note that there are induced maps [k] MV → Y [k] on ﬁbre products (all denoted by s in order to simplify the nota[k] tion), where MV is the disjoint union of all k-fold intersections of open sets [2] Vi . Now we pull back the line bundle L along s to a line bundle over MV , [3] and the isomorphism µ to an isomorphism of line bundles over MV . For a ∗ choice σij : Vi ∩ Vj → s L of local sections into the pull-back line bundle, we obtain smooth functions

by

gijk : Vi ∩ Vj ∩ Vk → C×

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s∗ µ(σij ⊗ σjk ) = gijk · σik.

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The associativity condition (8) leads to the equality gijk · gik = gij · gjk

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of functions on four-fold intersections Vi ∩ Vj ∩ Vk ∩ V . In other words, the ˇ element g = (gijk ) ∈ Cˇ 2 (V, C× M ) is a cocycle and deﬁnes a class in the Cech × 2 3 ˇ cohomology group H (V, CM ). Its image in the cohomology group H (M, Z) is called the Dixmier–Douady class dd(G) of the bundle gerbe G; it is analogous to the Chern class of a line bundle. To obtain a classiﬁcation result for isomorphism classes of bundle gerbes, we ﬁrst have to deﬁne morphisms between bundle gerbes. To simplify the notation, we work with the convention that we do not write pull-backs along canonical maps, as in (12) and (13) below. Definition 1.2. A morphism A : G1 → G2 between two bundle gerbes G1 = (π1 , L1 , µ1 ) and G2 = (π2 , L2 , µ2 ) is a pair A = (A, α) consisting of a vector bundle A over the ﬁbre product Z := Y1 ×M Y2 (whose surjective submersion to M is denoted by ζ) and an isomorphism α : L1 ⊗ ζ2∗ A → ζ1∗ A ⊗ L2

(12)

of vector bundles over Z [2] such that the diagram ∗ ∗ ζ12 L1 ⊗ ζ23 L1 ⊗ ζ3∗ A

µ1 ⊗id

∗ / ζ13 L1 ⊗ ζ3∗ A

(13)

∗ id⊗ζ23 α

∗ ∗ ζ12 L1 ⊗ ζ2∗ A ⊗ ζ23 L2

∗ ζ13 α

∗ α⊗id ζ12

∗ ∗ ζ1∗ A ⊗ ζ12 L2 ⊗ ζ23 L2

id⊗µ2

∗ / ζ1∗ A ⊗ ζ13 L2

of isomorphisms of vector bundles over Z [3] is commutative. The deﬁnition of the composition of two morphisms A : G1 → G2 and A : G2 → G3 is quite involved, and we omit its discussion for the purposes of this article; see [Ste00, Wal07] for more details. Bundle gerbes and their morphisms ﬁt into the structure of a 2-category rather than that of a category. This becomes obvious when comparing two morphisms A and A between the same bundle gerbes: since A and A themselves consist of vector bundles, the natural way to compare them is a morphism of vector bundles. Definition 1.3. Let A = (A, α) and A = (A , α ) be two morphisms from G1 = (π1 , L1 , µ1 ) to G2 = (π2 , L2 , µ2 ). A 2-morphism β : A ⇒ A

(14)

is an isomorphism β : A → A of vector bundles over Z, which is compatible with the isomorphisms α and α in the sense that the diagram

Gerbes and Lie Groups

L1 ⊗ ζ2∗ A 1⊗ζ2∗ β

L1 ⊗ ζ2∗ A

α

α

/ ζ1∗ A ⊗ L2

343

(15)

ζ1∗ β⊗1

/ ζ1∗ A ⊗ L2

of isomorphisms of vector bundles over Z [2] is commutative. The 2-categorical setup is also the appropriate context in which to address the question of which of the morphisms between two bundle gerbes are invertible. Proposition 1.4 ([Wal07]). A morphism A = (A, α) is invertible if and only if the vector bundle A is of rank 1. So, the invertible morphisms in the 2-category are the stable isomorphisms from [MS00], Section 3. Let us now return to the relation between bundle gerbes and the cohomology group H3 (M, Z). Let g and g be the cocycles extracted from two bundle gerbes G and G over M with respect to the same cover V of M and sections si : Vi → Y and si : Vi → Y . Now let A : G → G be an isomorphism, A = (A, α). Let t : MV → Z := Y ×M Y be the map sending a point x ∈ Vi to the pair (si (x), si (x)). Using t we pull back the line bundle A and choose non-zero sections σi : Vi → A. Then we obtain smooth functions hij : Vi ∩ Vj → C× by α(σi ⊗ ζ2∗ σij ) = hij · (ζ1∗ σij ⊗ σi ).

(16)

The compatibility condition (13) between α and the isomorphisms µ and µ of the bundle gerbes leads to the equation hij hjk , hik gijk = gijk

(17)

ˇ equivalently, in terms of the Cech coboundary operator, g = g ·δh. This means that the Dixmier–Douady classes [g] and [g ] of the isomorphic bundle gerbes G and G are equal. This is the main ingredient for the following classiﬁcation result. Theorem 1.5 ([MS00]). The Dixmier–Douady class deﬁnes a bijection between the set of isomorphism classes of bundle gerbes and the cohomology group H3 (M, Z). In particular, consider M = G a compact, simple, connected and simply connected Lie group. There is a canonical identiﬁcation H3 (G, Z) = Z, so that we have a canonical sequence of bundle gerbes associated to G. In Section 3, we give a geometric construction of these bundle gerbes. The 2-category of bundle gerbes admits several additional structures such as pull-backs, tensor products and duals. All these structures are compatible with the Dixmier–Douady class

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dd(G1 ⊗ G2 ) = dd(G1 ) + dd(G2 ) ,

dd(G ∗ ) = −dd(G)

(18)

and, for a smooth map f : X → M , dd(f ∗ G) = f ∗ dd(G).

(19)

To close, let us construct a bundle gerbe whose Dixmier–Douady class vanishes, representing the neutral element in H3 (M, Z). For this purpose, consider the identity idM : M → M as the surjective submersion and the trivial line bundle M × C over M . Now, the isomorphism µ can be chosen to be the identity idC , so that I := (idM , M × C, idC ) is a bundle gerbe. It is easy to verify that its Dixmier–Douady class vanishes, dd(I) = 0.

2 Connections on bundle gerbes and holonomy Before we discuss more examples of bundle gerbes in Section 3, we introduce several additional structures on bundle gerbes and the appropriate cohomology theory for their classiﬁcation. Again, we ﬁrst review similar additional structures for complex line bundles. One can equip every line bundle with a hermitian metric and a (unitary) connection ∇. In addition to the transition function gij : Vi ∩ Vj → C× , which now can be determined such that it takes values in U (1), we obtain local connection 1-forms Ai ∈ Ω(Vi ) by writing the covariant derivatives of the sections si : Vi → L as ∇si = Ai ⊗ si . The Leibniz rule implies the equality Aj − Ai − dlog(gij ) = 0.

(20)

As we shall see next, the local data (g, A) deﬁne a cocycle in the Deligne k complex DV (n) for n = 1. The ﬁrst cochain groups of this complex are 0 (1) = Cˇ 0 (V, U (1)M ) DV 1 1 (1) = Cˇ 1 (V, U (1)M ) ⊕ Cˇ 0 (V, ΩM ) DV 2 2 1 1 ˇ ˇ DV (1) = C (V, U (1)M ) ⊕ C (V, ΩM ),

(21) (22) (23)

and its diﬀerential is given by 1 2 (1) → DV (1) : (g, A) → (δg, δA − dlog(g)). D : DV

(24)

k ˇ (n) is the total complex of the Cech– In general, the Deligne complex DV ˇ Deligne double complex, whose rows are Cech cochain groups of the sheaf complex

0

/ U (1)M

dlog

/ Ω1 M

d

/ ...

d

n / ΩM ,

(25)

ˇ truncated in degree n, and the columns are the usual Cech complexes assok ciated to the sheaves U (1)M and ΩM . The truncation is necessary to obtain objects that are not ﬂat.

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Thus we can regard the local data (g, A) of a hermitian line bundle with 1 connection as an element of DV (1). By equations (4) and (20) it satisﬁes D(g, A) = 0, and thus deﬁnes a class in the ﬁrst cohomology group of the Deligne complex, denoted by H1 (M, D(1)). One can show that this deﬁnes a bijection Pic∇ (M ) ∼ (26) = H1 (M, D(1)) between the group Pic∇ (M ) of isomorphism classes of hermitian line bundles with connection and Deligne cohomology [Bry93]. Now we discuss bundle gerbes with similar additional structures. Definition 2.1. Let G = (π, L, µ) be a bundle gerbe over M. It can be equipped successively with the following additional structures. (a) A hermitian structure on G is a hermitian metric on the line bundle L, such that the isomorphism µ is an isometry. (b) A connection on the hermitian bundle gerbe G is a connection ∇ on the hermitian line bundle L, such that the isomorphism µ respects the induced connections. (c) A curving of a connection ∇ on the hermitian bundle gerbe G is a 2-form C ∈ Ω 2 (Y ), such that π2∗ C − π1∗ C = curv(∇),

(27)

where curv(∇) ∈ Ω 2 (Y [2] ) is the curvature of the connection ∇ on L. Every bundle gerbe admits all of these additional structures [Mur96]. Because the applications of bundle gerbes that we discuss later require all this additional structure, we work from now on only with hermitian bundle gerbes with connection and curving. An important feature of those gerbes is that they provide a notion of curvature. To see this, consider the derivative of equation (27): since the curvature of the connection ∇ is a closed form, we obtain π1∗ dC = π2∗ dC, which means that dC is the pull-back of a 3-form on M , dC = π ∗ H.

(28)

This 3-form H is uniquely determined and closed; it is called the curvature of the curving of the connection ∇ on the hermitian bundle gerbe G, and denoted by curv(C) := H. To provide a simple example of such additional structures, the bundle gerbe I = (idM , M × C, idC ) with vanishing Dixmier–Douady class becomes a hermitian bundle gerbe with connection by taking the canonical hermitian metric and the trivial ﬂat connection ∇ := d on the trivial line bundle M × C. Note that now any 2-form C ∈ Ω 2 (M ) satisﬁes the condition (27) for a curving on I, because ∇ is ﬂat and π1 = π2 = idM . So we have a canonical hermitian bundle gerbe IC := (idM , M × C, idC , d, C) with connection and curving for every 2-form C ∈ Ω 2 (M ). The curvature of its curving C is curv(C) = dC.

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Christoph Schweigert and Konrad Waldorf

Now we extend the cohomological classiﬁcation from bundle gerbes to hermitian bundle gerbes G = (π, L, µ, ∇, C) with connection and curving, using a good open cover V of M . Recall that we have extracted a transition function gijk : Vi ∩ Vj ∩ Vk → C× using a choice of sections si : Vi → Y and σi : Vi → s∗ L, deﬁning a representative g ∈ Cˇ 2 (V, C× ) of the Dixmier– Douady class of the bundle gerbe. Now that L is a hermitian line bundle, we choose the sections σi such that gijk is U (1)-valued. Furthermore, by using the connection s∗ ∇ on s∗ L, we obtain local connection 1-forms Aij ∈ Ω 1 (Vi ∩ Vj ). The condition that µ preserves connections implies Ajk − Aik + Aij + dlog(gijk ) = 0.

(29)

Finally, the curving C gives rise to local 2-forms Bi := s∗i C ∈ Ω 2 (Vi ), and the compatibility (27) with the curvature of ∇ implies Bj − Bi − dAij = 0.

(30)

Note that the curvature curv(C) of the curving can be computed from the local data by H|Vi = dBi . By (30), this indeed gives a globally deﬁned 3-form. ˇ In terms of Cech cohomology, we have extracted an element (g, A, B) in the second cochain group of the Deligne complex in degree 2, 2 DV (2) = Cˇ 2 (V, U (1)) ⊕ Cˇ 1 (V, Ω 1 ) ⊕ Cˇ 0 (V, Ω 2 ).

(31)

Here the diﬀerential is 2 3 (2) → DV (2) : (g, A, B) → (δg, δA + dlog(g), δB − dA), D : DV

(32)

so that the cocycle condition (27) and equations (29) and (30) imply the cocycle condition D(g, A, B) = 0. This way, a hermitian bundle gerbe with connection and curving deﬁnes a class in the cohomology of the Deligne complex in degree 2, H2 (M, D(2)). As an exercise, the reader may compute the Deligne class of the canonical hermitian bundle gerbe IC with connection and curving associated to any 2-form C ∈ Ω 2 (M ). 1 2 Note that both the Deligne cochain groups DV (1) from (22) and DV (2) from (31) have projections on the ﬁrst summand, which commute with the ˇ Deligne diﬀerential and the Cech coboundary operator, so that we get induced (surjective) group homomorphisms [Bry93] Hk (M, D(k)) → Hk+1 (M, Z).

(33)

This way we obtain the Chern class and the Dixmier–Douady class of a hermitian line bundle with connection and of a hermitian bundle gerbe with connection and curving, respectively, from their Deligne classes. The surjectivity of (33) means that Deligne cohomology reﬁnes the ordinary singular cohomology with Z coeﬃcients. To achieve a classiﬁcation result for hermitian bundle gerbes with connection and curving similar to the result (26) for hermitian line bundles with

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347

connection, we have to adapt the deﬁnition of a morphism between bundle gerbes to morphisms between hermitian bundle gerbes with connection and curving. Definition 2.2. A morphism A : G → G between two hermitian bundle gerbes G = (π, L, µ, ∇, C) and G = (π , L , µ , ∇ , C ) with connection and curving is a morphism A = (A, α) as in Deﬁnition 1.2, together with a connection on the vector bundle A, such that 1. the isomorphism α respects connections, 2. the curvature of is related to the curvings by 1 tr(curv()) = C − C. n

(34)

As in Proposition 1.4, a morphism is invertible precisely if the vector bundle is of rank 1. It is again an easy exercise to check that an isomorphism A : G → G of hermitian bundle gerbes with connection and curving with Deligne cocycles (g, A, B) and (g , A , B ), respectively, gives rise to a Deligne 1 cochain (h, W ) ∈ DV (2) which satisﬁes (g , A , B ) = (g, A, B) + D(h, W ).

(35)

This shows that isomorphism classes of hermitian bundle gerbes with connection and curving have well-deﬁned Deligne classes. Theorem 2.3 ([MS00]). Isomorphism classes of hermitian bundle gerbes with connection and curving correspond bijectively to the Deligne cohomology group H2 (M, D(2)). Particular examples of morphisms are trivializations: a trivialization of a hermitian bundle gerbe G with connection and curving is an isomorphism T : G → Iρ

(36)

for some 2-form ρ ∈ Ω 2 (M ). In terms of local data, this isomorphism corre1 sponds to a Deligne cochain (h, W ) ∈ DV (2) with (1, 0, ρ) = (g, A, B) + D(h, W ),

(37)

if (g, A, B) is local data of the bundle gerbe G. In particular, the existence of a trivialization implies that the Dixmier–Douady class of G vanishes. Many assertions about bundle gerbes and their isomorphisms can be proven either in a geometrical way or by computations in Deligne cohomology. As an illustration, we shall prove the following important lemma. Lemma 2.4. Two trivializations T1 : G → Iρ1

and

T2 : G → Iρ2

(38)

of the same hermitian bundle gerbe G over M with connection and curving determine a hermitian line bundle over M with connection of curvature ρ2 −ρ1 .

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Christoph Schweigert and Konrad Waldorf

Proof 1 (2-categorical). Using the features of the 2-category of bundle gerbes, we can give a quite conceptual proof: by taking the inverse and composition (which we have not explained in this article, but can be found in [Wal07]), we obtain an isomorphism T2 ◦ T1−1 : Iρ1 → Iρ2

(39)

of trivial bundle gerbes. From the deﬁnitions of isomorphisms and trivial bundle gerbes it follows immediately that T2 ◦ T1−1 is a line bundle with curvature ρ2 − ρ1 . Proof 2 (geometrical). The two isomorphisms Ti = (Ti , τi , i ) consist of hermitian line bundles Ti over Z := Y ×M M ∼ = Y , connections i of curvature curv(i ) = π ∗ ρi − C and isomorphisms τi : L ⊗ π2∗ Ti → π1∗ Ti of hermitian line bundles respecting the connections. They can be composed to an isomorphism τ2−1 ⊗ τ1∗ : π1∗ (T2 ⊗ T1∗ ) → π2∗ (T2 ⊗ T1∗ )

(40)

of hermitian line bundles with connection over Y [2] , which satisﬁes the obvious cocycle condition over Y [3] , due to the commutative diagram (13) for the τi . This is the condition for the hermitian line bundle T2 ⊗ T1∗ with connection 2 − 1 to descend from Y to M . The descent line bundle has the claimed curvature. Proof 3 (cohomological). If the isomorphisms T1 and T2 have local data (h1 , W1 ) and (h2 , W2 ), respectively, both satisfying equation (37), their difference satisﬁes D(h2 · h−1 (41) 1 , W2 − W1 ) = 0, which is the cocycle condition for a hermitian line bundle with connection of curvature d(W2 − W1 ) = curv(2 ) − curv(1 ) = ρ2 − ρ1 . One of the most important aspects of the theory of hermitian bundle gerbes with connection and curving is that they provide a notion of holonomy around surfaces. Definition 2.5 ([CJM02]). Let G be a hermitian bundle gerbe with connection and curving over M . For a closed oriented surface Σ and a smooth map φ : Σ → M , let T : φ∗ G → Iρ (42) be a trivialization of the pullback of G along φ. Then we deﬁne ρ holG (φ) := exp Σ

to be the holonomy of G around φ : Σ → M .

(43)

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349

Note that the Dixmier–Douady class of φ∗ G vanishes by dimensional reasons, so that the existence of the trivialization T is guaranteed. However, it is not unique, and diﬀerent trivializations may have diﬀerent 2-forms ρ. Now, by Lemma 2.4 we know that the diﬀerence ρ2 − ρ1 between two such 2-forms is the curvature of some line bundle over M , in particular: it is a closed form with integral class. Then, the calculation exp (44) ρ2 = exp ρ2 − ρ1 · exp ρ1 = exp ρ1 Σ

Σ

Σ

Σ

shows that the deﬁnition holG (φ) is independent of the choice of the trivialization. There also exist expressions for the holonomy holG (φ) in terms of local data of G. They generalize the local formulae for the holonomy of hermitian line bundles with connection. In Section 6 we describe the applications of holonomy of bundle gerbes in conformal ﬁeld theory.

3 Bundle gerbes over compact Lie groups Now that we have introduced bundle gerbes as geometric objects over arbitrary manifolds, we specialize to manifolds which are Lie groups. We describe in this section how the Lie group structure allows constructions of examples of bundle gerbes. First constructions of gerbes over diﬀerent types of compact Lie groups (in realizations diﬀerent from bundle gerbes) can be found in [Cha98, Bry]. Bundle gerbes (with connection and curving) have been constructed in [GR02, Mei02, GR03]. As already mentioned, for a compact, simple, connected and simply connected Lie group G, we have H3 (G, Z) = Z. The (up to isomorphism unique) bundle gerbe over G whose Dixmier–Douady class corresponds to 1 ∈ Z is called the basic bundle gerbe, and is denoted by G0 . The bundle gerbe with Dixmier–Douady class k ∈ Z can then be obtained from G0 or G0∗ by a k-fold tensor product. For the purposes of this article, we will restrict ourselves to the construction given in [GR02], by which we obtain the basic bundle gerbe over the special unitary groups SU(n) and the symplectic groups Sp(n). First we consider a general compact, simple and simply connected Lie group G with Lie algebra g. We choose a maximal torus T with Lie algebra t of rank r. We further choose a set of simple roots α1 , ..., αr and denote the associated positive Weyl chamber by C. Let α0 be the highest root and let A := {ξ ∈ C | α0 (ξ) ≤ 1}

(45)

be the fundamental alcove. It is bounded by the hyperplanes Hi perpendicular to the roots αi , and the additional hyperplane H0 consisting of elements

350

Christoph Schweigert and Konrad Waldorf

ξ ∈ t with α0 (ξ) = 1. Thus, it is an r-dimensional simplex in t with vertices µ0 , . . . , µr determined by the condition that µi ∈ Hj for all j = i. For simple and simply connected groups, the fundamental alcove parameterizes conjugacy classes of G in the sense that each conjugacy class contains a unique point exp ξ with ξ ∈ A. This deﬁnes a continuous map q : G → A.

(46)

Let Ai be the open complement of the face opposite to the vertex µi in A, and consider the open sets Ui := q −1 (Ai ). More generally, for any subset I ⊂ r = {0, ..., r} denote by UI the intersection of all Ui with i ∈ I, and similarly by AI the intersection of all Ai with i ∈ I. Of course, UI = q −1 (AI ). We use the open sets Ui to construct the surjective submersion Ui and π : Y → G : (i, x) → x. (47) Y := i∈r

Note that the k-fold ﬁbre products are disjoint unions of intersections UI . Y [k] =

(48)

|I|=k

The surjective submersion π : Y → M will serve as the ﬁrst ingredient of the bundle gerbe we want to construct. To construct the line bundle L over Y [2] , we show next that Y [2] projects onto a union of coadjoint orbits. For any I ⊂ r, all group elements exp ξ with ξ in the open face spanned by the vertices µi with i ∈ I have the same centralizer GI . For any inclusion I ⊂ J it follows that GJ ⊂ GI ; for I = r we obtain Gr = T . Let SI be the orbit of exp AI ⊂ T under the conjugation with GI . Consider the set G×GI SI consisting of equivalence classes of pairs (g, s) ∈ G × SI under the equivalence relation (g, s) ∼ (gh, h−1 sh) for h ∈ GI . We have the canonical projection ρI : G ×GI SI → G/GI and a smooth map u I : G × G I SI → U I

(49)

which sends a representative (g, s) to gsg −1 ∈ G. This is well deﬁned on equivalence classes, and for h ∈ GI and ξ ∈ AI with s := h exp ξ h−1 we ﬁnd q(gsg −1 ) = ξ and hence gsg −1 ∈ UI . The map uI is even a diﬀeomorphism: for g ∈ UI let ξ := q(g) ∈ AI and h ∈ G such that g = h exp ξ h−1 . Then, the inverse sends g to the equivalence class of (h, exp ξ). Since Gij ﬁxes the diﬀerence µij := µj − µi , the quotient G/Gij projects on the coadjoint orbit Oij of µij in g∗ . Now we specialize our construction to the case that µij is a weight. This is the case for SU(n) and Sp(n), where the vertices of the fundamental alcove are contained in the weight lattice. For µij being a weight there is a canonical line bundle Lij over the coadjoint orbit Oij . Let us recall the construction of the pull-back of this line bundle to G/Gij . Let χij : Gij → U (1) be the character associated to the weight µij .

Gerbes and Lie Groups

351

Then the line bundle is the bundle associated to the principal Gij -bundle G over G/Gij , namely, Lij := G ×Gij C. (50) The line bundles Lij over G/Gij can be pulled back along ∼

Ui ∩ Uj

/ G ×Gij Sij

/ G/Gij

(51)

to line bundles over Ui ∩ Uj , and their disjoint union gives a line bundle L over Y [2] . To close the deﬁnition of a bundle gerbe over G, it remains to construct the isomorphism µ of line bundles over Y [3] ; i.e., we need an isomorphism ∗ ∗ ∗ L ⊗ π23 L → π13 L µ : π12

(52)

of line bundles over Y [3] . Over each connected component Ui ∩ Uj ∩ Uk , this can be chosen as the pull-back of the canonical identiﬁcation Lij ⊗ Ljk ∼ = Lik

(53)

of line bundles over G/Gijk , which comes from the coincidence µik = µij +µjk of the weights from which the line bundles are constructed. This identiﬁcation is obviously associative. In order to calculate the Dixmier–Douady class of the bundle gerbe we have just constructed, we choose a connection and a curving. This procedure is analogous to the calculation of Chern classes of complex vector bundles using a choice of a hermitian metric and a connection; see [MS76]. Recall that this is based on the fact that the de Rham cohomology class of the curvature of a hermitian line bundle with connection equals the image of its Chern class under the induced map ι∗ : H2 (M, Z) → H2 (M, R). The same holds for a bundle gerbe G = (π, L, µ, ∇, C): [curv(C)] = ι∗ dd(G),

(54)

ˇ which can be proven by a zigzag-argument in the Cech–Deligne double complex [Bry93]. Note that—in a certain normalization of the Killing form [PS86]—the bi-invariant closed 3-form H :=

1 θ ∧ [θ ∧ θ] ∈ Ω 3 (G) 6

(55)

is a generator of the cohomology group H3 (G, Z). Here θ is the left invariant Maurer–Cartan form, a g-valued 1-form on G; at the unit element, the 3-form H takes the value H1 (X, Y, Z) = X, [Y, Z] on elements X, Y, Z ∈ g. Our goal is now to deﬁne a connection and a curving with curvature H. First note that the line bundle Lij from (50) inherits a hermitian metric from the standard metric on C, and that the isomorphism µ is an isometry.

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Christoph Schweigert and Konrad Waldorf

The line bundle Lij can also be equipped with a connection: we consider the 1-form Aij := µij , θ on the total space of the principal Gij -bundle G. It induces a connection on the associated line bundle Lij because µij is preserved under the action of Gij . In this way the bundle gerbe is a hermitian bundle gerbe with connection. To deﬁne a curving for this bundle gerbe, i.e., a 2-form C ∈ Ω 2 (Y ), we use the fact that the linear retraction of Ai to the vertex µi lifts to a smooth retraction of Ui to the conjugacy class Cµi [Mei02], ri : Ui × [0, 1] → Ui .

(56)

On the conjugacy class Cµi , the 3-form H becomes exact, ι∗i H = dωµi where ιi : Cµi → G is the inclusion, and Ad−1 + idg ∗ ιi θ ∈ Ω 2 (Cµi ) ωµi := ι∗i θ ∧ Ad−1 − idg

(57)

is an invariant 2-form on the conjugacy class [BRS01]. Here the notation is to be understood as follows: at some element h ∈ Cµi , the 2-form ωµi is obtained by considering the Maurer–Cartan forms in h and taking the inverse of the adjoint action (Adh )−1 : g → g. The endomorphism (Adh )−1 − idg becomes invertible when restricted to the image of ι∗i θ so that the fraction makes sense. By pull-back along ri and ﬁbre integration, one obtains a 2-form Ci ∈ Ω 2 (Ui ) with H|Ui = dCi . (58) One can now show that Cj − Ci = µij , dθ on Ui ∩ Uj , which is the condition on the curving. So, our construction realizes, for G = SU(n) and G = Sp(n), a hermitian bundle gerbe with connection and curving with Dixmier–Douady class 1 ∈ Z, the basic bundle gerbe. For the other compact, simple, connected and simply connected Lie groups, one can ﬁnd an integer k0 for which the vertices of k0 A are weights. Using the weights k0 µij in the construction of the line bundles Lij , and the 2-forms k0 Ci in the deﬁnition of the curving, we obtain bundle gerbes with Dixmier-Douady class k0 ∈ Z [Mei02]. The smallest such integer k0 is tabulated in [Bou68]: G SU(n) Spin(n) Sp(2n) E6 E7 E8 F4 G2 k0 1 2 1 3 12 60 6 2 The construction of the basic bundle gerbes on the groups with k0 > 1 requires more advanced techniques [Mei02, GR03]. Here it becomes in particular important that the deﬁnition of a bundle gerbes admits π : Y → G to be a surjective submersion, which is more general than just an open cover of G. Starting from a bundle gerbe G on a simply connected Lie group G, we now describe a method to obtain bundle gerbes on the non simply connected Lie groups G/Z which are quotients of G by a subgroup Z of the center of G. More generally, let Γ be a ﬁnite group acting on a manifold M by diﬀeomorphisms.

Gerbes and Lie Groups

353

Definition 3.1. A Γ -equivariant structure (Aγ , ϕγ1 ,γ2 ) on a bundle gerbe G over M consists of isomorphisms Aγ : G → γ ∗ G

(59)

for each γ ∈ Γ and of 2-isomorphisms ϕγ1 ,γ2 : γ1∗ Aγ2 ◦ Aγ1 =⇒ Aγ2 γ1

(60)

for each pair γ1 , γ2 ∈ Z, such that the diagram γ1∗ γ2∗ Aγ3 ◦ γ1∗ Aγ2 ◦ Aγ1

id◦ϕγ1 ,γ2

(61)

ϕγ2 γ1 ,γ3

γ1∗ ϕγ2 ,γ3 ◦id

γ1∗ Aγ3 γ2 ◦ Aγ1

+3 γ1∗ γ2∗ Aγ3 ◦ Aγ2 γ1

ϕγ1 ,γ3 γ2

+3 Aγ3 γ2 γ1

of 2-isomorphisms is commutative. Not every bundle gerbe G over M admits a Γ -equivariant structure, and if it does, it may not be unique. To obtain obstructions and classiﬁcations we use the cohomological language. For this purpose, we impose the structure n of a Γ -module on the Deligne cochain groups DV (k) [GSW]. We assume the existence of a good open cover V = {Vi }i∈I of M which is compatible with the group action in the sense that there is an induced action of Γ on the index set I such that γ(Vi ) = Vγi . For example, the open sets Ui that we have used in the construction of the basic bundle gerbe satisfy this condition. Then Γ n acts by pull-back on the cochain groups DV (k). For each Γ -module W , one can build the usual group cohomology complex, consisting of cochain groups CΓp (W ) := Map(Γ p+1 , W ) and the usual coboundary operator d : CΓk−1 (W ) → CΓk (W ). (62) A simple key observation is that the coboundary operator d and the Deligne diﬀerential D commute so that we have a double complex with cochain groups k CΓp (DV (n)) in degree (p, k). We denote the total cohomology of this double complex by HqΓ (M, D(n)). In particular, note that we have a natural group homomorphism eq : H2 (M, D(2)) → H2Γ (M, D(2)) : ξ → (γ ∗ ξ − ξ, 0, 0)

(63)

that includes an ordinary Deligne cohomology class into the cohomology of the total complex, that we just have deﬁned. Now let G be a hermitian bundle gerbe with connection and curving with Deligne class ξ, and let (Az , ϕz1 ,z2 ) be a Γ -equivariant structure on G. For 1 local data aγ ∈ CΓ1 (DV (2)) of the isomorphism Aγ , i.e.,

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Daγ = γ ∗ ξ − ξ,

(64)

0 and local data bγ1 ,γ2 ∈ CΓ2 (DV (2)) of the 2-isomorphisms ϕγ1 ,γ2 , i.e.,

Dbγ1 ,γ2 = (da)γ1 ,γ2 ,

(65)

the commutativity of diagram (61) imposes the condition (db)γ1 ,γ2 ,γ3 = 0. This means that, for a bundle gerbe with Deligne class ξ, the class eq(ξ) ∈ H2Γ (M, D(2)) is the obstruction class for G to admit Γ -equivariant structures. Furthermore, the cohomology group H1Γ (M, D(2)) classiﬁes the inequivalent choices. In the case of bundle gerbes over a compact, simple, connected and simply connected Lie group G, we consider the action of a subgroup Z of the center of G by multiplication. In this case the relevant cohomology groups reduce to the usual group cohomology of the ﬁnite group Z, so that there is an obstruction class in H3Grp (Z, U (1)), and the possible Z-equivariant structures are classiﬁed by H2Grp (Z, U (1)). For the bundle gerbes we have constructed above, all obstruction classes against Z-equivariant structures for all subgroups Z of the center of G can be calculated [GR02, GR03]. Let us now describe how a choice (Az , ϕz1 ,z2 ) of a Γ -equivariant structure on a given bundle gerbe G = (π, L, µ, ∇, C) with connection and curving over M deﬁnes a quotient bundle gerbe G = (π, L, µ, ∇, C) over M := M/Γ . Following [GR02], we set Y := Y and π := p ◦ π : Y → M , where p : M → M is the projection to the quotient. Note that the ﬁbre products are [2] [3] Y = Z γ and Y = Z γ1 ,γ2 (66) γ∈Γ

γ1 ,γ2 ∈Γ

for the smooth manifolds Z γ := Yγ ×M Y

and

Z γ1 ,γ2 = Yγ1 γ2 ×M Z γ1 ,

(67)

where Yγ := Y as manifolds but with projection γ −1 ◦ π instead of π. The manifolds Y

[2]

and Y

[3]

again have projections π i := πi and π ij := πij to

[2]

Y and to Y , respectively. The curving of the quotient bundle gerbe will be [2] C := C ∈ Ω 2 (Y ), and the line bundle L → Y will be L|Z γ := Aγ . Condition (27) for the curvature of L is curv(L)|Z γ = curv(Aγ ) = π2∗ C − π1∗ C = π ∗2 C − π ∗1 C, and hence is satisﬁed. The isomorphism µ over Y µ|Z γ1 ,γ2 := ϕZ γ1 ,γ2 ,

[3]

(68)

is deﬁned by (69)

and its associativity by means of diagram (8) is nothing but the condition on the isomorphisms ϕγ1 ,γ2 from Deﬁnition 3.1. So we have deﬁned a bundle gerbe G over M with connection and curving.

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Applying this procedure to the bundle gerbes over the simply connected Lie groups and their equivariant structures, one obtains examples of bundle gerbes over all compact, simple, connected and simply connected Lie groups.

4 Structure on loop spaces from bundle gerbes We describe a construction of a line bundle over the loop space LM of a manifold M from a given hermitian bundle gerbe over M with connection and curving. This construction is adapted from the one in [Bry93]. For preparation, we have to describe the set of isomorphisms between two ﬁxed hermitian bundle gerbes G1 and G2 with connection and curving. We denote the set of isomorphism classes of morphisms A : G1 → G2 by Iso(G1 , G2 ). The following lemma can easily be shown using the cohomological description. Lemma 4.1 ([SSW07]). The group Pic∇ 0 (M ) of isomorphism classes of ﬂat line bundles with connection over M acts freely and transitively on the set Iso(G1 , G2 ). Now let LM := C ∞ (S 1 , M ) be the free loop space of M , equipped with a smooth manifold structure as described in [Bry93]. Let G = (π, L, µ, ∇, C) be a hermitian bundle gerbe on M with connection and curving. The total space of the line bundle we are going to deﬁne is, as a set, L := Iso(γ ∗ G, I0 ). (70) γ∈LM

It comes with the evident projection to LM , and by Lemma 4.1 every ﬁbre is a torsor over the group 1 ∼ 1 ∼ Pic∇ 0 (S ) = Hom(π1 (S ), U (1)) = U (1).

(71)

Note that the canonical projection p : L → LM admits local sections: for a contractible open subset U ⊂ M we have an isomorphism T : G|U → Iρ which provides a section σ : LU → L : γ → (γ, γ ∗ T ). Proposition 4.2. There is a unique diﬀerentiable structure on L, such that the projection p and the sections σ are smooth, and L becomes a principal U (1)-bundle over LM . Proof. Since every gerbe over S 1 is trivializable, none of the ﬁbres p−1 (γ) is empty. Hence each ﬁbre is a U (1)-torsor. For the same reason, the image γ(S 1 ) of any loop γ has an open neighbourhood U ⊂ M (cf. the proof of Proposition 6.2.1 in [Bry93]), such that G|U admits a trivialization T : G|U → Iρ . The corresponding section σ identiﬁes p−1 (LU ) with LU × U (1), and thus deﬁnes a topology and a diﬀerentiable structure on each preimage p−1 (LU ). Let a

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topology on L be generated by all open subsets of all the ﬁbres p−1 (LU ). Now, for two intersecting subsets U1 and U2 and trivializations T1 and T2 , respectively, let N be the line bundle over U1 ∩ U2 from Lemma 2.4. The transition map LU1 × U (1) → LU2 × U (1) is then given by (γ, z) → (γ, z · holγ ∗ N (S 1 )−1 ), and is hence diﬀerentiable with respect to the loop γ : S 1 → U1 ∩ U2 . Instead of a principal U (1)-bundle, we will often and equivalently consider L as a hermitian line bundle. The construction just discussed also applies to the case when the bundle gerbe G is hermitian and has a connection. A curving deﬁnes a connection ∇ on the line bundle L, whose curvature is curv(∇) = ev∗ H, (72) S1

where H is the curvature of the gerbe G, and ev : LM × S 1 → M is the evaluation map. The hermitian line bundle L with connection is functorial in the following sense. Proposition 4.3. For a hermitian bundle gerbe G over M with connection and curving, denote the associated hermitian line bundle with connection by LG . i) Any isomorphism of gerbes G → G induces an isomorphism LG → LG of line bundles. ii) For a smooth map f : X → M denote the induced map on loop spaces by Lf : LX → LM . Then, the line bundles (Lf )∗ LG and Lf ∗ G are canonically isomorphic. iii) For the dual gerbe G ∗ we obtain a canonical isomorphism L∗G ∼ = LG ∗ . Let us also have a view on the cohomological counterpart of the construction of a line bundle over LM from a bundle gerbe over M . For this purpose, one has to extend the usual ﬁbre integration of diﬀerential forms, : Ω k+1 (X × S 1 ) → Ω k (X), (73) S1

to the Deligne cohomology groups. Here, X can be any smooth and possibly inﬁnite-dimensional manifold. Such extensions have been described in various ways [Gaw88, Bry93, GT01]. Then, for X = LM , the concatenation of this extension with the pull-back along the evaluation map ev : LM × S 1 → M gives a group homomorphism ◦ ev∗ : H3 (M, D(3)) → H2 (LM, D(2)). S1

(74)

(75)

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One can show that the image of the Deligne class [(g, A, B)] of a hermitian bundle gerbe G with connection and curving under this group homomorphism gives exactly the Deligne class of the line bundle L with the connection ∇ that we have constructed above in a direct geometric way. The construction of the line bundle L can in particular be applied to the bundle gerbes over compact Lie groups from Section 3. In this case, we obtain a sequence / U (1) / L p / LG /1 (76) 1 of smooth maps, where U (1) is mapped to the ﬁbre of L over the loop which is constantly 1 ∈ G. Now a natural question is whether one can equip the total space L with a group structure, such that the sequence (76) is an exact sequence of groups. This would provide a geometric construction of central extensions of loop groups. For simply connected groups there exist deﬁnitions of group structures on L [Bry93], while in the general case the geometric construction of loop group extensions from bundle gerbes is still an open problem.

5 Algebraic structures for gerbes There are several additional structures for bundle gerbes, some of which we introduce in this section. We describe the particular case of those additional structures on bundle gerbes over compact connected Lie groups. 5.1 Bundle gerbe modules Bundle gerbe modules, also known as twisted vector bundles, have been introduced in [BCM+ 02] in order to realize twisted K-theory geometrically. They are also the appropriate structure to extend the deﬁnition of holonomy to surfaces with boundary [CJM02]. Definition 5.1. Let G be a bundle gerbe over M . A G-module is a 1-morphism E : G → Iω for some 2-form ω ∈ Ω 2 (M ). The 2-form ω is called the curvature of the gerbe module. Let us compare this deﬁnition with the original deﬁnition of bundle gerbe modules in [BCM+ 02]. A left G-module E : G → Iω consists of a vector bundle E over Y and of an isomorphism : L ⊗ π2∗ E → π1∗ E of vector bundles over Y [2] which satisﬁes ∗ ∗ ∗ π13 ◦ (µ ⊗ id) = π23 ◦ π12 . (77) The similarity with an action of L on E justiﬁes the notion of gerbe module. The curvature of E is restricted by (34) to

with n the rank of E.

1 tr(curv(E)) = π ∗ ω − C n

(78)

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In terms of local data, a rank-n bundle gerbe module E: G → Iω is described by a collection (Gij , Πi ) of smooth functions Gij : Ui ∩ Uj → U (n) and u(n)-valued 1-forms Πi ∈ Ω 1 (Ui ) ⊗ u(n) which relate the local data of the bundle gerbes G and Iω in the following way: −1 1 = gijk · Gik G−1 jk Gij

G−1 ij Πi

0 = Aij + Πj − 1 ω = Bi + tr(dΠi ) n

on Ui ∩ Uj ∩ Uk , Gij −

G−1 ij

dGij

on Ui ∩ Uj ,

(79)

on Ui .

Note that the derivative of the last equality gives dω = dBi = curv(G).

(80)

Also note that if the bundle gerbe G is itself trivial, i.e., has local data (1, 0, C|Ui ) for a globally deﬁned 2-form C ∈ Ω 2 (M ), then (Gij , Πi ) are the local data of a rank-n vector bundle over M with curvature of trace n (ω−C). This explains the terminology “twisted” vector bundle in the non-trivial case. According to (80), a necessary condition for the existence of a bundle gerbe module is that the curvature is an exact form. However, this is not the case in many situations, for example, for the bundle gerbes on compact Lie groups we have constructed in Section 3, whose curvature is the canonical 3-form H. For this reason, one often considers a pair (Q, E) of a submanifold Q ⊂ M together with a gerbe module E : G|Q → Iω for the restriction of the gerbe to this submanifold. In conformal ﬁeld theory, the pair (Q, G) is also called a D-brane. In particular, we can consider this situation for the bundle gerbes over Lie groups G constructed in Section 3. In this case, the important submanifolds are conjugacy classes Q = Cλ , and we already know that the curvature curv(G) = H becomes exact when restricted to a conjugacy class, H|Cλ = dωλ . So the necessary condition (80) is satisﬁed. One can furthermore show [Gaw05] that precisely for integrable weights λ there exists a G|Cλ -module with curvature ωλ . This is the appropriate description of “ﬂux stabilization of D-branes” in string theory [BDS00]. 5.2 Bundle Gerbe bimodules Bundle gerbe bimodules generalize bundle gerbe modules for one bundle gerbe G to a structure for two bundle gerbes. Definition 5.2 ([FSW]). Let G1 and G2 be bundle gerbes over M . A G1 -G2 bimodule is a morphism D : G1 → G2 ⊗ I (81) for some 2-form ∈ Ω 2 (M ). The 2-form is called the curvature of the bimodule.

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This deﬁnition is related to the one of a gerbe module in the sense that—using the appropriate notion of duality for bundle gerbes [Wal07]—a G1 -G2 -bimodule is the same as a (G1 ⊗G2∗ )-module. We drop the discussion of the local data of a gerbe bimodule. However, it is clear that there is, analogously to equation (80), a necessary condition on the curvature of the two bundle gerbes, H1 = H2 + d.

(82)

It is again useful to consider bimodules for restrictions of the bundle gerbes to a submanifold Q ⊂ M . In particular, if M is the direct product of two manifolds M1 and M2 , each equipped with a bundle gerbe G1 and G2 , respectively, one considers p∗1 G1 |Q -p∗2 G2 |Q -bimodules, where pi : M → Mi is the projection on the ith factor. This is the setup to deﬁne holonomies around surfaces with defect lines [FSW]. Examples for such bimodules are again provided by compact Lie groups and the basic bundle gerbes thereon [FSW]. The study of such examples leads to the following relevant submanifolds Q of G × G, called the biconjugacy classes −1 Bh1 ,h2 := (x1 h1 x−1 (83) 2 , x1 h2 x2 ) ∈ G × G | x1 , x2 ∈ G for any pair (h1 , h2 ) ∈ G × G. Biconjugacy classes inherit two commuting actions of G from the diagonal left and diagonal right actions of G on G × G. One observes that the smooth map µ ˜ : G × G → G : (g1 , g2 ) → g1 g2−1 (84) intertwines the diagonal left and diagonal right action of G on G × G and the adjoint and trivial actions of G on itself, respectively. It now follows that a biconjugacy class in G × G is the preimage of a conjugacy class in G under the projection µ ˜: Bh1 ,h2 = µ (85) ˜−1 (Ch h−1 ) = (g1 , g2 ) ∈ G × G | g1 g2−1 ∈ Ch h−1 . 1

2

1

2

We introduce the 2-form ˜ ∗ ωh h1 ,h2 := µ

−1 1 h2

−

k ∗ p θ ∧ p∗2 θ 2 1

(86)

on Bh1 ,h2 , where both summands are restricted to the submanifold Bh1 ,h2 of G × G. From the intertwining properties of µ ˜ and the bi-invariance of ω, it follows that the 2-form is also bi-invariant. One can show that it satisﬁes p∗1 H = p∗2 H + dh1 ,h2

(87)

on a biconjugacy class Bh1 ,h2 , which is the necessary condition (82) [FSW].

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5.3 Jandl structures Another structure we want to introduce is a Jandl structure on a bundle gerbe G. Jandl structures extend the deﬁnition of holonomy we gave in Section 2 to unoriented and, in particular, unorientable surfaces. Definition 5.3 ([SSW07]). A Jandl structure J on a bundle gerbe G over M is an involution k of M together with an isomorphism

and a 2-morphism

A : k∗ G → G ∗

(88)

ϕ : k ∗ A ⇒ A∗

(89)

which satisﬁes the equivariance condition k ∗ ϕ = ϕ∗ .

(90)

To give an impression of the details of a Jandl structure, recall that an isomorphism such as A = (A, α) consists of a line bundle A over the space Z which is built up from the two surjective submersions of the bundle gerbes k ∗ G and G ∗ . In this particular situation, there is a canonical lift k˜ of the involution k into the space Z, and it is in fact easy to work out that the ˜ 2-morphism ϕ deﬁnes a k-equivariant structure on the line bundle A, which is compatible with the isomorphism α. Summarizing, a Jandl structure J on G is an isomorphism A : k∗ G → G ∗ (91) ˜ whose line bundle A is equivariant with respect to the involution k on Z [SSW07]. Recall that we introduced equivariant structures on bundle gerbes in Section 3 in order to produce bundle gerbes over quotients of a manifold M by a discrete group Z. One can combine equivariant structures and Jandl structures into Z-equivariant Jandl structures, leading to the mathematically appropriate description of what are called orientifolds in string theory [GSW]. The idea behind this combination is that a bundle gerbe G over M with a Γ equivariant Jandl structure deﬁnes a bundle gerbe G with Jandl structure over the quotient M/Z. For a cohomological description, one modiﬁes the action of Z on the Deligne cohomology group to an action of the semidirect product Γ := Z2 Z [GSW]. For bundle gerbes over compact Lie groups, where Z is a subgroup of the center of G, the relevant involutions are given by kz : G → G : g → (zg)−1

(92)

for any z in the center. Again, using the basic bundle gerbes constructed in Section 3, one can classify all equivariant Jandl structures over all these bundle gerbes [GSW].

