Connections, Curvature and Cohomology: 3

Connections, Curvature, and Cohomology Volume III Cohomology of Principal Bundles and Homogeneous Spaces

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Ellenberg and Hyman E a m r

Columbia University, New York

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In preparation I. MARTINISAACS. Character Theory of Finite Groups

Connections, Curvature, and Cohomology Werner Greub, Stephen Halperin, and Ray Vanstone DEPARTMENT OF MATHEMATICS UNIVERSITY OF TORONTO TORONTO, CANADA

VOLUME

rrr

Cohomology of Principal Bundles and Homogeneous Spaces

A C A D E M I C P R E S S New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich,Publishers

1976

COPYRIGHT 0 1976, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY A N Y MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR A N Y INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NW1

Library of Congress Cataloging in Publication Data Greub, Werner Hildbert, Date Connections, curvature, and cohomology. (Pure and applied mathematics; a series of monographs and textbooks, v. 47) Includes bibliographies. CONTENTS: v. 1. De Rham cohomology of manifolds and Lie groups, principal bundles, vector bundles.-v. 2. Cohomology of and characteristic classes.-v. 3. principal bundles & homogeneous spaces. 1. Connections (Mathematics) 2. Curvature. 1. Halperin, Stephen, joint 3. Homology theory. 11. Vanstone, Ray, joint author. 111. Title. author. IV. Series. QA3.P8 vol. 47 [QA649] 510'.8s [514'.2] 79-159608 ISBN 0-12-