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6 Applications to conformal field theory Let us explain the relation between conformal ﬁeld theory and Lie theory, which arises in the study of non-linear sigma models on a Lie group G. Such a model can be deﬁned by amplitudes A(φ) for some path integral, where φ is a map from a closed complex curve Σ—the world sheet—into the target space G of the model. In [Wit84], Witten gives the following deﬁnition for G = SU (2). Σ is the boundary of a three-dimensional manifold B, and because the homotopy groups πi (SU (2)) vanish for i = 1, 2, every map φ : Σ → M can be extended into the interior B to a map Φ : B → G. Witten showed that, due to the integrality of the canonical 3-form H, A(φ) := exp Skin (φ) + Φ∗ H (93) B

depends neither on the choice of B nor on the choice of the extension Φ, so that one obtains a well-deﬁned amplitude. Here Skin (φ) is a kinetic term, and, with a certain relative normalization of the two terms in (93), this model is called the Wess–Zumino–Witten model on G at level k. For non simply connected Lie groups, the extension Φ of φ to B does not exist in general. In these cases, the second summand of (93) has to be generalized. Proposition 6.1. Let G be a hermitian bundle gerbe over G with connection and curving of curvature H. For a three-dimensional oriented manifold B with boundary and a map Φ : B → G, we have holG (Φ|∂B ) = exp Φ∗ H . (94) B

Proof. Remember that for any trivialization T : Φ∗ G|∂B → Iρ we have Φ∗ H|∂B = dρ. The rest follows by Stokes’ theorem. This way we reproduce the amplitude of the coupling term of the Wess– Zumino–Witten model by A(φ) = exp (Skin (φ)) · holG (φ).

(95)

Notice that using bundle gerbes we did not impose any condition on the topology of the target space G. For compact, connected and simply connected Lie groups, it reproduces Witten’s original deﬁnition. However, for general target spaces there may be bundle gerbes with same curvature, which are not isomorphic. This occurs, for instance, for the Lie group Spin(4n)/(Z2 × Z2 ). Here, the theory of bundle gerbes over Lie groups has brought new insights into the Lagrangian description of Wess–Zumino–Witten models. In conformal ﬁeld theory, many applications require us to consider surfaces with boundary. For them, we are not able to apply Deﬁnition 2.5 of the

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holonomy of a bundle gerbe G: for a change of the chosen trivialization, a boundary term emerges which has to be compensated to achieve a holonomy independent of the choice of the trivialization. The compensating term is provided by the choice of a symmetric D-brane (Cλ , Eλ ): a conjugacy class Cλ for an integrable weight λ, together with a G|Cλ -module Eλ of curvature ωλ [Gaw05]. Another class of conformal ﬁeld theories involves unoriented world sheets Σ. Again, Deﬁnition 2.5 has to be generalized, since it involves the integral of a diﬀerential form over Σ. It has been shown in [SSW07] that the choice of a Jandl structure on the bundle gerbe G again makes the holonomy well deﬁned. The precise classiﬁcation of Jandl structures leads to a complete classiﬁcation of unoriented Wess–Zumino–Witten models [SSW07, GSW]. Let us also indicate the relevance of the line bundle L over the loop group LG that we have constructed in Section 4. The Hilbert space of holomorphic sections in L (completed with respect to its hermitian metric) serves as the space of states for the quantized theory [GR02]. The choice of additional structures, like bundle gerbe modules or Jandl structures, has implications on this space. For example, a Jandl structure on the bundle gerbe G implies by Proposition 4.3 an isomorphism ϕ : Lk ∗ L → L∗ which satisﬁes Lk ∗ ϕ = ϕ.

7 Open questions We conclude this contribution with the discussion of some lines for further research. One obvious direction is to extend the results explained here to gauged Wess–Zumino–Witten models, i.e., the coset theories. Those models having ﬁxed points under the action of the group implementing ﬁeld identiﬁcations for which the untwisted stabilizer is strictly smaller than the stabilizer [FRS04] should be particularly interesting: in this case, simple gerbe modules and bimodules of rank strictly bigger than one appear naturally. A precise understanding of such theories requires the notion of an H-equivariant gerbe (bi-)module on the ambient group G. Another subtle issue is the generalization of our results to non-compact Lie groups; this is partially due to the fact that much less is known about these theories in algebraic approaches. The following two points seem to be conceptually appealing questions: our initial motivating question in this paper was about central extensions of Lie groups. While hermitian bundle gerbes naturally account for a line bundle L on loop space, a more speciﬁc structure on the gerbe is needed to obtain a group structure on the line bundle L. Similarly, one wishes to ﬁnd a structure on a gerbe module E that endows the associated bundles E over loop and interval spaces with a natural L-module structure. Finally, we point out that gerbe bimodules have a natural operation of fusion which is very much in the spirit of a convolution of correspondences.

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Imposing additional conditions on gerbe bimodules over a compact Lie group (that in physical applications ensure the existence of enough conserved quantities), one should be able to single out interesting subcategories of gerbe bimodules that are semisimple tensor categories. If G is simply connected, the fusion ring of gerbe bimodules can be expected to be the corresponding Verlinde algebra; for non simply connected groups, we expect interesting cousins of the Verlinde algebra.

References [BCM+ 02] P. Bouwknegt, A. L. Carey, V. Mathai, M. K. Murray and D. Stevenson, Twisted K-Theory and K-Theory of Bundle Gerbes, Commun. Math. Phys. 228(1), 17–49 (2002), hep-th/0106194. [BDS00] C. Bachas, M. Douglas and C. Schweigert, Flux Stabilization of D-Branes, JHEP 0005(048) (2000), hep-th/0003037v2. ´ ements de Math´ematique. Fasc. XXXIV. Groupes et [Bou68] N. Bourbaki, El´ alg`ebres de Lie. Chapitre IV–VI, Hermann, Paris, 1968. [BRS01] P. Bordalo, S. Ribault and C. Schweigert, Flux Stabilization in Compact Groups, JHEP 0110(036) (2001). [Bry] J.-L. Brylinski, Gerbes on Complex Reductive Lie Groups, math/0002158. [Bry93] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, volume 107 of Progress in Mathematics, Birkh¨ auser, 1993. [Cha98] D. S. Chatterjee, On the Construction of Abelian Gerbes, Ph.D. thesis, Cambridge Univ., Cambridge, UK, 1998. [CJM02] A. L. Carey, S. Johnson and M. K. Murray, Holonomy on D-Branes, J. Geom. Phys. 52(2), 186–216 (2002), hep-th/0204199. [FRS04] J. Fuchs, I. Runkel and C. Schweigert, TFT Construction of RCFT Correlators III: Simple Currents, Nucl. Phys. B 694, 277–353 (2004), hep-th/0403157. [FSW] J. Fuchs, C. Schweigert and K. Waldorf, Bi-Branes: Target Space Geometry for World Sheet Topological Defects, hep-th/0703145, J. Geom. Phys. 58, 576–598 (2008). [Gaw88] K. Gaw¸edzki, Topological Actions in Two-Dimensional Quantum Field Theories, in Non-perturbative Quantum Field Theory, edited by G. Hooft, A. Jaﬀe, G. Mack, K. Mitter and R. Stora, pages 101–142, Plenum Press, New York, 1988. [Gaw05] K. Gaw¸edzki, Abelian and Non-Abelian Branes in WZW Models and Gerbes, Commun. Math. Phys. 258, 23–73 (2005), hep-th/0406072. [GR02] K. Gaw¸edzki and N. Reis, WZW Branes and Gerbes, Rev. Math. Phys. 14(12), 1281–1334 (2002), hep-th/0205233. [GR03] K. Gaw¸edzki and N. Reis, Basic Gerbe over Non Simply Connected Compact Groups, J. Geom. Phys. 50(1–4), 28–55 (2003), math.dg/0307010. [GSW] K. Gaw¸edzki, R. R. Suszek and K. Waldorf, WZW Orientifolds and Finite Group Cohomology, hep-th/0701071, Commun. Math. Phys. 284 1–49 (2008).

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[GT01] [Mei02] [MS76] [MS00] [Mur96] [PS86] [SSW07] [Ste00] [Wal07] [Wit84]

K. Gomi and Y. Terashima, Higher-Dimensional Parallel Transports, Math. Research Letters 8, 25–33 (2001). E. Meinrenken, The Basic Gerbe over a Compact Simple Lie Group, Enseign. Math., II. Sr. 49(3–4), 307–333 (2002), math/0209194. J. W. Milnor and J. D. Stasheﬀ, Characteristic Classes, Annals of Mathematical Studies, Princeton University Press, Princeton, NJ, 1976. M. K. Murray and D. Stevenson, Bundle Gerbes: Stable Isomorphism and Local Theory, J. Lond. Math. Soc. 62, 925–937 (2000), math/9908135. M. K. Murray, Bundle Gerbes, J. Lond. Math. Soc. 54, 403–416 (1996), dg-ga/9407015. A. Pressley and G. Segal, Loop Groups, Oxford Univ. Press, Oxford, 1986. U. Schreiber, C. Schweigert and K. Waldorf, Unoriented WZW Models and Holonomy of Bundle Gerbes, Commun. Math. Phys. 274(1), 31–64 (2007), hep-th/0512283. D. Stevenson, The Geometry of Bundle Gerbes, Ph.D. thesis, University of Adelaide, Australia, 2000, math.DG/0004117. K. Waldorf, More Morphisms Between Bundle Gerbes, Theory Appl. Categories 18(9), 240–273 (2007), math.CT/0702652. E. Witten, Nonabelian Bosonization in Two Dimensions, Commun. Math. Phys. 92, 455–472 (1984).

Part C

Representation Theory of Infinite-Dimensional Lie Groups

Functional Analytic Background for a Theory of Infinite-Dimensional Reductive Lie Groups Daniel Beltit¸a˘ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania, [email protected] Summary. Motivated by the interesting and yet scattered developments in representation theory of Banach–Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability. Key words: reductive Lie group, group decomposition, amenable group, operator ideal, triangular integral. 2000 Mathematics Subject Classiﬁcations: Primary 22E65. Secondary 22E46, 47B10, 47L20, 58B25.

1 Introduction: What a reductive Lie group is supposed to be We approach the problem of ﬁnding an appropriate inﬁnite-dimensional version of a reductive Lie group. The discussion is motivated by the need to have a reasonably general setting where the representation theoretic properties of the classical Lie groups associated with the Schatten ideals of Hilbert space operators—in the sense of [dlH72]—can be investigated in a systematic way. Thus the theory of operator ideals (see, e.g., [GK69], [GK70], [DFWW04], and [KW08]) provides the natural background for the present exposition. The ideas and methods of representation theory of ﬁnite-dimensional Lie groups cannot possibly be extended to the setting of Banach–Lie groups in a direct manner. Any attempt to do so fails because of some phenomena speciﬁc to the inﬁnite-dimensional Lie groups: there exist Lie algebras that do not arise from Lie groups (see, e.g., Section VI in [Ne06] or Section 3.3 in [Be06]); closed subgroups of Lie groups need not be Lie groups in the relative topology (see Example 3.3(i) in [Ho75] or Example 7.12 in [Up85]); one does not know how to construct smooth structures on homogeneous spaces unless one is able to ﬁnd a direct complement of the Lie subalgebra in the ambient Lie algebra (see

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_11, © Springer Basel AG 2011

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Corollary 8.3 in [Up85], the discussions in [BP07] and [Ga06], or Section 4.1 [Be06]); there exists no Haar measure on topological groups which are not locally compact ([We40]); and ﬁnally every inﬁnite-dimensional Banach space is the model space of some abelian Lie group without any nontrivial continuous representations (see Theorem 1 in [Ba83]). Nevertheless, the study of representation theoretic properties of some speciﬁc Banach–Lie groups has led to a number of interesting results; see for instance the papers [SV75], [Bo80], [Pi90], [Ne98], [NØ98], [BR07], or [BG08]. It seems reasonable to try to ﬁnd a class of Banach–Lie groups whose representations can be studied in a coherent fashion following the pattern of representation theory of ﬁnite-dimensional reductive Lie groups. The aim of the present paper is to survey some of the ideas and notions that might eventually lead to the description of a class of Banach–Lie groups appropriate for the purposes of such a representation theory. The exposition is streamlined by a number of phenomena that play a central role in the classical theory of reductive Lie groups: Cartan decompositions, Iwasawa decompositions, Harish-Chandra decompositions, and existence of invariant measures. By describing appropriate versions of these phenomena in inﬁnite dimensions, the paper provides a self-contained discussion of a number of functional analytic issues which should underlie the notion of an inﬁnite-dimensional reductive Lie group: triangular integrals, operator factorizations, and amenability. Finite-dimensional reductive Lie groups. In order to make clear the structures we shall be looking for in inﬁnite dimensions, we now recall the classical setting. Remarks 1.2 and 1.3 basically concern the matrix Lie algebras/groups, where reductivity means stability under the operation of taking adjoints of matrices. (These remarks will be our main motivation for the discussion of Φ-reductivity in Section 5.) The following general deﬁnition is the one used in [Kn96]. Definition 1.1. A ﬁnite-dimensional reductive Lie group is actually a 4-tuple (G, K, θ, B), where G is a ﬁnite-dimensional Lie group with the Lie algebra g, K is a compact subgroup of G with the Lie algebra k, θ : g → g is an involutive automorphism, and B : g × g → R is a nondegenerate symmetric bilinear form which is Ad(G)-invariant and θ-invariant, such that the following conditions are satisﬁed: (i) g is a reductive Lie algebra with the complexiﬁcation denoted gC ; (ii) k = Ker (θ − 1); (iii) if we denote p = Ker (θ + 1), then g = k p and we have B(k, p) = {0}, B is positive deﬁnite on p, and negative deﬁnite on k; (iv) the mapping K × p → G, (k, X) → k · expG X, is a diﬀeomorphism; (v) for every g ∈ G the automorphism Ad(g) : gC → gC belongs to the connected component of 1 ∈ Aut (gC ). We say that the Lie group G itself is a reductive Lie group if the other items K, θ, and B are clear from the context.

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Remark 1.2. Let g be a real ﬁnite-dimensional Lie algebra. Then g is a reductive Lie algebra if and only if it is isomorphic to a real Lie subalgebra g1 of the matrix algebra Mn (C) for some integer n ≥ 1 such that for every X ∈ g1 we have X ∗ ∈ g1 . In addition, if g were a complex Lie algebra, then g1 could be chosen to be a complex Lie subalgebra of Mn (C). Remark 1.3. Let G be a connected closed subgroup of GL(n, C) for some n ≥ 1 such that for every g ∈ G we have g ∗ ∈ G. Then G is a reductive Lie group with K = G ∩ U(n), θ(X) = −X ∗ for all X ∈ g = L(G) ⊆ Mn (C), and B(X, Y ) = Re (Tr (XY )) for all X, Y ∈ g (see Example 2 in Section 2 of Chapter VII of [Kn96]). It is not diﬃcult to prove that coverings with ﬁnitely many sheets of any matrix group G as above are reductive Lie groups —the covering groups of this type are precisely the connected reductive groups in the sense of Deﬁnition 2.5 of [Vo00]. (See Proposition 4.2 below for the way the Cartan decompositions lift to covering groups.) In particular, every connected semisimple Lie group with ﬁnite center is a reductive Lie group in the sense of Deﬁnition 1.1 above (Example 1 in Section 2 of Chapter VII of [Kn96]). Some conventions and notation. Throughout the paper we denote by H an inﬁnite-dimensional complex separable Hilbert space, by B(H) the set of all bounded linear operators on H, and by F the two-sided ideal of B(H) consisting of all ﬁnite-rank operators. Some convenient references for inﬁnitedimensional Lie groups with a functional analytic ﬂavor are [dlH72], [Up85], [Ne04], [Ne06], and [Be06]. As in the latter reference, we shall always denote by L(·) the Lie functor from the category of Banach–Lie groups into the category of Banach–Lie algebras. We also adopt the convention that Lie groups are denoted by Roman capitals and their Lie algebras are denoted by the corresponding Gothic lower case letters.

2 Triangular integrals and factorizations Norm ideals. The norm ideals will play a critical role for the present exposition. In fact, our candidates for inﬁnite-dimensional reductive groups will be Banach–Lie groups whose model spaces are norm ideals. Definition 2.1. By norm ideal we mean a two-sided ideal I of B(H) equipped with a norm · I satisfying T ≤ T I = T ∗ I and AT B I ≤ A T I B whenever A, B ∈ B(H). Definition 2.2. Let c be the vector space of all sequences of real numbers {ξj }j≥1 such that ξj = 0 for all but ﬁnitely many indices. A symmetric norming function is a function Φ : c → R satisfying the following conditions: (i) the function Φ(·) is a norm on the linear space c and Φ((1, 0, 0, . . . )) = 1; (ii) Φ({ξj }j≥1 ) = Φ({ξπ(j) }j≥1 ) whenever {ξj }j≥1 ∈ c and π : N\{0} → N\{0} is bijective.

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Any symmetric norming function Φ gives rise to two norm ideals SΦ and SΦ as follows. For every bounded sequence of real numbers ξ = {ξj }j≥1 deﬁne Φ(ξ) := supn≥1 Φ(ξ1 , ξ2 , . . . , ξn , 0, 0, . . . ) ∈ [0, ∞]. For all T ∈ B(H) denote T Φ := Φ({sj (T )}j≥1 ) ∈ [0, ∞], where sj (T ) = inf{ T − F | F ∈ B(H), rank F < j} whenever j ≥ 1. With this notation we can deﬁne (0)

SΦ := F

·Φ

⊆ {T ∈ B(H) | T Φ < ∞} =: SΦ ;

(0)

that is, SΦ is the · Φ -closure of the ideal of ﬁnite-rank operators F in SΦ . (0) Then · Φ is a norm making SΦ and SΦ into norm ideals (see Section 4 in Chapter III in [GK69]). We say that Φ is a mononormalizing symmetric (0) norming function if SΦ = SΦ . Let 1 ≤ p ≤ ∞ and Φp (ξ) = ξ p whenever ξ∈ c deﬁnes a symmetric norming function (mononormalizing if p < ∞) and (0) the corresponding norm ideal Sp := SΦp is the p-th Schatten ideal. In the special case p = 2 we call S2 the Hilbert–Schmidt ideal. (0)

Remark 2.3. Every separable norm ideal is equal to SΦ for some symmetric norming function Φ (see Theorem 6.2 in Chapter III in [GK69]). Remark 2.4. For every symmetric norming function Φ : c → R there exists a unique symmetric norming function Φ∗ : c → R such that ∞ 1 ξj ηj Φ∗ (η) = sup Φ(ξ) j=1

c and ξ1 ≥ ξ2 ≥ · · · ≥ 0 ξ = {ξj }j≥1 ∈

whenever η = {ηj }j≥1 ∈ c and η1 ≥ η2 ≥ · · · ≥ 0. The function Φ∗ is said to be adjoint to Φ and we always have (Φ∗ )∗ = Φ (see Theorem 11.1 in Chapter III in [GK69]). For instance, if 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1, Φp (ξ) = ξ p , and Φq (ξ) = ξ q whenever ξ ∈ c, then (Φp )∗ = Φq . If Φ is any symmetric norming (0) function, then the topological dual of the Banach space SΦ is isometrically (0) isomorphic to SΦ∗ by the duality pairing SΦ∗ × SΦ → C, (T, S) → Tr (T S) (see Theorems 12.2 and 12.4 in Chapter III in [GK69]). The next deﬁnition describes the Boyd indices of a symmetric norming function. Some convenient references for this notion are Section 3 in [Ar78] and subsections 2.17–19 in [DFWW04]. Definition 2.5. Let Φ be a symmetric norming function. For each m ≥ 1 deﬁne the linear operators Dm : c→ c and D1/m : c→ c by m 2m

1 1 Dm ξ = (ξ1 , . . . , ξ1 , ξ2 , . . . , ξ2 , . . . ) and D1/m ξ = ξi , ξi , . . . m i=1 m i=m+1 m times

m times

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for arbitrary ξ = (ξ1 , ξ2 , . . . ) ∈ c. We shall think of Dm and D1/m as linear operators on the space c equipped with the norm Φ(·), so that it makes sense to speak about the norms of these operators. The Boyd indices of the symmetric norming function Φ are deﬁned by log m m≥1 log Dm

pΦ = sup

and

log(1/m) m≥1 log D1/m

qΦ = inf

and we have 1 ≤ pΦ ≤ qΦ ≤ ∞. We shall say that these indices are nontrivial if 1 < pΦ ≤ qΦ < ∞. Remark 2.6. The Boyd indices of any symmetric norming function Φ are related to the Boyd indices of its adjoint Φ∗ by the equations 1/pΦ + 1/qΦ∗ = 1/pΦ∗ + 1/qΦ = 1. If 1 < r < ∞ and Φ(·) = · r then for all m ≥ 1 we have Dm = m1/r and D1/m = m−1/r , hence in this case pΦ = qΦ = r. The Boyd indices of symmetric norming functions are important for our present purposes because of the following interpolation theorem. Theorem 2.7. Let Φ be a mononormalizing symmetric norming function and assume that 1 ≤ p < pΦ ≤ qΦ < q ≤ ∞. Then the following assertions hold: (a) We have Sp ⊆ SΦ ⊆ Sq . (b) There exists a constant MΦ > 0 with this property: if T : Sq → Sq is a bounded linear operator such that T (Sp ) ⊆ Sp , then T (SΦ ) ⊆ SΦ and T |SΦ ≤ MΦ max{ T |Sp , T |Sq }. Proof. See Corollary 3.4(i) in [Ar78].

Triangular integrals. The triangular integral is a suitable inﬁnite-dimensional version of the operation of taking the upper triangular part (the lower diagonal part, or the diagonal, respectively) of a square matrix. The classical reference is [GK70] (see also [Er78]). We are going to describe this idea in a setting which is slightly more general than the classical one. Definition 2.8. Let B be a unital associative involutive Banach algebra and I a contractive B-bimodule; that is, I is a Banach space equipped with an involutive isometric antilinear map I → I, X → X ∗ and a trilinear map B × I × B → I,

(b1 , X, b2 ) → b1 Xb2 ,

such that I is an algebraic B-bimodule and in addition (b1 Xb2 )∗ = b∗2 X ∗ b∗1 and b1 Xb2 ≤ b1 X b2 whenever b1 , b2 ∈ B and X ∈ I. Also let E be a totally ordered set of self-adjoint idempotent elements in B such that 0, 1 ∈ E, and denote by Part(E) the set of all partitions of E, that is, the ﬁnite families P = {pi }0≤i≤n in E such that 0 = p0 < p1 < · · · < pn = 1. For such a partition P of E we deﬁne the diagonal truncation operator DP : I → I,

DP (X) =

n i=1

(pi − pi−1 )X(pi − pi−1 ),

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the strictly upper triangular truncation operator UP : I → I,

UP (X) =

n

pi−1 X(pi − pi−1 ),

i=1

and the strictly lower triangular truncation operator LP : I → I,

LP (X) =

n

(pi − pi−1 )Xpi−1 .

i=1

Now think of Part(E) as a directed set with respect to the partial order deﬁned by P ≤ Q if and only if P is a subfamily of Q. If X ∈ I and the net {UP (X)}P∈Part(E) is convergent in I, then the corresponding limit is denoted by U(X) and is called the strictly upper triangular integral of X. One similarly deﬁnes the strictly lower triangular integral L(X) and the diagonal integral D(X) whenever they exist. Proposition 2.9. Let B be a unital associative involutive Banach algebra, I a contractive B-bimodule, and E a totally ordered set of self-adjoint idempotents in B such that 0, 1 ∈ E. Assume that the following conditions are satisﬁed. (i) Either the Banach space underlying I is reﬂexive or the integrals D and U exist on dense subsets of I. (ii) Both families of operators {DP }P∈Part(E) and {UP }P∈Part(E) are uniformly bounded. Then the following assertions hold. (a) The strictly upper triangular integral, the strictly lower triangular integral, and the diagonal integral exist throughout I and the corresponding mappings are bounded linear idempotent operators U, L, D : I → I. (b) There exists the direct sum decomposition I = Ran L Ran D Ran U, and the corresponding decomposition of an arbitrary element X ∈ I is X = L(X) + D(X) + U(X). In addition, L(X ∗ ) = U(X)∗ and D(X ∗ ) = D(X)∗ for all X ∈ I. Proof. If the integrals D and U are convergent on dense subsets of I, then they exist everywhere because of hypothesis (ii). Now let us assume that the Banach space underlying I is reﬂexive. It is easy to check that for all P ∈ Part(E) all of the operators LP , DP , and UP are idempotent, their mutual products are equal to 0, and their sum is the identity mapping on I. In addition, the nets {UP (X)}P∈Part(E) and {LP (X)}P∈Part(E) are increasing, while {DP (X)}P∈Part(E) is decreasing, in the sense that if P, Q ∈ Part(E) and P ≤ Q, then UP UQ = UQ UP = UP , LP LQ = LQ LP = LP , and also

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DP DQ = DQ DP = DQ . Now hypotheses (i) and (ii) ensure that Theorem 2.1 of [Er78] applies, whence assertion (a) follows. For assertion (b) note that X = LP (X) + DP (X) + UP (X) and then take the limit with respect to P ∈ Part(E), for each X ∈ I. For all P, Q ∈ Part(E) we have LP DP = 0, whence by taking the limit we get LD = 0. Similarly, all of the mutual products of the operators L, D, and U, are equal to 0, whence the asserted direct sum decomposition follows. To complete the proof note that for all X ∈ I and P ∈ Part(E) we have LP (X ∗ ) = UP (X)∗ and DP (X ∗ ) = DP (X)∗ , and then again take the limit with respect to the partition P of E. Example 2.10. (See Theorem 4.1 in [Ar78].) Let Φ be a mononormalizing symmetric norming function whose Boyd indices are nontrivial and denote I = SΦ ⊆ B(H). Denote by E the set of orthogonal projections associated with a maximal, totally ordered set of closed subspaces of H. Then the corresponding strictly upper triangular, the strictly lower triangular, and the diagonal integral exist throughout I, and they deﬁne bounded linear idempotent operators U, L, D : I → I whose mutual products are equal to 0 and U + L + D = 1. Indeed, condition (i) in Proposition 2.9 is satisﬁed since the integrals of ﬁnite-rank operators are clearly convergent. Also, it follows by Theorem 3.2 in [Er78] that the families of operators {DP }P∈Part(E) and {UP }P∈Part(E) are uniformly bounded on each Schatten ideal Sp if 1 < p < ∞. Now Theorem 2.7 shows that both these families are uniformly bounded on I = SΦ as well, since the Boyd indices of Φ are nontrivial. Thus condition (ii) in Proposition 2.9 is satisﬁed as well, and it then follows that the integrals U, D, and L are convergent throughout I. Triangular factorizations. Our next purpose is to survey some of the methods that allow one to ﬁnd operator theoretic versions of the well-known LU factorization from linear algebra, that is, the fact that every invertible matrix factorizes as the product of a unitary matrix and a triangular one. Definition 2.11. Let E be the set of orthogonal projections associated with a maximal, totally ordered set of closed linear subspaces of H. Then the nest algebra of E is Alg E = {b ∈ B(H) | (∀e ∈ E) be = ebe}, that is, the set of all operators which leave invariant each subspace in the family {e(H)}e∈E . Definition 2.12. Let SI and SII be norm ideals. We shall say that (SI , SII ) is a pair of associated norm ideals if the following conditions are satisﬁed: (i) SI is separable; (ii) SI ⊆ SII ; (iii) for every maximal, totally ordered set of closed linear subspaces of H and every X ∈ SI , the corresponding strictly upper triangular integral is convergent in the contractive B(H)-bimodule I = B(H) and U(X) ∈ SII .

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(See [GK70] and [Er72] for the original version of this deﬁnition.)

Example 2.13. Let Φ be a mononormalizing symmetric norming function whose Boyd indices are nontrivial, and denote SI = SII = SΦ . Then Example 2.10 shows that (SI , SII ) is a pair of associated norm ideals. As a special case, it follows by Remark 2.6 that each pair (Sp , Sp ) consisting of the p-th Schatten ideal and itself is a pair of associated norm ideals if 1 < p < ∞. Theorem 2.14. Let (SI , SII ) be a pair of associated norm ideals and consider the triangular and diagonal integrals on the contractive B(H)-bimodule I = B(H) with respect to the set E of orthogonal projections associated with some maximal, totally ordered set of closed linear subspaces in H. Also assume that 0 ≤ a ∈ GL(H). Then the following conditions are equivalent: (i) U(a−1 ) exists and U(a−1 ) ∈ SI . (ii) D(a−1 ) exists and a−1 − D(a−1 ) ∈ SI . If one of these conditions holds, then there exist uniquely determined elements r ∈ SII ∩ Alg E and d = d∗ ∈ Alg E such that a = (1 + r)d(1 + r∗ ) and the spectrum of r is equal to {0}. Proof. See Theorem 4.2 and Lemma 2.5(i) in [Er72].

Here are two corollaries concerning the group GLΦ (H) of Example 5.2 below. In connection with these corollaries as well as with Theorem 2.14 above it is worth noting that for every element r whose spectrum in some unital associative Banach algebra is equal to {0}, the element 1 + r is invertible in that Banach algebra. This remark applies in particular for elements in the nest algebra Alg E. Corollary 2.15. Let Φ be a mononormalizing symmetric norming function whose Boyd indices are nontrivial, and denote by E the set of orthogonal projections associated with a maximal, totally ordered set of closed linear subspaces of H. If 0 ≤ a ∈ GLΦ (H), then there exist uniquely determined operators d, r ∈ B(H) such that 0 ≤ d ∈ GLΦ (H) ∩ Alg E, r ∈ SΦ ∩ Alg E, the spectrum of r is equal to {0}, and a = (1 + r∗ )d(1 + r). Proof. See Theorem 2.14 and Example 2.13.

Corollary 2.16. Assume the setting of Corollary 2.15. Then for every g ∈ GLΦ (H) there exist the operators b, u ∈ GLΦ (H) such that b ∈ Alg E, u∗ u = 1, and g = ub. Proof. Just apply Corollary 2.15 for a = g ∗ g. See for instance the proof of Corollary A.2 in [Be09] for more details.

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3 Invariant means on groups Amenable groups. We shall brieﬂy discuss the invariant means on topological groups. They can be thought of as weak versions of Haar measures, although they have two main drawbacks: they may not be faithful, in the sense that the mean of a nonzero function with nonnegative values can be equal to zero; and not every Lie group admits an invariant mean. On the other hand, we shall see that many Banach–Lie groups do have invariant means; see for instance Remark 5.6. Classical references for amenability are [Pa88] and [Ey72]. See [Pe06], [Ga06], and [BP07] for some recent developments. Definition 3.1. Let G be a topological group. Consider the commutative unital C ∗ -algebra ∞ (G) = {ψ : G → C | ψ ∞ := supG |ψ(·)| < ∞} and its automorphisms Lx , Rx : ∞ (G) → ∞ (G), (Lx ψ)(y) = ψ(xy), and (Rx ψ)(y) = ψ(yx) whenever y ∈ G and ψ ∈ ∞ (G), deﬁned for arbitrary x ∈ G. The space of right uniformly continuous bounded functions on G is RUCb (G) = {ψ ∈ ∞ (G) | the map G → ∞ (G), x → Lx ψ, is continuous}. Similarly, the space of left uniformly continuous bounded functions on G is the set LUCb (G) of all functions ψ ∈ ∞ (G) such that the mapping G → ∞ (G), x → Rx ψ, is continuous. We say that the topological group G is amenable if there exists a linear functional µ : RUCb (G) → C satisfying the following conditions: (i) µ(1) = 1, (ii) 0 ≤ µ(ψ) if 0 ≤ ψ ∈ RUCb (G), and (iii) µ(Lx ψ) = µ(ψ) for all ψ ∈ RUCb (G) and x ∈ G. In this case the linear functional µ is called a left invariant mean on G.

Remark 3.2. If the topological group G is amenable, then every left invariant mean µ is continuous on RUCb (G) and µ = 1. On the other hand, the space RUCb (G) is a unital C ∗ -subalgebra of ∞ (G) which consists only of continuous functions, and it is invariant under the automorphisms Lx for all x ∈ G. Thus the left invariant means on G are precisely the states of the commutative unital C ∗ -algebra RUCb (G) which are invariant under the automorphism group deﬁned by the mappings Lx for arbitrary x ∈ G. Example 3.3. If the topological group G is either compact or abelian, then it is amenable. In fact, let C(G) denote the space of all continuous functions on G. If G is compact, then the probability Haar measure deﬁnes a linear functional µ : C(G) → C. Compactness of G implies that RUCb (G) = C(G), and then the basic properties of the Haar measure show that µ is a left invariant mean on G. On the other hand, if G is abelian, denote by Gd the group G endowed with the discrete topology. Then RUCb (Gd ) = ∞ (G), and it follows by (0.15) in [Pa88] that there exists a left invariant mean µ : ∞ (Gd ) → C on the discrete group Gd . Now the restriction of µ to RUCb (G) deﬁnes a left invariant mean on G.

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Example 3.4. Assume that G is a topological group such that there exists a directed system of amenable topological subgroups {Gα }α∈A whose union is dense in G. Then G is amenable. To see this, we shall say that a linear functional µ : RUCb (G) → C is a mean on G if µ(1) = 1 and 0 ≤ µ(ψ) whenever 0 ≤ ψ ∈ RUCb (G). In this case µ is continuous and µ = 1. Now for every α ∈ A denote Λα = {µ | µ is a mean on G and µ ◦ Lx |RU Cb (G) = µ if x ∈ Gα }, which is a w∗ -compact subset of the unit ball in the topological dual space (RUCb (G))∗ . In addition, Λα = ∅. In fact, any left invariant mean µα on Gα gives rise to an element µ α ∈ Λα deﬁned by µ α (ψ) = µα (ψ|Gα ) for all ψ ∈ RUCb (G). On the other hand, if α, β ∈ A and Gα ⊆ Gβ , then Λα ⊇ Λβ . Thus {Λα }α∈A is a family of w∗ -compact subsets of the unit ball in (RUCb (G))∗ with the property that the intersection of each ﬁnite subfamily is nonempty.

Therefore Λα = ∅, and any element µ in this nonempty intersection is a α∈A

left invariant mean on G. In fact, we already know that µ is a mean on G. To check that it is left invariant, let x ∈ G be arbitrary. Since the union of the family {Gα }α∈A is dense in G, there exists a net {xi }i∈I in that union such that lim xi = x. Then for every ψ ∈ RUCb (G) we have lim Lxi ψ = Lx ψ i∈I

i∈I

in ∞ (G); hence µ(Lx ψ) = lim µ(Lxi ψ) = lim µ(ψ) = µ(ψ), where the second i∈I i∈I

equality follows since µ ∈ Λα and xi ∈ Gα for all i ∈ I. α∈A

α∈A

Note that the above reasoning leads to the same conclusion under a more general hypothesis—namely that {Gα }α∈A is a directed system of abstract subgroups of G whose union is dense in G, such that each of these subgroups is endowed with a group topology that makes it an amenable topological group continuously embedded into G such that the inclusion map Gα → Gβ is continuous if α ≤ β in A. Example 3.5. Let G be a ﬁnite-dimensional Lie group with ﬁnitely many connected components. Denote by R the radical of G (i.e., the connected subgroup of G corresponding to the largest solvable ideal of the Lie algebra of G) and by Gd the group G endowed with the discrete topology. (a) The group G is amenable if and only if it has either of the following properties: (i) the group G/R is compact; (ii) there exists no closed subgroup of G isomorphic to the free group F2 with two generators. (b) The discrete group Gd is amenable if and only if either of the following conditions is satisﬁed: (j) the group G/R is ﬁnite (i.e., the connected component of 1 ∈ G is a solvable Lie group); (jj) there exists no subgroup of Gd isomorphic to F2 . We refer to Theorems (3.8) and (3.9) in [Pa88] for proofs of these facts.

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→ G is a surjective homomorphism of topologRemark 3.6. Assume that φ : G is amenable ical groups such that Ker φ is an abelian group. Then the group G if and only if G is amenable. In fact, it follows by Example 3.3 that the topo is amenable, and then the assertion follows for logical subgroup Ker φ of G ◦ instance by remark 2 in Section 3 of Expos´e no .1 of [Ey72]. and every continuWe note that for every amenable topological group G ous homomorphism φ : G → G with dense range it follows that G is amenable → C is a left without any extra condition on Ker φ. In fact, if µ : RUCb (G) then it is straightforward to check that the corresponinvariant mean on G, dence f → µ (f ◦ φ) deﬁnes a left invariant mean on G. Mimicking the group algebras of compact groups Definition 3.7. Let G be a topological group and consider the duality pairing ·, · : (RUCb (G))∗ × RUCb (G) → C. There exists a bounded bilinear map (RUCb (G))∗ × RUCb (G) → RUCb (G),

(µ, ψ) → µ · ψ

deﬁned by (µ · ψ)(x) = µ, Lx ψ for all x ∈ G (by (2.11) in [Pa88]). The Arens-type product on (RUCb (G))∗ is the bounded bilinear mapping (RUCb (G))∗ × (RUCb (G))∗ → (RUCb (G))∗ ,

(µ, ν) → µ · ν

deﬁned by µ · ν, ψ := µ, ν · ψ for all ψ ∈ RUCb (G). Similarly, by using the duality pairing ·, · : (LUCb (G))∗ × LUCb (G) → C, one deﬁnes a bounded bilinear map (LUCb (G))∗ × LUCb (G) → LUCb (G),

(µ, ψ) → µ · ψ

by (µ · ψ)(x) = µ, Rx ψ for all x ∈ G (by the version of (2.11) in [Pa88] for left uniformly continuous functions). The Arens-type product on (LUCb (G))∗ is the bounded bilinear mapping (LUCb (G))∗ × (LUCb (G))∗ → (LUCb (G))∗ ,

(µ, ν) → µ · ν

deﬁned by µ · ν, ψ := ν, µ · ψ for all ψ ∈ LUCb (G).

Remark 3.8. Let G be a topological group, consider the space Cb (G) consisting of the continuous elements of ∞ (G), and deﬁne σ : Cb (G) → Cb (G), (σ(ψ))(x) = ψ(x−1 ) whenever x ∈ G and ψ ∈ Cb (G). The mapping σ is an antilinear isometric ∗-endomorphism of the unital C ∗ -algebra Cb (G) which satisﬁes σ 2 = idCb (G) and has the following additional properties: (1) For every x ∈ G we have Lx ◦ σ = σ ◦ Rx−1 . (2) We have σ(RUCb (G)) = LUCb (G) and σ(LUCb (G)) = RUCb (G).

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(3) If we deﬁne Σ : (RUCb (G))∗ → (LUCb (G))∗ as the anti-dual map of σ : LUCb (G) → RUCb (G), that is, Σ(µ), φ = µ, σ(φ) for all µ ∈ (RUCb (G))∗ and φ ∈ LUCb (G), then the diagram (RUCb (G))∗ × RUCb (G) −−−−→ RUCb (G) ⏐ ⏐ ⏐ ⏐σ Σ×σ (LUCb (G))∗ × LUCb (G) −−−−→ LUCb (G) is commutative,where the horizontal arrows are the maps introduced in Deﬁnition 3.7. (4) The mapping Σ : (RUCb (G))∗ → (LUCb (G))∗ is antilinear, isometric, bijective, and Σ(µ1 · µ2 ) = Σ(µ2 ) · Σ(µ1 ) for all µ1 , µ2 ∈ (RUCb (G))∗ . In fact, (1) follows by a straightforward computation, and it implies property (2) at once. For property (3) note that for all µ ∈ (RUCb (G))∗ and ψ ∈ RUCb (G) we have (Σ(µ) · σ(ψ))(x) = Σ(µ), Rx (σ(ψ)) = Σ(µ), σ(Lx−1 (ψ)) = µ, σ 2 (Lx−1 (ψ)) = µ, Lx−1 (ψ) = σ(µ · ψ)(x) whenever x ∈ G. To check property (4), compute (3)

Σ(µ2 ) · Σ(µ1 ), ϕ = Σ(µ1 ), Σ(µ2 ) · ϕ = Σ(µ1 ), σ(µ2 · σ(ϕ)) = µ1 , µ2 · σ(ϕ) = µ1 · µ2 , σ(ϕ) = Σ(µ1 · µ2 ), ϕ

for every ϕ ∈ LUCb (G).

The facts described in the following theorem can be found in Theorems 2.5 and 4.1 of [Gr]. See the paper [FN07] and the references therein for another line of research on these Banach algebras. Theorem 3.9. Every topological group G has the following properties. (a) The Arens-type products make (RUCb (G))∗ and (LUCb (G))∗ into associative Banach algebras. (b) Each continuous unitary representation π : G → B(Hπ ) gives rise to two representations of Banach algebras, π R : (RUCb (G))∗ → B(Hπ ) and π L : (LUCb (G))∗ → B(Hπ ), by means of the formulas πL (ν)ξ | η) = ν, (π(·)ξ | η) ( πR (µ)ξ | η) = µ, (π(·)ξ | η) and ( ∗

(1)

∗

for all ξ, η ∈ H, µ ∈ (RUCb (G)) , and ν ∈ (LUCb (G)) . These representations are related by the commutative diagram π

(RUCb (G))∗ −−−R−→ B(Hπ ) ⏐ ⏐ ⏐ ⏐ Σ S π

(LUCb (G))∗ −−−L−→ B(Hπ ) where S : B(Hπ ) → B(Hπ ), b → b∗ .

(2)

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Proof. Assertion (a) follows at once by (2.8) and (the left-sided version of) (2.11) in [Pa88]. See also Example (19.23)(b) in [HR63] and [Bu50]. For assertion (b), ﬁrstly note that the matrix coeﬃcients ψξ,η = (π(·)ξ | η) belong to the function space RUCb (G) ∩ LUCb (G) for arbitrary ξ, η ∈ Hπ . To see this, just note that for all x ∈ G we have Lx (ψξ,η ) = ψξ,π(x)∗ η and Rx (ψξ,η ) = ψπ(x)ξ,η , and then use the continuity of the representation π. Thus the right-hand sides of both equalities in (1) make sense, and then by means of the estimate ψξ,η ∞ ≤ ξ η we get π R (µ), π L (ν) ∈ B(Hπ ) and πR (µ) ≤ µ and πL (ν) ≤ ν . In addition, since σ(ψξ,η ) = ψη,ξ , we get πR (µ)η | ξ) = ( πL (Σ(µ))ξ | η) = Σ(µ), ψξ,η = µ, σ(ψξ,η ) = µ, ψη,ξ = ( ( πR (µ)∗ ξ | η), whence π L (Σ(µ)) = π R (µ)∗ , and thus the diagram (2) is commutative. To conclude the proof of (b), let µ1 , µ2 ∈ (RUCb (G))∗ . Then ( πR (µ1 · µ2 )ξ | η) = µ1 · µ2 , ψξ,η = µ1 , µ2 · ψξ,η = µ1 , ψπR (µ2 )ξ,η = ( πR (µ1 ) πR (µ2 )ξ | η), where the next-to-last equality holds since for every x ∈ G we have (µ2 · ψξ,η )(x) = µ2 , Lx (ψξ,η ) = µ2 , ψξ,π(x)∗ η = ( πR (µ2 )ξ | π(x)∗ η) = (π(x) πR (µ2 )ξ | η) = ψπR (µ2 )ξ,η (x). Thus π R is an algebra representation. Similarly, for ν1 , ν2 ∈ (LUCb (G))∗ we can check that ν1 · ψξ,η = ψξ,πL (ν1 )∗ η , whence ( πL (ν1 · ν2 )ξ | η) = ( πL (ν1 ) πL (ν2 )ξ | η) just as above, and the proof ends. Remark 3.10. In the setting of Theorem 3.9, RUCb (G) and LUCb (G) are commutative unital isomorphic C ∗ -algebras, hence there exist ∗-isomorphisms RUCb (G) C(M0 (G)) LUCb (G), where M0 (G) is a compact topological space. See the main theorem of [Tu95] for a description of this compact space (called the universal ambit of G) when G is the additive group (R, +) with the usual topology. In the special case when the group G is compact, we have M0 (G) = G, C(G) = RUCb (G) = LUCb (G), and the involutive Banach algebra (RUCb (G))∗ is the convolution measure algebra of G. In the general case, the topological duals of RUCb (G) and LUCb (G) still consist of the (not necessarily positive) measures on the compact space M0 (G); however, the Arens-type products may not be deﬁned by convolution formulas by (cf. the comment preceding Proposition (2.25) in [Pa88]) reasonable convolution formulas involving M0 (G). See however the general theory of convolutions of functionals developed in Chapter 5 of [HR63] or the other references mentioned in connection with the proof of Theorem 3.9(a) above. Another problem to be dealt with is to ﬁnd a method to distinguish, in the set of pairs of representations of the Banach algebras (RUCb (G))∗ and (LUCb (G))∗ that make the diagram (2) commutative, the ones that come from unitary representations of G by means of (1). In this connection, let us recall that there exists a promising approach to an axiomatic theory of group algebras by means of the host algebras, that is, C ∗ -algebras whose representations correspond in a one-to-one fashion with the unitary representations of a given group; see [Gr05], [GN09], and the references therein.

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Remark 3.11. Assume that G is an amenable topological group and pick a left invariant mean µ : RUCb (G) → C. Then µ is a state of the C ∗ -algebra RUCb (G), and the corresponding Gelfand–Naimark–Segal construction leads to a cyclic ∗-representation ιµ : RUCb (G) → B(H(µ) ). Recall that the Hilbert space H(µ) is obtained out of RUCb (G) as a quotient followed by a completion with respect to the nonnegative deﬁnite, sesquilinear form RUCb (G) × RUCb (G) → C, (ψ, χ) → µ(ψχ∗ ). For each x ∈ G we have µ ◦ Lx−1 |RU Cb (G) = µ; hence the mapping RUCb (G) → RUCb (G), ψ → Lx−1 ψ induces a unitary representation λµ : G → B(H(µ) ), which is easily seen to be continuous. In the special case when G is compact and µ is the probability Haar measure on G, we have H(µ) = L2 (G, µ) and λµ is the regular representation of G. For this reason, in the general case of an amenable group G, one can think of λµ as a regular representation associated with the left invariant mean µ. We note that the universal ambit M0 (G) mentioned in Remark 3.10 carries both a continuous action of G (see Sections 2.1 and 2.3 in [Pe06]) and a probability Borel measure µ induced by mean µ; this measure is invariant under the action of G, and the corresponding unitary representation of G on L2 (M0 (G), µ) is unitarily equivalent to the representation λµ . We have to point out that it may happen that dim H(µ) = 1. For instance, this is the case if G is an extremely amenable group and the left invariant mean µ is chosen to be multiplicative. (See [Pe06] for speciﬁc examples and for details on the latter notions.) It may also happen that the regular representation is trivial, in the sense that λµ (x) = 1 for all x ∈ G. This is the case for the exotic Banach–Lie groups (see Theorem 1 in [Ba83]), which are abelian topological groups that do not have any non-trivial continuous representation. It is not diﬃcult to verify that the regular representation λµ is trivial if and only if µ(χψ) = µ(χLx ψ) for all χ, ψ ∈ RUCb (G) and x ∈ G.

4 Lifting group decompositions to covering groups This section has a technical character, and its main purpose is to provide tools for enriching the class of reductive Banach–Lie groups to be set forth in the next section. In the proof of the following statements we use some ideas from the proofs of Theorem 6.31 and 6.46 in [Kn96]. are Banach–Lie groups, and e : G → G is Lemma 4.1. Assume that G and G a covering homomorphism. Let K be any Banach–Lie subgroup of G, denote K → G/K, → e( := e−1 (K), and deﬁne ψ : G/ gK g )K. Then the following K assertions hold. is a Banach–Lie subgroup of G and the mapping e| : K → K is a (i) K K covering homomorphism. (ii) The mapping ψ is a well-deﬁned diﬀeomorphism. is connected and the smooth homogeneous space G/K is (iii) If the group G and K are connected. simply connected, then both K

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→ L(G) is an isomorphism of Banach–Lie alProof. Since L(e) = T1 e : L(G) is a Banach–Lie subgroup gebras, it follows by Proposition 4.8 in [Be06] that K → T1 K. Then Remark C.13(b) in of G and the tangent map L(e)|T1 K : T1 K [Be06] shows that e|K : K → K is a covering map. This completes the proof of assertion (i). = e−1 (K), and ψ is To prove assertion (ii), note that ψ is injective since K surjective because e is surjective. On the other hand, we have a commutative diagram −−−e−→ G G ⏐ ⏐ ⏐ ⏐ ψ K −−− G/ −→ G/K

where the vertical arrows (which are the quotient maps) are submersions. Since the covering map e is a local diﬀeomorphism, it follows from this commutative diagram that ψ is a local diﬀeomorphism as well. Then ψ is actually a diﬀeomorphism since we have already seen that it is bijective. is a covering group of K, so it For assertion (iii), recall from (i) that K will be enough to show that K is connected. And the latter property follows from the long exact sequence of homotopy groups ← π0 (K) ← π1 (G/ K) ← π1 (G) ← π1 (K) ← ··· π0 (G) = {0} by the assumption on G, while π1 (G/ K) = {0} by the since π0 (G) K assumption on G/K along with the fact that G/K is homeomorphic to G/ according to assertion (ii). We now come to a proposition to the eﬀect that the Cartan decompositions lift to the covering groups. be two Banach–Lie groups, and assume that Proposition 4.2. Let G and G → G is a covering homomorphism. Now let K be any Banach–Lie sube: G := e−1 (K). Denote L(G) = g and L(K) = k, and group of G and denote K assume that p is a closed linear subspace of g such that g = k p and the mapping ϕ : K × p → G, (k, X) → k · expG X is a diﬀeomorphism. Moreover, × ( → denote p := L(e)−1 (p). Then the mapping ϕ : K p → G, k, X) k · expG X is a diﬀeomorphism as well. is a Banach–Lie subgroup of G by Lemma 4.1, Proof. First note that K → L(G) is an isomorphism of Banach–Lie algebras, so that and L(e) : L(G) L(e) k = k. Now note that there exists a commutative diagram ϕ × K p −−−−→ ⏐ e|K ×L(e)|p ⏐ ϕ

G ⏐ ⏐e

K × p −−−−→ G

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whose vertical arrows are covering maps (see Lemma 4.1(i)). Since ϕ is a diffeomorphism by assumption, it then follows that ϕ is a local diﬀeomorphism. To get the wished-for conclusion, we still have to prove that ϕ is a bijective map. j and X j ∈ To check that ϕ is injective, let kj ∈ K pj for j = 1, 2 such that k1 · expG X1 = k2 · expG X2 . By applying the map e to both sides of the latter equality, and using the commutation relation between the exponential maps and group homomorphisms (see, e.g., Remark 2.34 in [Be06]), we get 1 ) = e( 2 ). Since ϕ is injective, it then e( k1 ) · expG (L(e)X k2 ) · expG (L(e)X k2 ). The ﬁrst of these equalities follows that L(e)X1 = L(e)X2 and e(k1 ) = e( 1 = X 2 , whence j and implies that X k1 = k2 by the assumption on kj ∈ K j ∈ arbitrary. X pj for j = 1, 2. Now, to prove that ϕ is surjective, let g∈G Then e( g ) ∈ G, hence there exist k ∈ K and X ∈ p such that e( g ) = k·expG X := since ϕ is surjective. Further on, pick k0 ∈ e−1 (k) arbitrary and denote X −1 L(e) X. Then e( g) = e(k0 ) · expG (L(e)X) = e(k0 · expG X), so that by Thus · denoting z := k0 ·expG X k := z −1 g −1 we have z ∈ e−1 (1) ⊆ K. k0 ∈ K which concludes the proof. and we have g = k · expG X, The next proposition shows that the familiar integration of local Cartan involutions to global Cartan involutions carries over to the setting of Banach– Lie groups. Proposition 4.3. Let G be a connected Banach–Lie group with the Lie algebra L(G) = g, K a Banach–Lie subgroup of G, and assume that there exists an automorphism θ ∈ Aut (g) such that θ2 = idg , L(K) = Ker (θ − idg ), and the mapping ϕ : K × p → G, (k, X) → k · expG X is a diﬀeomorphism. Then there exists a unique automorphism Θ ∈ Aut (G) such that L(Θ) = θ and K = {g ∈ G | Θ(g) = g}. → G be the universal covering of G, and denote k = L(K), Proof. Let e : G := e−1 (K), p := Ker (θ + idg ), K g := L(G), p := L(e)−1 (p), and θ := −1 L(e) ◦ θ ◦ L(e) ∈ Aut ( g). Then Proposition 4.2 shows that the mapping is a diﬀeomorphism. ϕ : K × p → G, (k, X) → k · expG X is connected and simply connected, On the other hand, since the group G : G →G such it follows that there exists a unique smooth homomorphism Θ 2 that L(Θ) = θ. (See for instance Remark 3.13 in [Be06].) Since θ = idg , it 2 = id , and in particular Θ ∈ Aut (G). then follows that Θ G Now we use hypothesis (iii) to see that the mapping τ : p → G/K, X → (expG X)−1 K is a diﬀeomorphism. In fact, it is clear that this is a smooth map, and its inverse is the smooth well-deﬁned map G/K → p, gK → prp (g −1 ), where prp : G → p is the projection onto p deﬁned by means of the diﬀeomorphism ϕ−1 : G → K × p. Thus τ : p → G/K is a diﬀeomorphism, and in particular G/K is simply connected since p is. Then Lemma 4.1 are connected. Since θ| = id and K is connected, it shows that both K and K k k

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= id . In particular, the subgroup e−1 (1) is invariant under follows that Θ| K K −1 (since e (1) ⊆ K). Then there exists a unique automorphism Θ ∈ Aut (G) Θ it follows that Θ2 = Θ. In addition, since Θ 2 = Θ, such that Θ ◦ e = e ◦ Θ. It remains to prove that K = {g ∈ G | Θ(g) = g}. The inclusion ⊆ is clear = id and K = K/e −1 (1). Conversely, let g ∈ G such that Θ(g) = since Θ| K K g. By hypothesis (iii), there exist k ∈ K and X ∈ p such that g = k · expG X. Then k · expG X = g = Θ(g) = Θ(k) · expG (θ(X)) = k · expG (−X), so that expG (2X) = 1. Thus ϕ(1, 2X) = ϕ(1, 0), and then X = 0 since ϕ : K × p → G is injective. Consequently, g = k ∈ K, and the proof ends. We now come to a proposition that allows us to lift the global Iwasawa decompositions to covering groups. Proposition 4.4. Let G be a connected Banach–Lie group, and K, A, and N be connected Banach–Lie subgroups of G such that the multiplication map m : K × A × N → G is a diﬀeomorphism. In addition, assume that A and N are simply connected and AN = N A. is a connected Banach–Lie group with a covering hoNow assume that G → G, and deﬁne K := e−1 (K) and X the connected commomorphism e : G −1 are connected ponent of 1 ∈ e (X) for X = A or X = N . Then K, A, and N ×N →G : K ×A Banach–Lie subgroups of G, and the multiplication map m is a diﬀeomorphism. A, and N are Banach–Lie subgroups Proof. It follows by Lemma 4.1(i) that K, → X is a covering map when X is either of the groups K, of G and e|X : X A, N , or G. Since the groups A and N are simply connected, it then follows → X is actually a diﬀeomorphism if X = A or X = N that the map e|X : X and N (see for instance Theorem C.18 in [Be06]). In particular, the groups A are connected and simply connected. is connected as well, we ﬁrst show that G/K is homeTo prove that K omorphic to the simply connected group B := AN A × N , where the diﬀeomorphism A × N → B is deﬁned by the multiplication in G, as an easy consequence of the assumption. Now let prK : G → K and prB : G → B be the smooth projections given by the inverse of the diﬀeomorphism K × B → G. Then the continuous map τ : B → G/K, b → b−1 K is bijective, and its inverse is also continuous since it is given by τ −1 : G/K → B, gK → prB (g −1 ). Thus τ : B → G/K is a homeomorphism. By taking into account the homeomor K → G/K provided by Lemma 4.1(ii), it then follows that the phism ψ : G/ K is simply connected, since B ( A × N ) is simply connected. quotient G/ is connected. Then Lemma 4.1(iii) shows that K × A× N → G. :K We now prove the assertion on the multiplication map m Using locally deﬁned inverses of the covering map e : G → G, we see that m ×A × N. It remains to prove is a local diﬀeomorphism at every point of K arbitrary. Since is bijective. To prove that m is surjective, let g ∈ G that m m : K×A×N → G is surjective, there exist k ∈ K, a ∈ A, and n ∈ N such that

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e( g ) = kan. Denote a := (e|A)−1 (a) ∈ A and n := (e|N )−1 (n) ∈ N , and take −1 k0 ∈ e (k) arbitrary, so that e( k0 an ) = kan = e( g ). Denoting z := k0 an g −1 , Then and k := z −1 k0 ∈ K we have e( z ) = 1, so z ∈ e−1 (1) ⊆ e−1 (K) = K. : K ×A×N → G g = k an ∈ K AN . Since g ∈ G is arbitrary, it follows that m is surjective. we have is injective, ﬁrst note that for any To prove that m a ∈ A −1 −1 −1 e( aN a ) = e( a)N e( a) ⊆ N , since AN = N A. Thus aN a ⊆ e−1 (N ). −1 The ﬁrst of these sets is connected and contains 1 ∈ e (N ), hence actu . Consequently, A N =N A =: B is a subgroup of G. Since a−1 ⊆ N ally aN → X is a bijection if X = A or X = N and A ∩ N = {1}, we get e|X : X N = {1} as well. Now assume that and n for j = 1, 2, A∩ kj ∈ K, aj ∈ A, j ∈ N −1 −1 −1 ∩ B, so that it will and k1 k1 = a1 n 1 = k2 a2 n 2 . Then k2 a2 n 2 ( a1 n 1 ) ∈ K ∩B = {1}, that is, K ∩A N = {1}. To prove the be enough to check that K ∩A N arbitrary. Then e( latter equality, let x ∈K x) ∈ K ∩ AN = {1}. On and n , and e( the other hand, we have x = an for some a∈A ∈N a)e( n) = 1. Since A ∩ N = {1}, it follows that e( a) = e( n) = 1. Using the fact that → X is a bijective homomorphism if X = A or X = N , we get e|X : X ∩A N = {1}, and this completes the a=n = 1, whence x = an = 1. Thus K ×A ×N →G is bijective. :K proof of the fact that the map m

5 What a reductive Banach–Lie group could be Reductivity relative to a symmetric norming function. We are going to describe a class of connected Banach–Lie groups that is wide enough to include all the ﬁnite-dimensional connected reductive Lie groups of Deﬁniton 1.1 and the connected components of the identity in most of the classical Banach–Lie groups of operators (see [dlH72]). This is an attempt to provide an appropriate framework for the interesting developments from representation theory of these inﬁnite-dimensional Lie groups; see for instance the works [Bo80], [Ne98], or [NØ98]. We chose to single out a class of connected Banach– Lie groups that does not include certain interesting groups, like the restricted groups or unitary groups of operator algebras, because of certain unpleasant properties that they possess; see Section 6 in [BRT07] and Proposition 1.1 in [Be09]. As mentioned above, the following deﬁnition is suggested by Remarks 1.2 and 1.3. Definition 5.1. Let Φ be a symmetric norming function. By Φ-reductive Banach–Lie algebra we mean any closed real Lie subalgebra of SΦ satisfying the following conditions. (i) For every X ∈ g we have X ∗ ∈ g. (ii) The set g ∩ F of ﬁnite-rank operators in g is dense in g with respect to the (0) norm · Φ . Thus g ⊆ SΦ actually.

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(0)

In this case the connected Banach–Lie group G (⊆ 1 + SΦ ) corresponding to (0) the Lie subalgebra g of SΦ (H) is said to be a Φ-reductive linear Banach–Lie group. By Φ-reductive Banach–Lie group we shall mean any covering group of a Φ-reductive linear Banach–Lie group. In the above setting, the closure of g ∩ F with respect to the Hilbert– Schmidt norm · 2 will be called the L∗ -algebra associated with g and will be denoted by g2 . Example 5.2. (See [dlH72] and also Deﬁnitions 3.2 and 3.2 in [Be09].) Let Φ be any symmetric norming function and denote by GL(H) the group of all invertible bounded linear operators on the complex Hilbert space H. The classical complex Banach–Lie groups and Banach–Lie algebras associated with Φ are deﬁned as follows: (0)

(0)

(A) GLΦ (H) = GL(H)∩(1+SΦ (H)) with the Lie algebra glΦ (H) = SΦ (H); (B) OΦ (H) = {g ∈ GLΦ (H) | g −1 = Jg ∗ J −1 } (where J is a conjugation, that is, an antilinear isometry satisfying J 2 = 1) with the Lie algebra (0) oΦ (H) = {x ∈ SΦ (H) | x = −Jx∗ J −1 }; ∗ J −1 } (where J an anti-conjugation, (C) SpΦ (H) = {g ∈ GLΦ (H) | g −1 = Jg i.e., an antilinear isometry satisfying J 2 = −1), with the Lie algebra (0) ∗ J −1 }. spΦ (H) = {x ∈ SΦ (H) | x = −Jx The classical real Banach–Lie groups and Banach–Lie algebras associated with the symmetric norming function Φ are the following: (AI) GLΦ (H; R) = {g ∈ GLΦ (H) | gJ = Jg} with the Lie algebra (0) glΦ (H; R) = {x ∈ SΦ (H) | xJ = Jx}, where J : H → H is a conjugation; with the Lie algebra (AII) GLΦ (H; H) = {g ∈ GLΦ (H) | g J = Jg} (0) glΦ (H; H) = {x ∈ SΦ (H) | xJ = Jx}, where J : H → H is an anticonjugation; (AIII) UΦ (H+ , H− ) = {g ∈ GLΦ (H) | g ∗ V g = V } with the Lie algebra (0) uΦ (H ) = {x ∈ SΦ (H) | x∗ V = −V x}, where H = H+ ⊕ H− and + , H− 1 0 with respect to this orthogonal decomposition of H; V = 0 −1 (BI) OΦ (H+ , H− ) = {g ∈ GLΦ (H) | g −1 = Jg ∗ J −1 and g ∗ V g = V } with (0) ∗ −1 oΦ (H+ , H− ) = {x ∈ S and x∗ V = −V x}, where | x = −Jx J Φ (H) 1 0 with respect to this decomposition of H, H = H+ ⊕ H− , V = 0 −1 and J : H → H is a conjugation such that J(H± ) ⊆ H± ; with the Lie (BII) O∗Φ (H) = {g ∈ GLΦ (H) | g −1 = Jg ∗ J −1 and g J = Jg} (0) algebra o∗Φ (H) = {x ∈ SΦ (H) | x = −Jx∗ J −1 and xJ = Jx}, where J : H → H is a conjugation and J : H → H is an anti-conjugation such that J J = JJ;

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∗ J −1 and gJ = Jg} with the (CI) SpΦ (H; R) = {g ∈ GLΦ (H) | g −1 = Jg (0) ∗ J −1 and xJ = Lie algebra spΦ (H; R) = {x ∈ SΦ (H) | −x = Jx Jx}, where J : H → H is any anti-conjugation and J : H → H is any conjugation such that J J = JJ; ∗ J −1 and g ∗ V g = V } with (CII) SpΦ (H+ , H− ) = {g ∈ GLΦ (H) | g −1 = Jg (0) ∗ J −1 and x∗ V = −V x}, spΦ (H+ , H− ) = {x ∈ SΦ (H) |x = −Jx 1 0 with respect to this decomposition where H = H+ ⊕ H− , V = 0 −1 ± ) ⊆ H± . of H, and J : H → H is an anti-conjugation such that J(H As a by-product of the classiﬁcation of the L∗ -algebras (see for instance Theorems 7.18 and 7.19 in [Be06]), every (real or complex) topologically simple L∗ -algebra is isomorphic to one of the classical Banach–Lie algebras associated with the Hilbert–Schmidt ideal S2 (H). We refer to [dlH72] and [Ne02a] for information on the homotopy groups of the classical Banach–Lie groups associated with the Schatten ideals. The corresponding description of homotopy groups actually holds true for the classical Banach–Lie groups associated with any symmetric norming function Φ. It is is clear that the connected 1-component of any classical Banach–Lie group associated with Φ is a Φ-reductive linear Banach–Lie group. It would be interesting to understand how the classical Lie algebras associated with an operator ideal ﬁt in the framework of Φ-reductive Lie algebras. Here is a proposition in this connection. Recall that a subset A ⊆ B(H) is irreducible if {0} and H are the only closed linear subspaces of H which are invariant under all operators in A. Proposition 5.3. Let Φ be a symmetric norming function, g a Φ-reductive Lie algebra, and g2 the L∗ -algebra associated with g. Then the Lie algebra g is irreducible if and only if g2 is irreducible. In addition, if SΦ = S1 , then g is one of the classical (real or complex) Banach–Lie algebras associated with the norm ideal SΦ if and only if g2 is one of the classical (real or complex) Banach–Lie algebras associated with the Hilbert–Schmidt ideal S2 . If this is the case, then g and g2 are classical Lie algebras of the same type. Proof. Both algebras g and g2 are closed under taking the adjoints, hence any of them is irreducible if and only if every operator that commutes with that algebra is a scalar multiple of the identity operator on H. Therefore it will be enough to prove the following assertion: (∀T ∈ B(H))

[T, g] = {0} ⇐⇒ [T, g2 ] = {0}.

(3)

In fact, if [T, g] = {0}, then in particular [T, g ∩ F] = {0}. Since the set of ﬁnite-rank operators g ∩ F is dense in g2 , it follows that [T, g2 ] = {0}. Conversely, if the operator T satisﬁes the latter condition, then we have in particular [T, g ∩ F] = {0}. By property (ii) of a Φ-reductive Lie algebra, it then follows that [T, g] = {0}.

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Now assume SΦ = S1 and let g be the closure of g2 ∩ F with respect to the norm · Φ . Since g ∩ F ⊆ g2 ∩ F and g ∩ F is dense in g, it then follows that g ⊆ g. We are going to prove that g is actually a closed ideal of the Lie algebra g. For this purpose it will be enough to show that [g ∩ F, g2 ∩ F] ⊆ g ∩ F. In fact, let X ∈ g ∩ F and Y ∈ g2 ∩ F arbitrary. According to the deﬁnition of g2 , there exists a sequence {Xj }j≥1 in g ∩ F such that Xj − Y 2 → 0 as j → ∞. Then [X, Xj ] − [X, Y ] 2 → 0 as j → ∞. On the other hand, the commutators {[X, Xj ]}j≥1 have the ranks uniformly bounded by 2(rank X) since the dimension of linear spaces decreases under linear transformations, hence the rank of a product of operators is less than the rank of any of the factors. Now Lemma 4.3 in [BR05] implies that in the topology of · Φ we also have [X, Xj ] → [X, Y ] as j → ∞, whence [X, Y ] ∈ g ∩ F. Now it is easy to see that the assertion holds. In fact, if g2 is one of the classical (real or complex) Banach–Lie algebras associated with the Hilbert– Schmidt ideal S2 , then g is the similar classical Banach–Lie algebra associated with the norm ideal SΦ . Since SΦ = S1 , it follows as in Proposition 8 on page 92 in [dlH72] that g has no non-trivial closed ideals, whence g = g. Conversely, if g is one of the classical (real or complex) Banach–Lie algebras associated with the norm ideal SΦ , then it is obvious that the L∗ -algebra associated with g is one of the classical L∗ -algebras. Group decompositions. The following theorem supplies a Cartan decomposition for some Φ-reductive Banach–Lie groups. In this connection, we refer to [Ne02b] for a systematic investigation of polar decompositions in inﬁnite dimensions (that is, Cartan decompositions for linear groups). Theorem 5.4. Assume that Φ is a symmetric norming function and G is a Φ-reductive Banach–Lie group whose Lie algebra g ⊆ SΦ (H) is one of the classical Lie algebras associated with Φ. Denote k = {X ∈ g | X ∗ = −X}, p = {X ∈ g | X ∗ = X}, and let K be the connected Banach–Lie group corresponding to the Lie subalgebra k of g. Then K is a Banach–Lie subgroup of G and the mapping ϕ : K × p → G, (k, X) → k expG X, is a diﬀeomorphism. Moreover, there exists a unique automorphism Θ ∈ Aut (G) such that L(Θ)X = −X ∗ for all X ∈ g and K = {g ∈ G | Θ(g) = g}. Proof. It follows by Proposition 4.2 that we may assume that G is a Φ(0) reductive linear Banach–Lie group. Thus let G ⊆ 1 + SΦ (H) be the connected 1-component of the classical Banach–Lie group associated with the classical Lie algebra g ⊆ SΦ (H). In this case the assertion can be proved by a method similar to the one used in Proposition III.8 of [Ne02a] in the case of the classical groups associated with the Schatten ideals. The existence of the automorphism Θ ∈ Aut (G) as asserted follows by an application of Proposition 4.3. The following theorem states that there exist Iwasawa decompositions for the Φ-reductive Banach–Lie groups corresponding to the classical Lie algebras,

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provided that the symmetric norming function Φ satisﬁes some reasonable conditions. Theorem 5.5. Let Φ be a mononormalizing symmetric norming function whose Boyd indices are nontrivial and let G be a Φ-reductive Banach–Lie group whose Lie algebra g ⊆ SΦ (H) is one of the classical Lie algebras associated with Φ. Denote k = {X ∈ g | X ∗ = −X}, p = {X ∈ g | X ∗ = X}, and let K be the connected Banach–Lie group corresponding to the Lie subalgebra k of g. Then there exists X0 ∈ p with the following properties. (a) The set EX0 of spectral projections of the self-adjoint operator X0 corresponding to the intervals of the form (0, t] with t ∈ R determines a direct sum decomposition g = k aX0 nX0 , where aX0 = {X ∈ p | [X, X0 ] = 0} and nX0 = {X ∈ g ∩ Alg (EX0 ) | the spectrum of X is equal to {0}}. (b) If we denote by A and N the connected Banach–Lie groups corresponding to the closed subalgebras aX0 and nX0 of g, then the multiplication map m : K × A × N → G is a diﬀeomorphism. Moreover, A and N are simply connected Banach–Lie subgroups of G and AN = N A. Proof. Proposition 4.4 allows us to assume that G is actually a Φ-reductive linear Banach–Lie group. The idea of the proof in this case is to start by studying the group G = GLΦ (H), which is the largest classical group. For this group, the construction of a local Iwasawa decomposition of its Lie algebra (assertion (a)) relies on local spectral theory (see [BS01]) along with the properties of triangular integrals established in Example 2.10. As regards the corresponding global Iwasawa decomposition (assertion (b)), one uses Corollary 2.16. See Proposition 4.3 and Theorem 4.5 in [Be09] for details of the proof in the case G = GLΦ (H). The construction of Iwasawa decompositions for all types of Φ-reductive linear Banach–Lie groups is discussed in Sections 4–6 of the paper [Be09]. Remark 5.6. The fundamental groups of topological groups are always abelian; see for instance Corollary 1.7.10 in [Sp81] for the more general statement on homotopy groups of H-spaces. (See also [dlH72] and [Ne02a] for a description of the fundamental groups of the classical Banach–Lie groups associated with Φ.) It then easily follows by Remark 3.6 along with Examples 3.3 through 3.5 that all of the groups K, A, and N that occur in Theorem 5.5 are amenable, although the group G itself may not be amenable; see Example 3.5(a). Harish-Chandra decompositions. We are going to draw a little closer to representation theory, which was the main motivation of the present exposition. For this purpose we borrow the following deﬁnition of inﬁnitedimensional Lie groups of Harish-Chandra type from [NØ98]. Some good references for such Harish-Chandra decompositions in the setting of ﬁnitedimensional Lie groups are [Sa80], [Kn96], and [Ne00]. Definition 5.7. By Banach–Lie group of Harish-Chandra type we actually mean a 4-tuple (G, GC , K, K C ) consisting of a connected complex Banach–Lie

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group GC and three connected Banach–Lie subgroups G, K, and K C of GC such that the following conditions are satisﬁed: (i) the Lie algebra gC of GC is the complexiﬁcation of the Lie algebra g of G; (ii) K C is a complex Banach–Lie subgroup of GC and the Lie algebra kC of K C is the complexiﬁcation of the Lie algebra k of K; (iii) there exist connected complex Banach–Lie subgroups P ± of GC whose Lie algebras p± have the properties (ad p± )n gC = {0} for some integer n ≥ 1 and p± ∩ z(g) = {0}, where z(gC ) denotes the center of gC , and moreover: (HC1) we have the direct sum decomposition gC = p+ kC p− , and in addition [kC , p± ] ⊆ p± and p− = p+ , where X → X, gC → gC , is the antilinear involutive map whose ﬁxed-point set is g; (HC2) the multiplication P + × K C × P − → GC , (p+ , k, p− ) → p+ kp− is a biholomorphic diﬀeomorphism onto its open image; (HC3) we have G ⊆ P + K C P − and G ∩ K C P − = K. If the groups GC , K, and K C are singled out in a certain context, then we may say that the group G itself is a Banach–Lie group of Harish-Chandra type. Example 5.8. Let Φ be a symmetric norming function and H = H+ ⊕ H− . Then the corresponding classical real Banach–Lie group of type (AIII)—in the terminology of Example 5.2—provides an example of a group of HarishChandra type. Speciﬁcally, we mean the 4-tuple (G, GC , K, K C ), where G = UΦ (H+ , H− ), C

G = GLΦ (H),

K = {k ∈ UΦ (H+ , H− ) | k(H± ) ⊆ H± } K C = {g ∈ GLΦ (H) | g(H± ) ⊆ H± }.

Then the conditions of Deﬁnition 5.7 are satisﬁed with the connected complex Banach–Lie subgroups P ± = {g ∈ GLΦ (H) | (g − 1)H∓ ⊆ H± }. If we write the operators on H as 2 × 2 block matrices with respect to the orthogonal decomposition H = H+ ⊕ H− , then AB + C − ∈ GLΦ (H) | D ∈ GL(H− ) , G⊆P K P = g= CD and every element in this set can be factorized as 1 BD−1 1 0 AB A − BD−1 C 0 = , D−1 C 1 CD 0 D 0 1 where the factors on the right-hand side belong to P + , K C , and P − , respectively. We refer to [NØ98] for more details as well as for similar examples provided by groups of type (BII) and (CI) (again in the terminology of Example 5.2). Remark 5.9. As noted in [NØ98] (see also [Ne00]), the objects involved in the deﬁnition of a group of Harish-Chandra type (Deﬁnition 5.7 above) have the following additional properties:

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(a) If we denote L(P ± ) = p± , then the exponential maps expP ± : p± → P ± are biholomorphic diﬀeomorphisms and the complex Banach–Lie groups P ± are nilpotent and simply connected. In particular, there exist the logarithm maps logP ± = (expP ± )−1 : P ± → p± . (b) There exists an open connected K-invariant subset Ω ⊆ p+ such that the mapping Ω × K C P − → GK C P − , (Z, p) → (expGC Z)p, is a well-deﬁned biholomorphic diﬀeomorphism. (c) Denote Ξ = {(g, Z) ∈ GC × p+ | g expGC Z ∈ P + K C P − }, which is an open neighborhood of G × Ω in GC × p+ . Let ζ ± : P + K C P − → P ± and κ : P + K C P − → K C be the natural projections (see condition (HC2) in Deﬁnition 5.7), and deﬁne Ξ → p+ , (g, Z) → g.Z = logP + (ζ + (g expGC Z)) and J : Ξ → K, (g, Z) → J(g, Z) = κ(g expGC Z). Then the mapping (g, Z) → g.Z deﬁnes a transitive action of G upon Ω by biholomorphic diﬀeomorphisms and J has a cocycle property with respect to this action. (d) We have 0 ∈ Ω and the isotropy group of 0 is equal to K. Thus the aforementioned action leads to a G-equivariant diﬀeomorphism G/K Ω. These are some of the ideas that allow one to deﬁne as in [NØ98] natural reproducing kernels on Ω out of certain representations of the Banach–Lie group K. In this way one ends up with the corresponding reproducing kernel Hilbert spaces of holomorphic functions on Ω which carry representations of the bigger Banach–Lie group G of Harish-Chandra type. Remark 5.10. Applications of a diﬀerent type of reproducing kernels in the representation theory of Banach–Lie groups can be found in [BR07] and [BG08]. The viewpoint held in the ﬁrst of these papers is somehow dual to the one of Remark 5.9 (in the sense of duality theory of symmetric spaces); i.e., the bases of the corresponding vector bundles are homogeneous spaces of compact type. The complexiﬁcation of this picture of compact type is analyzed in [BG08] along with the relationship to Stinespring dilation theory, which eventually leads to geometric models for representations of operator algebras. Acknowledgements: We wish to thank Professor Jos´e Gal´e for drawing our attention to some pertinent references, Professor Hendrik Grundling for kindly sending us his unpublished manuscript, Professor Mihai S ¸ abac and Professor Gary Weiss for several useful comments, and ﬁnally Professor Karl-Hermann Neeb for kind encouragement and valuable suggestions and the referee for many helpful remarks. Partial support from the grant GR202/2006 (CNCSIS code 813) is acknowledged.

References [Ar78] J. Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces, Integral Equations Operator Theory 1 (1978), no. 4, 453–495. [Ba83] W. Banaszczyk, On the existence of exotic Banach–Lie groups, Math. Ann. 264 (1983), no. 4, 485–493.

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[Be06] D. Beltit¸˘ a, Smooth Homogeneous Structures in Operator Theory, Monogr. and Surveys in Pure and Appl. Math., 137. Chapman & Hall/CRC Press, Boca Raton-London-New York-Singapore, 2006. [Be09] D. Beltit¸˘ a, Iwasawa decompositions of some infinite-dimensional Lie groups. Trans. Amer. Math. Soc. 361 (2009), no. 12, 6613–6644. [BG08] D. Beltit¸˘ a, J.E. Gal´e, Holomorphic geometric models for representations of C ∗ -algebras. J. Funct. Anal. 255 (2008), no. 10, 2888–2932. [BP07] D. Beltit¸a ˘, B. Prunaru, Amenability, completely bounded projections, dynamical systems and smooth orbits, Integral Equations Operator Theory 57 (2007), no. 1, 1–17. [BR05] D. Beltit¸˘ a, T.S. Ratiu, Symplectic leaves in real Banach Lie-Poisson spaces, Geom. Funct. Anal. 15 (2005), no. 4, 753–779. [BR07] D. Beltit¸˘ a, T.S. Ratiu, Geometric representation theory for unitary groups of operator algebras, Adv. Math. 208 (2007), no. 1, 299–317. [BRT07] D. Beltita, T.S. Ratiu, A.B. Tumpach, The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits, J. Funct. Anal. 247 (2007), no. 1, 138–168. [BS01] D. Beltit¸˘ a, M. S ¸ abac, Lie Algebras of Bounded Operators, Operator Theory: Advances and Applications, 120. Birkh¨ auser Verlag, Basel, 2001. [Bo80] R.P. Boyer, Representation theory of the Hilbert-Lie group U(H)2 , Duke Math. J. 47 (1980), no. 2, 325–344. [Bu50] R.C. Buck, Generalized group algebras, Proc. Nat. Acad. Sci. USA 36 (1950), 747–749. [DFWW04] K. Dykema, T. Figiel, G. Weiss, M. Wodzicki, Commutator structure of operator ideals, Adv. Math. 185 (2004), no. 1, 1–79. [Er72] J.A. Erdos, The triangular factorization of operators on Hilbert space, Indiana Univ. Math. J. 22 (1972/73), 939–950. [Er78] J.A. Erdos, Triangular integration on symmetrically normed ideals, Indiana Univ. Math. J. 27 (1978), no. 3, 401–408. [Ey72] P. Eymard, Moyennes Invariantes et Repr´esentations Unitaires, Lecture Notes in Math. 300, Springer-Verlag, Berlin, 1972. [FN07] S. Ferri, M. Neufang, On the topological centre of the algebra LUC(G)∗ for general topological groups, J. Funct. Anal. 244 (2007), no. 1, 154–171. [Ga06] J.E. Gal´e, Geometr´ıa de ´ orbitas de representaciones de grupos y ´ algebras promediables, Rev. R. Acad. Cienc. Exactas F´ıs. Qu´ım. Nat. Zaragoza (2) 61 (2006), 7–46. [GK69] I.C. Gohberg, M.G. Kre˘ın, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., 18, Amer. Math. Soc., Providence, RI, 1969. [GK70] I.C. Gohberg, M.G. Kre˘ın, Theory and Applications of Volterra Operators in Hilbert Space, Transl. Math. Monogr., 24, Amer. Math. Soc., Providence, RI, 1970. [Gr] H. Grundling, unpublished manuscript. [Gr05] H. Grundling, Generalising group algebras, J. London Math. Soc. (2) 72 (2005), no. 3, 742–762. [GN09] H. Grundling, K.-H. Neeb, Full regularity for a C ∗ -algebra of the canonical commutation relations. Rev. Math. Phys. 21 (2009), no. 5, 587–613. [dlH72] P. de la Harpe, Classical Banach–Lie Algebras and Banach–Lie Groups of Operators in Hilbert Space, Lecture Notes in Math. 285, Springer-Verlag, Berlin, 1972.

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[HR63] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. I. Grundl. Math. Wiss., Bd. 115. Springer-Verlag, Berlin, 1963. [Ho75] K.H. Hofmann, Analytic groups without analysis, Inst. Nat. Alta Mat. Symp. Math. 16 (1975), 357–374. [KW08] V. Kaftal, G. Weiss, A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem, in: Hot Topics in Operator Theory, Theta Ser. Adv. Math., 9, Theta, Bucharest, 2008, pp. 101–135. [Kn96] A.W. Knapp, Lie Groups Beyond an Introduction, Progr. Math., 140, Birkh¨ auser-Verlag, Boston-Basel-Berlin, 1996. [Ne98] K.-H. Neeb, Holomorphic highest weight representations of infinite-dimensional complex classical groups, J. Reine Angew. Math. 497 (1998), 171–222. [Ne00] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, 28. Walter de Gruyter & Co., Berlin-New York, 2000. [Ne02a] K.-H. Neeb, Classical Hilbert-Lie groups, their extensions and their homotopy groups, in: Geometry and Analysis on Finite and Inﬁnite-Dimensional Lie Groups (B¸edlewo, 2000), Banach Center Publ., 55, Polish Acad. Sci. Warsaw, 2002, pp. 87–151. [Ne02b] K.-H. Neeb, A Cartan-Hadamard theorem for Banach-Finsler manifolds, Geom. Dedicata 95 (2002), 115–156. [Ne04] K.-H. Neeb, Infinite-dimensional groups and their representations, in: Lie Theory, Progr. Math., 228, Birkh¨ auser, Boston, MA, 2004, pp. 213–328. [Ne06] K.-H. Neeb, Towards a Lie theory of locally convex groups, Japan. J. Math. (3rd series) 1 (2006), no. 2, 291–468. [NØ98] K.-H. Neeb, B. Ørsted, Unitary highest weight representations in Hilbert spaces of holomorphic functions on infinite-dimensional domains, J. Funct. Anal. 156 (1998), no. 1, 263–300. [Pa88] A.L.T. Paterson, Amenability, Mathematical Surveys and Monographs, 29. American Mathematical Society, Providence, RI, 1988. [Pe06] V. Pestov, Dynamics of Inﬁnite-Dimensional Groups. The Ramsey-Dvoretzky-Milman Phenomenon, University Lecture Series, 40. Amer. Math. Soc. Providence, RI, 2006. [Pi90] D. Pickrell, Separable representations for automorphism groups of infinite symmetric spaces, J. Funct. Anal. 90 (1990), no. 1, 1–26. [Sa80] I. Satake, Algebraic Structures of Symmetric Domains, Kano Mem. Lectures, 4. Publ. Math. Soc. Japan, 14. Iwanami Shoten & Princeton Univ. Press, Tokyo-Princeton, NJ, 1980. [Sp81] E.H. Spanier, Algebraic Topology. Corrected reprint. Springer-Verlag, New York-Berlin, 1981. [SV75] S ¸ . Str˘ atil˘ a, D. Voiculescu, Representations of AF-algebras and of the Group U(∞), Lecture Notes in Math., 486. Springer-Verlag, Berlin-New York, 1975. [Tu95] S. Turek, Universal minimal dynamical system for reals, Comment. Math. Univ. Carolin. 36 (1995), no. 2, 371–375. [Up85] H. Upmeier, Symmetric Banach Manifolds and Jordan C ∗ -Algebras, Math. Stud., 104, Notas de Mat., 96, North-Holland, Amsterdam, 1985. [Vo00] D.A. Vogan, Jr., The method of coadjoint orbits for real reductive groups, in: Representation Theory of Lie Groups (Park City, 1998), IAS/ Park City Math. Ser. 8, Amer. Math. Soc., Providence, RI, 2000, pp. 177-238. [We40] A. Weil, L’Int`egration dans les Groupes Topologiques et ses Applications, Actual. Sci. Ind., 869. Hermann et Cie., Paris, 1940.

Heat Kernel Measures and Critical Limits Doug Pickrell Mathematics Department, University of Arizona, Tucson, AZ, USA 85721, [email protected]

Summary. This article is an exposition of several questions linking heat kernel measures on inﬁnite-dimensional Lie groups, limits associated with critical Sobolev exponents, and Feynman–Kac measures for sigma models. The ﬁrst part of the article concerns existence and invariance issues for heat kernel measure classes. The main examples are heat kernel measures on groups of the form C 0 (X, F), where X is a Riemannian manifold and F is a ﬁnite-dimensional Lie group. These measures depend on a smoothness parameter s > dim(X)/2. The second part of the article concerns the limit s ↓ dim(X)/2, especially dim(X) ≤ 2, and how this limit is related to issues arising in quantum ﬁeld theory. In the case of X = S 1 , we conjecture that heat kernel measures converge to measures which arise naturally from the Kac–Moody–Segal point of view on loop groups, as s ↓ 1/2. Key words: heat kernel measure, abstract Wiener space, abstract Wiener group, critical Sobolev exponent, loop group, sigma model, Feynman–Kac measure. 2000 Mathematics Subject Classifications: 58D20, 22E65, 22E67.

1 Introduction Given a ﬁnite-dimensional real Hilbert space f , there is an associated Lebesgue measure λf , a (positive) Laplace operator ∆f , and a convolution semigroup of heat kernel measures t

1

2

νtf = e− 2 ∆f δ0 = (2πt)− dim(f )/2 e− 2t |x|f dλf (x).

(1)

The map f → {νtf } is functorial, in the sense that if P : f1 → f2 is an orthogonal projection, then P∗ νtf1 = νtf2 . (2) More generally, given a ﬁnite-dimensional Lie group F, with a ﬁxed inner product on its Lie algebra f , there is aninduced left invariant Riemannian metric, a Laplace type operator ∆F = − Xi2 , where {Xi } is an orthonormal K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_12, © Springer Basel AG 2011

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basis of left invariant vector ﬁelds, and a convolution semigroup of heat kernel measures, f νtF = e−t∆F /2 δ1 = lim (exp∗ (νt/N ))N , (3) N →∞

where the power refers to convolution of measures on F (see Sections 4.7–4.8 of [McK]). Unlike the other structures, the heat kernel measures are always inversion invariant. The map F → {νtF } is functorial, in the sense that if P : F1 → F2 is a map of Lie groups, and the induced map of Lie algebras f1 → f2 is an orthogonal projection, then P∗ νtF1 = νtF2 .

(4)

In Section 2 we will recall how this construction can be generalized to inﬁnite dimensions. The form of the linear generalization is well known: given a separable real Hilbert space h, there is an associated convolution semigroup of Gaussian measures {νth }. An essential complication arises when dim(h) = ∞, namely the νth are only ﬁnitely additive, viewed as cylinder measures on h (see [St] for orientation on this point). To overcome this, following Gross ([Gr]), we consider an abstract Wiener space h → b; in this framework, the Gaussians are realized as countably additive measures νth⊂b on b, a Banach space completion of h. Geometrically speaking, one imagines that the support of the heat kernel measures diﬀuses into the enveloping space b. In the larger group category, the objects are abstract Wiener groups. Such an object consists of an inclusion of a separable Hilbert–Lie group into a Banach–Lie group, H → G, (5) with a ﬁxed compatible inner product on the Lie algebra h, such that the induced map of Lie algebras h → g is an abstract Wiener space. A morphism in this category is a map of pairs of Banach–Lie groups P : (H1 , G1 ) → (H2 , G2 ) such that the induced map of Lie algebras P : h1 → h2 is an orthogonal projection. With some possible restrictions, the corresponding heat kernel measures νtH⊂G can be constructed as in the ﬁnite-dimensional case, and they form a convolution semigroup of inversion invariant probability measures on G (this use of Ito’s ideas in an inﬁnite-dimensional context apparently originated in [DS], and more concretely in [M]). This construction is also functorial, a point which we will emphasize. In Section 3 we consider invariance properties of heat kernel measure classes. The Cameron–Martin–Segal theorem asserts that for t > 0, the measure class [νth⊂b ] is invariant with respect to translation by h ∈ b if and only if h ∈ h. In the larger group category, we conjecture that [νtH⊂G ] is invariant with respect to translation by h ∈ G if and only if h ∈ H0 and Ad(h)∗ Ad(h) − 1 is a Hilbert–Schmidt operator on h. The evidence in favor of this conjecture is based on deep results of Driver. In Section 4, and in the remainder of the paper, we focus on groups of maps. Suppose that X is a compact manifold and W is a real Hilbert–Sobolev

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space corresponding to degree of smoothness s > dim(X)/2. In this context the Sobolev embedding W → C 0 (X; R) (6) is a canonical example of an abstract Wiener space. If F is a ﬁnite-dimensional Lie group with a ﬁxed inner product on its Lie algebra f , then in a natural way H = W (X, F) → G = C 0 (X, F) (7) is an abstract Wiener group. In this example the measure νtH⊂G is determined by its distributions corresponding to evaluation at a ﬁnite number of points of X; these distributions involve a nonlocal Green’s function interaction between pairs of points. When X = S 1 , F = K, a simply connected compact Lie group with Ad(K)-invariant inner product, and s = 1, Driver and collaborators have proven that the measure class of νtH⊂G equals the measure class of a Wiener measure. This latter measure has a heuristic expression in terms of a local energy functional 1 exp(− g −1 dg ∧ ∗g −1 dg) dλ(g(θ)) (8) 2t X X

(see [Dr2]). When dim(X) = 2, s = 1 is the critical exponent, the Sobolev embeddings (6)–(7) and associated heat kernel measures are deﬁned only for s > 1, and the heuristic expression (8) is the Feynman–Kac measure for the sigma model with target K. This motivates the following question, considered in Sections 5 and 6: is there a sense in which heat kernel measures have nontrivial limits as s ↓ dim(X)/2, the critical exponent? In the case of X = S 1 , for various reasons (e.g., the Kac–Moody–Segal point of view on loop groups) one is led to consider, in place of (7), the hyperfunction completion W 1/2 (S 1 , K) → Hyp(S 1 , G),

(9)

where G denotes the complexiﬁcation of K. A generic point in the hyperfunction completion is represented by a formal Riemann–Hilbert factorization g = g− · g0 · g+ ,

(10)

where g− ∈ H 0 (∆∗ , ∞; G, 1) (i.e., a holomorphic function on ∆∗ with values in G sending ∞ to 1 ∈ G), g0 ∈ G, g+ ∈ H 0 (∆, 0; G, 1), and ∆ (∆∗ ) is the open unit disk centered at 0 (∞). There is a family of measures on Hyp(S 1 , G), indexed by a level l, having heuristic expressions involving Toeplitz determinants. The associated measure classes are invariant with respect to left and right translations, by real analytic loops in K, and conjecturally with respect to real analytic reparameterization. Whereas heat kernel measures (and

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Wiener measures) are determined by distributions corresponding to evaluating continuous K-valued loops at points on S 1 , these latter measures are determined by distributions corresponding to g0 , and to evaluating the holomorphic functions g± at points oﬀ S 1 . It seems likely that these distributions are computable, but at present very little is known rigorously. We conjecture that the heat kernel measures parameterized by s and t converge to a measure with level l = 1/2t as s ↓ 1/2. This is nontrivial for non-abelian K, because the change of variables from values of a loop around S 1 to coeﬃcients of the Riemann–Hilbert factorization is complicated. a closed Riemannian surface, because The case of most interest is X = Σ, of (8). In Section 6 I have adopted the point of view of quantum ﬁeld theory in trying to understand the right questions to ask about the critical limit as s ↓ 1. This point of view stresses that the two-dimensional case is tightly inter → W 1/2 (S), twined with the one-dimensional case, by the trace map W 1 (Σ) when S is a 1-manifold (space) embedded in Σ (a Euclidean space-time). This critical limit is also possibly related to longstanding questions concerning the irreducibility of energy representations in two dimensions (see Section 3.5 of [A]). Surveys of related topics, with more extensive bibliographies, include [Dr2] (for Wiener measure and heat kernel measures on loop groups), [A] (for energy representations), and [Ga2] (for sigma models). I should note that there are important examples of heat-kernel-like measures which do not ﬁt into a Banach space framework; see [AM] and the references in that paper. Notation 1 (1) Throughout this paper, all spaces are assumed to be separable and real, unless explicitly noted otherwise. (2) We will frequently encounter continuous maps of Hilbert spaces h → f such that the quotient Hilbert space equals the Hilbert space structure on the target. If f is a subspace of h itself, then such a map is an orthogonal projection. In this paper we will refer to such a map h → f as a projection. (3) Suppose that h is a Hilbert space. Let L2 denote the symmetrically normed ideal of Hilbert–Schmidt operators on h. For an invertible operator A on h, the two conditions AAt ∈ 1 + L2 ,

and

A − A−t ∈ L2

(11)

are equivalent. We let GL(h)(L2 ) denote the group of invertible operators satisfying these two conditions. In the same way, we can define a restricted general linear group GL(h)(I) , for any symmetrically normed ideal I. (4) Given a measure ν on a space X, and a Borel space Y , a map f : X → Y will be said to be ν-measureable if f −1 (E) is ν-measureable for each Borel set E ⊂ Y . The measure class of ν will be denoted by [ν]. (5) Lebesgue measure will be denoted by dλ, and (left) Haar measure will be denoted by dλG , for a group G.

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2 General constructions 2.1 Abstract Wiener spaces and Gaussian measures Suppose that h is a Hilbert space. The corresponding convolution semigroup of Gaussian measures, {νth }t>0 , has the heuristic expression 1

2

dνth = (2πt)− dim(h)/2 e− 2t |x|h dλh (x),

(12)

where λh denotes the “Lebesgue measure for the Hilbert space h.” The measure λh and the expression (12) have literal meaning only when h is ﬁnite dimensional. To go beyond this, following Gross, suppose that h → b is a continuous dense inclusion of h into a Banach space. We can consistently deﬁne a ﬁnitely additive probability measure νth⊂b on b in the following way. A cylinder set is a Borel subset of b of the form p−1 (E), where p : b → f , p|h : h → f is a ﬁnite rank projection, and E is a Borel subset of f . The set of cylinder sets is an algebra; it is a σ-algebra if and only if h = b is ﬁnite dimensional. The functorial property of ﬁnite-dimensional Gaussian measures, (2), implies that there exists a well-deﬁned ﬁnitely additive measure νth⊂b on the algebra of cylinder sets satisfying p∗ νth⊂b = νtf ,

(13)

for all p as above. Definition 2.1. (a) The inclusion h → b is called an abstract Wiener space if the finitely additive measure νth⊂b has a (necessarily unique) countably additive extension to a Borel probability measure √ on b, for some t > 0 (and hence all t > 0, because νth⊂b (E) = ν1h⊂b (E/ t)). (b) A map of abstract Wiener spaces, (h1 ⊂ b1 ) → (h2 ⊂ b2 )

(14)

is a map b1 → b2 such that the restriction to h1 is a projection h1 → h2 . For a map of abstract Wiener spaces, the corresponding Gaussian semigroups push forward. In many ways, the restriction to Banach space completions of h is artiﬁcial. However, the Banach framework seems elegant and natural, in part because of the following characterization and examples. Theorem 2.2. The inclusion h → b is an abstract Wiener space if and only if for each > 0 there exists a finite-dimensional subspace f ⊂ h such that νth⊂b {x ∈ b : |px|f > } <

(15)

for all p : b → f vanishing on f , where p : h → f is a finite rank projection. If b is itself a Hilbert space, this is equivalent to the condition that the inclusion h → b is a Hilbert–Schmidt operator.

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For a discussion of this theorem, see Section 3.9 of [B]. Example 2.3. (a) If f is a ﬁnite-dimensional Hilbert space, then f = h = b is an abstract Wiener space, and νtf is given literally by (12). (b) Suppose that X is a compact manifold, possibly with boundary, and W is a Hilbert Sobolev space of degree of smoothness s > dim(X)/2. Then the Sobolev embedding W → C 0 (X; R) (16) is an abstract Wiener space (see [CL]). This is closely related to the fact that W → W 0 (X) = L2 (X) is a Hilbert–Schmidt operator precisely when s is above the critical exponent. (c) For W as in (b), if h → b is an abstract Wiener space, then W (X, h) = W ⊗ h → C 0 (X, b)

(17)

is also an abstract Wiener space (see 3.11.29 of [B]). A special case of (c) is the Wiener space analogue of the path functor, W ⊗ h = W 1 ([0, T ], 0; h, 0) → C 0 ([0, T ], 0; b, 0)

(18)

(i.e., continuous b-valued paths beginning at 0 ∈ b) with inner product T x, yW 1 = x, ˙ y ˙ L2 = x(τ ˙ ), y(τ ˙ )h dτ. (19) 0

To stay within the Banach category, we have chosen T < ∞. By letting T → ∞, we obtain a Gaussian semigroup on the semi-inﬁnite path space C 0 ([0, ∞), 0; b, 0).

(20)

For notational simplicity, we will write W ⊗h⊂C 0 ([0,∞),0;b,0)

ν h⊂b = ν1

,

(21)

and we will refer to this as the Brownian motion associated to h ⊂ b. Given a partition V : 0 < t1 < · · · < tn , there is an evaluation map EvalV : C 0 ([0, ∞), 0; b, 0) →

n

b : x → (xi ),

xi = x(ti ).

(22)

1

In terms of the coordinates xi , (EvalV )∗ ν h⊂b = dνth⊂b (x1 ) × dνth⊂b (x2 − x1 ) × · · · × dνth⊂b (xn − xn−1 ). 1 1 −t2 n −tn−1 (23) In particular, the distribution of ν h⊂b at time t is νth⊂b . The semigroup property of νth⊂b is equivalent to the consistency of these distributions for ν h⊂b when the partition is reﬁned. The path space construction can be iterated. Thus given h ⊂ b, there is a Brownian motion, a Brownian sheet, etc., associated to h ⊂ b.

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2.2 Abstract Wiener groups and heat kernel measures Definition 2.4. (a) An abstract Wiener group is an inclusion of a separable Hilbert–Lie group H into a Banach–Lie group G, with a fixed compatible inner product on the Lie algebra h, such that the associated Lie algebra map h → g is an abstract Wiener space. (b) A map of abstract Wiener groups is a map of pairs of Lie groups (H1 , G1 ) → (H2 , G2 ) such that the induced Lie algebra map h1 → h2 is a projection. (c) An abstract Wiener group H ⊂ G with the property that finite rank maps of Wiener groups p : G → F separate the topology of G is called local. Given an abstract Wiener group H ⊂ G, there is an induced left invariant Riemannian metric on H, and a left invariant Finsler metric on G. These induce separable complete metrics generating the topologies of H and G, respectively. Example 2.5. (a) If F is a ﬁnite-dimensional Lie group, with an arbitrary inner product on its Lie algebra f , then F = H = G is an abstract Wiener group. (b) If W is as in (b) of Example 2.3, given an abstract Wiener group H ⊂ G, W (X, H) ⊂ C 0 (X, G) (24) is a local abstract Wiener group. (c) The construction in (b) can be adapted to local gauge transformations of a nontrivial principal bundle P → X, provided that the structure group K is a compact Lie group with a ﬁxed Ad(K)-invariant inner product. (d) There are also nonlocal examples involving inﬁnite classical matrix groups; see [Go]. The following would be the ideal existence result. Conjecture 2.6. For ﬁxed t ≥ 0, the sequence of probability measures h⊂g h⊂g exp∗ (νt/N ) ∗ · · · ∗ exp∗ (νt/N )

(25)

(an N -fold convolution of probability measures on G) has a weak limit with respect to BC(G), bounded continuous functions. The limits, denoted νtH⊂G , form a convolution semigroup of inversion invariant probability measures on G. Given a map P : (H1 ⊂ G1 ) → (H2 ⊂ G2 ) (26) of abstract Wiener groups, P∗ νtH1 ⊂G1 = νtH2 ⊂G2 , for each t ≥ 0. This conjecture is known to be true if one additionally assumes that the inclusion h ⊂ g is 2-summable (see [BD]). I have not stated this result in a standard form, and this obscures an important idea. To explain this idea, ﬁx t > 0. Consider the map C 0 ([0, ∞), 0; g, 0) → G : x → gN (t) = ex1 ex2 −x1 · · · exN −xN −1 ,

(27)

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where xi = x(it/N ). The gN (t)-distribution of the Brownian motion measure ν h⊂g is the N -fold convolution (25), because the xi − xi−1 are independent (g-valued) Gaussian variables with variance t/N . Thus the conjecture is equivalent to the assertion that gN (t) has a limit as N → ∞, say g(t), with ν h⊂g -probability one. The latter statement implies that the heat kernel measure is an image of a Gaussian measure, an insight due to Ito. Example 2.7. (a) If F is a ﬁnite-dimensional real Lie group, with an arbitrary inner product on the Lie algebra f , νtF can be expressed as in (3). This measure is absolutely continuous with respect to the Haar measure, and it has a real analytic density. (b) Consider the path construction H = W 1 (I, 0; F, 1) → G = C 0 (I, 0; F, 1),

(28)

where I = [0, T ], and h has the norm (19) (with f in place of h in (18)). In this case νtH⊂G is the Brownian motion for F, with variance t. The path space for G is the double path space C 0 (I × I, ({0} × I) ∪ (I × {0}); F, 1),

(29)

and ν H⊂G is the F-valued Brownian sheet . The Brownian sheet is relevant to physics in the following way. Supppose that F = K, a compact Lie group, and the inner product is Ad(K)-invariant. Suppose that Σ is a closed surface with an area element. The Yang–Mills measure on the space of K-connections has the heuristic expression 1

e− 2

FA ∧∗FA

dλ(A)

(30)

for A ∈ Ω 1 (Σ; k), where FA is the curvature for the connection d + A, and dλ denotes a heuristic Lebesgue measure on the linear space of k-valued one forms. This expression can be interpreted as a ﬁnitely additive measure in the following way. Let pt (g)dλ(g) denote the heat kernel measure for K. Given a triangulation of Σ, one considers the projection Ω 1 (Σ, k) → K E : A → (ge ),

(31)

where E denotes the set of edges for the triangulation, and ge ∈ K represents parallel translation for the connection d + A. The image of the Yang–Mills measure is pt Area(f ) (g∂f ) dge , (32) F

E

where g∂f denotes the holonomy around the face f ∈ F . The fact that these measures are coherent with respect to reﬁnement of the triangulation follows directly from the semigroup property of the heat kernel measures. When projected to gauge equivalence classes, this measure can be expressed as a ﬁnite-dimensional conditioning of the Brownian sheet (see [Sen]).

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401

3 Invariance questions The linear invariance properties of Gaussian measure classes are summarized by the following. Theorem 3.1. Suppose that h → b is an abstract Wiener space. Let ν = νth⊂b for some t > 0. (a) The subset of translations in b which fix [ν] is h. (b) The group of linear transformations which fix [ν] is GL(h)(L2 ) , in the following sense: given a ν-measureable linear map T : b → b which fixes [ν], T = T|h ∈ GL(h)(L2 ) ; conversely, given T ∈ GL(h)(L2 ) , there exists a ν-measureable linear map T : b → b which fixes [ν] and satisfies T|h = T . These results, and their history, are discussed in a very illuminating way in [B] (where one can also ﬁnd expressions for the Radon–Nikodym derivatives). Now suppose that H ⊂ G is an abstract Wiener group, and g ∈ G. When is [(Lg )∗ νtH⊂G ] = [νtH⊂G ], (33) where Lg is left translation? Necessarily g ∈ G0 , the identity component (because νtH⊂G is supported in the identity component). The abelian case suggests that g ∈ H. Because νtH⊂G is inversion invariant, translation invariance implies that [νtH⊂G ] is g-conjugation invariant. Thus if (33) holds for all t, this suggests that [νth⊂g ] is Ad(g)-invariant, and this implies that, as an operator on h, Ad(g) ∈ GL(h)(L2 ) . These considerations suggest the following. Conjecture 3.2. For T > 0 and h ∈ G, [νTH⊂G ] is left and right h-invariant if and only if h ∈ H0 and Ad(h) ∈ GL(h)(L2 ) . Hilbert–Schmidt criteria are universal in unitary representation-theoretic questions of this type. In the next section we will consider evidence in favor of this conjecture. Before leaving this abstract setting, I will mention two other related questions. When t → ∞, a Gaussian measure is asymptotically invariant in the following sense introduced in [MM]: given νt = νth⊂b and h ∈ h, for each p < ∞, dνt (b + h) p (34) dνt (b) → 0 1 − dνt (b) as t → ∞ (in this linear context, this integral reduces to a one-dimensional integral, which is easily estimated). Now suppose that Conjecture 3.2 is true, and let νt = νtH⊂G and h satisfy the conditions in Conjecture 3.2. Is it true that for each p < ∞, dνt (gh) p (35) 1 − dνt (g) → 0 dνt (g)

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as t → ∞? In [MM] it is proved that the Wiener measure on C 0 (S 1 , K) is asymptotically invariant in this sense (see Chapter 4, Part III, of [Pi1] for a quantitative version of this result, and an example of how this is used to prove existence of an invariant measure for the loop group). The second question is the following. Suppose that we change the inner product on h in a way which (by (b) of Theorem 3.1) does not change the measure class of νth⊂g : x, y1 = Ax, y,

x, y ∈ h

(36)

where A − 1 ∈ L2 (h). Write h1 for h with this inner product depending on A. Is [νtH1 ⊂G ] = [νtH⊂G ], for each t?

4 The Example of M ap(X, F) (see [M]) Let X denote a compact Riemannian manifold, and ﬁx a Sobolev space (with a ﬁxed inner product) of continuous real-valued functions W → C 0 (X),

(37)

necessarily corresponding to some degree of smoothness s > dim(X)/2. For clarity, we will suppose that ∂X is empty, and the inner product for W is of the form φ1 , φ2 W = (P s/2 (φ1 ))(P s/2 (φ2 ))dV, (38) X

where P denotes a positive elliptic pseudodiﬀerential operator of order 2, e.g., P = (m20 + ∆), where ∆ is a (positive) Laplace-type operator. We will write G(x, y) for the Green’s function of P s ; thus Pys (G(x, y)dV (y)) = δx (y)

(39)

in the sense of distributions. Evaluation at x ∈ X deﬁnes a continuous linear functional δx : W → R : φ → φ(x); (40) this functional is represented by G(x, ·) ∈ W : φ(x) = φ, G(x, ·)W .

(41)

Let F denote a ﬁnite-dimensional abstract Wiener group. Then H = W (X, F) → G = C 0 (X, F)

(42)

is an abstract Wiener group, where h = W ⊗ f , as a Hilbert space. It is local because we can evaluate at points of X. Given a ﬁnite set of points V ⊂ X, let EvalV : C 0 (X, F) → FV : g → (g(v))

(43)

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403

denote the evaluation map. This is a map of abstract Wiener groups, where f V has the Hilbert space structure induced by the Lie algebra map EvalV : W (X, f ) → f V .

(44)

It follows from (41) that this inner product on f V is given by Gv,w xv , yw f , (xv ), (yw ) =

(45)

v,w∈V

where (Gv,w ) is the matrix inverse to the covariance matrix (G(v, w))v,w∈V . More generally, given an embedded submanifold S ⊂ X, restriction induces a map of abstract Wiener groups C 0 (X, F) → C 0 (S, F),

(46)

where the inner product on W (S, f ) corresponds to a degree of smoothness s − codim(S)/2. This is very reminiscent of functorial properties which are exploited at a heuristic level for Feynman–Kac measures (see Section 6). (v) If f1 , . . . , fN denotes an orthonormal basis for f , and if fi denotes the V left invariant vector ﬁeld on F corresponding to the v-coordinate, then the left invariant Laplacian on FV determined by the inner product (45) is given by N (v) (w) ∆V = G(v, w)fj fj , (47) v,w∈V j=1

and

(FV )

(EvalV )∗ νtH⊂G = e−t∆V /2 δ1

.

(48) νtH⊂G .

We will These ﬁnite-dimensional distributions determine the measure compare this with the Wiener measure below, when X = S 1 . We now want to understand the content of Conjecture 3.2. Suppose that g ∈ W (X, F). We need to compute the adjoint for Ad(g), denoted Ad(g)∗W , acting on the Hilbert–Lie algebra W (X, f ). Let Ad(g)t denote the adjoint for Ad(g) acting on L2 (X, dV ) ⊗ f . Then ∗W (49) Ad(g) (f1 ), f2 h = P s f1 Ad(g)(f2 )dV = Thus and

P s (P −s Ad(g)t P s )(f1 )f2 dV.

(50)

Ad(g)∗W = P −s ◦ Ad(g)t ◦ P s ,

(51)

Ad(g) Ad(g)∗W = Ad(g)P −s Ad(g)t P s .

(52)

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In general this is simply a zeroth order operator. However, suppose that Ad(g) has values in O(f ), the orthogonal group. Then Ad(g) Ad(g)∗W = (Ad(g)P −s − P −s Ad(g) + P −s Ad(g)) Ad(g)−1 P s = [Ad(g), P −s ] Ad(g)−1 P s + 1.

(53) (54)

Because of the commutator, this diﬀers from the identity by an operator of order −1 (see [Fr]). To formulate this in operator language, let L+ d denote the ideal of compact operators T on h satisfying N

sup N

1 sn (T )d < ∞ log(N ) 1

(55)

where s1 ≥ s2 ≥ · · · are the eigenvalues of |T |. The fact that (54) has order −1 implies that it belongs to L+ d . This implies the following statement. Theorem 4.1. Let d = dim(X). (a) Suppose that g ∈ W (X, f ) and Ad(g(x)) ∈ O(f ), for each x ∈ X. Then Ad(g) ∈ GL(h)(L+ ) . d

(56)

(b) Suppose that F = K, or F = K C , where K is a compact Lie group, and the inner product on f is Ad(K)-invariant. Assuming the truth of Conjecture 3.2, [νtH⊂G ] is W (X, K)-invariant when X = S 1 , and noninvariant for dim(X) > 1. The important point is that dim(X) = 2 is the critical case, where the criterion marginally fails. We now want to consider evidence in support of Conjecture 3.2. ∂ 2 Suppose that X = S 1 , and P = m20 − ( ∂θ ) . For s > 1/2, G(θ1 , θ2 ) = G(θ1 − θ2 ) =

1 2 (m0 + n2 )−s cos(n(θ1 − θ2 )). π

(57)

n≥0

We will write νts,m0 for the corresponding heat kernel measures. When s = 1, we can compare νts,m0 with Wiener measure ωt . This latter measure has distributions −1 (EvalV )∗ ωt = pt (gv gw ) dλK (gv ), (58) E

V

where E denotes the set of edges (between vertices in V ), and νtK = pt dλK . The important point is that this is a nearest neighbor interaction, which is far more elementary than the interaction involving all pairs of points for heat kernel measures, as in (48). The following summarizes some of the deep results of Driver and collaborators on heat kernel measures for loop groups.

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Theorem 4.2. Suppose that X = S 1 and F = K, a simply connected compact Lie group with Ad(K)-invariant inner product on f . (a) For s = 1, [νts,m0 ] = [ωt ]. (b) For all s > 1/2, [νts,m0 ] is W (S 1 , K)-biinvariant. A relatively short and illuminating discussion of (a) can be found in [Dr2]. Part (b) is a long story. For s = 1, part (b) follows from (a) and the relatively well-understood fact that [ωt ] is W 1 (S 1 , K)-biinvariant. Driver gave a direct proof of (b), for s = 1 ([Dr1]), and the ideas were shown to extend to all s > 1/2 in [I]. In support of Conjecture 3.2, one can also directly analyze the Brownian motion construction (b) of Example 2.7, using a stochastic Ito map (which essentially linearizes the problem). When X is a closed Riemannian surface, we expect that the heat kernel measure classes for W s,m0 (X, K) → C 0 (X, K) do just miss being translation invariant, for all s > 1. If this is the case, it is interesting to ask whether there is another natural construction which yields an invariant measure class. This should be compared with Shavgulidze’s construction of invariant measure classes for Dif f (X), for X of any dimension (see page 312 of [B]).

5 The critical Sobolev exponent and X = S 1 For s = dim(X)/2, the Sobolev embedding fails, and in general a W dim(X)/2 function on X is not bounded. This leads to a subtle situation. On the one hand, while W dim(X)/2 (X, k) is a Hilbert space, unless k is abelian, it is not a Lie algebra. On the other hand, because K is assumed compact, a map g ∈ W dim(X)/2 (X, K) is automatically bounded. Consequently, W dim(X)/2 (X, K), with the Sobolev topology (we do not impose uniform convergence), is a topological group, but it is not a Lie group (for the algebraic topology of mapping spaces with topologies deﬁned by critical Sobolev norms, see [Br]). For non-abelian k, in general it is simply not clear how to form an analogue of (42), when s = dim(X)/2. In the rest of this section we will focus on the case X = S 1 . We will freely use facts about loop groups, as presented in [PS]. For simplicity of exposition, we will assume that K is simply connected, k is simple, and the inner product (on the dual) satisﬁes θ, θ = 2, where θ is a long root (the abelian case is essentially trivial, but requires qualiﬁcations). The complexiﬁcation of K will be denoted by G. For each s > 1/2, there is a (Kac–Moody) universal central extension of Lie groups s (S 1 , K) → W s (S 1 , K) → 0. 0→T→W (59) This extension exists for s = 1/2 as well (as an extension of a topological group), and this is the maximal domain for the extension. This extension is

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realized using operator methods in [PS]. If we ﬁx a faithful representation K ⊂ U (N ), a loop g : S 1 → K can be viewed as a unitary multiplication operator Mg on H = L2 (S 1 , CN ). Relative to the Hardy polarization H = H+ ⊕ H− , where H+ consists of functions with holomorphic extension to the disk, AB Mg = , (60) CD where B (or C) is the Hankel operator, and A (or D) is the Toeplitz operator, associated to the loop g. The condition g ∈ W 1/2 is equivalent to B ∈ L2 . The Toeplitz operator deﬁnes a map A : W 1/2 (S 1 , K) → Fred(H+ ).

(61)

The lift of the left action of W 1/2 (S 1 , K) on itself to A∗ Det, the pull-back of the determinant line bundle Det → Fred(H+ ), induces (a power of) the extension (59). The upshot is that, from an analytic point of view, Kac–Moody theory is intimately related to the critical exponent and Toeplitz determinants. The group C ω (S 1 , G) (of real analytic loops) is a complex Lie group. An open, dense neighborhood of the identity consists of those loops which have a unique (triangular or Birkhoﬀ or Riemann–Hilbert) factorization g = g− · g0 · g+ ,

(62)

where g− ∈ H 0 (D∗ , ∞; G, 1), g0 ∈ G, g+ ∈ H 0 (D, 0; G, 1), and D and D∗ denote the closed unit disks centered at 0 and ∞, respectively. A model for this neighborhood is H 1 (D∗ , g) × G × H 1 (D, g),

(63)

−1 (∂g+ ), θ− = where the linear coordinates are determined by θ+ = g+ −1 (∂g− )g− . The (left or right) translates of this neighborhood cover C ω (S 1 , G). The hyperfunction completion, Hyp(S 1 , G), is modelled on the space

H 1 (∆∗ , g) × G × H 1 (∆, g),

(64)

where ∆ and ∆∗ denote the open disks centered at 0 and ∞, respectively, and the transition functions are obtained by continuously extending the transition functions for the analytic loop space. The group C ω (S 1 , G) acts naturally from both the left and right of Hyp(S 1 , G) (see Chapter 2, Part III of [Pi1]). There is a holomorphic line bundle L → Hyp(S 1 , G) with a map 1/2 (S 1 , K) → W L ↓ ↓ W 1/2 (S 1 , K) → Hyp(S 1 , G)

(65)

ω (S 1, K). which is equivariant with respect to natural left and right actions by C ∗ The line bundle L has a distinguished holomorphic section σ0 ; restricted to

Heat Kernel Measures

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W 1/2 loops, L∗m = A∗ Det and σ0m = det A, the pull-back of the canonical holomorphic section of Det → Fred(H+ ), where m is the ratio of the CN -trace form and the normalized inner product on k. We will write |L|2 for the line ¯ we can form real powers of this bundle, because the transition bundle L ⊗ L; functions are positive. In the following statement, note that (σ0 ⊗ σ ¯0 )l has ∗ 2l values in the dual bundle |L | . Theorem 5.1. For each l ≥ 0, there exists a C ω (S 1 , K)-biinvariant measure 2l dµ|L| with values in the line bundle |L|2l → Hyp(S 1 , G) such that ¯0 )l dµ|L| dµl = (σ0 ⊗ σ

2l

(66)

is a probability measure. In particular, there exists a C ω (S 1 , K)-biinvariant 0 probability measure dµ = dµ|L| on Hyp(S 1 , G). This is a reﬁnement of the main result in [Pi1] (see the Introduction of [Pi1], and also [Pi2], for further background information and motivation). These bundle-valued measures are conjecturally unique. Uniqueness would imply invariance with respect to the natural action of analytic reparameterizations of S 1 . This independence of scale is the hallmark of the critical exponent. 2l One motivation for constructing the measure µ|L| is to prove a Peter– Weyl theorem of the schematic form H 0 ∩ L2 (L∗⊗l ) = H(Λ) ⊗ H(Λ)∗ , (67) level(Λ)=l

where the H(Λ) are the positive energy representations of level l ∈ Z+ . (This statement requires more explanation. Here we will simply say, Kac and Peterson proved an algebraic generalization of the Peter–Weyl theorem (see Section 1.7, Part I of [Pi1]), and we are seeking an analytic version, suitable for application to sewing rules discussed below.) This remains incomplete. Because |σ0 |2 = det |A|2/m , restricted to W 1/2 (S 1 , K), µl has a heuristic expression 1 dµl = det |A|2(l/m) dµ. (68) Z This begs two questions: (1) why do Toeplitz determinants have anything to do with (limits of) heat kernel measures, and (2) how do we write the background µ in a way which suggests how to think about it analytically? It is instructive to ﬁrst consider the abelian case. When K = T, the theorem is valid provided l > 0 (the background µ does not exist, reﬂecting a lack of “compactness” in this ﬂat case), and we consider identity components. An element of Hyp(S 1 , C∗ )0 can be written uniquely as xn z n ) · exp(x0 ) · exp( xn z n ), (69) exp( n<0

n>0

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where the sums represent holomorphic functions in the open disks. If (69) represents a loop g ∈ W 1/2 (S 1 , T)0 , then det |A(g)|2l = exp(−l n|xn |2 ) (70) n>0

(the Helton–Howe formula). Consequently, dµl = dλT (exp(x0 ))

1 exp(−lk|xk |2 )dλ(xk ), Zk

(71)

k>0

and there is no diﬀusion in noncompact directions: x−k = −x∗k a.e. Since T dνts,m0 = dνm −2s (exp(x0 )) t 0

1 1 exp(− (k 2 + m20 )s |xk |2 )dλ(xk ), Zk 2t

(72)

k>0

(71) is the limit of heat kernel measures (with l = 1/2t) as s ↓ 1/2, provided that we also send the mass m0 to zero. Now suppose that K is simply connected. Since K and its loop group ﬁt into the Kac–Moody framework, it is natural to compare µ to Haar measure dλK for K. Fix a triangular decomposition g = n − ⊕ h ⊕ n+ .

(73)

A generic g ∈ K (or G) can be written uniquely as g = lmau,

(74)

where l ∈ N − = exp(n− ), m ∈ T = exp(h ∩ k), a ∈ A = exp(hR ), and u ∈ N + = exp(n+ ). This implies that there is a K-equivariant isomorphism of homogeneous spaces, K/T → G/B + , and l ∈ N − is a coordinate for the top stratum. Harish-Chandra discovered that, in terms of this coordinate, the unique K-invariant probability measure on K/T can be written as a4δ dλ(l) =

|σi (g)|2 dλ(l) =

1 dλ(l), |l · vδ |4

(75)

where (74) implicitly determines a = a(gT ) as a function of l, the σi are the fundamental matrix coeﬃcients, dλ(l) denotes a properly normalized Haar measure for N − , 2δ denotes the sum of the positive complex roots for the triangular factorization (73), and vδ is a highest weight vector in the highest weight representation corresponding to the dominant integral functional δ. The point of the third expression is that it shows that the denominator for the density is a polynomial in l; hence the integrability of (75) is a delicate issue. It also follows from work of Harish-Chandra that for λ ∈ h∗R 2δ, α (76) a−iλ dλK g = 2δ − iλ, α α>0

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(the right-hand side is Harish-Chandra’s c-function for G/K). In particular, this determines the integrability of powers of a: for > 0, a(1+)2δ dλ(l) = −(d−r)/2 < ∞, (77) d and r denote the dimension and rank of g, respectively. The formula (75) can be derived by calculating a Jacobian in a straightforward way. The formula (76) can be deduced from the Duistermaat–Heckman localization principle (in particular, log(a) is a momentum map), using a (Drinfeld–Evens–Lu) Poisson structure which generalizes to the Kac–Moody framework (see [Pi3]). Now consider the loop situation. Recall that θ− is a coordinate for H 0 (∆∗ , 0; G, 1) (which is similar to N − ). The loop analogue of the formula (75) leads to the following heuristic expression for the θ− distribution of µl , where initially we think of θ− as corresponding to a unitary loop g ∈ W 1/2 (S 1 , K): 1 ˙ |σ0 (g)|2(2g+l) dλ(θ− ) Z 1 ˙ = det |A(g)|2(2g+l)/m dλ(θ− ), Z

(θ− )∗ µl =

(78) (79)

where g˙ is the dual Coxeter number, and dλ(θ− ) is a heuristic Lebesgue background. The important point is the shift by the dual Coxeter number, which reﬂects a regularization of the “sum over all positive roots”. In a similar manner the Duistermaat–Heckman approach to (76) applies in a heuristic way to compute the g0 distribution of µl :

a(g0 )−iλ dµl =

1 sin( 2(g+l) 2δ, α) ˙

α>0

1 sin( 2(g+l) 2δ − iλ, α) ˙

.

(80)

The function on the right-hand side is an aﬃne analogue of Harish-Chandra’s c-function. The heuristic expression (79) explains why µl is expected to be invariant with respect to P SU (1, 1) (the Toeplitz determinant and the background Lebesgue measure are conformally invariant). It also points to a way of constructing the θ− (or g− ) distribution of µl , by imposing a cutoﬀ and taking a limit: 1 ˙ det |A(g(PN θ− ))|2(2g+l)/m dλ(PN θ− ), N ↑∞ ZPN

d((g− )∗ µl )(θ− ) = lim

(81)

where PN θ− is the projection onto the ﬁrst N coeﬃcients, g− is related to −1 P θ− by P θ− = (∂g− )g− , and g is a unitary loop having Riemann–Hilbert factorization (62). Many, but not all, of the details of this construction have been worked out. The main idea is that log(det |A(g)|2 ) is part of a momentum map. Consequently, localization can be used to compute integrals involving

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the determinant against a symplectic volume element, on ﬁnite-dimensional approximations. The formula (80) for the g0 distribution remains a conjecture, but it seems to point in a fruitful direction. The zero-mode g0 roughly arises from the projection 0, ∞; G, 1) → Hyp(S 1 , G)generic → G. H 0 (C, (82) More generally, given a closed Riemann surface Σ and a real analytic embed a G-hyperfunction induces a holomorphic G-bundle on Σ, and ding S 1 → Σ, consequently there is a bundle \ S 1 , G) → Hyp(S 1 , G) → H 1 (Σ, OG ), H 0 (Σ

(83)

OG ) denotes the set of isomorphism classes of holomorphic Gwhere H 1 (Σ, OG ) identify with the irreducible points bundles. The stable points of H 1 (Σ, 1 of H (Σ, K), which has a canonical symplectic structure (see [AB]). This suggests the following. Question 5.2. Does dµ project to the (normalized) symplectic volume on the OG )? stable points of H 1 (Σ, Continuous loops have Riemann–Hilbert factorizations, and as a consequence there is a diagram W s (S 1 , K) → C 0 (S 1 , K) ↓ ↓ 1/2 1 W (S , K) → Hyp(S 1 , G) for each s > 1/2. We consider the W s norm (m20 + n2 )s |xn |2 ,

(84)

(85)

n

and νts,m0 , the corresponding heat kernel measures, which we view as measures on Hyp(S 1 , G). Conjecture 5.3. The measures νts,m0 have a limit as s ↓ 1/2, and the measure class of this limit equals the measure class of dµl , where l = 1/2t. When m0 ↓ 0, this limit converges to µl . The intuition is simply that when m0 = 0, the heat kernel measures should relax to a distribution which is conformally invariant, as s ↓ 1/2. As we saw above, the conjecture is true in the abelian case, and in this case the limit is expressed naturally in terms of Toeplitz determinants. In general the heat kernel measure class [νts,m0 ] is biinvariant with respect to W s (S 1 , K), s > 1/2. It is expected that [µl ] will be biinvariant with respect to W 1/2 (S 1 , K), in the sense that the natural induced unitary representation C ω (S 1 , K × K) → U (L2 (dµl ))

(86)

will extend continuously to W 1/2 (S 1 , K × K). These symmetry considerations ﬁt nicely with the conjecture.

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6 2D quantum field theory The functorial properties of heat kernel measures (see (46)) are reminiscent of properties generally ascribed to path integrals. We will illustrate these properties using what is called a P (φ)2 model. We will then use this as a guide in deciding what is reasonable to ask about heat kernel measures as s ↓ 1. The relevant aspect of quantum ﬁeld theory (QFT) has been formalized by Segal in a very attractive way. Fix a dimension d, thought of as the dimension of (Euclidean) space-time. As in Section 4 of [Se], let Cmetric denote the category for which the objects are oriented closed Riemannian (d − 1)manifolds, and the morphisms are oriented compact Riemannian d-manifolds with totally geodesic boundaries. Definition 6.1. A primitive d-dimensional unitary QFT is a representation of Cmetric by separable Hilbert spaces and Hilbert–Schmidt operators such that disjoint union corresponds to tensor product, orientation reversal corresponds to adjoint, and Cmetric -isomorphisms correspond to natural Hilbert space isomorphisms. To my knowledge there does not exist an example of a nonfree QFT which has been shown to satisfy Segal’s primitive axioms in dimension d > 2. However, in two dimensions, the space of all theories is deﬁnitely large (from one point of view, this space is the conﬁguration space of string theory, and it is expected to have a remarkable geometric structure (see [Ma])). We will begin by recalling how a “generic” theory is constructed. The rationale is that these generic theories are rigorously deﬁned by path integrals (see [Pi4]). Fix (a bare mass) m0 > 0, and a polynomial P : R → R which is bounded the P (φ)2 -action is the local below. Given a closed Riemannian surface Σ, functional 1 →R:φ→ A : F (Σ) ( (|dφ|2 + m20 φ2 ) + P (φ))dA, (87) 2 Σ is the appropriate domain of R-valued ﬁelds on Σ for A. where F (Σ) A heuristic expression for the P (φ)2 -Feynman–Kac measure is exp(−A(φ)) dλ(φ(x)). (88) x∈Σ

In this two-dimensional setting, there is a rigorous interpretation of (88) as a ﬁnite measure on generalized functions, 1 e− detζ (m20 + ∆)1/2

Σ

:P (φ):C0

dφC ,

(89)

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where C = (m20 + ∆)−1 , dφC is the Gaussian probability measure with

: covariance C (corresponding to the critical Sobolev space W 1 (Σ)), P (φ) :C0 denotes a regularization of the nonlinear interaction, and detζ denotes the zeta function determinant. Suppose that S is a closed Riemannian 1-manifold. Given an inner product for W 1/2 (S), there is an associated Gaussian measure on generalized functions. Associated to the measure class of this Gaussian measure, say C(S), there is a Hilbert space H(S), the space of half-densities associated to C(S) (see the appendix to [Pi4]). This space is independent of the mass and the length scales of the components of S. Suppose that Σ is a Riemannian surface with totally geodesic boundary = Σ ∗ ◦ Σ obtained by gluing S. We consider the closed Riemannian surface Σ ∗ Σ to its mirror image Σ along S. Because dφC corresponds to the critical exponent s = 1, typical conﬁgurations for the Feynman measure (89) are not ordinary functions. However, typical conﬁgurations are suﬃciently regular so that it makes sense to restrict them to S. Consequently, the Feynman measure can be pushed forward to a ﬁnite measure on generalized functions along S. Essentially because the underlying map of Hilbert spaces is the trace map → W 1/2 (S), this ﬁnite measure lands in the measure class C(S). By W 1 (Σ) we obtain a taking a square root (which is required because Σ is half of Σ), half-density Z(Σ) ∈ H(S). The main result of [Pi4] is the following. Theorem 6.2. The maps S → H(S) and Σ → Z(Σ) define a representation of Segal’s category. This construction extends to RN -valued ﬁelds. But severe problems arise for QFTs with classical ﬁelds having values in nonlinear targets. The geometric constraints have to be relaxed to accommodate quantum ﬂuctuations. For example, for the target S n−1 , it is believed that the corresponding quantum sigma model is a limit of theories (in the same universality class) involving potentials of the form P (φ) = λ(|φ|2 − a2 )2 , where φ has values in Rn . These potentials retain the symmetry, but relax the geometry, of the target. This point of view explains some qualitative features of the sigma model (see [K]), but it does not seem to shed light on how to construct the sigma model itself. A basic assumption in what follows is that there should be an intrinsic geometric construction. A naive approach to constructing a projective theory, which might approximate the sigma model with target K, is described by the following rough outline. Fix a degree of smoothness s > 1 and a time t > 0. For a closed Riemannian 1-manifold S, deﬁne H(S) to be the space of half-densities corresponding to the measure class of a heat kernel measure (for t) corresponding to W s−1/2 (S, K) (as in the P (φ)2 case, the details of the inner product on the Lie algebra should not aﬀect this measure class and the Hilbert space). the Given a Riemannian surface Σ with geodesic boundary S and double Σ, functorial properties of heat kernel measures imply that the heat kernel mea K) will push forward to a measure in the sure (for t) corresponding to W s (Σ,

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measure class corresponding to W s−1/2 (S, K). We can thus deﬁne Z(Σ) to be the line determined by the square root of this pushforward. While these deﬁnitions make sense, one suspects that the homomorphism property fails to hold (projectively), because the heat kernel measures do not have a local character. (It is not clear whether there is some reasonable “why” to quantify this failure.) It is undoubtedly essential to let s ↓ 1 in some sense. The meaning of the Feynman–Kac measure for the sigma model is described in terms of renormalization group analysis (see Lecture 3 of [Ga2]). In this analysis the parameter t in the Feynman–Kac measure, and perhaps other details of the measure, depend dynamically on a momentum scale, which one can think of as s in the present context. This suggests the following question. Suppose that we consider the pushforward of the heat kernel measures cor K) to the hyperfunction completion Hyp(S, G) (or the responding to W s (Σ, larger formal completion considered in [Pi1] and [Pi2]). If we let t depend on s, do these measures have a limit as s ↓ 1? (in the sigma model analysis, t is proportional to ln(s)). It is important to note that the action for the sigma model can be augmented to include a Wess–Zumino–Witten term, so that the Feynman–Kac measure is of the form 1 exp(− g −1 dg ∧ ∗g −1 dg + 2πilW ZW (g)) dλK (g(x)). (90) 2t Σ x When the level l is a positive integer, and t = 1/2l, this (W ZWl ) model is solvable in many senses (see Lecture 4 of [Ga2], and [We]). It is believed that the model satisﬁes Segal’s axioms (see the Foreword to [Se]). It is also believed that there is a one-parameter family of completely integrable massive theories which interpolates between W ZWl and the original sigma model (see [ORW] and [Sm]). The heuristic expression (90) is not known to have a mathematical interpretation as a measure. However, when t = 1/2l, (90) can be plausibly manipulated, along the same lines as the P (φ)2 path measure. This is explained in [Ga1] and [Ga2]. The upshot is that the Hilbert space of the W ZWl model is given by the right-hand side of (67), which one can interpret as a surrogate for the space of half-densities corresponding to W 1/2 (S, K), taking into account the twisting by the WZW term. The vacuum of the theory, the vector corresponding to a disk, is det(A)l , which one can interpret as a “holomorphic square root” of dµl (the limit of heat kernel measures, according to Conjecture 5.3). For a general compact surface Σ, Z(Σ) can similarly be interpreted as a square root (see (14) and (15) of [Ga1]). The proof of sewing (in the holomorphic sectors, involving the W ZWl modular functor, see page 468 of [Se]), ultimately hinges on a generalized Peter–Weyl theorem; the conjectural analytical version (67) would ﬁt perfectly with Segal’s global approach.

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References [A] [AB] [AM] [BD] [B] [Br] [DS] [CL] [Dr1] [Dr2] [Fr] [Ga1] [Ga2] [Go] [Gr] [I] [K] [MM] [M]

S. Albeverio, R. Hoegh-Krohn, J. Marion, D. Testard, B. Torresani (1993) Noncommutative Distributions, Pure and Applied Mathematics, Vol. 175, Marcel Dekker, New York. M. Atiyah and R. Bott (1982) The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A 308, 523–615. H. Airault and P. Malliavin (2006) Quasi-invariance of Brownian measures on the group of circle homeomorphisms and inﬁnite dimensional Riemannian geometry, J. Funct. Anal. 241, 99–142. Ya. I. Belopolskaya and Yu. L. Dalecky (1990) Stochastic Equations and Diﬀerential Geometry, Kluwer, Dordrecht. V. I. Bogachev (1998) Gaussian Measures, Mathematical Monographs 62, Amer. Math. Soc. H. Brezis (2006) New questions related to topological degree. In: Prog. Math. 244, Birkhauser, 137–154. Yu. L. Daletskii and Ya. I. Snaidermann (1969) Diﬀusion and quasiinvariant measures on inﬁnite dimensional Lie groups, Funct. Anal. Appl. 3, no. 2, 156–158. P. Colella and O. E. Lanford (1973) Sample ﬁeld behavior for the free Markov random ﬁeld, in Constructive Quantum Field Theory, edited by G. Velo and A. Wightman, Springer, New York, 44–70. B. Driver (1997) Integration by parts and quasi-invariance for heat kernel measures on loop groups, J. Funct. Anal. 149, no. 2, 470–547. B. Driver (2003) Analysis of Wiener measure on path and loop groups. In: Finite and Inﬁnite Dimensional Analysis in Honor of Leonard Gross, Contemp. Math. 317, Amer. Math. Soc., 57–85. D. Freed (1987) The geometry of loop groups, J. Diﬀ. Geom. 28, 223–276. K. Gawedzki (1990) Wess–Zumino–Witten conformal ﬁeld theory. In: Constructive Quantum Field Theory, II (Erice, 1988) NATO Adv. Sci. Inst. Ser. B Phys. 234, Plenum, 89–120. K. Gawedzki (1999) Lectures on conformal ﬁeld theory. In: Quantum Fields and Strings: A Course for Mathematicians, Vol. 2, Amer. Math. Soc., 727– 805. M. Gordina (2005) Heat kernel analysis on inﬁnite dimensional groups. In: Inﬁnite Dimensional Harmonic Analysis III, World Sci. Publ., Hackensack, NJ, 71–81. L. Gross (1965) Abstract Wiener spaces, Proc. 5th Berkeley Symp. Math. Stat. Prob., Part 1, Univ. CA Press, Berkeley, 31–41. Y. Inahama (2001) Logarithmic Sobolev inequality on free loop groups for heat kernel measures associated with the general Sobolev space, J. Funct. Anal. 179, no. 1, 170–213. C. Kopper (1999) Mass generation in the large N-nonlinear sigma model, Comm. Math. Phys. 202, no. 1, 89–126. M.P. Malliavin and P. Malliavin (1992) Integration on loop groups, III. Asymptotic Peter–Weyl orthogonality, J. Funct. Anal. 108, 13–46. P. Malliavin (1991) Hypoellipticity in inﬁnite dimensions. In: Diﬀusion Process and Related Problems in Analysis, edited by M.A. Pinsky, Birkhauser, Boston, MA.

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E. Martinec (1990) The space of 2d quantum ﬁeld theories. In: The Interface of Mathematics and Particle Physics, Oxford University Press, Oxford, 107– 115. [McK] H. McKean (1969) Stochastic Integrals, Probability and Mathematical Statistics, No. 5, Academic Press. [ORW] E. Olgievtsky, N. Reshetikhin, P. Weigmann (1987) The principal chiral ﬁeld in two dimensions on classical Lie algebras. The Bethe-ansatz solution and factorized theory of scattering, Nuclear Phys. B 280, no. 1, 45–96. [Pi1] D. Pickrell (2000) Invariant measures for unitary forms of Kac-Moody groups, Memoirs Amer. Math. Soc., Number 693, 1–144. [Pi2] D. Pickrell (2006) An invariant measure for the loop space of a simply connected compact symmetric space, J. Funct. Anal. 234, 321–363. [Pi3] D. Pickrell (2006) The diagonal distribution for the invariant measure of a unitary type symmetric space, Transf. Groups, vol. 11, no. 4, 705–724. [Pi4] D. Pickrell (2008) P (φ)2 quantum ﬁeld theories and Segal’s axioms, Comm. Math. Phys. 280, 403–425. [PS] A. Pressley and G. Segal (1986) Loop Groups, Oxford University Press, Oxford. [Se] G. Segal (2003) The deﬁnition of conformal ﬁeld theory. In: Topology, Geometry and Quantum Field Theory, L.M.S. Lect. Note Sers. 308, Cambridge University Press, Cambridge, UK, 423–576. [Sen] A. Sengupta (1997) Gauge theory on compact surfaces, Memoirs Amer. Math. Soc., Number 600, 1–84. [Sm] F. A. Smirnov (1992) Dynamical symmetries of massive integrable models. In: Inﬁnite Analysis, A. Tsuchiya, T. Eguchi and M. Jimbo (eds.), Adv. Ser. Math. Phys. 16, World Scientiﬁc, 813–838. [St] D. Stroock (1996) Gaussian measures in traditional and not so traditional contexts, Bull. Amer. Math. Soc. 33, no. 2, 135–155. [We] R. Wendt (2001) A symplectic approach to certain functional integrals and partition functions, J. Geom. Phys. 40, 65–99.

Coadjoint Orbits and the Beginnings of a Geometric Representation Theory Tudor S. Ratiu Section de Math´ematiques, Ecole Polytechnique F´ed´erale de Lausanne, CH-1015 Lausanne, Switzerland, [email protected] Summary. This is a review of Banach Poisson manifolds with special emphasis on Banach–Lie–Poisson spaces. Unlike the ﬁnite-dimensional case, the existence of the (weak) symplectic leaves is not guaranteed even for coadjoint orbits in the predual of the associated Lie algebra. A signiﬁcant part of the paper is hence devoted to examples of such coadjoint orbits. A special class of K¨ ahler orbits is isolated for which the classical Borel–Weil theorem extends to the Banach case by the use of reproducing kernels. Key words: Banach Poisson manifold, Banach–Lie–Poisson space, C ∗ -algebra, W ∗ algebra, predual, coadjoint orbit, operator ideals, restricted unitary algebra, Grassmannian, GNS representation, reproducing kernel, geometric representation theory, Borel–Weil theorem. 2000 Mathematics Subject Classiﬁcations: Primary 22E46, 46T05, 53D17, 53D20. Secondary 22E46, 22E65, 22E67, 46E22, 46L10, 46L30, 46L55, 46L65, 47L20, 53D50, 53Z05, 58B12.

1 Introduction Symplectic and Poisson geometry together with Lie group actions on it lie at the foundation of the geometric formulation of physical phenomena. The theory is in very good shape in ﬁnite dimensions, and there are numerous books and articles that present this material from various points of view and emphasize diﬀerent kinds of applications in both pure and applied mathematics, as well as physics, chemistry, biology, robotics, and various engineering sciences (see, for example, [AM78], [Ar89], [AKN06], [Bl03], [BL05], [CJ04], [Co02], [CB97], [CMOI], [Da07], [DCC], [DG05], [Ga83], [GRKSD], [GS90], [HLW], [HZ94], [JSa98], [Ju97], [LM87], [LiR97], [LiM02], [Ma92], [MMOPR], [MMR], [MR94], [McDS], [McIO], [MoR05], [OB98], [ODK], [O02], [Ol86], [OrR04], [PD07], [ST07], [SDM], [SCRa], [SCRb], [Si01], [Si05], [Su03], [T00], [WS88], [YKMK], and references therein). K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_13, © Springer Basel AG 2011

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In inﬁnite dimensions, the situation is very unsatisfactory. There are essentially three levels of generalizations of symmetry that have permeated all applications. The simplest ones are Lie algebras and Lie groups whose elements are maps from a manifold to a given ﬁnite-dimensional Lie algebra or Lie group. The operations are pointwise. Often one considers certain natural extensions of such objects. The typical, and by far the best developed, such example is the family of Kac–Moody Lie algebras and of the various objects associated to them (underlying Lie groups, representations, theory of coadjoint orbits, etc.) The past decades have seen an explosive growth of work in this area with fundamental achievements that generalize the ﬁnite-dimensional theory in a spectacular way. The second level of inﬁnite-dimensional Lie groups and Lie algebras are constructed from operator algebras. At this stage of generalization, all ﬁnite-dimensional vestiges have been eliminated, and one has to deal with genuine inﬁnite-dimensional problems using the full power of functional analysis. Nevertheless, these objects still retain one key property: they are genuine Lie algebras and Lie groups and the ﬁnite-dimensional theory does serve as a guide to formulate reasonable conjectures regarding their structure and properties of their (linear and non-linear) actions. The third level, and by far the most complicated one, is formed by manifolds of maps, diﬀeomorphism groups, classes of vector ﬁelds, and their actions. Here, one is immediately faced with the necessity to specify topologies and smooth structures. The results heavily depend on these choices. Unfortunately, there is no universal selection of a manifold structure, and the guide is to take the one that is most natural for the examples one wants to consider. Thus, for certain geometric properties, a speciﬁc choice may seem very appropriate, but one may ﬁnd out later, when a partial diﬀerential equation of physical signiﬁcance appears in the study, that a diﬀerent smooth structure is better adapted to it. The present review is aimed at the second level described above, that is, the study of symplectic and Poisson geometric properties of Lie groups, Lie algebras, and their actions constructed from various algebras of operators. The key applications of such objects lie in quantum mechanics. Due to the heavy functional analytic machinery available, such objects are in the process of being understood. The point of view of this work is geometric, that is, it tries to answer typical symplectic and Poisson geometric questions that one poses to classical functional analytic objects. This review is based on joint work with D. Beltit¸a˘, A. Odzijewicz, and A.B. Tumpach [OR03], [BR05], [BR07], [BRT07] and is written closely following my presentation at the Oberwolfach meeting Inﬁnite Dimensional Lie Theory in December 2006. In §2, the notion of Banach Poisson manifold is introduced and discussed. The special case of Banach–Lie–Poisson spaces is investigated in §3 and particular attention is given to W ∗ -algebras. The symplectic leaves and, in particular, coadjoint orbits of Banach–Lie groups whose Lie algebras admit preduals, are presented in §4. The rest of the paper is

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entirely devoted to examples. In §5 various coadjoint orbits in certain families of operator algebras are presented. In particular, orbits of the restricted unitary group with special emphasis on Grassmannians and their invariant complex structures are discussed. The beginning of a geometric representation theory modeled on the Borel–Weil theorem is formulated in §6. Acknowledgements: This paper could not have been written without my collaboration with D. Beltit¸˘a, A. Odzijewicz, and A.B. Tumpach. It has been a privilege to work with them on these interesting problems. The partial support of the Swiss National Science Foundation is gratefully acknowledged.

2 Banach Poisson manifolds Sections 2–4 review some of the key results in [OR03]. We begin here with the deﬁnition and elementary properties of Banach Poisson manifolds. Recall that in ﬁnite dimensions (P, {·, ·}) is said to be a Poisson manifold if the ring of smooth real-valued functions C ∞ (P ) has a Lie algebra bracket {·, ·} satisfying the Leibniz property: {f g, h} = f {g, h} + {f, h}g for all smooth functions on P . For the theory of various aspects of Poisson manifolds see, e.g., [We83], [MR86], [LM87], [MR94], [Va94], [We98], [OrR04], and references therein. 2.1 The definition One could try to adopt the same deﬁnition if P is an inﬁnite-dimensional Banach manifold. However, this leads immediately to several problems. Indeed, by the Leibniz property, the value of the Poisson bracket at a given point p ∈ P depends only on the diﬀerentials df (p), dg(p) ∈ Tp∗ P , which implies that there is a section of the vector bundle ∧2 T ∗∗ P satisfying {f, g} = (df, dg). We shall assume throughout this paper that is a smooth section, that is, for each p ∈ P the map p : Tp∗ P × Tp∗ P → R is a continuous bilinear antisymmetric map that depends smoothly on the base point p. Let : T ∗ P → T ∗∗ P be the bundle map covering the identity deﬁned by p (dh(p)) := (·, dh)(p), that is, p (dh(p))(dg(p)) = {g, h}(p), for any locally deﬁned functions g and h. The symbol d denotes the exterior derivative on forms and hence on functions. Denote Xf := (·, df ) = (df ), or, as a derivation, Xf = {·, f }. Then Xf is a smooth section of T ∗∗ P and hence is not, in general, a vector ﬁeld on P . In analogy with the ﬁnite-dimensional case, we want Xf to be the Hamiltonian vector ﬁeld deﬁned by the function f . In order to achieve this, there are several ways to proceed. The simplest one is to assume that the Poisson bracket on P satisﬁes the condition (T ∗ P ) ⊂ T P ⊂ T ∗∗ P , which leads to the following deﬁnition.

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Definition 2.1. A Banach Poisson manifold is a pair (P, {·, ·}) consisting of a smooth Banach manifold P and a bilinear operation {·, ·} : C ∞ (P ) × C ∞ (P ) → C ∞ (P ) satisfying the following conditions: (i) (C ∞ (P ), {·, ·}) is a Lie algebra; (ii) {·, ·} satisﬁes the Leibniz identity on each factor; (iii) the vector bundle map : T ∗ P → T ∗∗ P covering the identity satisﬁes (T ∗ P ) ⊂ T P . By (iii), every h ∈ C ∞ (P ) deﬁnes a Hamiltonian vector field Xh ∈ X(P ) by Xh [f ] := df, Xh = {f, h}, where f is an arbitrary smooth locally deﬁned function on P . A smooth map ϕ : (P1 , {·, ·}1 ) → (P2 , {·, ·}2 ) is canonical or Poisson if ϕ∗ {f, g}2 = {ϕ∗ f, ϕ∗ g}1 for any two smooth locally deﬁned functions f and g on P2 . Using (iii), this is equivalent to Xf2 ◦ ϕ = T ϕ ◦ Xf1◦ϕ for any smooth locally deﬁned function f on P2 . Therefore, the ﬂow of a Hamiltonian vector ﬁeld is a Poisson map and Hamilton’s equations in Poisson bracket formulation f˙ = {f, h} are valid. Classical Poisson reduction in the spirit of [MR86] extends to Banach Poisson manifolds if one takes the closure of the relevant subspaces appearing in the reduction method. For details see [OR03]. We shall not discuss this construction in the present review. A second way to deal with the problem that has values in the bidual tangent bundle instead of the tangent bundle itself is to work with densely deﬁned Hamiltonian vector ﬁelds and to deal with harder analytic phenomena that are the non-linear analogues of unbounded operators. The only fundamental attempt along these lines was made in [CM74] and, unfortunately, this approach, eminently suited for the theory of non-linear partial diﬀerential equations, has not been further developed in a systematic way, although physically interesting examples abound; see [KP03], [Ku00], and references therein for the rigorous treatment of certain aspects of this program. We shall not comment on this point of view further here because this approach is in need of major development. 2.2 Banach symplectic manifolds Let ω ∈ Ω 2 (P ) be a 2-form on the Banach manifold P . If the continuous linear map vp ∈ Tp P → ω(p)(vp , ·) ∈ Tp∗ P is • injective, ω is said to be weak, • bijective, ω is said to be strong.

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A strong (weak) symplectic form is a closed strong (weak) form on P . Any strong symplectic Banach manifold (P, ω) is a Banach Poisson manifold: for any f ∈ C ∞ (P ) there exists a unique vector ﬁeld Xf , the Hamiltonian vector field associated to f , deﬁned by df = ω(Xf , ·). The Poisson bracket is deﬁned by {f, g} = ω(Xf , Xg ) = df, Xg ; thus df = Xf , so (T ∗ P ) ⊂ TP. If (P, ω) is weak, this construction breaks down. This shows that Deﬁnition 2.1 generalizes the notion of a strong symplectic manifold. It should be emphasized right away that there is a huge diﬀerence between strong and weak symplectic manifolds. For example, the usual Lie transform method (the “Moser trick”) extends without any diﬃculties from the ﬁnite-dimensional case to the strong Banach case to prove the Darboux theorem: any strong symplectic form is locally constant. For weak symplectic manifolds this statement is no longer true; see, for example, Exercise 3.2H in [AM78] for Marsden’s classical counterexample. This and other major diﬀerences with the strong symplectic case impose in the weak case, as already mentioned above, a diﬀerent approach such as restricting the function spaces and working with vector ﬁelds that are only densely deﬁned. Thus, it comes as a surprise that the restricted notion of Banach Poisson manifold in Deﬁnition 2.1 yields large classes of very interesting examples. One of the goals of this review is to present some of them in the framework of Banach–Lie groups and algebras.

3 Banach–Lie–Poisson spaces This section presents the simplest Banach Poisson manifolds that are not strong symplectic manifolds, namely those whose underlying manifold is a Banach space and whose Poisson bracket is linear. 3.1 Characterization A Banach–Lie algebra (g, [·, ·]) is a Banach space g that is also a Lie algebra such that the Lie bracket [·, ·] : g × g → g is bilinear and continuous. The adjoint and coadjoint maps associated to a given element x ∈ g are denoted, as usual, by adx : y ∈ g → [x, y] ∈ g and ad∗x : g∗ → g∗ , where y, ad∗x b = [x, y], b for all y ∈ g and b ∈ g∗ ; here ·, · : g × g∗ → R denotes the duality pairing between g and its Banach space dual g∗ . The maps adx and ad∗x are linear and continuous. With these preparations we can deﬁne Banach–Lie– Poisson spaces. Definition 3.1. A Banach–Lie–Poisson space (b, {·, ·}) is a Banach Poisson manifold with b a Banach space and such that the dual b∗ ⊂ C ∞ (b) is a Banach–Lie algebra under the Poisson bracket operation.

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For (b, {·, ·}) a Banach–Lie–Poisson space, denote by [·, ·] the restriction of the Poisson bracket {·, ·} from C ∞ (b) to the Lie subalgebra b∗ , that is, [x, y] := {x, y} for all x, y ∈ b∗ ⊂ C ∞ (b). Banach–Lie–Poisson spaces are characterized in the following manner. Theorem 3.2. The Banach space b is a Banach–Lie–Poisson space (b, {·, ·}) if and only if its dual b∗ is a Banach–Lie algebra (b∗ , [·, ·]) satisfying ad∗x b ⊂ b ⊂ b∗∗ for all x ∈ b∗ . Moreover, the Lie–Poisson bracket of f, g ∈ C ∞ (b) is given by {f, g}(b) = [Df (b), Dg(b)], b, (1) where b ∈ b and D denotes the Fr´echet derivative. If h is a smooth function on b, the associated Hamiltonian vector ﬁeld is given by Xh (b) = − ad∗Dh(b) b.

(2)

For example, if b is a reﬂexive Banach–Lie algebra, then its dual b∗ is a Banach–Lie–Poisson space. In particular, this recovers the classical situation of ﬁnite-dimensional Lie–Poisson spaces. A morphism between two Banach–Lie–Poisson spaces b1 and b2 is a continuous linear map φ : b1 → b2 that preserves the Poisson bracket structure, that is, {f ◦ φ, g ◦ φ}1 = {f, g}2 ◦ φ for any f, g ∈ C ∞ (b2 ); φ is also called a linear Poisson map. Note that a continuous linear map φ : b1 → b2 is a morphism of Banach–Lie–Poisson spaces if and only if φ∗ : b∗2 → b∗1 is a Banach–Lie algebra homomorphism. At this point the category of Banach–Lie–Poisson spaces has been constructed, and one can investigate various properties of its objects and morphisms. For details see [OR03]. 3.2 Examples As already mentioned, there are large classes of examples of Banach–Lie– Poisson spaces. We begin by describing the simplest ones, namely various families of operator algebras. One of the earliest references where concrete examples of Banach–Lie groups and Banach–Lie algebras of operators in Hilbert space were studied in a systematic way is [Ha72]. Let H be a complex Hilbert space. Denote by • • • •

S(H) = S(H)1 , the trace class operators on H, HS(H) = S(H)2 , the Hilbert–Schmidt operators on H, K(H), the compact operators on H, B(H) = gl(H), the bounded operators on H.

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All of them are involutive Banach algebras. In addition, S(H), HS(H), K(H) are self-adjoint ideals in B(H) and we have the continuous inclusions S(H) ⊂ HS(H) ⊂ K(H) ⊂ B(H). On the other hand, the following duality relations exist between these spaces: K(H)∗ ∼ = S(H),

HS(H)∗ ∼ = HS(H),

S(H)∗ ∼ = B(H).

The right-hand sides of these Banach space isomorphisms are all Banach– Lie algebras. These dualities are implemented by the strong non-degenerate pairing x, ρ = Trace (xρ), where x ∈ S(H), ρ ∈ K(H) for the ﬁrst isomorphism, ρ, x ∈ HS(H) for the second isomorphism, and x ∈ B(H), ρ ∈ S(H) for the third isomorphism. Therefore, the Banach spaces S(H), HS(H), and K(H) are Banach–Lie– Poisson spaces. The Lie–Poisson bracket becomes in this case {F, H}(ρ) = Trace ([DF (ρ), DH(ρ)]ρ), where ρ is an element of S(H), HS(H), or K(H), respectively. The bracket [DF (ρ), DH(ρ)] denotes the commutator bracket of operators. The Hamiltonian vector ﬁeld associated to H is given by the Lax equation ρ˙ = XH (ρ) = [DH(ρ), ρ]. Some of these examples generalize to a larger family, namely, the preduals of W ∗ -algebras. Recall that a W ∗ -algebra is a C ∗ -algebra m which possesses a predual Banach space m∗ , i.e., m = (m∗ )∗ ; this predual is unique. Since m∗ = (m∗ )∗∗ , the predual Banach space m∗ canonically embeds into the Banach space m∗ dual to m. Thus, we shall always think of m∗ as a Banach subspace of m∗ . Recall that a net {xα }α∈A ⊂ m converges to x ∈ m in the σ-topology if, by deﬁnition, limα∈A xα , b = x, b for all b ∈ m∗ . The σ-topology is Hausdorﬀ. The Alaoglu theorem states that the unit ball of m is compact in the σtopology. A fundamental result in the theory of W ∗ -algebras characterizes the predual space in the following way: the predual space m∗ is the subspace of m∗ consisting of all σ-continuous linear functionals (see, for example, [Sa71], [Ta79]). The link between W ∗ -algebras and Banach–Lie–Poisson spaces is given by the following result. Theorem 3.3. The predual m∗ of a W ∗ -algebra is a Banach–Lie–Poisson space.

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Proof. To show this, we need to check the conditions characterizing a Banach– Lie–Poisson space, namely, that m is a Banach–Lie algebra and ad∗a m∗ ⊂ m∗ for each a ∈ m. Since m is an associative Banach algebra, it is a Banach–Lie algebra relative to the commutator bracket [x, y] = xy − yx of x, y ∈ m. So the ﬁrst condition is satisﬁed. To verify the second condition, we proceed in the following way. It is known that left and right multiplication by a ∈ m deﬁne uniformly and σ-continuous maps La : m x → ax ∈ m and Ra : m x → xa ∈ m. Let L∗a : m∗ → m∗ and Ra∗ : m∗ → m∗ denote their duals. If v ∈ m∗ , then L∗a (v) and Ra∗ (v) are σ-continuous functionals and therefore, by the characterization of the predual m∗ as the subspace of σ-continuous functionals in m∗ , it follows that L∗a (v), Ra∗ (v) ∈ m∗ . Since ada = [a, ·] = La − Ra we conclude that ad∗a v = L∗a (v) − Ra∗ (v) ∈ m∗ for every v ∈ m∗ . Corollary 3.4. Let a be a C ∗ -algebra. Then its dual a∗ is a Banach–Lie– Poisson space. Proof. The bidual a∗∗ is isomorphic to the universal enveloping W ∗ -algebra of a. The C ∗ -algebra a can be thought of as a C ∗ -subalgebra of a∗∗ by the canonical inclusion a → a∗∗ . Since a∗ is predual to a∗∗ , the previous theorem guarantees that it is a Banach Lie–Poisson space. Let m∗ be the predual of the W ∗ -algebra m and ι : m∗ → m∗ be the canonical inclusion. Then ι is an injective linear Poisson map, and the Poisson structure induced by it from m∗ coincides with the original Lie–Poisson structure on m∗ . The theory of Banach–Lie–Poisson spaces is intimately connected to coherent states quantization. The literature on this subject is vast, and we shall not review it here. The papers that are closest in spirit to this Poisson geometric approach are [Od88], [Od92], [OR03], [OS97]. The references cited in these articles are an excellent guide to the literature.

4 Symplectic leaves The study of the internal structure of Banach–Lie–Poisson spaces is the subject of the rest of this review. For a ﬁnite-dimensional Poisson manifold P , the smooth characteristic distribution S whose ﬁber at a given point p ∈ P is given by the characteristic subspace Sp := {Xf (p) | f ∈ C ∞ (P )} is involutive in the sense of the Stefan–Sussmann theorem (see [St74a, St74b, Su73] or for a discussion of these results [LM87, Va94, OrR04]). Thus, this generalized distribution is integrable and hence P admits a generalized foliation

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determined by the condition that the tangent space to each leaf is precisely the value at that point of the characteristic distribution. The leaves of this generalized foliation are called symplectic leaves since the Poisson structure on P induces on each one of them a symplectic structure whose Poisson bracket coincides with the given one. The symplectic leaf through a point p ∈ P is the set of all points that can be reached from p by a broken continuous path formed by a ﬁnite number of pieces of integral curves of Hamiltonian vector ﬁelds, that is, it is the accessible set of p. For Banach Poisson manifolds this approach no longer works, essentially due to the lack of a reasonable inﬁnite-dimensional extension of the Stefan–Sussmann theorem. There have been attempts at generalizing the Stefan–Sussmann theorem to Banach manifolds (see, e.g., [Nu92]), but the few theorems that are available in the literature all have hypotheses that are violated for many examples of Banach Poisson manifolds. There is a generalization of the classical Frobenius theorem to Banach manifolds if the distribution is given by a vector subbundle ([Bo71], [Bo72], [AMR]); in particular, the ﬁber at each point (that is, the characteristic subspace) is a Banach subspace of the tangent space at that point. As discussed below, this hypothesis is almost never veriﬁed for Banach Poisson manifolds, so even Lie’s classical proof giving the decomposition into symplectic leaves of Poisson manifolds whose characteristic distributions are subbundles, does not generalize to the Banach manifold setting. As we shall see below, the situation is quite bad and not much extends from the ﬁnite-dimensional situation to the category of Banach Poisson manifolds, in general. Even reasonable examples do not behave well. 4.1 Attempt at constructing symplectic leaves Already the notion of immersion has three possible generalizations to Banach manifolds. A smooth map f : M → N between Banach manifolds is called (i) an immersion if for every m ∈ M the tangent map Tm f : Tm M → Tf (m) N is injective with closed split range; (ii) a quasi immersion if for every m ∈ M the tangent map Tm f : Tm M → Tf (m) N is injective with closed range; (iii) a weak immersion if for every m ∈ M the tangent map Tm f : Tm M → Tf (m) N is injective. Obviously, any immersion is a quasi immersion and any quasi immersion is a weak immersion. The reverse implications clearly do not hold, in general. The notion of immersion is by far the best one, and it is used in all books dealing with Banach manifolds (such as [Bo67], [Bo71], [Bo72], [AMR]). The usual properties of immersions in ﬁnite dimensions generalize without any diﬃculties to Banach manifolds. For example, every immersion is locally an inclusion in one factor of the model of the manifold N which necessarily splits.

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Quasi immersions retain some of the properties of immersions, but are far less convenient. They appear in modern inﬁnite-dimensional Lie theory when one tries to deﬁne the notion of a Lie subgroup of a not necessarily split Lie subalgebra (see §4.2). Weak immersions are rarely used because they do not have good properties. As we shall see, in the category of Banach Poisson manifolds, unfortunately, one needs to deal with this weakest notion. If (P, {·, ·}P ) is a Banach Poisson manifold, we shall adopt the same deﬁnitions, conventions, and terminology as in the ﬁnite-dimensional case. Thus, the vector subspace Sp := {Xf (p) | f ∈ C ∞ (U ), U open in P, p ∈ U } of Tp P is called the characteristic subspace at p; it is not closed in Tp P , in general. The subset S := ∪p∈P Sp ⊂ T P , is called the characteristic distribution of the Poisson structure on P ; it is not a subbundle of T P , in general. However, S is always a smooth generalized distribution, that is, for every vp ∈ Sp ⊂ Tp P there is a locally deﬁned smooth vector ﬁeld that is a section of S (namely some Xf ) and whose value at p is vp . Suppose that the characteristic distribution is integrable. Recall that for ﬁnite-dimensional manifolds this is always the case. In the Banach Poisson manifold setting this is not automatic, so it has to be assumed. This is the ﬁrst major diﬀerence with the ﬁnite-dimensional case. More diﬀerences will become apparent below. Let L be a leaf of the characteristic distribution, that is, (i) (ii) (iii) (iv)

L is a connected smooth Banach manifold, the inclusion ι : L → P is a weak injective immersion, Tq ι(Tq L) = Sq for each q ∈ L, if the inclusion ι : L → P is another weak injective immersion satisfying the three conditions above and L ⊂ L , then necessarily L = L, that is, L is maximal. If we assume, in addition, that on the leaf L

(v) there is a weak symplectic form ωL consistent with the Poisson structure on P , then L is called a symplectic leaf. The term “consistent” in point (v) needs an explanation. The linear continuous map p : Tp∗ P → Tp P induces a bijective continuous map [p ] : Tp∗ P/ ker p → Sp . By deﬁnition, ωL is consistent with the Poisson structure on P if ωL (q)(uq , vq ) = (ι(q)) ([ι(q) ]−1 ◦ Tq ι)(uq ), ([ι(q) ]−1 ◦ Tq ι)(vq ) , (3) for all q ∈ L, uq , vq ∈ Tq L, where ι : L → P is the inclusion (which is a weak immersion) and is the Poisson structure on P . Recall that is a smooth section of the vector bundle ∧2 T ∗∗ P deﬁned by {f, g} = (df, dg) for any smooth locally deﬁned functions f and g on P . The arguments on

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the right-hand side of (3) are in Tp∗ P/ ker p , so the expression has to be understood in the following way. Recall that, by deﬁnition of a Banach Poisson manifold, p : Tp∗ P → Tp P is deﬁned by αp , p (βp ) := (p) (αp , βp ) for any αp , βp ∈ Tp∗ P . Therefore, (p) induces a continuous bilinear skew-symmetric real form on Tp∗ P/ ker p , which we will denote, by abuse of notation, by the same symbol (p). So, on the right-hand side of (3), (ι(q)) is this bilinear ∗ form on the quotient Tι(q) P/ ker ι(q) . Note that formula (3) shows that the weak symplectic form ωL on L, consistent with the Poisson structure on P , is unique, if it exists. For ﬁnite-dimensional Poisson manifolds all leaves are automatically symplectic so assumption (v) above is not necessary. In the inﬁnite-dimensional case it is not known, in general, if the leaves are (weak) symplectic, even for Banach–Lie–Poisson spaces, a situation that will be discussed in great detail in the rest of this review beginning with §4.2. Proposition 4.1. Let ι : (L, ωL ) → P be a symplectic leaf of the Banach Poisson manifold (P, {·, ·}) and U ⊂ P an open subset. Then (i) for any f ∈ C ∞ (U ), q ∈ ι−1 (U ) ∩ L, one has −1 d (f ◦ ι)|ι−1 (U ) (q) = ωL (q) (Tq ι) (Xf (ι(q))), · ; (ii) the subspace ι∗ (C ∞ (P )) of C ∞ (L) consisting of functions that are obtained as restrictions of smooth functions from P is a Poisson algebra relative to the bracket {·, ·}L given by −1 −1 {f ◦ ι, g ◦ ι}L (q) := ωL (q) (Tq ι) (Xf (ι(q))), (Tq ι) (Xg (ι(q))) ; (iii) ι∗ : (C ∞ (P ), {·, ·}P ) → (ι∗ (C ∞ (P )), {·, ·}L ) is a Poisson homomorphism, that is, {f ◦ ι, g ◦ ι}L = {f, g}P ◦ ι, for any f, g ∈ C ∞ (P ). Part (i) of this proposition is remarkable since it guarantees the existence of Hamiltonian vector ﬁelds on the weak symplectic manifold L for a large class of functions, namely those that are pull-backs to the symplectic leaf. 4.2 Coadjoint orbits in Banach–Lie–Poisson spaces We begin by recalling a few standard facts from the theory of Banach–Lie groups. A homomorphism f : H → G of Banach–Lie groups is a smooth map that is a group homomorphism. A subset H ⊂ G is a split Lie subgroup of the Banach–Lie group G if H is a submanifold and a subgroup of G. In Chapter 3 of [Bo72] these are called Lie subgroups, but we shall use this term for a weaker notion discussed below. It follows that, relative to the induced group and manifold structures, H is also a Lie group. In addition, it is known ([Bo72], Chapter 3, §1, Proposition 5) that H is necessarily closed and that the

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inclusion H → G is a Banach–Lie group homomorphism and an embedding (not just an injective immersion or initial map [OrR04]). If H is a subgroup of the Banach–Lie group G, then it is a split Lie subgroup of G if and only if there is an open neighborhood V in G of the identity element such that H ∩ V is a submanifold of G ([Bo72], Chapter 3, §1, Proposition 6). The following criterion for H to be a split Lie subgroup of G is often useful. If K is another Banach–Lie group, a map g : K → H satisfying g(K) ⊂ H is a Lie group homomorphism from K to H if and only if it is a Lie group homomorphism from K to G ([Bo72], Chapter 3, §1, Proposition 5). The quotient G/H of a Banach–Lie group G by a split Banach–Lie subgroup H carries a unique manifold structure making the canonical projection G → G/H a surjective submersion ([Bo72], Chapter 3, §1, Proposition 11). If G is a ﬁnite-dimensional Lie group, it is well known that every closed subgroup H is necessarily a Lie subgroup ([Bo72], Chapter 3, §6, Proposition 2). For inﬁnite-dimensional Banach–Lie groups this result is no longer true, in general; counterexamples are given in [Bo72], Chapter 3, §8, Exercise 2, [Ho75], and [Ne06]. Because of this, one introduces the following deﬁnition. A closed subgroup H of a Banach–Lie group G which is itself a Lie group relative to the induced topology is called a Lie subgroup. Note that the Lie algebra of a Lie subgroup H is not necessarily split in the Lie algebra of G. The closed subgroups of G which are Banach–Lie groups in their own right are precisely the Lie subgroups of G (this was ﬁrst observed in [Ho75]; see also Theorem 4.3.3 in [Ne06]). It is not known if quotients by Lie subgroups are smooth manifolds. However, all other major properties for split Lie subgroups also hold for Lie subgroups (see the review [Ne06]). Let us address the question of the existence of symplectic leaves for Banach–Lie–Poisson spaces. To do this, we shall need additional hypotheses that imitate the ﬁnite-dimensional situation. Let G be a Banach–Lie group whose Banach–Lie algebra is g. We shall assume that (i) g admits a predual g∗ , that is, (g∗ )∗ = g ; (ii) the coadjoint action of G on the dual g∗ leaves the predual g∗ invariant, that is, Ad∗g (g∗ ) ⊂ g∗ , for any g ∈ G; (iii) if ρ ∈ g∗ , the closed subgroup Gρ := {g ∈ G | Ad∗g ρ = ρ} of G is a split Lie subgroup of G (a submanifold of G and not just injectively immersed). In (iii) one has to assume that Gρ is a split Lie subgroup since, as was discussed previously, closed subgroups of inﬁnite-dimensional-Banach–Lie groups are not necessarily split Lie subgroups. If Gρ is a split Lie subgroup, then one can easily show that the Lie algebra of Gρ equals gρ := {ξ ∈ g | ad∗ξ ρ = 0}. In addition, as mentioned earlier, the quotient topological space G/Gρ := {gGρ | g ∈ G} admits a unique smooth Banach manifold structure, making the canonical projection π : G → G/Gρ a surjective submersion.

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Theorem 4.2. Under the hypotheses above, the manifold G/Gρ is weak symplectic relative to the 2-form ωρ given by ωρ ([g])(Tg π(Te Lg ξ), Tg π(Te Lg η)) := ρ, [ξ, η] , where ξ, η ∈ g, g ∈ G, [g] := π(g) = gGρ , and ·, · : g∗ × g → R is the canonical duality pairing between g∗ and g. The 2-form ωρ is invariant under the left action of G on G/Gρ given by g · [h] := [gh], for g, h ∈ G. The weak symplectic structure of the coadjoint orbits of G through points of g∗ is given by the following theorem. Theorem 4.3. Let the Banach–Lie group G and the element ρ ∈ g∗ satisfy the hypotheses of the previous theorem. Then the map ι : [g] ∈ G/Gρ → Ad∗g−1 ρ ∈ g∗ is an injective weak immersion of the quotient manifold G/Gρ into the predual space g∗ . Endow the coadjoint orbit O := {Ad∗g ρ | g ∈ G} with the smooth manifold structure making ι into a diﬀeomorphism. The push forward ωO := ι∗ (ωρ ) of the weak symplectic form ωρ ∈ Ω 2 (G/Gρ ) to O has the expression ωO (Ad∗g−1 ρ) ad∗Adg ξ Ad∗g−1 ρ, ad∗Adg η Ad∗g−1 ρ = ρ, [ξ, η], for g ∈ G, ξ, η ∈ g, and ρ ∈ g∗ . Relative to this orbit symplectic form, the connected components of the coadjoint orbit O are symplectic leaves of the Banach–Lie–Poisson space g∗ . There is a remarkable particular situation that ensures that the symplectic form is strong. Theorem 4.4. Let the Banach–Lie group G and the element ρ ∈ g∗ satisfy the previous hypotheses. The following conditions are equivalent: (i) ι : G/Gρ → g∗ is an injective immersion; (ii) the characteristic subspace Sρ := {ad∗ξ ρ | ξ ∈ g} is closed in g∗ ; (iii) Sρ = g◦ρ , where g◦ρ is the annihilator of gρ in g∗ . Endow the coadjoint orbit O := {Ad∗g ρ | g ∈ G} with the manifold structure making ι a diﬀeomorphism. Then, under any of the hypotheses (i)–(iii), the weak symplectic form ωO is strong. Next, we turn to the study of concrete examples.

5 Coadjoint orbits in operator spaces In this section we shall give examples of coadjoint orbits of Banach–Lie groups that satisfy the hypotheses of the previous theorems. In addition, we shall inquire when some of these coadjoint orbits are K¨ahler. Orbits of the restricted unitary group will also be considered.

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5.1 Symplectic leaves in preduals of W ∗ -algebras The ﬁrst subsections give concrete examples of coadjoint orbits in various families of operator algebras. They review some of the major results in [BR05]. We begin by recalling standard notation and deﬁnitions from the theory of C ∗ -algebras. Let a be a unital C ∗ -algebra. If a ∈ a and ϕ ∈ a∗ denote: {a} = {b ∈ a | ab = ba}, the centralizer of a ∈ a, aϕ = {a ∈ a | ϕ(ab) = ϕ(ba) for all b ∈ a}, the centralizer of ϕ ∈ a∗ , Ga = {g ∈ a | g invertible} ⊂ gl(a), open, so it is a Banach–Lie group, Ua = {u ∈ a | uu∗ = u∗ u = 1} ⊆ Ga is a split Lie subgroup of Ga , ua = {a ∈ a | a = −a∗ } is the Banach–Lie algebra of Ua , asa = {a ∈ a | a = a∗ }, the self-adjoint elements of a, (a∗ )sa = {ϕ ∈ a∗ | ϕ(a∗ ) = ϕ(a)for all a ∈ a}, the self-adjoint functionals. In the special case of a W ∗ -algebra m we will also use the notation: m∗ = {ϕ ∈ m∗ | ϕ is w∗ -continuous}, the (unique) predual of m, ∗ sa msa ∗ = m∗ ∩ (m ) . Recall that an element ϕ ∈ m∗ is called faithful if ϕ(a∗ a) > 0 whenever 0 = a ∈ m and self-adjoint if ϕ = ϕ∗ , in the sense that ϕ(a∗ ) = ϕ(a) for all a ∈ m. As noted in §5.2 in a more general setting, faithful functionals are necessarily self-adjoint. The following statement (Theorem 2.2 in [BR05]) gives information about the unitary orbit of a faithful element in m∗ . It is based on several results in [AV99], [AV02], [Be06], [HK77], [Ta72]. Theorem 5.1. Let m be a W ∗ -algebra and ϕ ∈ m∗ faithful. The unitary orbit Uϕ = {ϕ ◦ Ad(u) | u ∈ Um } Um /Umϕ of ϕ, where Umϕ := {a ∈ Um | ϕ(ab) = ϕ(ba) for all b ∈ m} is the unitary group of the centralizer algebra mϕ , i.e., the unitary elements of mϕ . Then: (i) Uϕ ⊆ m∗ , (ii) The unitary group Umϕ of the centralizer algebra mϕ is a split Lie subgroup of Um , (iii) The unitary orbit Uϕ has a natural structure of weakly immersed submanifold of m∗ and Um acts on it smoothly on the left via (u, ψ) ∈ UM × Uϕ → ψ ◦ Ad(u−1 ) ∈ Uϕ , (iv) The smooth manifold Uϕ is simply connected. If m∗ is separable, it is known (see Remark A.2.2 in [JS97]) that there are faithful elements in m∗ .

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In particular, if m = B(H) for some complex inﬁnite-dimensional Hilbert space H, then, for any faithful state ϕ ∈ m∗ = S(H) (that is, ϕ = 1) it is known (see [AV99]) that the orbit Uϕ is not locally closed in S(H). Thus, the weakly immersed submanifolds Uϕ are never embedded submanifolds of S(H). Theorem 5.1 concerns the unitary orbits Uϕ of Um passing through a special type of elements ϕ ∈ m∗ , namely the faithful ones. The next theorem extends this result to the unitary orbits through all elements in (um )∗ . It is important to note that the predual (um )∗ consists of all self-adjoint (not skew-adjoint) elements of m∗ . Theorem 5.2. Let m be a W ∗ -algebra and ϕ∗ = ϕ ∈ m∗ . Then: (i) Umϕ is a split Lie subgroup of Um , (ii) The coadjoint orbit of the Lie group Um through ϕ ∈ (um )∗ ⊆ (um )∗ has the structure of a Um -homogeneous weak symplectic manifold which is weakly immersed into (um )∗ . Corollary 5.3. Let m be an arbitrary W ∗ -algebra and consider the corre∗ sponding real Banach–Lie–Poisson space msa ∗ = {ϕ ∈ m∗ | ϕ = ϕ }. Then sa the characteristic distribution of m∗ is integrable, and all its maximal integral manifolds are symplectic leaves and coincide with the Um -coadjoint orbits through all elements of msa ∗ . They are hence weakly immersed weak symplectic submanifolds of msa . ∗ Sometimes these weak symplectic manifolds are strong symplectic. For instance, if m = B(H) for some complex Hilbert space H, the coadjoint orbits of rank-one projections are strong symplectic because one can show that each such orbit is symplectically diﬀeomorphic to the projectivized Hilbert space P(H), which is known to be strong symplectic. Which coadjoint orbits are embedded? To my knowledge, even in ﬁnite dimensions this question is not answered. It is known that for algebraic groups this is the case and there are examples of Lie groups where this is not true (see [MR94], §14.1, Example(f) which presents Kirillov’s classical example). In inﬁnite dimensions we have the following examples of embedded coadjoint orbits due to Bona [Bon00], [Bon04]. Theorem 5.4. The embedded coadjoint orbits of GL(H) through elements of S(H) are precisely those containing ﬁnite-rank operators. On unitary orbits of ﬁnite-rank operators on Hilbert spaces, the natural quotient topology coincides with the trace-class topology. A similar characterization of the embedded unitary orbits in B(H) and in an arbitrary C ∗ -algebras can be found in [AS89] and [AS91], respectively.

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5.2 Symplectic leaves in C ∗ -algebras Certain results from the previous subsection can be extended to C ∗ -algebras. We present them below. For the proofs see [BR07] (which uses results from [Be02], [Be05], [Be06], [BR05], [KR97], [OR03]). Let a be a unital C ∗ -algebra. A continuous linear functional ϕ on a is said to be positive (respectively faithful ) if ϕ(a∗ a) ≥ 0 for every a ∈ a (respectively ϕ(a∗ a) > 0 for every 0 = a ∈ a). Continuous positive linear functionals are necessarily self-adjoint. A continuous positive linear functional ϕ ∈ a∗ is tracial if ϕ(a∗ a) = ϕ(aa∗ ) for every a ∈ a. A continuous positive functional ϕ ∈ a∗ with ϕ = 1 is called a state. Let a be a unital C ∗ -algebra and b a unital C ∗ -subalgebra of a. A bounded linear idempotent map E : a → b of norm one is called a conditional expectation. A theorem of Tomiyama (Theorem 1 in [To57] or Theorem 9.1 in [Str81]) states that E has the additional properties: E(a∗ ) = E(a)∗ ,

0 ≤ E(a)∗ E(a) ≤ E(a∗ a),

E(b1 ab2 ) = b1 E(a)b2 ,

for all a ∈ A and b1 , b2 ∈ B. We have the following result. Proposition 5.5. Let a be a unital C ∗ -algebra having a faithful tracial state τ : a → C. Deﬁne Θτ : a ∈ a → Θaτ ∈ a∗ , where Θaτ : b ∈ a → τ (ab) ∈ C. Then: τ

(i) We have ker Θτ = {0} and aΘa = {a} for all a ∈ a. (ii) The map Θτ is Ga -equivariant with respect to the adjoint action of Ga on a and the coadjoint action of Ga on a∗ . In particular, the mapping Θτ |asa : asa → (a∗ )sa is Ua -equivariant with respect to the adjoint action of Ua on asa and the coadjoint action of Ua on (a∗ )sa . (iii) For each a ∈ a the mapping Θτ induces a bijection of the adjoint orbit Ga ·a onto the coadjoint orbit Ga ·Θaτ . Thus, if a ∈ asa and there exists a conditional expectation of a onto {a} then UaO ·a L LLL τ LLΘL LLL % / Ua ·Θaτ Ua / U{a} is a commutative diagram of Ua -equivariant diﬀeomorphisms of Banach manifolds. (iv) If, moreover, a is a W ∗ -algebra and the faithful tracial state τ is normal (that is, τ is continuous on a relative to the weak ∗ topology), then im Θτ ⊆ a∗ and the hypothesis on conditional expectation in (iii) holds for each a ∈ asa . This proposition, together with the results of the previous section, has the following consequences.

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Corollary 5.6. Let a be a unital C ∗ -algebra having a faithful tracial state τ : a → C and let a = a∗ ∈ a be such that there exists a conditional expectation of a onto {a} . Then the unitary orbit of a is naturally a Ua -homogeneous weak symplectic manifold. Corollary 5.7. In a W ∗ -algebra m that admits a faithful normal tracial state the unitary orbit of each self-adjoint element has a natural structure of a Um -homogeneous weak symplectic manifold. The result in Corollary 5.6 can be strengthened using other hypotheses. Proposition 5.8. Let a be a unital C ∗ -algebra and a = a∗ ∈ a with ﬁnite spectrum. Then the unitary orbit of a has a natural structure of a Ua -homogeneous complex Banach manifold. Adding to the hypothesis above the existence of a faithful tracial state yields a much stronger result. Theorem 5.9. If a unital C ∗ -algebra a possesses a faithful tracial state, then the unitary orbit of each self-adjoint element with ﬁnite spectrum has a natural structure of a Ua -homogeneous weak K¨ ahler manifold. 5.3 Symplectic leaves in preduals of operator ideals There are results similar to the ones in the previous two subsections for preduals of operator ideals proved in [BR05]. The present subsection reviews some of them. Let H be a separable complex Hilbert space. A Banach ideal is a twosided ideal I of B(H) equipped with a norm · I satisfying T ≤ T I = T ∗ I and AT BI ≤ A T I B whenever A, B ∈ B(H). For every Banach ideal J of B(H) deﬁne UJ = U(H) ∩ (1 + J), which is a Banach–Lie group uI = u(H) ∩ J, its Banach–Lie algebra. Proposition 5.10. Let J be a Banach ideal whose underlying Banach space is reﬂexive and let T ∈ uJ . Then UJ,T := {U ∈ UJ | U T U −1 = T } is a split Lie subgroup of UJ . This proposition has the following important consequence. Corollary 5.11. Let (B, J) be a pair of Banach ideals in B(H) and assume that BJ ⊂ S(H). Suppose that the trace pairing B × J → C,

(T, S) → Trace(T S)

induces a topological isomorphism of J onto the topological dual B∗ .

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(i) If the underlying Banach spaces of B and J are reﬂexive, then the characteristic distribution of the real Banach–Lie–Poisson space uB = (uJ )∗ is integrable and all its maximal integral manifolds are symplectic leaves. (ii) Assume that the Banach–Lie group UJ is connected. Let F denote the ideal of all ﬁnite-rank operators on H. If T ∈ uB ∩ F, then the homogeneous space UJ /UJ,T has a UJ -invariant weak K¨ ahler structure and this homogeneous space is weakly immersed into uB . Here UJ,T := {U ∈ UJ | U T U −1 = T }. A typical example for which part (i) applies is a pair of Schatten ideals (Sp , Sq ) with p, q ∈ (1, ∞) and 1/p + 1/q = 1. If in part (ii) we take B = J = HS (the Hilbert–Schmidt ideal), then the weak K¨ ahler structure on the homogeneous space UJ /UJ,T guaranteed by the corollary is always strong, as shown in [Ne04]. To understand the next statement we need to introduce the concept of symmetric norming functions. Let c be the vector space of all real sequences {ξj }j≥1 such that ξj = 0 for all but ﬁnitely many indices. A symmetric norming function is a function Φ : c → R satisfying the following conditions: (i) (ii) (iii) (iv) (v)

Φ(ξ) > 0 whenever 0 = ξ ∈ c, Φ(αξ) = |α|Φ(ξ) whenever α ∈ R and ξ ∈ c, Φ(ξ + η) ≤ Φ(ξ) + Φ(η) whenever ξ, η ∈ c, Φ((1, 0, 0, . . . )) = 1, Φ({ξj }j≥1 ) = Φ({ξπ(j) }j≥1 ) whenever {ξj }j≥1 ∈ c and π : {1, 2, . . . } → {1, 2, . . . } is bijective.

Any symmetric norming function Φ gives rise to two Banach ideals SΦ (0) and SΦ as follows. For every bounded sequence of real numbers ξ = {ξj }j≥1 deﬁne Φ(ξ) := sup Φ(ξ1 , ξ2 , . . . , ξn , 0, 0, . . . ) ∈ [0, ∞]. n≥1

For all T ∈ B(H) denote T Φ := Φ({sj (T )}j≥1 ) ∈ [0, ∞], where sj (T ) = inf{T − F | F ∈ B(H), rank F < j} whenever j ≥ 1. Deﬁne SΦ := {T ∈ B(H) | T Φ < ∞}, (0)

SΦ := F (0)

·Φ

⊆ SΦ ,

that is, SΦ is the · Φ -closure of the ﬁnite-rank operators F in SΦ . Then (0) · Φ is a norm making SΦ and SΦ Banach ideals. Actually, every separable (0) Banach ideal equals SΦ for some symmetric norming function Φ. Typical examples of such ideals are the Schatten class ideals Sp for 1 ≤ p ≤ ∞. In this case the symmetric norming function is given by Φp (ξ) = ξp for ξ ∈ c.

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Every symmetric norming function Φ : c → R induces another unique symmetric norming function Φ∗ : c → R, the adjoint of Φ, by ⎧ ⎫ ∞ ⎨ 1 ⎬

ξj ηj ξ = {ξj }j≥1 ∈ c and ξ1 ≥ ξ2 ≥ · · · ≥ 0 Φ∗ (η) := sup ⎩ Φ(ξ) ⎭ j=1

whenever η = {ηj }j≥1 ∈ c and η1 ≥ η2 ≥ · · · ≥ 0. The relation (Φ∗ )∗ = Φ always holds. If Φ is any symmetric norming function, then the topological (0) dual of the Banach space SΦ is isometrically isomorphic to SΦ∗ by means of the duality pairing (0)

SΦ∗ × SΦ → C,

(T, S) → Trace(T S).

For more information on this subject see Sections 11 and 12 of Chapter III in [GK69]. For example, if 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1, Φp (ξ) = ξp , and Φq (ξ) = ξq whenever ξ ∈ c, then (Φp )∗ = Φq . Theorem 5.12. Let Φ and Ψ be symmetric norming functions, J = SΦ , T = T ∗ ∈ F, and UJ (T ) := {V ∗ T V | V ∈ UJ }. Then the following assertions hold: (i) The orbit map π : UJ → F,

V → V ∗ T V,

induces a diﬀeomorphism of the homogeneous space UJ /UJ,T onto the submanifold UJ (T ) of SΨ . (ii) If, moreover, Ψ ∗ = Φ and the Banach–Lie group UJ is connected, then the orbit UJ (T ) is a UJ -homogeneous weak K¨ ahler manifold. Note that part (i) involves two (completely unrelated to each other) symmetric norming functions. Topologically, this corresponds to the fact that any two symmetric norming functions deﬁne the same topology (namely the norm topology) on any unitary orbit of a ﬁnite-rank operator. There are many symmetric norming functions that deﬁne various types of operator ideals like Schatten, Lorentz, Orlicz and so on. See [DFWW] for a survey of this subject. 5.4 The restricted unitary algebra and its central extension In this subsection we continue to construct examples of weak symplectic leaves in Banach–Lie–Poisson spaces. The Banach–Lie groups and Lie algebras we consider here have a more intricate internal structure. The statements of the next three subsections review the major results in [BRT07]. Let H be a complex Hilbert space and assume that H = H+ ⊕ H− is the orthogonal sum of two closed subspaces H± . This decomposition is ﬁxed throughout the subsequent discussion. Sometimes, additional hypotheses are imposed on H such as separability or the inﬁnite-dimensionality of both direct

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summands H± . Let p± : H → H± denote the corresponding orthogonal projections. Denote, as usual, the Banach ideal of trace class operators in B(H) by S(H) and the Hilbert ideal of Hilbert–Schmidt operators by HS(H). The norms on these spaces are denoted by · 1 and · 2 , respectively. The constructions that follow involve the Banach–Lie group U(H) = {u ∈ B(H) | u∗ u = uu∗ = I} of unitary operators on H whose Banach– Lie algebra u(H) = {a ∈ B(H) | a∗ = −a} is formed by skew-Hermitian operators. The element d := i(p+ − p− ) ∈ u(H) allows us to introduce the restricted Banach algebra Bres : = {a ∈ B(H) | [d, a] ∈ HS(H)} = {a ∈ B(H) | ares := a + [d, a]2 < ∞}, and the restricted unitary group Ures = {u ∈ U(H) | [d, u] ∈ HS(H)} = U(H) ∩ Bres . The Banach–Lie algebra of Ures is ures = {a ∈ u(H) | [d, a] ∈ HS(H)} = u(H) ∩ Bres . Proposition 5.13. The Banach–Lie algebra (ures )∗ := {ρ ∈ u(H) | [d, ρ] ∈ HS(H), p± ρ|H± ∈ S(H± )} is a predual of the restricted unitary Banach–Lie algebra ures , the duality pairing · , · : (ures )∗ × ures → R being given by (b, c) → Trace(bc). A connected Banach–Lie group with Lie algebra (ures )∗ is U1,2 = {a ∈ U(H) | a − I ∈ HS(H), p± a|H± ∈ I + S(H± )}. Deﬁne the Banach–Lie group U1 = {a ∈ U(H) | a − I ∈ S(H)} with Banach–Lie algebra u1 = u(H) ∩ S(H) and the Hilbert–Lie group U2 = {a ∈ U(H) | a − I ∈ HS(H)} with Hilbert–Lie algebra u2 = u(H) ∩ HS(H).

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˜res = ures ⊕ R of ures relative to Finally, we introduce the central extension u the continuous two-cocycle s(A, B) := Trace(A[d, B]). ˜res is a Banach–Lie algebra endowed with the bracket Thus, u [(A, a), (B, b)]d = ([A, B], −s(A, B)) = ([A, B], − Trace(A[d, B])) . Proposition 5.14. The cohomology class [s] is a generator of the continuous Lie algebra cohomology space H 2 (ures , R). ˜res admits as predual the Banach space (˜ ures )∗ = The Banach–Lie algebra u (ures )∗ ⊕ R. The duality pairing · , ·d between (˜ ures )∗ = (ures )∗ ⊕ R and ˜res = ures ⊕ R is given by u (µ, γ), (A, a)d = µ, A + γa = Trace(µA) + γa for any A ∈ ures , µ ∈ (ures )∗ , and a, γ ∈ R. Theorem 5.15. The Banach space (˜ ures )∗ is a Banach–Lie–Poisson space for the Poisson bracket {f, g}d(µ, γ) := µ, [Dµ f (µ), Dµ g(µ)] − γs(Dµ f, Dµ g), where f, g ∈ C ∞ ((˜ ures )∗ ), (µ, γ) ∈ (˜ ures )∗ , and Dµ denotes the partial Fr´echet derivative with respect to µ ∈ (ures )∗ . The Hamiltonian vector ﬁeld deﬁned by h ∈ C ∞ ((˜ ures )∗ ) is given by Xh (µ, γ) = − ad∗(Dµ h,Dγ h) (µ, γ) = − ad∗Dµ h µ − γ[Dµ h, d], 0 . For each γ ∈ R, (ures )∗ ⊕ {γ} is a Poisson submanifold of (˜ ures )∗ for the Poisson bracket {·, ·}d . More general central extensions of Banach–Lie–Poisson spaces were constructed in [OR04]. The previous central extension (˜ ures )∗ of the Banach–Lie– Poisson space (ures )∗ is a particular case of this more general theory. The considerations above show that (˜ ures )∗ is simultaneously a Banach–Lie algebra and a Banach–Lie–Poisson space. It is therefore reasonable to search res of the Banach–Lie group Ures whose Banach– for a central extension U ˜res and to conjecture, as in the ﬁnite-dimensional case, that Lie algebra is u res is a Poisson–Lie group. Part of the problem is to this Banach–Lie group U deﬁne in precise functional analytic terms the concept of a Banach Poisson–Lie group, where duals appearing in the ﬁnite-dimensional theory of Poisson–Lie groups are replaced by preduals. A second major challenge in this respect is to work with a well-deﬁned Poisson tensor that is a section of the second wedge product of the tangent bundle and not of the second wedge product of the bidual tangent bundle.

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Theorem 5.16. The restricted unitary group Ures acts on the Banach–Lie– Poisson space (ures )∗ ⊕ {γ} ⊂ (˜ ures )∗ by aﬃne coadjoint action as follows. For g ∈ Ures , g · (µ, γ) := Ad∗ (g −1 )(µ) − γσ(g), γ , where µ ∈ (ures )∗ , γ ∈ R, and where σ : g ∈ Ures → gdg −1 − d ∈ (ures )∗ . The isotropy group of (0, γ) ∈ (ures )∗ ⊕ {γ} for the Ures -aﬃne coadjoint action is a split Lie subgroup of Ures . The smooth aﬃne coadjoint orbits of Ures are tangent to the characteristic distribution of the Banach Poisson manifold (˜ ures )∗ . For every γ = 0, the connected components of the Ures -aﬃne coadjoint orbit O(0,γ) of (0, γ) ∈ (ures )∗ ⊕{γ} are strong symplectic leaves in the Banach– Lie–Poisson space (˜ ures )∗ . Next we turn to the study of the Banach–Lie–Poisson space (ures )∗ . As we have seen above, this is also a Banach–Lie algebra and hence, as before, we conjecture that Ures is a Banach Poisson-Lie group. From the point of view of the theory of Banach–Lie–Poisson spaces, note that (ures )∗ → ures is a continuous inclusion and that [(ures )∗ , ures ] ⊆ (ures )∗ , which implies that the predual (ures )∗ is left invariant by the coadjoint representation of the Banach–Lie algebra ures and hence, by Theorem 3.2, we conclude the following. • •

The predual Banach space (ures )∗ has a natural structure of Banach–Lie– Poisson space. If ρ ∈ (ures )∗ has the property that the isotropy subgroup Ures,ρ := {u ∈ Ures | uρu−1 = ρ} is a split Lie subgroup of Ures , then the connected components of the coadjoint orbit Oρ are integral manifolds of the characteristic distribution of (ures )∗ . Moreover, Oρ is a weak symplectic manifold when equipped with the orbit symplectic structure.

One expects, therefore, that the characteristic distribution of the Banach– Lie–Poisson space (ures )∗ is integrable. Therefore, achieving this reduces to the proof that the isotropy group Ures,ρ of any ρ ∈ (ures )∗ is a split Lie subgroup of Ures . This is in turn equivalent to showing that its Lie algebra is complemented in ures . To prove this equivalence one needs to appeal to the Harris–Kaup theorem [HK77], which is applicable in this case because the isotropy subgroup Ures,ρ is an algebraic subgroup of the invertible elements in the unital Banach algebra glres := Bres ⊂ B(H), whose elements can be alternatively described as operators of the form ρ++ ρ+− ρ= ∈ B(H) ρ−+ ρ−−

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satisfying ρ±± := p± ρp± ∈ B(H± ), ρ+− := p+ ρp− ∈ HS(H− , H+ ), and ρ−+ := p− ρp+ ∈ HS(H+ , H− ). It turns out that showing that the Lie algebra of Ures,ρ is complemented in ures is not an easy task. Nevertheless, using the averaging method developed in [Ba90] and [BP07] it is possible to construct complements to the Lie algebra of Ures,ρ in ures in many important situations. Proposition 5.17. If ρ ∈ (ures )∗ and [d, ρ] = 0, then the coadjoint isotropy group of ρ is a split Lie subgroup of Ures and the connected components of the corresponding Ures -coadjoint orbit Oρ are smooth leaves of the characteristic distribution of (ures )∗ . So, the Ures -coadjoint orbit of every element ρ ∈ (ures )∗ which commutes with d is a smooth manifold; its connected components are therefore the weak symplectic leaves of the characteristic distribution of (ures )∗ . It follows that the same conclusion holds for every element ρ ∈ (ures )∗ which is Ures -conjugate to an element commuting with d, or equivalently to a diagonal operator with respect to a Hilbert basis compatible with the eigenspaces of d. The set of elements with the latter property is not equal to the whole (ures )∗ ; however, it is dense in (ures )∗ . It turns out that this property is the key modiﬁcation in the proof of the previous theorem to yield the following result. Proposition 5.18. If ρ ∈ (ures )∗ is a ﬁnite-rank operator, then the coadjoint isotropy subgroup of ρ is a split Lie subgroup of Ures , and the connected components of the corresponding Ures -coadjoint orbit Oρ are smooth leaves of the characteristic distribution of (ures )∗ . Recall that every element in the Lie algebra u(n) of skew-Hermitian matrices, the Lie algebra of the unitary group U(n), is U(n)-conjugate to a diagonal matrix with respect to a given basis of Cn . Thus, U(n) acts transitively on the set of Cartan subalgebras of u(n). This is no longer true in the inﬁnitedimensional case. Proposition 5.19. The restricted unitary group Ures does not act transitively on the set of maximal abelian subalgebras of either its Lie algebra ures or its predual (ures )∗ . Since every compact skew-Hermitian operator admits an orthonormal basis of eigenvectors, the set of conjugacy classes of maximal abelian subalgebras in (ures )∗ bijectively corresponds to U(H)/ Ures which is an inﬁnite set. The conjugacy classes of maximal abelian subalgebras are, of course, related to the conjugacy classes of maximal tori. In view of the result above, it is a diﬃcult question to decide whether a given operator ρ in (ures )∗ or ures has the property of being Ures -conjugate to a diagonal operator with respect to a basis adapted to the decomposition H = H+ ⊕ H− . The result below gives a criterion that ensures this property. Proposition 5.20. Assume that ρ ∈ (ures )∗ and that there exist an orthonormal basis {en }n≥1 and real numbers t ∈ (0, 1) and s ∈ (0, 3(1 − t)/100] such that the following conditions are satisﬁed.

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(i) We have {en | n ≥ 1} ⊆ H+ ∪ H− . (ii) The matrix (ρmn )m,n≥1 of ρ with respect to the basis {en }n≥1 has the properties |ρm+1,n+1 | ≤ t|ρm,n | whenever m, n ≥ 1, and |ρm,n |2 ≤

s2 |ρmm ρnn | whenever m, n ≥ 1 and m = n. (mn)2

Then the coadjoint isotropy group of ρ is a split Lie subgroup of Ures and the corresponding Ures -coadjoint orbit Oρ is a smooth leaf of the characteristic distribution of (ures )∗ . The following fact is worth noting. For ρ ∈ B(H) each of the following two conditions is equivalent to the existence of a unitary operator u ∈ Ures such that [d, uρu−1 ] = 0. (i) There exists p ∈ B(H) such that p = p∗ = p2 , p − p+ ∈ HS(H), and ρp = pρ. (ii) There exists an element W ∈ Grres such that ρ(W) ⊆ W. In this statement, Grres is the restricted Grassmannian which will be deﬁned and studied in §5.5. Even though it is not known if the characteristic distribution of (ures )∗ is integrable, the situation for the adjoint orbits of Ures in its Lie algebra ures is considerably better, as the following result shows. Proposition 5.21. There exists an open Ures -invariant neighborhood V of d ∈ ures such that V is a union of smooth adjoint orbits of the Banach–Lie group Ures . Now recall that u1,2 , the Lie algebra of the Banach–Lie group U1,2 , is the predual of ures (see Proposition 5.13), so one can pose the same question for the coadjoint orbits of the Banach–Lie group U1,2 . It turns out that a similar result also holds in this case as well as for (˜ ures )∗ . Proposition 5.22. For any γ ∈ R \ {0} there exists an open U1,2 -invariant neighborhood V of γd ∈ ures = u∗1,2 such that V is a union of smooth coadjoint orbits of the Banach–Lie group U1,2 . Proposition 5.23. For any γ ∈ R \ {0} there exists an open Ures -invariant neighborhood W of (0, γ) ∈ (˜ ures )∗ such that W is a union of smooth aﬃne coadjoint orbits of the Banach–Lie group Ures . In view of these properties we can conjecture that Ures is a Poisson–Lie group. As we have seen, all these algebras of operators do not behave quite like their ﬁnite-dimensional analogues. The basic reason for this is that the Banach ∗-algebra Bres that contains them is not a C ∗ -algebra. In fact, there are several major diﬀerences between them.

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Proposition 5.24. All of the unitary groups (1 + F ) ∩ U(H), U1,2 , and Ures are unbounded subsets of the unital associative Banach algebra Bres . It is well known that every self-adjoint normal functional in the predual of a W ∗ -algebra can be written as the diﬀerence of two positive normal functionals. A similar property holds for the preduals of numerous operator ideals. For example, if J and B are Banach operator ideals in B(H) such that BJ ⊂ S(H) and the trace pairing B × J → C,

(T, S) → Trace(T S)

induces a topological isomorphism of J onto the topological dual B∗ , then for every T = T ∗ ∈ B there exist T1 , T2 ∈ B such that T1 ≥ 0, T2 ≥ 0, and T = T1 − T2 . In fact, we can take T1 = (|T | + T )/2 and T2 = (|T | − T )/2, and we have T1 , T2 ∈ B since |T | ∈ B. (The latter property follows since if T = W |T | is the polar decomposition of T , then |T | = W ∗ T ∈ B.) It turns out that the predual (ures )∗ of the Lie algebra of Ures does not have the similar property of being spanned by its elements ρ with i ρ ≥ 0. In fact, the linear span of these elements is the proper subspace u1 of (ures )∗ . More precisely, we can state the following result. Proposition 5.25. The following assertions hold. (i) If a ∈ (ures )∗ and i a ≥ 0, then a ∈ S(H) and a1 ≤ a(ures )∗ ≤ √ (1 + 2)a1 . (ii) If ρ ∈ (ures )∗ \ u1 , then there exist no ρ1 , ρ2 ∈ (ures )∗ such that i ρ1 ≥ 0, i ρ2 ≥ 0, and ρ = ρ1 − ρ2 . 5.5 The restricted Grassmannian The restricted Grassmannian Grres is deﬁned as the set of subspaces W of the Hilbert space H such that the orthogonal projection from W to H+ (respectively to H− ) is a Fredholm operator (respectively a Hilbert–Schmidt operator). This Hilbert manifold was studied extensively in [PS90] and [Wu01]. Since Grres = Ures / (U(H+ ) × U(H− )), it follows that Grres is a homogeneous space under the natural action of Ures . The connected components of Ures are the sets

U++ U+−

k ∈ Ures index(U++ ) = k for k ∈ Z. Ures = U−+ U−− The pairwise disjoint sets

Grkres = W ∈ Grres index(p+ |W : W → H+ ) = k ,

k∈Z

are the images of Ukres by the projection Ures → Grres , and thus they are the connected components of Grres . So, the connected component of Grres containing H+ is Gr0res .

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The K¨ ahler form ωGr of Grres is the Ures -invariant 2-form whose value at H+ is given by ωGr (X, Y ) = 2 Im Trace(X ∗ Y ), where X, Y ∈ HS(H+ , H− ) TH+ (Grres ) and Im z denotes the imaginary part of z ∈ C. Equivalently, ωGr is the quotient of the real-valued antisymmetric bilinear form ΩGr on ures which vanishes on u(H+ ) ⊕ u(H− ) and is invariant under the U(H+ ) × U(H− )-action: ΩGr (A, B) = −2s(A, B) = −2 Trace(A[d, B]), where A, B ∈ ures . In this paper, an element A++ −A∗−+ A= ∈ ures A−+ A−− is identiﬁed with the vector X = A−+ ∈ HS(H+ , H− ) TH+ (Grres ). The Poisson geometric relationship between the restricted Grassmannian and the characteristic distribution of (˜ ures )∗ is given by the following result. Theorem 5.26. The connected components of the restricted Grassmannian are strong symplectic leaves in the Banach–Lie–Poisson space (˜ ures )∗ . More precisely, for every real γ = 0, the Ures -aﬃne coadjoint orbit O(0,γ) of (0, γ) ∈ (ures )∗ ⊕{γ} is isomorphic to the restricted Grassmannian Grres via the smooth diﬀeomorphism Φγ : W ∈ Grres → 2 i γ(pW − p+ ) ∈ O(0,γ) , where pW denotes the orthogonal projection on W . The pull-back by Φγ of the orbit symplectic form on O(0,γ) equals (γ/2)-times the symplectic form ωGr on Grres . 5.6 Hilbert–Schmidt skew-Hermitian operators If instead of ures one considers the Hilbert–Lie algebra u2 = u(H) ∩ HS(H), whose underlying Hilbert–Lie group is U2 = {a ∈ U(H) | a − I ∈ HS(H)}, the results of the previous subsections can be signiﬁcantly improved. Denote by ˜ u2 := u2 ⊕R the central extension of u2 deﬁned by the restriction ˜2 implies of the two-cocycle s to u2 × u2 . The natural isomorphism (˜ u2 )∗ u ˜2 is a Banach–Lie–Poisson space, for the Poisson bracket given by that u {f, g}d(µ, γ) := µ, [Dµ f (µ), Dµ g(µ)] − γs(Dµ f, Dµ g), where f, g ∈ C ∞ (˜ u2 ), (µ, γ) ∈ ˜ u2 , and Dµ denotes the partial Fr´echet derivative with respect to µ ∈ u2 . Our ﬁrst result is the following.

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Theorem 5.27. The characteristic distribution of the Hilbert–Lie–Poisson space (˜ u2 )∗ ˜ u2 (˜ u2 )∗ is integrable. The (co)adjoint orbits of U2 are the weak symplectic leaves of this distribution. The following has been shown in Theorem 3.5 of [Ca85] and Proposition V.7 in [Ne02a]. Proposition 5.28. The Hilbert–Lie group U2 acts transitively on the connected component Gr0res of the restricted Grassmannian Grres . The action of the split Lie subgroup U1,2 of U2 on Gr0res has been studied in Section 1.3.4 of [Tu05], in which the following result was established; see also [BRT07] for a geometric proof of this statement. Proposition 5.29. The split Lie subgroup U1,2 ⊂ U2 acts transitively on Gr0res . As was the case before, the Grassmannians are important special aﬃne coadjoint orbits. Theorem 5.30. The connected component Gr0res of the restricted Grassman˜2 . More nian is a strong symplectic leaf in the Hilbert–Lie–Poisson space u ˜(0,γ) of (0, γ) ∈ u ˜2 precisely, for every γ = 0, the U2 -aﬃne coadjoint orbit O is diﬀeomorphic to Gr0res via the map Φγ : W ∈ Gr0res → 2 i γ(pW − p+ ) ∈ O(0,γ) , where pW denotes the orthogonal projection on W . The pull-back by Φγ of the ˜(0,γ) equals (γ/2)-times the symplectic form ωGr orbit symplectic form on O 0 on Grres . The existence of invariant complex structures on the coadjoint orbits is given by the following result. Theorem 5.31. Every symplectic leaf of the Hilbert–Lie–Poisson space ˜u2 is transitively acted upon by U2 by means of the aﬃne coadjoint action and is U2 -equivariantly covered by some complex homogeneous space of U2 . In particular, every simply connected symplectic leaf of the Banach–Lie– ˜2 has a U2 -invariant complex structure. For instance, this Poisson space u is the case for the connected component Gr0res of the restricted Grassmannian ˜2 . viewed as a symplectic leaf of u Remark. Even though not presented here, the theory of coadjoint orbits in L∗ -algebras is well developed and gives rise to a large class of examples with remarkable additional properties. For the development of this subject see [Ne04], [Tu05], [Tu06], and [Tu09] and references therein.

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6 Geometric representation theory In view of the results of the previous section, one can hope for an inﬁnitedimensional extension of the classical techniques of geometric representation theory. In order to obtain this, one could proceed as in the theory of geometric realizations of Lie group representations (see, e.g., [DF95], [Do97], [N200], [ES02]) by trying to build representation spaces as spaces of sections of certain vector bundles. The basic ingredient in this construction is the reproducing kernel Hilbert space (see, e.g., [Ar50], [Sc64], [Ha82], [Sai88], [Arv98], [BH98], [N200], [CG01], [CG06]). This method can be applied to the case of group representations obtained by restricting Gelfand–Na˘ımark–Segal (GNS) representations to unitary groups of C ∗ -algebras as shown in [BR07]. The goal of this subsection is to review these results. The idea is to build for these representations one-to-one intertwining operators from the representation spaces onto reproducing kernel Hilbert spaces of sections of certain Hermitian vector bundles. The construction of these Hermitian vector bundles is based on a choice of a C ∗ -subalgebra that is related in a very precise manner to the state involved in the GNS construction. In the case of normal states of W ∗ -algebras, this subalgebra is chosen to be the centralizer subalgebra of the state in question, and the base of the corresponding vector bundle is one of the symplectic leaves presented in the previous section. As we have seen, these symplectic leaves are unitary orbits of states. Therefore, the geometric representation theory reviewed here shows that the equivalence class of an irreducible GNS representation only depends on the unitary orbit of the corresponding pure state (see [GK60]). A lot of work has been done on the classiﬁcation of unitary group representations of operator algebras (see, for example, [Boy80], [Boy88], [Boy93], and references therein). As will be discussed below, some of these representations, namely the ones obtained by restricting GNS representations to unitary groups, can be realized geometrically in the spirit of the classical Borel–Weil theorem for compact groups. 6.1 GNS unital ∗-representation Let a be a unital C ∗ -algebra, and ϕ : a → C a state, that is, a continuous positive linear functional with ϕ = 1. Deﬁne the Hilbert space H as the completion of the quotient of a by the null space of the non-negative Hermitian sesquilinear form a × a (a1 , a2 ) → ϕ(a∗2 a1 ) ∈ C. For each a ∈ a, the linear continuous map a ∈ a → aa ∈ a induces a linear continuous map ρ(a) : H → H. This deﬁnes the norm-continuous GNS unital ∗-representation ρ : a → B(H).

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6.2 The fundamental construction Let a be a unital C ∗ -algebra, b a unital C ∗ -subalgebra of a, and ϕ : a → C a state such that there exists a conditional expectation E : a → b with ϕ◦E = ϕ. Let ρ : a → B(H) and ρϕ : b → B(Hϕ ) be the GNS unital ∗representations of a and b corresponding to ϕ and ϕ|b , respectively. Then Hϕ is a closed subspace of H, and for all b ∈ b the diagram ρ(b)

a → H −−−−→ ⏐ ⏐ ⏐PH ⏐ E ϕ

H ⏐ ⏐PH ϕ

ρϕ (b)

b → Hϕ −−−−→ Hϕ commutes, where PHϕ : H → Hϕ denotes the orthogonal projection, while a → H and b → Hϕ are the natural maps. The conditional expectation E : a → b is continuous with respect to the GNS scalar products on a and b: if a ∈ a then E(a)∗ E(a) ≤ E(a∗ a), so ϕ(E(a)∗ E(a)) ≤ ϕ(E(a∗ a)) = ϕ(a∗ a). Its “extension” by continuity is PHϕ . Restricting, we get a norm-continuous unitary representation ρ|Ub : Ub → U(Hϕ ) of the unitary group Ub of b. Since Ub is a Lie group with the topology inherited from the unitary group Ua of a, and the self-adjoint mapping E gives a continuous projection of the Lie algebra of Ua onto the Lie algebra of Ub , it follows that we have a principal Ub -bundle πb : u ∈ Ua → u · Ub ∈ Ua / Ub . Let Πϕ,b : [u, f ] ∈ Dϕ,b := Ua ×Ub Hϕ → u·Ub ∈ Ua / Ub be the associated vector bundle, i.e., (u1 , f1 ) ∼ (u2 , f2 ) if and only if there is some v ∈ Ub such that u1 = u2 v and f1 = ρϕ (v −1 )f2 . This associated bundle Πϕ,b : Dϕ,b = Ua ×Ub Hϕ → Ua / Ub has additional structure. (i) The real analytic Ua -actions on Ua / Ub and on Dϕ,b given by αb : (u , u · Ub ) ∈ Ua × Ua / Ub → u u · Ub ∈ Ua / Ub βϕ,b : (u , [u, f ]) ∈ Ua ×Dϕ,b → [u u, f ] ∈ Dϕ,b make the diagram Ua ×Dϕ,b ⏐ ⏐ idUa ×Πϕ,b

βϕ,b

−−−−→

Dϕ,b ⏐ ⏐Π ϕ,b

α

Ua × Ua / Ub −−−b−→ Ua / Ub commutative. (ii) The vector bundle Πϕ,b : Dϕ,b → Ua / Ub is homogeneous Hermitian. Indeed, there is a bundle Hermitian metric relative to which its ﬁbers are complex Hilbert spaces and for each u ∈ Ua the mapping βϕ,b (u, ·) : Dϕ,b → Dϕ,b is (bounded linear and) ﬁber-wise unitary.

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For example, if b = a and E = ida , then the homogeneous vector bundle associated with ϕ and b is just the vector bundle with the base reduced to a single point and with the ﬁber equal to H. The other extreme case is obtained when b = C1 and E(·) = ϕ(·)1. Let T = {z ∈ C | |z| = 1} be the unit circle in C. In this case, it follows that the homogeneous vector bundle associated with ϕ and b is a line bundle whose base is the Banach–Lie group Ua /T1. If the C ∗ -algebra a is a W ∗ -algebra, then there are natural choices that guarantee the hypotheses of the previous construction. Theorem 6.1. If m is a W ∗ -algebra and 0 ≤ ϕ ∈ m∗ , then there always exists a conditional expectation Eϕ : m → mϕ satisfying ϕ ◦ Eϕ = ϕ. 6.3 Reproducing kernels We need to work with vector bundles in diﬀerent categories; here we establish terminology and notation. A vector bundle Π : D → T is called (i) continuous (smooth) if D and T are topological spaces (smooth manifolds), Π is continuous (smooth), the ﬁbers Dt := Π −1 (t) for t ∈ T are complex Banach spaces, and Π is locally trivial with ﬁberwise bounded continuous (smooth) linear trivializations; (ii) Hermitian if the continuous vector bundle D has a continuous Hermitian structure (· | ·) : D ⊕ D → C that makes each Dt into a complex Hilbert space; (iii) holomorphic Hermitian if, in addition, D and T are complex Banach manifolds, Π is holomorphic, and the Hermitian structure is smooth. We denote by C(T, D), C ∞ (T, D), and O(T, D) the spaces of continuous, smooth, and holomorphic sections of the bundle Π : D → T . Let Π : D → T be a continuous Hermitian vector bundle, p1 , p2 : T ×T → T the projections on the ﬁrst and second factors, and p∗j Π : p∗j D → T × T the pull-back of Π by pj , j = 1, 2, which are in turn Hermitian vector bundles. Also, consider the continuous vector bundle Hom(p∗2 Π, p∗1 Π) → T × T , whose ﬁber over (s, t) ∈ T ×T is the Banach space of bounded linear maps B(Dt , Ds ) between the corresponding ﬁbers. Definition 6.2. A positive definite reproducing kernel on Π is a continuous section K ∈ C(T × T, Hom(p∗2 Π, p∗1 Π)) having the property that for every integer n ≥ 1 and all choices of t1 , . . . , tn ∈ T and ξ1 ∈ Dt1 , . . . , ξn ∈ Dtn , we have n j,l=1

(K(tl , tj )ξj | ξl )Dtl ≥ 0.

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If Π is a holomorphic vector bundle, we say that the reproducing kernel K is holomorphic if for each t ∈ T and every ξ ∈ Dt we have K(·, t)ξ ∈ O(T, D). The deﬁnition immediately implies that (i) for all t ∈ T we have K(t, t) ≥ 0 in B(Dt ); (ii) for all t, s ∈ T we have K(t, s) ∈ B(Ds , Dt ), K(s, t) ∈ B(Dt , Ds ), and K(t, s)∗ = K(s, t). The following statement is Theorem 4.2 in [BR07]. Theorem 6.3. Let Π : D → T be a continuous Hermitian vector bundle and K ∈ C(T × T, Hom(p∗2 Π, p∗1 Π)). Then K is a reproducing kernel on Π if and only if there exists a linear mapping ι : H → C(T, D), where H is a complex Hilbert space, such that evιt := (ι(·))(t) : H → Dt is a bounded linear operator for all t ∈ T and K(s, t) = evιs (evιt )∗ : Dt → Ds , for all t ∈ T . If this is the case, then the mapping ι can be chosen to be injective. If Π is a holomorphic vector bundle, then the reproducing kernel K is holomorphic if and only if any mapping ι as above has the range contained in O(T, D). 6.4 Reproducing kernels and GNS-representations Let a be a unital C ∗ -algebra, b a unital C ∗ -subalgebra of a, and ϕ : a → C a state such that there exists a conditional expectation E : a → b satisfying ϕ◦E = ϕ. Let ρ : a → B(H) and ρϕ : b → B(Hϕ ) be the GNS representations associated to ϕ and ϕ|b . Let Πϕ,b : Dϕ,b = Ua ×Ub Hϕ → Ua / Ub be the homogeneous Hermitian vector bundle associated to ϕ and b. Deﬁne the realization operator ιϕ,b : H → C(Ua / Ub , Dϕ,b ) associated with ϕ and b by ιϕ,b (h)(u Ub ) = [u, PHϕ (ρ(u)−1 h)] for all h ∈ H and u ∈ Ua and the reproducing kernel Kϕ,b ∈ C (Ua / Ub ) × (Ua / Ub ), Hom(p∗2 (Πϕ,b ), p∗1 (Πϕ,b )) associated with ϕ and b by Kϕ,b (u1 Ub , u2 Ub )[u2 , f ] = [u1 , PHϕ (ρ(u−1 1 u2 )f )], where u1 , u2 ∈ Ua , f ∈ Hϕ , and p1 , p2 : (Ua / Ub ) × (Ua / Ub ) → Ua / Ub are the projections on the ﬁrst and second factors, respectively.

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Theorem 6.4. The realization operator ιϕ,b : H → C(Ua / Ub , Dϕ,b ) associated with ϕ and b is injective, has the property that (ιϕ,b (·))(s) : H → (Dϕ,b )s is bounded linear for all s ∈ Ua / Ub , and the corresponding reproducing kernel (given by the general theory) is precisely the reproducing kernel Kϕ,b associated with ϕ and b. Moreover, ιϕ,b intertwines the unitary representation ρ|Ua of Ua on H and the natural representation of Ua by linear mappings on C(Ua / Ub , Dϕ,b ). So the representation space has been realized as a space of sections of a homogeneous Hermitian vector bundle. The following result presents the important case when the homogeneous vector bundle occurring in the theorem is holomorphic and the reproducing kernel Hilbert space consists only of holomorphic sections. Corollary 6.5. Let m be a W ∗ -algebra with a faithful normal tracial state τ : m → C. Let a ∈ m have ﬁnite spectrum and satisfy 0 ≤ a, τ (a) = 1. Deﬁne ϕ : m → C by ϕ(b) = τ (ab). Then the homogeneous vector bundle Πϕ,mϕ : Dϕ,mϕ → Um / Umϕ associated with ϕ and mϕ is holomorphic and the reproducing kernel Kϕ,mϕ is holomorphic as well. Also, if ρ : m → B(H) is the GNS representation corresponding to the normal state ϕ, then the image of the realization operator ιϕ,mϕ : H → C(Um / Umϕ , Dϕ,mϕ ) consists of holomorphic sections. The techniques of the proof of this theorem (see [BR07]) are based on results in [Be02] and [Be05]. For completeness, let us also mention the following related statement, which in some sense provides a complexiﬁcation of the picture drawn in Theorem 6.4. This is a special case of a theorem proved in [BG07] for Stinespring dilations of completely positive maps. However, in order to avoid introducing extra constructions, we are going to state it in the setting of Theorem 6.4, for the special case of GNS representations (which are just Stinespring dilations of C ∗ -algebra states). Theorem 6.6. In the setting of the beginning of the present subsection, let Ga be the group of invertible elements in the unital C ∗ -algebra a with its split Lie subgroup Gb of invertible elements in the C ∗ -subalgebra b, and deﬁne the inclusion mapping θ : Ua /Ub → Ga /Gb by uUb → uGb . If we deﬁne p = {X ∈ a | X ∗ = −X}, then the following assertions hold. (i) There exists a real analytic diﬀeomorphism Ga → Ua × p × G+ b,

a → (u(a), X(a), b(a)),

where G+ b denotes the set of positive elements in Gb , such that a = u(a) exp(iX(a))b(a) for all a ∈ a, which induces a polar decomposition in Ga /Gb given by aGb = u(a) exp(iX(a))Gb for every a ∈ Ga .

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(ii) The mapping −∗ : Ga /Gb → Ga /Gb , u exp(iX)Gb → u exp(−iX)Gb , is an anti-holomorphic involutive diﬀeomorphism of Ga /Gb whose ﬁxedpoint set is precisely the image of Ua /Ub under the aforementioned embedding θ. (iii) The projection Ga /Gb → Ua /Ub , u exp(iX)Gb → uUb , has the structure of a vector bundle isomorphic to the tangent bundle of the homogeneous space Ua /Ub , which is in turn isomorphic to the homogeneous vector bundle Ua ×Ub p → Ua /Ub . The corresponding isomorphism is given by u exp(iX)Gb → [(u, X)] for all u ∈ Ua , X ∈ p. (iv) Let HU (E, ϕ) ⊆ C(Ua /Ub , D) be the range of the realization operator ιϕ,b and H(E, ϕ) ⊆ O(Ga /Gb , Ga ×Gb Hϕ ) the range of the holomorphic realization operator γϕ,b : H → O(Ga /Gb , Ga ×Gb Hϕ ),

γϕ,b (h)(aGb ) = [(a, PHϕ (ρ(a)−1 h))].

Denote by µ(a) both operators on the section spaces C(Ua /Ub , D) and O(Ga /Gb , Ga ×Gb Hϕ ) deﬁned by natural multiplication by a ∈ Ua and a ∈ Ga , respectively. Then HU (E, ϕ) and H(E, ϕ) are Hilbert spaces isometric to H by means of the realization operators ιϕ,b and γϕ,b , and the diagram ιϕ,b

HU (E, ϕ) ←−−−− ⏐ ⏐ µ(a) ιϕ,b

γϕ,b

H −−−−→ H(E, ϕ) ⏐ ⏐ ⏐ρ(a) ⏐µ(a) γϕ,b

HU (E, ϕ) ←−−−− H −−−−→ H(E, ϕ) is commutative for every a ∈ Ua . (v) If we denote by T1 µ the tangent map of µ(·)|H(E,ϕ) at the identity 1 ∈ Ga , then T1 µ is a ∗-representation of the C ∗ -algebra a which extends the group representation µ : Ga → B(H(E, ϕ)). Moreover, there exists a bundle section h0 ∈ H(E, ϕ) such that ϕ(a) = (T1 µ)(a)h0 , h0 for every a ∈ a, where ·, · stands for the scalar product in the Hilbert space H(E, ϕ) consisting of holomorphic sections in the vector bundle Ga ×Gb Hϕ → Ga /Gb . 6.5 Example Let us analyze Corollary 6.5 in a very simple ﬁnite-dimensional situation in order to get a feeling of what it really says. Let n ≥ 1, m = gl(n, C) (which is a W ∗ -algebra), and let Cn be the complex n-dimensional Hilbert space with the usual scalar product (· | ·). Next, denote ⎛ ⎞ ⎛ ⎞ 1 0 ... 0 1 ⎜0 0 . . . 0⎟ ⎜0⎟ ⎜ ⎟ ⎜ ⎟ and p = ⎜ . . . . ⎟ ∈ m. h = ⎜.⎟ . . . . . ⎝. . . .⎠ ⎝.⎠ 0

0 0 ... 0

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Now deﬁne ϕ : m → C by ϕ(a) = (ah|h) = Trace(ap) = a11 for a = (aij )1≤i,j≤n ∈ m. It is clear that ϕ is a pure normal state of m. We begin by constructing the GNS representation of m with respect to ϕ. The kernel of the sesquilinear form m × m (a1 , a2 ) → ϕ(a∗2 a1 ) ∈ C is ⎧⎛ ⎞⎫ 0 ∗ ··· ∗ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨⎜0 ∗ · · · ∗⎟⎪ ⎜ ⎟ ∗ m0 := {a ∈ m | ϕ(a a) = 0} = {a ∈ m | ah = 0} = ⎜ . . . . ⎟ ⎪ ⎝ . . . . ⎠⎪ ⎪ ⎪ ⎪ . . . . ⎪ ⎭ ⎩ 0 ∗ ··· ∗ and ⎛

ϕ(a∗ a) = |a11 |2 + |a21 |2 + · · · + |an1 |2

a11 ⎜ a21 ⎜ for a = ⎜ . ⎝ ..

0 ··· 0 ··· .. . . . .

⎞ 0 0⎟ ⎟ .. ⎟ . .⎠

an1 0 · · · 0 Hence, the completion H of m/m0 with respect to the scalar product induced by (a, b) → ϕ(b∗ a) is just the Hilbert space Cn with the usual scalar product, viewed as the set of column vectors, and the natural mapping m → H is ⎛ ⎞ ⎛ ⎞ a11 a11 a12 · · · a1n ⎜ a21 ⎟ ⎜ a21 a22 · · · a2n ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ .. .. . . .. ⎟ → ⎜ .. ⎟ . ⎝ . ⎠ ⎝ . . ⎠ . . an1 an1 an2 · · · ann Moreover, according to the way matrices multiply, it follows that for each a ∈ m the operator ρ(a) : H → H that makes the diagram b →ab

m −−−−→ ⏐ ⏐

m ⏐ ⏐

ρ(a)

H −−−−→ H commutative is just the natural action of a ∈ m = gl(n, C) on Cn . Thus, the GNS representation of m associated to ϕ is just the natural representation of m = gl(n, C) on Cn . We have ⎧⎛ ⎞⎫ ⎪ ⎪ ⎪ ∗ 0 ··· 0 ⎪ ⎪

⎬ ⎨⎜0 ∗ · · · ∗⎟⎪ z 0

⎟ ⎜ ϕ m = {p} = ⎜ . . . . ⎟ =

z ∈ C, W ∈ gl(n − 1, C) 0W ⎪ ⎝ .. .. . . .. ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 ∗ ··· ∗ C × gl(n − 1, C).

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Hence, mϕ ∩ m0 = (1 − p)m(1 − p) gl(n − 1, C), and thus the space of the GNS representation ρϕ : mϕ → B(Hϕ ) of mϕ corresponding state to the z 0 ϕ is the ϕ|mϕ : m → C is one dimensional, i.e., Hϕ = C. Moreover, ρϕ 0W multiplication-by-z operator on Hϕ for all z ∈ C and W ∈ gl(n − 1, C). Now Um / Umϕ = U(n)/(U(1) × U(n − 1)) = Pn−1 (C), and it follows at once that the homogeneous vector bundle Πϕ,mϕ : Dϕ,mϕ → Um / Umϕ associated with ϕ is dual to the tautological line bundle over the complex projective space Pn−1 (C). Thus, in this special case, the theorem says that the natural representation of U(n) on Cn can be geometrically realized as a representation in the ﬁnitedimensional vector space of global holomorphic sections of the dual to the tautological line bundle over the U(n)-homogeneous compact K¨ ahler manifold Pn−1 (C), which is a special case of the Borel–Weil theorem.

References [AM78]

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Tudor S. Ratiu Olver, PJ (1986) Applications of Lie Groups to Diﬀerential Equations. Volume 107 of Graduate Texts in Mathematics. Springer-Verlag, Berlin Ortega, JP, Ratiu, TS (2004) Momentum Maps and Hamiltonian Reduction. Volume 222 of Progress in Mathematics. Birkh¨ auser, Boston, MA Ostrowski, J, Burdick, J (1998) The geometric mechanics of undulatory robotic locomotion. The International Journal of Robotics Research 17(7): 683–701 Ostrowski, JP, Desai, JP, Kumar, V (2000) Optimal gait selection for nonholonomic locomotion systems. The International Journal of Robotics Research 19(3): 225–237 Peng J, Dabiri, JO (2007) A potential-ﬂow, deformable-body model for ﬂuid-structure interactions with compact vorticity: application to animal swimming measurements. Experiments in Fluids 43(5): 655–664 Pressley, A, Segal, G (1990) Loop Groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, Oxford Saitoh, S (1988) Theory of Reproducing Kernels and Its Applications. Volume 189 of Pitman Research Notes in Mathematics Series. Longman Scientiﬁc & Technical, Harlow Sakai, S (1971) C ∗ -Algebras and W ∗ -Algebras. 1998 reprint of the 1971 edition. Volume 60 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, New York Schwartz, L (1964) Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux associ´es (noyaux reproduisants). J. Analyse Math. 13: 115–256 Sfakiotakis, M, Tsakiris, DP (2007) Biomimetic centering for undulatory robots. The International Journal of Robotics Research 26(11-12): 1267–1282 Shadden, SC, Dabiri, JO, Marsden, JE (2006) Lagrangian analysis of entrained and detrained ﬂuid in vortex rings. Physics of Fluids 18, 1–11 Shammas, EA, Choset, H, Rizzi, AA (2007) Geometric motion planning analysis for two classes of underactuated mechanical systems. The International Journal of Robotics Research 16(10): 1043–1073 Shammas, EA, Choset, H, Rizzi, AA (2007) Towards a uniﬁed approach to motion planning for dynamic underactuated mechanical systems with non-holonomic constraints. The International Journal of Robotics Research 26(10): 1075–1124 Singer, SF (2001) Symmetry in Mechanics. A Gentle, Modern Introduction. Birkh¨ auser, Boston, MA Singer, SF (2005) Linearity, Symmetry, and Prediction in the Hydrogen Atom. Undergraduate Texts in Mathematics. Springer-Verlag, New York Stefan, P (1974a) Accessible sets, orbits and foliations with singularities. Proc. London Math. Soc. 29: 699–713 Stefan, P (1974b) Accessibility and foliations with singularities. Bull. Amer. Math. Soc. 80: 1142–1145 Str˘ atil˘ a, S ¸ (1981) Modular Theory in Operator Algebras. Editura Acad., Bucure¸sti; Abacus Press, Tunbridge Wells, 1981 Suris, Y (2003) The Problem of Integrable Discretization: Hamiltonian Approach. Volume 219 in Progress in Mathematics. Birkh¨ auser-Verlag, Basel

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Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces Joseph A. Wolf Department of Mathematics, University of California, Berkeley, CA 94720–3840, USA, [email protected] Summary. We study direct limits (G, K) = lim (Gn , Kn ) of compact Gelfand pairs. −→ First, we develop a criterion for a direct limit representation to be a multiplicityfree discrete direct sum of irreducible representations. Then we look at direct limits G/K = lim Gn /Kn of compact riemannian symmetric spaces, where we combine −→ our criterion with the Cartan–Helgason theorem to show in general that the regular representation of G = lim Gn on a certain function space lim L2 (Gn /Kn ) is −→ −→ multiplicity-free. That method is not applicable for direct limits of nonsymmetric Gelfand pairs, so we introduce two other methods. The ﬁrst, based on “parabolic direct limits” and “deﬁning representations”, extends the method used in the symmetric space case. The second uses some (new) branching rules from ﬁnite-dimensional representation theory. In both cases we deﬁne function spaces A(G/K), C(G/K), and L2 (G/K) to which our multiplicity-free criterion applies. Key words: Lie group, Gelfand pair, commutative space, direct limit representation, multiplicity-free representation. 2000 Mathematics Subject Classifications: 20G05, 22E45, 22E65, 43A85, 43A90.

1 Introduction Gelfand pairs (G, K), and the corresponding “commutative” homogeneous spaces G/K, form a natural extension of the class of riemannian symmetric spaces. We recall some of their basic properties. Let G be a locally compact topological group, K a compact subgroup, and M = G/K. Then the following conditions are equivalent; see [W2007, Theorem 9.8.1]. 1. (G, K) is a Gelfand pair, i.e., L1 (K\G/K) is commutative under convolution. 2. If g, g ∈ G then µKgK ∗ µKg K = µKg K ∗ µKgK (convolution of Dirac measures on K\G/K). 3. Cc (K\G/K) is commutative under convolution.

K.-H. Neeb and A. Pianzola (eds.), Developments and Trends in Infinite-Dimensional Lie Theory, Progress in Mathematics 288, DOI 10.1007/978-0-8176-4741-4_14, © Springer Basel AG 2011

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4. The measure algebra M(K\G/K) is commutative. 5. The representation of G on L2 (M ) is multiplicity-free. If G is a connected Lie group one can also add 6. The algebra of G-invariant diﬀerential operators on M is commutative. When we drop the requirement that K be compact, conditions 1, 2, 3, and 4 lose their meaning because integration on M or K\G/K no longer corresponds to integration on G. Condition 5 still makes sense as long as K is unimodular in G. Condition 6 remains meaningful (and useful) whenever G is a connected Lie group; there one speaks of “generalized Gelfand pairs”. In this paper we look at some cases where G and K are not locally compact, in fact are inﬁnite dimensional, and show in those cases that the multiplicityfree condition 5 is satisﬁed. We ﬁrst discuss a multiplicity-free criterion that can be viewed as a variation on some of the combinatoric considerations of [DPW2002]; it emerged from some discussions with Ivan Penkov in another context. We then apply the criterion in the setting of symmetric spaces, proving that direct limits of compact symmetric spaces are multiplicity-free. This applies in particular to inﬁnite-dimensional real, complex, and quaternionic Grassmann manifolds, and it uses some basic symmetric space structure theory. In particular, our argument for direct limits of compact riemannian symmetric spaces makes essential use of the Cartan–Helgason theorem, and thus does not extend to direct limits of nonsymmetric Gelfand pairs. In order to extend the multiplicity-free result to at least some direct limits of nonsymmetric Gelfand pairs, we deﬁne the notion of “deﬁning representation” for a direct system {(Gn , Kn )}, where the Gn are compact Lie groups and the Kn are closed subgroups. We show how a deﬁning representation for {(Gn , Kn )} leads to a direct system {A(Gn /Kn )} of C-valued polynomial function algebras, a continuous function completion {C(Gn /Kn )}, and a Lebesgue space completion {L2 (Gn /Kn )}. The direct limit spaces A(G/K), C(G/K), and L2 (G/K) are the function spaces on G/K = lim Gn /Kn which −→ we study as G-modules. Next, we prove the multiplicity-free property, for the action of G on A(G/K), C(G/K), and L2 (G/K), when {(Gn , Kn )} is one of several families of Gelfand pairs related to spheres and Grassmann manifolds. We prove the multiplicity-free property for three other types of direct limits of Gelfand pairs. Finally we summarize the results, extending them slightly by including the possibility of enlarging the Kn within their Gn -normalizers without losing the property that {Kn } is a direct system. Our proofs of the multiplicity-free condition, for some direct limits of nonsymmetric Gelfand pairs, use a number of branching rules, new and old, for ﬁnite-dimensional representations. This lends a certain ad hoc ﬂavor which I hope can be avoided in the future. Direct limits (G, K) = lim(Gn , Kn ) of riemannian symmetric spaces were −→ studied by Ol’shanskii from a very diﬀerent viewpoint [Ol1990]. He viewed the

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Gn inside dual reductive pairs and examined their action on Hilbert spaces of Hermite polynomials. Ol’shanskii made extensive use of factor representation theory and Gaussian measure, obtaining analytic results on limit-spherical functions. See Faraut [Fa2006] for a discussion of spherical functions in the setting of direct limit pairs. In contrast to the work of Ol’shanskii and Faraut, we use the rather simple algebraic method of renormalizing formal degrees of representations to obtain isometric embeddings L2 (Gn /Kn ) → L2 (Gn+1 /Kn+1 ). That leads directly to our multiplicity-free results. Acknowledgements: I am indebted to Ivan Penkov for discussions of multiplicities in direct limit representations, which are formalized in Theorem 2 below. I also wish to acknowledge hospitality from the Mathematisches Forschungsinstitut Oberwolfach and support from NSF Grant DMS 04 00420.

2 Direct limit groups and representations We consider direct limit groups G = lim Gn and, their direct limit represen−→ tations π = lim πn . This means that πn is a representation of Gn on a vector −→ space Vn , that the Vn form a direct system, and that π is the representation of G on V = lim Vn given by π(g)v = πn (gn )vn whenever n is suﬃciently large −→ that Vn → V and Gn → G send vn to v and gn to g. The formal deﬁnition amounts to saying that π is well deﬁned. It is clear that a direct limit of irreducible representations is irreducible, but there are irreducible representations of direct limit groups that cannot be formulated as direct limits of irreducible ﬁnite-dimensional representations. This is a combinatoric matter and is discussed extensively in [DPW2002]. The following deﬁnition is closely related to those combinatorics but applies to a somewhat simpler situation. Definition 1 We say that a representation π of G is limit aligned if it has form lim πn in such a way that (i) each πn is a direct sum of primary representations, −→ and (ii) the corresponding representation spaces V = lim Vn have the property −→ that every primary subspace of Vn is contained in a primary subspace of Vn+1 . Theorem 2 A limit-aligned representation π = lim πn of G = lim Gn is a −→ −→ direct sum of primary representations. If the πn are multiplicity free, then π is a multiplicity-free direct sum of irreducible representations. Proof. Let V = lim Vn be the representation spaces. Decompose Vn = −→ α∈In Vn,α , where the Vn,α are the subspaces for the primary summands of πn . Write πn,α for the representation of Gn on Vn,α , so πn = α∈In πn,α . Since π is limit aligned, i.e., since each Vn,α ⊂ Vn+1,β for some β ∈ In+1 , we may assume In ⊂ In+1 in such a way that each V n,α ⊂ Vn+1,α for every α ∈ In . Now V = α∈I Vα , discrete sum, where I = In and Vα = Vn,α . The sum is direct, for if u1 + u2 + · · · + ur = 0 where ui ∈ Vαi for distinct

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indices α1 , . . . , αr , then we take n suﬃciently large so that each ui ∈ Vn,αi and conclude that u1 = u2 = · · · = ur = 0. Thus π is the discrete direct sum of the representations πα = lim πn,α of G on Vα . −→ Let Cα = {X : Vα → Vα linear | Xπα (g) = πα (g)X for all g ∈ G}, the commuting algebra of πα . If πα fails to be primary, then Cα contains nontrivial commuting ideals Cα and Cα . Then for n large, the stabilizer NCα (Vn,α ) of Vn,α in Cα contains nontrivial commuting ideals NCα (Vn,α ) and NCα (Vn,α ). That is impossible because πn,α is primary. We have proved that π is the discrete direct sum of primary representations πα . If the πn are multiplicity free, then the πn,α are irreducible and it is immediate that the πα = lim πn,α are irreducible. This completes the proof of −→ Theorem 2. A direct limit of irreducible representations is irreducible, but it is not immediate that every irreducible direct limit representation can be rewritten as a direct limit of irreducible representations. With this and Theorem 2 in mind, we extend Deﬁnition 1 as follows. Definition 3 A representation π of G = lim Gn is lim irreducible if it has form −→ π = lim πn where each πn is an irreducible representation of Gn . Similarly, π is −→ lim primary if it has form π = lim πn where each πn is a primary representation −→ of Gn . Theorem 4 Consider a representation π = lim πn of G = lim Gn with −→ −→ representation space V = lim Vn . Suppose that each πn is a multiplicity-free −→ direct sum of irreducible highest weight representations. Suppose for n > > 0 that the direct system map Vn−1 → Vn sends Gn−1 -highest weight vectors to Gn -highest weight vectors. Then π is a multiplicity-free direct sum of lim-irreducible representations of G. Proof. By hypothesis each πn is a direct sum of primary representations which, in fact, are irreducible highest weight representations. We recursively choose highest weight vectors so that πn−1 = πλ,n−1 , where πλ,n−1 has highest weight vector vλ,n−1 ∈ Vn−1 that maps to a highest weight vector vλ,n ∈ Vn of an irreducible constituent πλ,n of πn . This exhibits π as a limit-aligned direct sum because it embeds the summand Vλ,n−1 of Vn−1 into the irreducible summand of Vn that contains vλ,n . Now Theorem 2 shows that π is a multiplicity-free direct sum of lim-irreducible representations of G.

3 Limit theorem for symmetric spaces We now apply Theorems 2 and 4 to direct limits of compact riemannian symmetric spaces. Fix a direct system of compact connected Lie groups Gn and subgroups Kn such that each (Gn , Kn ) is an irreducible riemannian symmetric pair. Suppose that the corresponding compact symmetric spaces

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Mn = Gn /Kn are connected and simply connected. Up to renumbering and passage to a common coﬁnal subsequence, the only possibilities are as given in the following table.

1 2 3 4 5 6 7 8 9 10 11

compact irreducible riemannian symmetric Mn = Gn /Kn Gn Kn RankMn DimMn SU (n) × SU (n) diagonal SU (n) n−1 n2 − 1 Spin(2n + 1) × Spin(2n + 1) diagonal Spin(2n + 1) n 2n2 + n Spin(2n) × Spin(2n) diagonal Spin(2n) n 2n2 − n Sp(n) × Sp(n) diagonal Sp(n) n 2n2 + n (5) SU (p + q), p = pn , q = qn S(U (p) × U (q)) min(p, q) 2pq (n−1)(n+2) SU (n) SO(n) n−1 2 SU (2n) Sp(n) n − 1 2n2 − n − 1 SO(p + q), p = pn , q = qn SO(p) × SO(q) min(p, q) pq SO(2n) U (n) [ n2 ] n(n − 1) Sp(p + q), p = pn , q = qn Sp(p) × Sp(q) min(p, q) 4pq Sp(n) U (n) n n(n + 1)

Fix one of the direct systems {(Gn , Kn )} of Table 5. Then we have involutive automorphisms θn of Gn such that the Lie algebras decompose into ±1 eigenspaces of the θn , gn = kn + sn in such a way that kn = gn ∩ kn+1 and sn = gn ∩ sn+1 . Then we recursively construct a system of maximal abelian subspaces an : maximal abelian subspace of sn such that an = gn ∩ an+1 . The restricted root systems Σn = Σn (gn , an ) : the system of an -roots on gn form an inverse system of linear functionals: Σ = Σ(g, a) is the system lim Σn ←− of linear functionals on a = lim an . In this inverse system, the multiplicities of −→ the restricted roots will increase without bound, but we can make consistent choices of positive subsystems Σn+ = Σn+ (gn , an ) : system of positive an -roots on gn + so that Σn+ ⊂ Σm |an for m n n0 . Consider the reduced root system

Σ0,n = {α ∈ Σn | 2α ∈ Σn } + and its positive subsystem Σ0,n := Σ0,n ∩ Σn+ . Examining the tables of Araki ([Ar1962], or referring to [He1978, pp. 532–534] or [W1980, pp. 90–93]), we see the following.

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Lemma 6 Suppose that Gn is simple. Then there are only two possibilities. (a) Σ0,n = Σn ; in other words, if α ∈ Σn then 2α ∈ / Σn . (b) Σ0,n = Σn ; there is exactly one simple root ψ1,n for Σn+ such that 2ψ1,n ∈ Σn , and ψ1,n is at the end of the Dynkin diagram of Σn+ opposite to + the end where roots are added to obtain the diagram of Σn+1 . + , which we denote Then the corresponding simple root systems for Σ0,n

Ψn = Ψn (gn , an ) = {ψ1,n , . . . , ψrn ,n } : simple reduced an -roots on gn satisfy Ψn ⊂ Ψm |an for m n n0 as well. Here rn = dim an , rank of Mn . In case (a) of Lemma 6, Ψn is a simple root system for Σn+ , but in case (b) the corresponding simple root system for Σn+ is { 12 ψ1,n , ψ2,n , . . . , ψrn ,n }. In both cases Ψn ⊂ Ψm |an for m n n0 . More precisely, if ψj,n ∈ Ψn and m n, then there is just one element ψ ∈ Ψm with ψ|an = ψj,n . In other words, we may (and do) recursively enumerate the simple root systems Ψn so that if m n and ψj,n ∈ Ψn , then ψj,m ∈ Ψm satisﬁes ψj,m |an = ψj,n , retaining the convention that in case (b) of Lemma 6 the Later we will use the fact that

1 2 ψ1,n

are roots.

+ in case (b) of Lemma 6, if m n and 12 ψ1,n ∈ Σn+ , then 12 ψ1,m ∈ Σm . (7)

Recursively deﬁne θn -stable Cartan subalgebras of hn = tn + an of gn with hn = gn ∩ hn+1 . Here tn is a Cartan subalgebra of the centralizer mn of an in kn . Now recursively construct positive root systems Σ + (mn , tn ) such that if α ∈ Σ + (mn+1 , tn+1 ), then either α|tn = 0 or α|tn ∈ Σ + (mn , tn ). Then we have positive root systems Σ + (gn , hn ) = {α ∈ ih∗n | α|an = 0 or α|an ∈ Σn+ (gn , an )}, the corresponding simple root systems, and the resulting systems of fundamental highest weights. The Cartan–Helgason theorem says that the irreducible representation πλ of gn of highest weight λ gives a summand of the representation of Gn on L2 (Mn ) if and only if (i) λ|tn = 0, so we may view λ as an element of ia∗n , and λ,α (ii) if α ∈ Σn+ (gn , an ) then α,α is an integer 0. Condition (i) persists under restriction λ → λ|hn−1 because tn−1 ⊂ tn . Given (i), condition (ii) says that 1 2 λ belongs to the weight lattice of gn , so its restriction to hn−1 exponentiates to a well-deﬁned function on the corresponding maximal torus of Gn−1 and thus belongs to the weight lattice of gn−1 . Given condition (i) now (7) says that condition (ii) persists under restriction λ → λ|hn−1 . With this in mind, we deﬁne linear functionals ξn,j ∈ ia∗n by ξn,i ,ψn,j ψn,j ,ψn,j

= δi,j for 1 j rn , except that

ξn,1 ,ψn,1 ψn,1 ,ψn,1

= 2 if 2ψn,1 ∈ Σn .

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The weights ξn,j are the class 1 fundamental highest weights for (gn , kn ). We denote Ξn = Ξn (gn , tn , an ) = {ξn,1 , . . . , ξn,rn } . Deﬁne Λn = Λ(gn , kn , an ) =

nk ξk | ξk ∈ Ξn and nk ∈ Z, nk 0 .

This is the set of highest weights for representations of Gn on L2 (Mn ), and we have just veriﬁed that Λn |an−1 ⊂ Λn−1 . Lemma 8 For n suﬃciently large, and passing to a coﬁnal subsequence, if ξ ∈ Ξn−1 there is a unique ξ ∈ Ξn such that ξ |an−1 = ξ. Proof. In the group manifold cases, lines 1, 2, 3, and 4 of Table 5, express Gn = Ln × Ln , and note that the complexiﬁcation (Ln−1 )C is the semisimple component of a parabolic subgroup of (Ln )C . The restricted root and weight systems of (Gn , Kn ) are the same as the unrestricted root and weight systems of Ln , and the assertion follows. In the Grassmann manifold cases, lines 5, 8, and 10 of Table 5, we ﬁrst consider the case where {pn } is bounded. Then we may assume pn = p constant and qn increasing for n > > 0. Thus an−1 = an , Ψn−1 = Ψn (though the multiplicities of the restricted roots will increase), and Ξn−1 = Ξn . The assertion now is immediate. In the Grassmann manifold cases we may now assume that both pn and qn are unbounded. If pn = qn on a coﬁnal sequence of indices n we may assume pn = qn for all n, so Ψn is always of type Crn . Then we interpolate pairs and renumber so that pn = qn = pn−1 + 1 = qn−1 + 1 for all n and notice that the Dynkin diagram inclusions Cr−1 ⊂ Cr are uniquely determined by the integer r. If pn = qn for only ﬁnitely many n and pn < qn on a coﬁnal sequence of indices n we may assume that rn = pn < qn for all n, so Ψn is always of type Brn . Then we interpolate (pn−1 , qn − 1), (pn−1 , qn ), (pn−1 +1, qn ), . . . , (pn , qn ) and renumber so that we always have rn = rn−1 or rn = rn−1 + 1 and notice that the Dynkin diagram inclusions Br−1 ⊂ Br are uniquely determined by the integer r. If pn = qn for only ﬁnitely many n and also pn = qn for only ﬁnitely many n, then pn > qn on a coﬁnal sequence of indices n, and we may assume pn > qn = rn for all n. We interpolate as before, exchanging the rˆ oles of p and q , and we note again that the Dynkin diagram inclusions Br−1 ⊂ Br are uniquely determined by the integer r. Thus in all cases the fundamental highest weights restrict as asserted. In the lower rank cases, lines 6 and 7 of Table 5, Ψn is of type An−1 , so again restriction to an−1 has the required property. In the hermitian symmetric case, line 11 of Table 5, an is a Cartan subalgebra of gn and gn−1 complexiﬁes to the semisimple part of a parabolic subalgebra of (gn )C , so the assertion follows as in the group manifold cases. In the remaining case, line 9 of Table 5, Ψn is of type Cn/2 for n even, type B(n−1)/2 for n odd. Passing to a coﬁnal subsequence

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we may assume n always even or always odd, and we may interpolate as necessary by pairs so that n increases in steps of 2. Then, again, there is no choice about the restriction, and the assertion follows. In view of Lemma 8, after passage to a coﬁnal subsequence and renumbering, we may assume the sets Ξn ordered so that Ξn = Ξ(gn , kn , an ) ={ξ1,n , . . . , ξrn ,n } with ξ,n−1 = ξ,n |an−1 for 1 rn−1 .

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Now deﬁne In : all rn -tuples I = (i1 , . . . , irn ) of non-negative integers, I = lim In where In → Im by (i1 , . . . , irn ) → (i1 , . . . , irn , 0, . . . , 0), −→ πI,n : rep of Gn with highest weight ξI = i1 ξ1 + · · · + ip ξrn ,

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πI = lim πI,n for I ∈ I. −→ According to the Cartan–Helgason theorem, the πI,n exhaust the representations of Gn on L2 (Mn ). Denote VI,n : representation space for the abstract representation πI,n .

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Then VI,n occurs with multiplicity 1 in the representation of Gn on L2 (Mn ). In eﬀect, the representation of Gn on L2 (Mn ) is multiplicity-free, and L2 (Mn ) ∼ = V as a G -module. However, in the following we must distinguish n I∈I I,n between I∈I VI,n as a Gn -module and L2 (Mn ) as a space of functions. Let U(gn ) denote the (complex) universal enveloping algebra of gn . Let vn+1 be a highest weight unit vector in VI,n+1 for the action of Gn+1 . Then we have the Gn -submodule U(gn )(vn+1 ) ⊂ VI,n+1 ⊂ L2 (Mn+1 ). If u, v ∈ VI,n we write fu,v;I,n for g → u, πI,n (g)v , the matrix coeﬃcient function on Gn . These matrix coeﬃcient functions span a space EI,n that is invariant under left and right translations by elements of Gn . As a (Gn × Gn )∗ module EI,n ∼ . If u∗n is the (unique up to scalar multiplication) = VI,n VI,n ∗ Kn -ﬁxed unit vector in VI,n , then the right Kn -ﬁxed functions in EI,n form Kn ∼ the left Gn -module EI,n = VI,n . = VI,n ⊗ u∗n C ∼ In the following, it is crucial to distinguish between the abstract represenKn tation space VI,n and the space EI,n of functions on Gn /Kn . We normalize the Haar measure on Gn (and the resulting measure in Mn ) to total mass 1. If u, v, u , v ∈ VI,n , then we have the Schur orthogonality relation fu,v;I,n , fu ,v ;I,n |L2 (Gn ) = (deg πI,n )−1 u, u v, v . Kn Theorem 12 The space EI,n of functions on Gn /Kn is Gn -module equivaK

K

Kn n+1 n+1 . We map EI,n into EI,n+1 as follows. lent to U(gn )(vn+1 ⊗ u∗n+1 ) ⊂ EI,n+1 Let {wj } be a basis of VI,n and deﬁne

ψn+1,n

Inﬁnite-Dimensional Multiplicity-Free Spaces, I

cj fwj ,u∗n ;I,n

= (deg πI,n+1 / deg πI,n )1/2

K

n+1 . cj fwj ,u∗n+1 ;I,n+1 ∈ EI,n+1

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K

Kn n+1 : EI,n → EI,n+1 is Gn -equivariant and is isometric for L2 Then ψn+1,n norms on Gn /Kn and Gn+1 /Kn+1 . In particular, as I varies with n ﬁxed, ψn+1,n : L2 (Gn /Kn ) → L2 (Gn+1 /Kn+1 ) is a Gn -equivariant isometry.

Proof. We have a(vn+1 ) = ξI (a)vn+1 for all a ∈ a. The inclusion Gn → Gn+1 is Gn -equivariant, so restriction of functions is Gn -equivariant and thus is Aequivariant, and (vn+1 ⊗ u∗n+1 )|Mn is a ξI -weight vector in L2 (Mn ). If α is a positive restricted root for Gn+1 and eα ∈ gn+1 is an α root vector, then eα (vn+1 ) = 0. If α is already a root for Gn and if eα ∈ gn , then we have eα ((vn+1 ⊗ u∗n+1 )|Mn ) = 0. Thus either the restriction (vn+1 ⊗ u∗n+1 )|Mn = 0 Kn or (vn+1 ⊗ u∗n+1 )|Mn is a highest weight vector in EI,n . Suppose that (vn+1 ⊗ u∗n+1 )|Mn = 0 as a function on Mn = Gn /Kn . Denote Vn = U(gn )(vn+1 ). It is a cyclic highest weight module for Gn with highest weight ξI , and (Vn ⊗ u∗n+1 C)|Mn = 0, and it contains a unique (up to scalar multiple) Kn -invariant unit vector un . The coeﬃcient function ϕ(g) := un , πI,n+1 (g)un Vn = Gn (un ⊗ u∗n )(x)(un ⊗ u∗n )(x−1 g)dx is identically zero because the un (x) factor in the integrand vanishes for x ∈ Gn . But ϕ|Gn is the positive deﬁnite (Gn , Kn )-spherical function on Gn for the representation πI,n , and in particular ϕ(1) = 1. That is a contradiction. We conclude that Kn . (vn+1 ⊗ u∗n+1 )|Mn = 0, so (vn+1 ⊗ u∗n+1 )|Mn is a highest weight vector in EI,n K Kn ∼ n+1 ∗ In particular, EI,n = (Vn ⊗ un+1 C)|Mn ⊂ EI,n+1 |Mn . That is the equivariant map assertion. The unitary map assertion follows by Schur orthogonality. Theorem 12 gives isometric embeddings ψm,n : L2 (Mn ) → L2 (Mm ) for n m. By construction, ψm,n is Gn -equivariant. Deﬁne } : direct limit in the L2 (G/K) = lim{L2 (Gn /Kn ), ψm,n −→ category of Hilbert spaces and unitary injections.

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We emphasize the renormalizations of Theorem 12. Without those renormalizations we lose the Hilbert space structure of L2 (G/K). Theorem 15 The left regular representation of G on L2 (G/K) is a multiplicity-free discrete direct sum of lim-irreducible representations. Specif ically, that left regular representation is π , where πI = lim πI,n is I∈I I − → the irreducible representation of G with highest weight ξI := ir ξr . This applies to all the direct systems of Table 5. In particular, we have the thirteen inﬁnite-dimensional multiplicity-free spaces

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1. SU (∞) × SU (∞)/diag SU (∞), group manifold SU (∞), 2. Spin(∞) × Spin(∞)/diag Spin(∞), group manifold Spin(∞), 3. Sp(∞) × Sp(∞)/diag Sp(∞), group manifold Sp(∞), 4. SU (p + ∞)/S(U (p) × U (∞)), Cp subspaces of C∞ , 5. SU (2∞)/[S(U (∞) × U (∞))], C∞ subspaces of inﬁnite codim in C∞ , 6. SU (∞)/SO(∞), 7. SU (2∞)/Sp(∞), 8. SO(p + ∞)/[SO(p) × SO(∞)], oriented Rp subspaces of R∞ , 9. SO(2∞)/[SO(∞) × SO(∞)], R∞ subspaces of inﬁnite codim in R∞ , 10. SO(2∞)/U (∞), 11. Sp(p + ∞)/[Sp(p) × Sp(∞)], Hp subspaces of H∞ , 12. Sp(2∞)/[Sp(∞) × Sp(∞)], H∞ subspaces of inﬁnite codim in H∞ , 13. Sp(∞)/U (∞). Proof. λ is limit aligned by Theorem 12. Denote VI = VI,n = lim VI,n . −→ Then G acts irreducibly on it by πI = lim πI,n , and the various πI are mutu−→ ally inequivalent because they have diﬀerent highest weightsξI := ir ξr , and are lim irreducible by construction. Now let V = V . Then I∈I I V = lim L2 (Gn /Kn ) = L2 (G/K). −→

4 Gelfand pairs and defining representations In this section we set the stage for the extension of Theorem 15 to a number of direct systems {(Gn , Kn )} of compact nonsymmetric Gelfand pairs. A glance at [Ya2004] or [W2007] reveals many such pairs, but here we will only consider those for which the compact groups Gn are simple. The following table shows the Kr¨ amer classiﬁcation of Gelfand pairs corresponding to compact simple Lie groups (see [Kr1979] or [Ya2004] or [W2007, Table 12.7.1]).

1 2 3 4 5 6 7 8 9 10 11 12

Mn = Gn /Hn weakly symmetric Gn Hn Conditions SU (m + n) SU (m) × SU (n)] n > m 1 SO(2n) SU (n) n odd, n 3 E6 Spin(10) SU (2n + 1) Sp(n) n1 SU (2n + 1) Sp(n) × U (1) n1 Spin(7) G2 G2 SU (3) SO(10) Spin(7) × SO(2) SO(9) Spin(7) Spin(8) G2 SO(2n + 1) U (n) n2 Sp(n) Sp(n − 1) × U (1) n 1

Gn /Kn symmetric Kn with Hn ⊂ Kn ⊂ Gn S[U (m) × U (n)] U (n) Spin(10) · Spin(2) U (2n) = S[U (2n) × U (1)] U (2n) = S[U (2n) × U (1)] (there is none) (there is none) SO(8) × SO(2) SO(8) Spin(7) SO(2n) Sp(n − 1) × Sp(1)

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469

This gives us the nonsymmetric direct systems {(Gn , Kn )}, where (a) Gn = SU (pn + qn ) and Kn = SU (pn ) × SU (qn ), pn < qn (b) Gn = SO(2n) and Kn = SU (n), n odd, n 3 (c) Gn = SU (2n + 1) and Kn = Sp(n), n 1 (d) Gn = SU (2n + 1) and Kn = U (1) × Sp(n), n 1 (e) Gn = SO(2n + 1) and Kn = U (n), n 2

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(f) Gn = Sp(n) and Kn = U (1) × Sp(n − 1), n 2. Definition 18 Let {(Gn , Kn )} be a direct system of Lie groups and closed subgroups. Suppose that π = lim πn is a lim-irreducible representation of −→ G = lim Gn , with representation space V = lim Vn , such that (i) πn (Kn ) is −→ −→ the πn (Gn )-stabilizer of a vector vn ∈ Vn and (ii) each vn+1 = vn +wn+1 where πn (Gn ) leaves wn+1 ﬁxed. (Thus the vn give a coherent system of embeddings of the Gn /Kn .) Suppose further that for n > > 0 the πn have the same highest weight vector. Then we say that π = lim πn is a deﬁning representation for −→ {(Gn , Kn )}. Now let’s consider some important examples of deﬁning representations. We will use these examples later. Example 19 Gn = SU (pn + qn ) and Kn = SU (pn ) × SU (qn ), pn < qn , in (17). Let {e1 , . . . , epn +qn } denote the standard orthonormal basis of Cpn +qn . Then Kn is the Gn -stabilizer of e1 ∧· · ·∧epn in the representation πn = Λpn (τ ), where τ is the standard (vector) representation of SU (pn + qn ) on Cpn +qn . In the usual notation, e1 ∧ · · · ∧ epn also is the highest weight vector, and the highest weight is ε1 + · · · + εpn . If the pn are bounded, so that we may assume each pn = p < ∞, then π = lim πn is well deﬁned and is a deﬁning −→ representation for {(Gn , Kn )}. ♦ Example 20 Gn = SU (2n + 1) and Kn = U (1) × Sp(n), n 1, in (17). Again, {e1 , . . . , e2n+1 } is standard orthonormal basis of C2n+1 . Now Kn the n 2 is the Gn -stabilizer of =1 e2 ∧ e2+1 in the representation πn = Λ (τ ), 2n+1 where τ is the standard (vector) representation of SU (2n + 1) on C . Here e1 ∧ e2 is the highest weight vector and the highest weight is ε1 + ε2 . Thus π = lim πn is well deﬁned and is a deﬁning representation for {(Gn , Kn )}. ♦ −→ Example 21 Gn = SO(2n + 1) and Kn = U (n), n 2, in (17). Let 2n+1 {e . Let J =

01 , .1.. , e2n+1 } denote the standard orthonormal basis of R . Then K is the G -stabilizer of diag{0, J, . . . , J} ∈ g in the adjoint n n n −1 0 representation of G ; in other words (in this case), it is the G -stabilizer of n n n 2 e ∧ e in the representation π = Λ (τ ), where τ is the standard 2 2+1 n =1 (vector) representation of SO(2n + 1) on R2n+1 . As in the previous example, e1 ∧ e2 is the highest weight vector and the highest weight is ε1 + ε2 . Thus π = lim πn is well deﬁned and is a deﬁning representation for {(Gn , Kn )}. ♦ −→

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Example 22 Gn = Sp(n) and Kn = U (1) × Sp(n − 1), n 2, in (17). In quaternion matrices, Kn is the Gn -commutator of diag{i, 0, 0, . . . , 0}. In 2n × 2n complex

0 1 matrices, it is the Gn -commutator of diag{J, 0, 0, . . . , 0}, where J = −1 0 . There, Gn consists of all elements g ∈ U (2n) such that t = J, where J = diag{J, J, . . . , J}. Thus gn is given by xJ + J xt = 0, g Jg and in particular diag{J, 0, 0, . . . , 0} ∈ gn . Now Kn is the Gn -stabilizer of diag{J, 0, 0, . . . , 0} in the adjoint representation πn of Gn . That adjoint representation is the symmetric square of the standard (vector) representation of Gn on C2n , so it has highest weight 2ε1 and highest weight vector e21 . Thus π = lim πn is well deﬁned and is a deﬁning representation for {(Gn , Kn )}. ♦ −→

5 Function algebras Fix a deﬁning representation π = lim πn for {(Gn , Kn )}. We are going to −→ deﬁne algebras

A(Gn ) and A(G) = A(Gn );

A(Gn /Kn ) A(Gn /Kn ) and A(G/K) = of complex-valued polynomial functions and look at their relations to square integrable functions. Let dn = dimR Vn . Then we can consider Gn to be a group of real dn × dn matrices. Since the Gn are reductive linear algebraic groups, this lets us deﬁne A(Gn ) : the algebra of all C-valued functions f |Gn where f : Rdn ×qn → C is a polynomial, rn : A(Gn ) → A(Gn−1 ) : restriction of functions, Sn : kernel of the algebra homomorphism rn ,

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Tn : Gn−1 -invariant complement to Sn in A(Gn ). The following is immediate. Lemma 24 The restriction rn |Tn : Tn → A(Gn−1 ) is a Gn−1 -equivariant vector space isomorphism. In other words we have a Gn−1 -equivariant injection (rn |Tn )−1 : A(Gn−1 ) → A(Gn ) of vector spaces with image complementary to the kernel of the restriction rn : A(Gn ) → A(Gn−1 ) of functions. Lemma 24 gives us A(G) = lim A(Gn ) = −→

A(Gn ).

Taking the right-invariant functions we arrive at

Inﬁnite-Dimensional Multiplicity-Free Spaces, I

A(Gn /Kn ) := {h ∈ A(Gn ) | h(xk) = h(x) for x ∈ Gn , k ∈ Kn },

A(G/K) = A(Gn /Kn )

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= {h ∈ A(G) | h(xk) = h(x) for x ∈ G, k ∈ K}. These are our basic function algebras. The algebra A(Gn ) contains the constants, separates points on Gn , and is stable under complex conjugation. The Stone–Weierstrass theorem is the main component of the following lemma. Lemma 26 The algebra A(Gn ) is dense in C(Gn ), the algebra of continuous functions Gn → C with the topology of uniform convergence. Let Sn and Tn denote the uniform closures of Sn and Tn in C(Gn ). Then rn extends by continuity to the restriction map rn : C(Gn ) → C(Gn−1 ), that extension rn restricts to a Gn−1 -equivalence Tn ∼ = C(Gn−1 ), C(Gn ) is the vector space direct sum of closed Gn−1 -invariant subspaces Sn and Tn , and this identiﬁes C(Gn−1 ) as a Gn−1 -submodule of C(Gn ). Proof. The density is exactly the Stone–Weierstrass theorem in this setting. Since Sn and Tn involve diﬀerent sets of variables, so do Sn and Tn . Now rn extends to rn as asserted and Sn is the kernel of rn . Similarly, Sn ∩ Tn = 0, and the induced algebra homomorphism rn : C(Gn ) → C(Gn−1 ) restricts to a Gn−1 -equivariant map rn : Tn ∼ = C(Gn−1 ). Finally, Sn + Tn is closed in C(Gn ) and contains A(Gn ). Thus C(Gn ) = Sn ⊕ Tn and we can identify C(Gn−1 ) with the closed Gn−1 -invariant subspace Tn of C(Gn ). Weuse the identiﬁcations C(Gn−1 ) ⊂ C(Gn ) of Lemma 26 to form the union C(Gn ). Note that C(Gn ) is the algebra of continuous functions on G that depend on only ﬁnitely many variables. Now use the sup norm, and thus the topology of uniform convergence, and deﬁne a Banach algebra

C(G) : functions f : G → C in the uniform limit closure of C(Gn ) with sup norm and topology of uniform convergence. Passing to the right Kn -invariant functions we have Banach function algebras C(Gn /Kn ) := {h ∈ C(Gn ) | h(xk) = h(x) for x ∈ Gn , k ∈ Kn and

C(G/K) = C(Gn /Kn )

(27)

= {h ∈ C(G) | h(xk) = h(x) for x ∈ G, k ∈ K}. Here A(Gn /Kn ) is the subalgebra consisting of all Gn -ﬁnite functions in C(Gn /Kn ), and consequently A(G/K) is the subalgebra consisting of all Gﬁnite functions in C(G/K). We pass to L2 limits more or less in the same way as in (25) and (27), except that we must rescale to preserve L2 norms as in Theorem 12. For this we need some machinery from [W2009]. Let {Gn } be a strict direct system

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of compact connected Lie groups, and {(Gn )C } the direct system of their complexiﬁcations. Suppose that, for each n, the semisimple part [(gn )C , (gn )C ] of the reductive algebra (gn )C is the semisimple component of a parabolic subalgebra of (gn+1 )C .

(28)

Then we say that the direct systems {Gn } and {(Gn )C } are parabolic and that lim Gn and lim(Gn )C are parabolic direct limits. This is a special case of the −→ −→ deﬁnition of parabolic direct limit in [W2005]. Now let {Gn } be a strict direct system of compact connected Lie groups that is parabolic. We recursively construct Cartan subalgebras tn ⊂ gn with t1 ⊂ t2 ⊂ · · · ⊂ tn ⊂ tn+1 ⊂ . . . and simple root systems Ψn = Ψ ((gn )C , (tn )C ) such that each simple root for (gn )C is the restriction of exactly one simple root for (gn+1 )C . Then we may assume that Ψn = {ψn,1 , . . . , ψn,p(n) } in such a way that each ψn,j is the (tn )C -restriction of ψn+1,j and of no other element of Ψn+1 . The corresponding sets Ξn = {ξn,1 , . . . , ξn,p(n) } of fundamental highest weights can be ordered so that they satisfy: ξn+1,j is the unique element of Ξn+1 whose (tn )C -restriction is ξn,j , for 1 j p(n). Exactly as in Theorem 12 this gives us isometric Gn -equivariant injections ψm,n : L2 (Gn ) → L2 (Gm ) for n m. The associated direct limit maps ψn : L2 (Gn ) → lim{L2 (Gn ), ψm,n } deﬁne the direct limit in the category of −→ Hilbert spaces and unitary maps as the Hilbert space completion completion L2 (G) = lim {L2 (Gn ), ψm,n } = ψn (L2 (Gn )) . −→ unitary

Lemma 29 Let {(Gn , Kn )} be one of the systems of Examples 19, 20, 21, or 22. Then {Gn } is parabolic and the Gn -equivariant maps ψm,n : L2 (Gn ) → L2 (Gm ) send right-Kn -invariants to right-Kn+1 -invariants, resulting in Gn -equiva riant unitary injections ψm,n : L2 (Gn /Kn ) → L2 (Gm /Km ). Proof. We use the deﬁning relations given in Examples 19, 20, 21, and 22. In each case we look at the subspaces of L2 given by polynomials of degree d; those are ﬁnite-dimensional invariant subspaces of the A(Gn /Kn ). We observed above that A(Gn ) → A(Gn+1 ) maps right-Kn -invariants to rightKn+1 -invariants. On each irreducible summand, the L2 (Gn ) → L2 (Gn+1 ) diﬀer only by scale from the corresponding summands of A(Gn ) and A(Gn+1 ), so they also map right-Kn-invariants to right-Kn+1-invariants. Now we have some L2 analogues of (25) and (27). L2 (Gn /Kn ) := {h ∈ L2 (Gn ) | h(xk) = h(x) for x ∈ Gn , k ∈ Kn }, completion L2 (G/K) = ψn (L2 (Gn /Kn )) = {h ∈ L2 (G) | h(xk) = h(x) for x ∈ G, k ∈ K}.

(30)

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We have A(G/K) ⊂ C(G/K) ⊂ L2 (G/K) for these spaces, and A(G/K) is the set of polynomial elements in L2 (G/K). Theorem 31 Let {(Gn , Kn )} be one of the direct systems of nonsymmetric Gelfand pairs given by Examples 19, 20, 21, and 22. Then the left regular representations of G on A(G/K), C(G/K), and L2 (G/K) are multiplicityfree discrete direct sums of lim-irreducible representations. In the notation of (9), (10), and (11), those left regular representations are I∈I πI , where πI = lim πI,n is the irreducible representation of G with highest weight ξI := ir ξr . −→ Thus we have the inﬁnite-dimensional multiplicity-free spaces (1) SU (p + ∞)/(SU (p) × SU (∞)) for 1 p ∞, (2) SU (1 + 2∞)/(U (1) × Sp(∞)), (3) SO(1 + 2∞)/U (∞), and (4) Sp(1 + ∞)/(U (1) × Sp(∞)) Proof. Examples 19, 20, 21, and 22 have deﬁning representations and welldeﬁned function spaces A(G/K) and C(G/K). The same holds for L2 (G/K) by Lemma 29. In these examples {Gn } is parabolic, so the left regular representations are limit aligned by Theorem 12. Now the proof of Theorem 15 holds for these four examples, resulting in the multiplicity-free property for their left regular representations.

6 Pairs related to spheres and Grassmann manifolds In dealing with nonsymmetric Gelfand pairs we have to be very speciﬁc about the embeddings Gn−1 → Gn , so we review a few of those embeddings. Orthogonal groups. Let Gn = SO(n0 + 2n), the special orthogonal group n +2n ui vi . The embeddings are given by for the bilinear form h(u, v) = 1 0 x 00 0 10 0 01

. Then G = lim Gn is the classical direct −→ limit group SO(∞). It doesn’t matter what n0 is here, but sometimes we have to distinguish between the cases of even or odd n0 , and in any case we want {Gn } to be parabolic, so we jump by two 1’s instead of just one. Speciﬁcally, this direct system consists either of groups of type B (when the n0 + 2n are odd) or of type D (when the n0 + 2n are even). In this section Kn = { ( 10 x0 )| x ∈ SO(n0 + 2n − 1)} ⊂ Gn . Then Gn /Kn is the sphere S n0 +2n−1 , G = lim Gn = SO(∞), and we express K = lim Kn as SO(1) × SO(∞ − 1) to −→ −→ indicate the embedding K → G. A deﬁning representation for {(Gn , Kn )} is given by the family of standard (vector) representations πn of SO(n0 + 2n) on Rn0 +2n . Here {SO(n0 + 2n)} is a parabolic direct system. In the standard orthonormal basis the πn all have the same highest weight vector e1 and highest weight ε1 . Following the considerations of Section 5, this deﬁning representation π = lim πn −→ Gn → Gn+1 given by x →

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deﬁnes the function spaces A(Gn /Kn ), C(Gn /Kn ), and L2 (Gn /Kn ). The πn share a highest weight vector so we have natural equivariant inclusions A(Gn−1 /Kn−1 ) → A(Gn /Kn ), C(Gn−1 /Kn−1 ) → C(Gn /Kn ), and L2 (Gn−1 /Kn−1 ) → L2 (Gn /Kn ), and thus the limits A(G/K), C(G/K), and L2 (G/K). Thus we have the regular representation of G = SO(∞) on those limit spaces. Unitary groups. Fix p > 0 and deﬁne Gn = SU (p + n), the special p+n unitary group for the complex hermitian form h(u, v) = ui v¯i . The 1 embedding Gn → Gn+1 is given by x → ( x0 10 ). Then G = lim Gn is −→ the classical parabolic direct limit group SU (∞). In this section Kn = { ( 10 x0 )| x ∈ SU (p), y ∈ SU (n)}. Then Gn /Kn is a circle bundle over the Grassmann manifold of p-dimensional linear subspaces of Cp+n , G = lim Gn = −→ SU (∞), and we sometimes express K = lim Kn as SU (p) × SU (∞ − p) to −→ indicate the embedding K → G. If p = 1 then Gn /Kn is the sphere S 2n+1 , the complex Grassmann manifold is a complex projective space, and the circle bundle projection is the Hopf ﬁbration. Here the deﬁning representation is essentially that of Example 19. Let πξ1 denote the usual vector representation of Gn on Cp+n . Write πξp for its pth alternating power, the representation of Gn on Λp (Cp+n ); it is the ﬁrst representation of Gn with a vector ﬁxed under Kn . That vector is e1 ∧ · · · ∧ ep relative to the standard basis {e1 , . . . , en } of Cn , and Kn is its Gn -stabilizer. Thus the πn = πξp give a deﬁning representation for {(Gn , Kn )}. Note that the πn all have the same highest weight vector e1 ∧ · · · ∧ ep and highest weight ε1 + · · · + εp . Following the considerations of Section 5, this deﬁning representation π = lim πn deﬁnes the function spaces A(Gn /Kn ), C(Gn /Kn ), and −→ L2 (Gn /Kn ). The πn share a highest weight vector, so we have natural equivariant inclusions A(Gn−1 /Kn−1 ) → A(Gn /Kn ), C(Gn−1 /Kn−1 ) → C(Gn /Kn ), and L2 (Gn−1 /Kn−1 ) → L2 (Gn /Kn ), and thus the limits A(G/K), C(G/K), and L2 (G/K). That gives us the regular representation of G = SU (∞) on those limit spaces. Symplectic groups. Here nSp(n) is the unitary group of the quaternionhermitian form h(u, v) = 1 ui v¯i on the quaternionic vector space Hn . We then have Gn = Sp(n) × Sp(1), where the Sp(1) acts by quaternion scalars on Hn . We will also look at its subgroup Sp(n) × U (1), where U (1) is any (they are all conjugate) circle subgroup of Sp(1), say {eiθ | θ ∈ R}. In both cases the embeddings Gn → Gn+1 are speciﬁed by the maps Sp(n) → Sp(n + 1) given by x → ( x0 10 ). (We are using quaternionic matrices.) Then G = lim Gn is the −→ classical direct limit group Sp(∞) × Sp(1) and G = lim Gn is Sp(∞) × U (1). −→ (We need the Sp(1) or the U (1) factor because otherwise, as we will see below, the multiplicity-free property will fail.) Symplectic 1. First consider the parabolic direct system given by Gn = Sp(n) × Sp(1). Given n we have two Sp(1) groups to deal with at the same time, so we avoid confusion by denoting the Sp(1) factor of Gn as Sp(1)ext,n (ext for external) and the identity component of the centralizer of Sp(n − 1)

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in Sp(n) by Sp(1)int,n (int for internal). In our matrix descriptions of Gn , the group Sp(1)diag,n is the diagonal subgroup in Sp(1)int,n × Sp(1)ext,n . Then Gn = Sp(n) × Sp(1)ext,n and G = lim Gn = Sp(∞) × Sp(1). Now let −→ Kn = { ( 10 x0 )| x ∈ Sp(n − 1)} × Sp(1)diag,n and K = lim Kn . Then Gn /Kn −→ is the sphere S 4n−1 , in other words, the Hopf ﬁbration 3-sphere bundle over quaternion projective space P n−1 (H). In order to indicate the embedding K → G we express K as {1} × Sp(∞ − 1) × Sp(1). A deﬁning representation for {(Gn , Kn )} is given by the family of standard (vector) representations πn of Sp(n) on C2n tensored with the standard 2-dimensional representation of Sp(1) on C2 . That representation has an invariant real form R4n . Consider the standard orthonormal basis {ei ⊗ fj } of C2n ⊗ C2 . The representations πn of Gn there have the same highest weight vector e1 ⊗ f1 and highest weight (ε1 )Sp(n) + (ε1 )Sp(1) . They give a deﬁning representation for {(Gn , Kn )}. Following the considerations of Section 5, this deﬁning representation π = lim πn deﬁnes the function spaces A(Gn /Kn ), −→ C(Gn /Kn ), and L2 (Gn /Kn ). The πn have the same highest weight vector so we have natural equivariant inclusions A(Gn−1 /Kn−1 ) → A(Gn /Kn ), C(Gn−1 /Kn−1 ) → C(Gn /Kn ), and L2 (Gn−1 /Kn−1 ) → L2 (Gn /Kn ), and thus the limits A(G/K), C(G/K), and L2 (G/K). So we have the regular representation of G = Sp(∞) × Sp(1) on those limit spaces. Symplectic 2. Next consider the parabolic direct system given by Gn = Sp(n) × U (1), where the Sp(1) factor of Sp(n) × Sp(1) is replaced by the circle subgroup {eiθ | θ ∈ R}. Given n we have two U (1) groups, the U (1)ext,n that is the U (1) factor of Gn and the corresponding circle subgroup U (1)int,n of Sp(1)int,n . Then of course we have the diagonal U (1)diag,n . As above we deﬁne Kn to be the product group { ( 10 x0 )| x ∈ Sp(n − 1)} × U (1)diag,n and we set K = lim Kn . Then Gn /Kn again is the sphere S 4n−1 . We express K −→ as {1} × Sp(∞ − 1) × U (1). A deﬁning representation for {(Gn , Kn )} is given by the family of standard (vector) representations πn of Sp(n) on C2n tensored with the standard 1-dimensional representation of U (1) on C. The representations πn of Gn there have the same highest weight vector e1 ⊗f1 . The corresponding highest weight is (ε1 )Sp(n) +(ε1 )U (1) , and the πn give a deﬁning representation for {(Gn , Kn )}. Following the considerations of Section 5, this deﬁning representation π = lim πn deﬁnes the function spaces A(Gn /Kn ), C(Gn /Kn ), and L2 (Gn /Kn ). −→ The πn have the same highest weight vector, so we have natural equivariant inclusions A(Gn−1 /Kn−1 ) → A(Gn /Kn ), C(Gn−1 /Kn−1 ) → C(Gn /Kn ), and L2 (Gn−1 /Kn−1 ) → L2 (Gn /Kn ), and thus the limits A(G/K), C(G/K), and L2 (G/K). So we have the regular representation of G = Sp(∞) × U (1) on those limit spaces. Symplectic 3. A variation on the case just considered is where Kn = { ( z0 x0 )| z ∈ U (1), x ∈ Sp(n − 1)} × U (1), and K = lim Kn . Then the U (1) −→ factor of Gn is contained in Kn so it acts trivially on Gn /Kn . Thus Gn /Kn is a 2-sphere bundle over P n−1 (H) exactly as in the “Symplectic 2” case.

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We express K as U (1) × Sp(∞ − 1) × U (1). The groups Kn are larger than the case “Symplectic 2” just considered, so the present function spaces A(Gn /Kn ), C(Gn /Kn ), and L2 (Gn /Kn ) are subspaces of those of “Symplectic 2”, and the same holds for their limits A(G/K), C(G/K), and L2 (G/K). Now we have the regular representation of G = Sp(∞) × Sp(1) on those limit spaces. Symplectic 4. A variation on the “Symplectic 1” case is where Gn = Sp(n) × Sp(1) and Kn = { ( z0 x0 )| z ∈ U (1), x ∈ Sp(n − 1)} × Sp(1) and K = lim Kn . Then the Sp(1) factor of Gn is contained in Kn , so it acts trivially −→ on Gn /Kn . Thus Gn /Kn = Sp(n)/[U (1) × Sp(n − 1)] is a 2-sphere bundle over P n−1 (H), exactly as in the “Symplectic 3” case above. We express K as U (1) × Sp(∞ − 1) × Sp(1) and we note that the function spaces A(Gn /Kn ), C(Gn /Kn ), and L2 (Gn /Kn ) are exactly the same as those of “Symplectic 3”, so the same holds for their limits A(G/K), C(G/K), and L2 (G/K). Thus we have the regular representation of G = Sp(∞) × Sp(1) on those limit spaces. The classiﬁcations of Kr¨amer [Kr1979] and Yakimova [Ya2004] (see [W2007]) show that the six direct systems just described, one orthogonal, one unitary, and four symplectic, all consist of Gelfand pairs.

7 Limits related to spheres and Grassmann manifolds In this section we prove the multiplicity-free property for the direct limits of Gelfand pairs described in Section 6. Theorem 32 Let (G, K) = lim{(Gn , Kn )}, where {(Gn , Kn )} is one of the −→ six systems described in Section 6. Let A(G/K), C(G/K), and L2 (G/K) be as described there. Then the regular representations of G on A(G/K), C(G/K), and L2 (G/K) are multiplicity-free discrete direct sums of lim-irreducible representations. Proof. We run through the proof of Theorem 32 for the three types of limit groups G. In each case we do this by examining the representation of Gn on A(Gn /Kn ), verifying the limit-aligned condition, and applying Theorem 4 to the regular representation of G on A(G/K). We already know the result for the orthogonal group case, where the (Gn , Kn ) are symmetric pairs, but we need the representation-theoretic information from that case in order to deal with the other cases. Orthogonal group case. Here we shift the index so that Gn = SO(n) and Kn = SO(n − 1). Then Gn /Kn is the unit sphere in Rn . The Gn -ﬁnite functions on Gn /Kn are just the restrictions of polynomial functions on Rn . Let ψ1;n denote the usual representation of Gn on Rn and let ξ denote its highest weight. Choose orthonormal linear coordinates {x1 , . . . , xn } of that Rn such that the monomial x1 is a highest weight vector. Then the representation of Gn on the space of polynomials of pure degree is of the form ψ;n ⊕ γ;n ,

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where ψ;n is the irreducible representation of highest weight ξ and highest weight vector x1 . Then γ;n is the sum of the ψ−2j;n for 1 j [/2], and the representation space of that ψ−2j;n consists of the polynomial functions on Rn divisible by ||x||2j but not by ||x||2j+2 . Write E;n for the space of functions on Gn /Kn obtained by restricting those polynomials of degree contained in the representation space for ψ;n . Then A(Gn /Kn ) = 0 E;n . We now verify that the inclusions A(Gn /Kn ) → A(Gn+1 /Kn+1 ) send E;n into E;n+1 , so that the representation of G on A(G/K) is limit aligned and Theorem 4 shows that lim A(Gn /Kn ) is the multiplicity-free direct sum of −→ lim-irreducible G-modules E = lim E;n . For that, note that the restriction −→ A(Gn+1 /Kn+1 ) → A(Gn /Kn ) is obtained by setting xn+1 equal to zero. Thus the inclusions E,n → A(Gn+1 /Kn+1 ) are given by identifying the function x1 : Rn → R with the function x1 : Rn+1 → R and applying Gn -equivariance. Now A(G/K) = lim A(Gn /Kn ) is the direct sum of the E = lim E;n , and the −→ −→ representations of G on the E are the mutually inequivalent lim-irreducible lim ψ;n . That gives an elementary proof for the case G = SO(∞) and K = −→ SO(∞ − 1). Unitary group cases. Here we shift the index so that Gn = SU (n) and Kn = SU (p)×SU (n−p), n > p. So G = SU (∞) and K = SU (p)×SU (∞−p). Without loss of generality assume n > 2p so that the (Gn , Kn ) are Gelfand pairs. Recall the deﬁning representation π = lim πn where πn = πξp , the pth −→ exterior power of the vector representation of Gn on Cn . So Kn is the Gn stabilizer of eI0 := e1 ∧ · · · ∧ ep , resulting in the map Gn /Kn → Λp (Cn ) by gKn → g(eI0 ). We have C-linear functions zI on Λp (Cn ) dual to the basis of Λp (Cn ) consisting of the eI with I = (i1 , . . . , ip ), where 1 i1 < · · · < ip n. (Here I0 = (1, 2, . . . , p).) Their real and imaginary parts generate the algebra A(Gn /Kn ). Relative to the diagonal Cartan subalgebra of gn the eI are weight vectors, and eI0 is the highest weight vector, for πξp . Now the action of Gn on the polynomials of degree in the zI and the zI is r+s= πrξp +sξn−p , where πrξp +sξn−p has highest weight rξp + sξn−p and highest weight vector zIr0 zIs0 . Those representations are mutually inequivalent, using n > 2p, and A(Gn /Kn ) = E 0 r+s= r,s;n , where Gn acts on Er,s;n by πrξp +sξn−p . The A(Gn /Kn ) → A(Gn+1 /Kn+1 ) are given on the level of Er,s;n → Er,s;n+1 by identifying zIr0 zIs0 : Λp (Cn ) → C with zIr0 zIs0 : Λp (Cn+1 ) → C. In view of Theorem 4, it follows that the representation of G on A(G/K) is a limitaligned discrete direct sum of mutually inequivalent lim-irreducible representations. We will need the case p = 1 when we look at the symplectic group cases. There Gn = SU (n) and Kn = {1} × SU (n − 1), and the Gn -ﬁnite functions on Gn /Kn are just the restrictions of ﬁnite linear combinations of the functions z r z¯s . We saw how to decompose A(S 2n−1 ) into irreducible modules for SO(2n): it is the sum of the spaces E;2n described above with highest weight

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ξ and highest weight vector x1 , where, of course, xj = 12 (zj + z¯j ). In terms of the Dynkin diagram that representation is c ψ : c c q q q c HH Hc and ψ;2n |U (n) = r+s= ψr,s;n , where ψr,s;n has diagram r c

s−r × . r s ¯1 . Let Both ψr,s;n and ψr,s;n |SU(n) have highest weight vector z 1z Er,s;n denote the representation space for ψr,s;n . Now A(Gn /Kn ) = 0 r+s= Er,s;n . Symplectic group cases. First suppose Gn = Sp(n) × U (1). There are two cases: (i) Kn = {1} × Sp(n − 1) × U (1)diag,n and (ii) Kn = U (1) × Sp(n − 1) × U (1). The assertions for case (i) will imply them for case (ii), so we may assume that Kn = {1} × Sp(n − 1) × U (1)diag,n . Then (G, K) = lim{(Gn , Kn )} has deﬁning representation π = lim πn where πn −→ −→ 1 1 b qqq b < b × of Gn on C2n . is the representation b Note that πn factors through the vector representation of U (2n) on C2n . We saw how U (2n) acts irreducibly on the space Er,s;2n by the representation ψr,s;2n , which has diagram r s s−r c c q q q c × . sends the U (1) factor of Gn We now need two facts. First, Gn → U (2n) to the center of U (n). Second, ψr,s;2n |Sp(n) = 0mmin(r,s) ϕr,s,m;n , where c

r + s − 2m m

qqq

s c

c c q q q c < c . That gives us ϕr,s,m;n has diagram ψr,s;2n |Sp(n)U (1) = 0mmin(r,s) ϕr,s,m;n , where ϕr,s,m;n is the representation r + s − 2m m s−r c c q q q c < c of Sp(n)U (1) with diagram × , and ϕr,s,m;n has the same representation space (call it Er,s,m;n ) as ϕr,s,m;n . The Er,s,m;n are irreducible and inequivalent under Sp(n)U (1); in other words, the irreducible representations ϕr,s,m;n all are mutually inequivalent. Note, however, that ϕr,s,m;n ϕr+t,s−t,m;n for all t such that r + t, s − t 0; this reﬂects the fact that (Sp(n), Sp(n − 1)) is not a Gelfand pair. To trace the inclusions let {z1 , . . . , z2n } be the coordinates of C2n , all weight vectors, where z1 is the highest weight vector, z2 = e−α1 z1 is the next highest, and so and the antisymmetric bilinear invariant of Sp(n) on C2n on, n is vn (z, w) = 1 (z2i−1 w2i − z2i w2i−1 ). Then z1 is the highest weight vector b b qqq b < b and Λ2 C2n is the sum Λ20 C2n ⊕ vn C of its of irreducible component and trivial under the action of Sp(n). component its 0 1 0 1 Here vn has matrix diag −1 . . . and we work with the maximal 0 −1 0 toral subalgebra that consists of all matrices diag {a1 , −a1 ; . . . ; an , −an }; thus the highest weight vector on Λ20 C2n is sn (z, w) = z1 w3 − z3 w1 . Now sm n is the m b b qqq b < b . The corresponding highest weight vector of

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highest weight vector of ϕr,s,m;n is z1r−m z¯1s−m sm n . Now the restriction A(Gn+1 /Kn+1 ) → A(Gn /Kn ) maps the highest weight vector z1r−m z¯1s−m sm n+1 of ϕr,s,m;n+1 to the highest weight vector z1r−m z¯1s−m sm n . This proves that the representation of G on A(G/K) is limit aligned. Theorem 4 shows that lim A(Gn /Kn ) is the multiplicity-free direct sum of lim-irreducible G-modules −→ Er,s,m := lim Er,s,m;n . −→ Finally, we suppose Gn = Sp(n) × Sp(1). Again there are two cases: (i) that Kn = {1}×Sp(n−1)×Sp(1)diag,n and (ii) Kn = U (1)×Sp(n−1)×Sp(1). The function algebras and group actions in case (ii) are exactly the same as those of the setting (Gn , Kn ) = (Sp(n) × U (1), U (1) × Sp(n − 1) × U (1)) above, where the assertions are proved. Thus we need only consider case (i), Kn = {1} × Sp(n − 1) × Sp(1)diag,n . Then (G, K) = lim{(Gn , Kn )} −→ has deﬁning representation π = lim πn described in “Symplectic 1” above. −→ Those πn satisfy the condition of Theorem 4 because Sp(n) × Sp(1) simply puts together representation spaces Er−m,s−m,m;n of Sp(n) × U (1) on A(Sp(n)U (1)/Sp(n − 1)U (1)). This assembly maintains total degree = (r − m) + (s − m) + 2m, views the U (1) factor of Sp(n) × U (1) as a maximal s−r torus of the Sp(1) factor of Sp(n)× Sp(1), and sums the spaces for the × to form the space for the irreducible representation (call it β ) of Sp(1) of degree b . Now the irreducible spaces for Sp(n) × Sp(1) are + 1. It has diagram the F,m;n := r+s= E r−m,s−m,m;n and the corresponding representations are the ϕ,m,n := r+s= ϕr−m,s−m,m;n . This proves that the representation of G on A(G/K) is limit aligned. Theorem 4 shows that lim A(Gn /Kn ) is the −→ multiplicity-free direct sum of lim-irreducible G-modules F,m := lim F,m;n . −→ We have proved Theorem 32. Remark 33 Alternatively, the systems (d), (e), and (f) from the list (17), and also (a) when the {pn } are bounded, can be treated by the method of Sections 6 and 7. That gives an alternative proof of the multiplicity-free property for the pairs (1) SU (p + ∞)/(SU (p) × SU (∞)) for 1 p ∞, (2) SU (1 + 2∞)/(U (1) × Sp(∞)), (3) SO(1 + 2∞)/U (∞), and (4) Sp(1 + ∞)/(U (1) × Sp(∞)) of Theorem 31.

♦

8 Conclusions We have proved that the regular representations of G on A(G/K), C(G/K), and L2 (G/K), are multiplicity-free discrete direct sums of lim-irreducible representations in the following cases. In addition, in these cases it is always permissible to enlarge the groups Kn , say to F · Kn where F is a closed subgroup

480

Joseph A. Wolf

of the normalizer NGn (Kn ), because A(Gn /[F · Kn ]) is a Gn -submodule of A(Gn /Kn ). Limits of riemannian symmetric spaces. We have the multiplicity-free property for the thirteen cases described in Theorem 15, as well as some obvious variations. The latter include SO(∞) × SO(∞)/diag SO(∞) = lim SO(n) × SO(n)/diag SO(n) and −→ SO(p + ∞)/[S(O(p) × O(∞))] = lim SO(n)/[S(O(p) × O(n − p))] . −→ Limits of a few systems of Gelfand pairs. We have the multiplicity-free property for the four cases described in Theorem 31, (1) SU (p + ∞)/(SU (p) × SU (∞)) for 1 p ∞, (2) SU (1 + 2∞)/(U (1) × Sp(∞)), (3) SO(1 + 2∞)/U (∞), and (4) Sp(1 + ∞)/(U (1) × Sp(∞)). We also have the multiplicity-free property for spaces that interpolate between (SU (p + ∞), SU (p) × SU (∞)) and the limit Grassmannian (U (p + ∞), lim U (p) × U (n). −→ Fix a closed subgroup F of U (1). Then we have the multiplicity-free property for the pairs (G, K) = lim{(Gn , Kn )}, where Gn = SU (p + n) and −→ kn 0 k ∈ U (p), k ∈ SU (n), det k ∈ F . Kn = n n n 0 k n

Limits of Gelfand pairs related to spheres and Grassmann manifolds. We have the multiplicity-free property for the six cases described in Theorem 32, four of which are nonsymmetric, as well as some obvious variations. Fix a closed subgroup F of U (1); it can be any ﬁnite cyclic group or the entire circle group U (1). As a result we have the multiplicity-free property for the nonsymmetric pairs SU (∞)/[SU (p) × SU (∞ − p)] = lim SU (n)/[SU (p) × SU (n − p)] , −→ [Sp(∞) × U (1)]/[F × Sp(∞ − 1) × U (1)diag ] = lim[Sp(n) × U (1)]/[F × Sp(n − 1) × U (1)diag ] , −→ [Sp(∞) × Sp(1)]/[{1} × Sp(∞ − 1) × Sp(1)diag ] = lim[Sp(n) × Sp(1)]/[{1} × Sp(n − 1) × Sp(1)diag ] , and −→ [Sp(∞) × Sp(1)]/[{±1} × Sp(∞ − 1) × Sp(1)diag ] = lim[Sp(n) × Sp(1)]/[{±1} × Sp(n − 1) × Sp(1)diag ] . −→ What we don’t have. There is a huge number of direct systems {(Gn , Kn )} of Gelfand pairs where the Gn are compact connected Lie groups. We have

Inﬁnite-Dimensional Multiplicity-Free Spaces, I

481

only veriﬁed the multiplicity-free condition for a few of them. We have not, for example, checked it for the interesting cases Gn = SU (2n + 1) and Kn = F × Sp(n), F ⊂ U (1) ﬁnite cyclic, and Gn = SO(2n) and Kn = F × SU (n), n odd, n 3. Also, we have not checked it for the very interesting case Gn = Sp(an ) × Sp(bn ) and Kn = Sp(an − 1) × Sp(1) × Sp(bn − 1), which is a prototype for nonsymmetric irreducible direct systems {(Gn , Kn )} with the Gn semisimple → Gn is given by not simple. In that case Kan −1

k 0 a but 0 1 n (k1 , a, k2 ) → , , so G /K ﬁbers over P (H) × P bn −1 (H) n n 0 k 0 a 2 3 with ﬁber (Sp(1) × Sp(1))/(diagonal) = S .

References [Ar1962]

S. Araki, On root systems and an inﬁnitesimal classiﬁcation of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. [DPW2002] I. Dimitrov, I. Penkov & J. A. Wolf, A Bott–Borel–Weil theory for direct limits of algebraic groups, Amer. J. of Math. 124 (2002), 955–998. [Fa2006] J. Faraut, Inﬁnite dimensional harmonic analysis and probability, in “Probability Measures on Groups: Recent Directions and Trends,” ed. S. G. Dani & P. Graczyk, Narosa, New Delhi, 2006. [He1978] S. Helgason, “Diﬀerential Geometry, Lie Groups, and Symmetric Spaces,” Academic Press, New York, 1978 [Kr1979] M. Kr¨ amer, Sph¨ arische Untergruppen in kompakten zusammenh¨ angenden Liegruppen, Compositio Math. 38 (1979), 129–153. [Ol1990] G. I. Ol’shanskii, Unitary representations of inﬁnite dimensional pairs (G, K) and the formalism of R. Howe, in “Representations of Lie Groups and Related Topics,” ed. A. M. Vershik & D. P. Zhelobenko, Advanced Studies Contemp. Math. 7, Gordon and Breach, New York, 1990. [W1980] J. A. Wolf, Foundations of representation theory for semisimple Lie groups, in “Harmonic Analysis and Representations of Semisimple Lie groups,” ed. Wolf, Cahen & DeWilde, Mathematical Physics and Applied Mathematics, vol. 5 (1980), Reidel, Dordrecht, pp. 69–130. [W2005] J. A. Wolf, Direct limits of principal series representations, Compositio Mathematica, 141 (2005), 1504–1530. [W2007] J. A. Wolf, Harmonic Analysis on Commutative Spaces, Math. Surveys & Monographs vol. 142, Amer. Math. Soc., 2007. [W2009] J. A. Wolf, Inﬁnite dimensional multiplicity-free spaces II: Limits of commutative nilmanifolds, Contemporary Mathematics 491 (2009), 179–208. [Ya2004] O. S. Yakimova, Weakly symmetric Riemannian manifolds with reductive isometry group, Math. USSR Sbornik 195 (2004), 599–614.

Index

(C • (g, V ), dg ); Chevalley–Eilenberg complex of g, 286 C k -regular Lie group, 269 Fω ; ﬂux homomorphism, 329 M (k, σ1 , . . . , σr ); multi loop algebra, 324 Pρ ; ﬂat bundle, 287 V (k), 283 W ∗ -algebra, 423 [A, A], 78 [a b], 76 Aut(P ), 282, 286 Bun(M, K), 328 Γ E; sections of the bundle E, 286 Gau(P ), 282, 286 aut(P ) = V(P )K , 282 V, 315 z ⊕ω g; central extension deﬁned by ω, 294 gau(P ), 287, 314 Ad(P ), 316 λg , 285 1 Ω (M, V), 315 p Ω (M, V ), 294 [p] ω ; cocycles deﬁned by a diﬀerential form, 302 ωη , 290 ωκ , 290, 315 ωκ,η , 290 1 Ω (M, V ), 283 perκ , 297 perκ ; κ invariant bilinear form, 297 perω ; period homomorphism, 329

ρg , 285 σ-topology, 423 Diﬀ(M ), 287 ∧κ , 298 kω -space, 256 xl , left invariant vector ﬁeld, 285 xr , right invariant vector ﬁeld, 285 A(P ); aﬃne space of connections on P , 321 V(M ), 282 (LB)-space, 260 (LF)-space, 260 2-morphism betw. morph. of bun. ger., 342

˙ I (root system of type A), 62 A An (ﬁnite root system of type A), 63 abstract Wiener group, 399 abstract Wiener space, 397 accessible set, 425 adjoint action, 432 adjoint map, 421 adjoint of a symmetric norming function, 435 adjoint orbit, 432 admissible Cartan matrix, 173 aﬃne coadjoint action, 438, 443 aﬃne coadjoint orbit, 442 aﬃne form, 74 aﬃne Kac–Moody Lie groups, 282 aﬃne Kac–Moody superalgebras, 183 aﬃne Lie algebra, 65 extended, 104

484

Index

locally extended, 108 toral type extended, 110 aﬃne reﬂection Lie algebra, 96 isomorphic, 96 nullity, 97 aﬃne reﬂection system, 72 aﬃne form, 74 discrete, 75 morphism, 72 nullity, 72 aﬃne root system, 65 extended, 75 locally extended, 75 Saito’s extended, 75 twisted, 67 untwisted, 65 algebra, 76 k-algebra, 76 base ﬁeld extension, 79 central, 77 central-simple, 77 centre (associative), 77 centre (Lie), 78 centroid, 76 derivation, 76 graded, see graded algebra, 77 invariant bilinear form, 76 perfect, 76 simple, 76 almost local map, 262 alternative torus, 23, 29 isotope, 29 A(u1 ,u2 ) , 29 isotopic, 30 opposite, 31 amenable group, 375 anisotropic roots, 100 Arens-type product, 377 ARLA, see aﬃne reﬂection Lie algebra, 96 associated norm ideals, 373 associated vector bundle, 445 associative torus, 23, 24 associative torus with involution, 23, 35 isotope, 37 (A, ι(h) ), 37 isotopic, 39 mod-2 quadratic form, 36 automatic smoothness, 275

automorphic forms on orthogonal groups, 151, 157–160, 162–165, 167 automorphic product, 157–160, 162–165, 167 classiﬁcation, 164–165, 167 reﬂective, 160, 163–165, 167 BI (root system of type B), 62 Bn (ﬁnite root system of type B), 63 Banach ideal, 433, 434 Banach Poisson manifold, 420 Banach symplectic manifold, 420 Banach–Lie algebra, 421 Φ-reductive, 384 classical complex, 385 classical real, 385 Banach–Lie group Φ-reductive, 385 Φ-reductive linear, 385 classical complex, 385 classical real, 385 of Harish-Chandra type, 388 Banach–Lie–Poisson space, 421 base, 174 BCI (root system of type BC), 62 BCn (ﬁnite root system of type BC), 63 biconjugacy class, 359 bilinear form graded, 78 invariant, 61, 76 radical of, 61 Borel–Weil Theorem, 419, 444, 451 Bott–Thurston cocycle, 309 bounded operator, 422 boundedly regular, 249 box, 257 box topology, 257 Boyd indices, 371 Brownian motion, 398 Brownian sheet, 400 bump function, 268 bundle automorphism group; Aut(P ), 282 bundle gerbe, 323, 341, 349, 354 basic, 347 bundle gerbe bimodule, 358 bundle gerbe module, 357

Index CI (root system of type C), 62 Cn (ﬁnite root system of type C), 63 canonical map, 420 canonical projection, 72 Cartan map, 290 Cartan matrix, 174 Cartan subalgebra splitting, 64 Cartan–Helgason Theorem, 459 Cartan–Helgason theorem, 464 CDer (centroidal derivations), 82 Cent(A) (centroid of A), 76 (Cent A) ( centroidal transformations of degree ), 79 central, 77 central extension, 90, 339, 357, 435, 437, 442 covering, 90 graded, 90 graded covering, 90 central-simple, 77 centralizer, 432 centralizer of a functional, 430 centralizer of an element, 430 centreless core, 96 Lie algebra, 77 centroid, 9, 76 centroidal grading group, 81 centroidal derivation, 82 centroidal grading group, 9, 81 Γ (L), 9 character, 194 character formula, 151, 153, 154 characteristic distribution, 424, 426, 431, 434, 438–440, 442, 443 characteristic subspace, 424, 426, 429 class 1 fundamental highest weight, 465 classical root systems, 62 classiﬁcation of locally ﬁnite root systems, 62 2-cocyle group, 80 invariant toral, 102 coadjoint action, 428, 432, 438 coadjoint map, 421 coadjoint orbit, 350, 427, 431, 432, 438, 440, 443 coeﬃcient function, 467

485

coherent pre-reﬂection system, 60 commutative space, 459 compact operator, 422 compact regularity, 246 compactly regular, 246 compatible grading, 77 complex analytic map, 252 complex projective space, 451 conditional expectation, 432, 445–447 cone, 252 conjugacy class, 352, 358 connected component of Re(R), 60 connected real part, 60 connected roots, 60 connection on a bundle gerbe, 345 on a line bundle, 344 consistent, 426 continuous vector bundle, 446 contractive bimodule, 371 contragredient Lie algebra, 169 contragredient Lie superalgebra, 170 core, 96 coroot system, 62 covering, 90 crossed homomorphism of Lie algebras, 304 crossed homomorphism of Lie groups, 306 crossed product algebra, 80 curving, 345 DI (root system of type D), 62 Dn (ﬁnite root system of type D), 63 D (degree derivations), 83 D-brane, 362 Dahmen’s Theorem, 261 Darboux Theorem, 421 ∆(E, H), 5 defect, 211 deﬁning representation, 469 degree derivation, 83 Deligne cohomology, 344, 346, 347, 355 denominator identity, 151, 153, 154, 160–163, 165, 166 Der derivation algebra, 76 (Der A) (derivations of degree ), 79 derivation centroidal derivation, 82

486

Index

degree derivation, 83 diﬀeomorphism group, 243 direct limit, 252, 459 direct limit chart, 246 direct limit properties, 245 direct limit property, 268 direct limit representation, 461 direct limit topology, 254 direct sum of pre-reﬂection systems, 60 direct system, 252 discrete extended aﬃne Lie algebra, 107 reﬂection system, 75 divisible root, 62 division-(R, Λ)-graded Lie algebra, 88 division-R-graded Lie algebra, 89 division-graded associative algebra, 79 division-root-graded Lie algebra, 88 Dixmier–Douady class, 340 dual reductive pair, 461 duality pairing, 423, 433, 435–437, 441

isotopy, 14–22 isotropic root, 6 locally, 108 nullity, 6 toral type, 110 type, 6 extended aﬃne root system, 75 locally, 75 Saito’s, 75 extension central extension, 90 graded, 90 datum, 69 of locally ﬁnite root system, 73 locally ﬁnite root system, 72 canonical projection, 72 of pre-reﬂection systems, 69 universal central extension, 90 extension datum, 69 of a locally ﬁnite root system, 73 extremely amenable group, 380

Eij (matrix unit), 112 EALA, see extended aﬃne Lie algebra, see extended aﬃne Lie algebra EARS, see extended aﬃne root system endomorphism of degree , 78 equivalent crossed homomorphisms, 306 equivariant structure on a bundle gerbe, 353, 360 Euler element, 225 ev (evaluation map), 83 even reﬂection, 179 evolution map, 247 exact invariant bilinear map, 290 exponential function, 247 exponential function (of a Lie group), 286 exponential map, 247 extended aﬃne Lie algebra, 5, 104 centreless core, 6 construction, 10–13 core, 6 discrete, 107 E(L, D, τ ), 12 family, 13 bijectively isomorphic, 22 isomorphic, 6, 105

faithful, 448 faithful functional, 430, 432 fake monster algebra, 151, 160–163, 166 fgc, 77 ﬁber integral, 310 ﬁnite growth, 173 ﬁnite root system, 62 ﬁnite-dimensional superalgebras, 182 ﬁnite-rank operator, 434 ﬂag geometries, 235 ﬂux homomorphism, 329 formal degree, 461 formal degree renormalization, 466 Fq (the quantum torus associated to q), 85 fragmentation map, 258 function algebra, 470 Fundamental Formula, 227 fundamental highest weight, 464 Γ (centroidal grading group), 81 gl (general linear Lie algebra), 112 gl (general Lie algebra of row- and column-ﬁnite matrices), 119 gauge group Gau(P ), 282 Gelfand pair, 459 generalized foliation, 424

Index generalized Kac–Moody algebra, 151–156, 160–166 classiﬁcation, 164–165, 167 construction, 160–161, 165–167 generalized reductive Lie algebra, 109 root system, 75 geometric representation theory, 444 GIF (graded invariant bilinear forms), 78 GNS representation, 444, 445, 447, 450 grCDer (graded centroidal derivations), 82 grCent (graded centroid), 79 grDer (graded derivation algebra), 79 grEnd (graded endomorphism algebra), 79 grSCDer (graded skew centroidal derivations), 82 graded algebra, 77 compatible grading, 77 crossed product algebra, 80 division-graded (associative), 79 division-graded (Lie), 89 full support, 78 graded-central, 79 graded-central-simple, 79 graded-isomorphic, 78, 89 graded-simple, 78 twisted, 80 homogeneous space, 77 isograded-isomorphic, 78, 89 Lie torus, 89 multiloop, 81 predivision-graded (associative), 79 predivision-graded (Lie), 89 quantum torus, 85 root-graded, 88, 89 division, 88 invariant, 89 predivision, 88, 89 support, 77 torus, 79 Lie, 89 graded-central algebra, 79 graded-central-simple algebra, 79 graded-isomorphic, 5, 78, 89 graded-simple, 78 grading subalgebra, 94

487

Grassmann manifold, 476 Grassmannian, 233 group algebra, 77 twisted, 80 group cohomology, 353 group manifold, 465 group of germs of diﬀeomorphisms, 244 GRRS, see generalized reductive root system, 75 HC1 (C) (ﬁrst cyclic homology group of C), 92 Hamilton equations, 420 Hamiltonian vector ﬁeld, 420–423, 427, 437 Harish-Chandra decompositions, 388 Heisenberg group, 320 Hermitian vector bundle, 446 highest vector, 195 highest weight, 195 highest weight module, 131, 132, 138, 141, 142 highest weight representation, 462 highest weight vector, 462 Hilbert–Schmidt ideal, 370, 434 Hilbert–Schmidt operator, 422, 436 holomorphic Hermitian vector bundle, 442 holomorphic reproducing kernel, 447 holonomy of a bundle gerbe, 348 homogenous space, 77 homomorphism, 427 homotopy group, 246, 273 IDer (inner derivations), 84 IDer (completed inner derivations), 119 IARA, see invariant aﬃne reﬂection algebra, 100 imaginary root, 59, 181 immersion, 425, 429 indecomposable module, 128, 135, 139, 143 indecomposable pre-reﬂection system, 60 indivisible root, 62 inﬁnite dimensional Lie group, 285 inﬂation map, 331 initial Lie subgroup, 274

488

Index

inner ideal, 238 integrable module, 194 integrable root, 64 integrable weight, 196, 209 integral diagonal, 372 strictly lower triangular, 372 strictly upper triangular, 372 integral pre-reﬂection system, 60 intrinsic subspace, 237 invariant aﬃne reﬂection algebra, 100 isomorphic, 101 bilinear form, 61, 76 root-graded Lie algebra, 89 toral 2-cocycle, 102 invariant 2-cocycle, 11 invariant diﬀerential operator, 460 inverse of an invertible element, 88 invertible, 227 invertible element, 88 irreducible, 62 isograded-isomorphic, 5, 78, 89 ηgr , 5 isomorphic aﬃne reﬂection Lie algebras, 96 extended aﬃne Lie algebras, 105 invariant reﬂection algebras, 101 isotope, 90 isotopic, 90 Isotrivilaity, 46 isotropic root, 173 iterated loop algebra, 324 Jandl structure, 360 Jordan algebra, 227 Jordan pair, 225 Jordan torus, 23, 25 isotope, 25 A(u) , 25 isotopic, 25 Jordan triple system, 226 Jordan-H¨ older series, 134 Jordan-Lie functor, 226, 233 k-algebra, 76 K¨ ahler form, 442 K¨ ahler manifold, 451 Kac–Moody Lie superalgebra, 170, 179

M algebras, 169 Kantor pair, 226 k[Λ] (group algebra of Λ), 77 kt [Λ] (twisted group algebra of Λ), 80 Kr¨ amer classiﬁcation, 468 k(S), 65 Lα (α-root space of L), 64 Lagrangian geometry, 235 LB-space, 258 leaf, 426 leaf of the characteristic distribution, 426 LEALA, see locally extended aﬃne Lie algebra, 108 LEARS, see locally extended aﬃne root system, 75 Leibniz property, 419 LF-space, 258 Lie algebra aﬃne, 65 aﬃne reﬂection, 96 discrete extended aﬃne, 107 extended aﬃne, 104 generalized reductive, 109 locally extended aﬃne, 108 root-graded, 89 simply connected, 90 Steinberg, 116 toral type extended, 110 Lie algebra (ﬁltered), 236 Lie algebra (graded), 224 Lie algebra (symmetric), 224 Lie group compact, 357 simply-connected, 349, 357 Lie subgroup, 428 Lie torus, 7 bi-isograded-isomorphic, 8 ϕr and ϕe , 8 bi-isomorphic, 8 centreless, 8 coordinatization theorems, 22–24 grading pair, 8 isotope, 13 L(s) , 13 isotopic, 14 multiloop, 41 root grading and external grading, 7

Index type, 7 untwisted, 9 Lie triple system, 224 Lie-(R, Λ)-torus, 89 Lie–Poisson bracket, 422 limit aligned representation, 461 limit irreducible representation, 458 line bundle, 340, 345 over a coadjoint orbit, 350 over the loop space, 355 linear Poisson map, 422 Littlewood-Richardson coeﬃcients, 129 local map, 262 locally kω space, 256 locally convex direct limit, 248 locally convex direct limit topology, 258 locally exponential Lie group, 275, 286 locally extended aﬃne Lie algebra, 108 aﬃne root system, 75 locally ﬁnite Lie algebra, 127, 144, 264 locally ﬁnite root system, 62 classical, 62 classiﬁcation, 62 coroot system, 62 divisible root, 62 extension datum, 73 indivisible root, 62 irreducible, 62 normalized form, 63 root basis, 62 Loewy length, 129, 133, 139, 142 logarithmic derivative, (left/right) δ l/r (f ), 286 long root, 65 loop algebra, 324 multiloop 81 twisted, 67 untwisted, 66 loop group, 281 loop space, 355 Mat (ﬁnitary matrices), 112 Mat (row- and column-ﬁnite matrices), 119 Mackey complete, 247 mean, 376 left invariant, 375 Meyberg’s Theorem, 227

489

mononormalizing, 370 moonshine for Conway’s group, 160–163 morphism, 422 between bundle gerbes, 342, 347 of aﬃne reﬂection systems, 72 multiloop algebra, 49, 81, 324 multiplicity, 201 multiplicity-free, 459 natural representation, 127, 128, 130, 131, 137, 138, 141, 144, 145 negative root, 172 nest algebra, 373 non-isotropic root, 173 non-regular Lie group, 272 non-symmetrizable superalgebra, 186, 193 nondegenerate pre-reﬂection system, 60 nontrivial Boyd indices, 371 norm ideal, 369 normal, 432, 441, 448 normal functional, 432 normalized form, 63 normalized matrices, 172 null roots, 100 null-system, 234 nullity of an aﬃne reﬂection Lie algebra, 97 of an aﬃne reﬂection system, 72 octonion torus, 29 odd reﬂection, 176 orbit symplectic form, 429 psl (projective special linear Lie algebra), 114 pair geometry (linear, aﬃne), 231 parabolic direct limit, 472 parabolic direct system, 472 partial section, 68 partition, 129 perfect algebra, 76 period homomorphism, 329 pointed reﬂection subspace, 70 Poisson bracket, 421 Poisson Lie group, 440 Poisson manifold, 419 Poisson map, 420 Poisson-Lie group, 437, 438

490

Index

polar decomposition, 448 positive deﬁnite reproducing kernel, 446 positive functional, 432 positive root, 172 pre-reﬂection system, 59 aﬃne form, 74 coherent, 60 direct sum, 60 extension, 69 indecomposable, 60 integral, 60 invariant bilinear form, 61 morphism, 60 partial section, 68 nondegenerate, 60 quotient, 69 real part of, 60 connected, 60 connected component, 60 reduced, 60 root string, 61 unbroken, 61 strictly invariant bilinear form, 61 symmetric, 60 tame, 60 predivision-(R, Λ)-graded Lie algebra, 88 predivision-R-graded Lie algebra, 89 predivision-graded associative algebra, 79 predivision-root-graded Lie algebra, 88, 89 predual, 423, 428, 430, 433, 436–438 principal root, 176 product integral, 247 product map, 258 projective elementary Lie algebra, 28 pe3 (A), 28 projective geometry (generalized), 230 projective group (elementary), 230 quantum matrix, 85 quantum torus, 24, 85 quasi immersion, 425 quasisimple, 173 quotient pre-reﬂection system, 69 quotient root system, 72 (R, Λ)-graded Lie algebra, 88

graded-isomorphic, 89 isograded-isomorphic, 89 R-graded Lie algebra, 89 (pre)division, 89 Ran (anisotropic roots), 100 R char (coroot system of R), 62 R div (divisible roots of R), 62 R im (imaginary roots of R), 59 Rind (indivisible roots of R), 62 R (integrable roots of R), 64 Rre (real roots of R), 59 R× (the non-zero roots of R), 88 × (the non-zero indivisible roots of Rind R), 88 R0 (null roots), 100 Rad (radical of a bilinear form), 61 Re(R) (the real part of R), 60 real analytic map, 252 real part of a pre-reﬂection system, 60 real root, 59, 181 realization operator, 447–449, 451 reduced pre-reﬂection system, 60 reduced root system, 7 reductive Lie algebra, 369 reductive Lie group, 368 reﬂection space, 70 reﬂection subspace, 70 pointed, 70 symmetric, 70 reﬂection system, 59 aﬃne, 72 associated to bilinear forms, 63 reﬂective root, 59 regular Cartan matrix, 173 regular contragredient superalgebra, 176 regular Lie group, 247, 285 regular root, 173 regular weight, 205, 209 reproducing kernel, 390, 446, 447 restricted Banach algebra, 436 restricted general linear group, 396 restricted Grassmannian, 441–443 restricted unitary algebra, 435, 436 restricted unitary group, 436, 438, 439 restriction map, 331 riemannian symmetric space, 462 (R, Λ)-graded Lie algebra isotope, 90

Index isotopic, 90 root, 172 anisotropic, 100 divisible, 62 divisible, 62 imaginary, 59 integrable, 64 long, 65 null, 100 real, 59 reﬂective, 59 short, 65 root basis, 62 root space decomposition, 64 root string, 61 unbroken, 61 root system, aﬃne, 65 twisted, 67 untwisted, 65 extended aﬃne, 75 ﬁnite, 62 generalized reductive, 75 locally ﬁnite, 62 quotient, 72 root-graded Lie algebra, 88, 89 grading subalgebra, 94 invariant, 89 root-reductive Lie algebra, 144 S lg (long roots), 65 Ssh (short roots), 65 SS(β, α) (α-root sting through β), 61 slI (A) (special linear Lie Lie algebra), 112 st (Steinberg Lie algebra), 116 sl2 -triple, 64 sα (reﬂection in α), 59 Saito’s extended aﬃne root system, 75 Schatten ideal, 370, 434 Schur functor, 129 Schur Orthogonality, 466 Schur-Weyl Duality, 133, 135, 136, 146 SEARS, see Saito’s extended aﬃne root system, 75 section, 446 self-adjoint element, 430 self-adjoint functional, 430 Serre’s relations, 180

491

set of weights, 194 short root, 65 Silva space, 260 simple algebra, 76 simple roots, 172 simply connected, 90 singular root, 173 small subgroup, 274 small subgroups, 248 smooth bump function, 268 smooth generalized distribution, 426 smooth vector bundle, 446 smoothly regular space, 268 Sobolev–Lie group, 244 socle ﬁltration, 128, 129 special linear Lie algebra, 32 slr+1 (A), 32 special symplectic Lie algebra, 34 ssp2r (A, ι), 34 spherical function, 467 spin factor, 229 split Lie subgroup, 427 splitting Cartan subalgebra, 64 squeezed bundle, 314 standard base, 175 standard imbedding, 224 state, 237, 431, 432, 444, 445, 447, 448 Steinberg Lie algebra, 116 Stinespring dilation, 390 Stone–Weierstrass theorem, 471 strange twisted aﬃne superalgebra, 185, 193 strict direct system, 254 strictly invariant bilinear form, 61 string, 178 strong symplectic form, 420, 429, 442, 443 strong symplectic leaf, 434, 438, 439 strong symplectic manifold, 421, 431 strongly C k -regular Lie group, 269 subalgebra ad-diagonalizable, 64 toral, 64 subordinated cohomology class, 302 subsystem, 60 support, 77 full, 78 symmetric norming function, 369, 434, 435

492

Index

symmetric pre-reﬂection system, 60 symmetric reﬂection subspace, 70 symmetric space, 232 symplectic leaf, 425, 426, 429, 434 tr (trace), 112 tame, 217 tame pre-reﬂection system, 60 tame toral pair, 96 tensor algebra, 128 tensor representation, 128, 130–132, 135–141, 144 test function group, 244 tier number, 65 Tits-Kantor-Koecher Lie algebra, 24 TKK(A), 24 topological group with Lie algebra, 275 toral pair, 96 centreless core of, 96 core of, 96 tame, 96 toral subalgebra, 64 toral type extended aﬃne Lie algebra, 110 toroidal Lie algebra, 9, 314 Torsor, 46 torus associative, 79 Lie, 89 quantum, 85 trace class operator, 422, 436 tracial, 432, 448 transversal, 230, 235, 238 triangular decomposition, 172 trivialization of a bundle gerbe, 347 truncation diagonal, 371 strictly lower triangular, 372 strictly upper triangular, 372 twisted aﬃne Kac–Moody Lie groups, 282 twisted aﬃne root system, 67 twisted aﬃne superalgebras, 184 twisted loop algebra, 67

two-cocycle, 437 types I and II, 174 typical weight, 200, 209 uce (universal central extension), 90 unbroken root string, 61 uniformly continuous, left, 375 right, 375 unitary group, 430, 436 unitary orbit, 430, 431, 433 universal ambit, 379 universal central extension, 90 untwisted aﬃne root system, 65 untwisted loop algebra, 66 Verma module, 195 vertex algebra, 151, 153, 155, 157, 160, 162, 165, 166 lattice, 156, 160, 162 vertex operator algebra, 155–157, 165, 166 WZW model, 156 Virasoro group, 282, 308 W (R), 60 weak direct limit chart, 246 weak direct product, 244, 257 weak immersion, 425, 429–431, 434 weak K¨ ahler, 433–435 weak symplectic form, 420, 429, 438 weak symplectic leaf, 435, 439, 443 weak symplectic manifold, 421, 431, 433 weight module, 194 Weil representation, 158, 159, 164 well-ﬁlled chart, 272 Wengenroth’s Theorem, 265 Wess–Zumino–Witten model, 361 Weyl group, 60, 179 Weyl groupoid, 176 Yamasaki’s Theorem, 257 Young projector, 129 Z(β, α), 61 Z(.) (the centre of an algebra), 77, 78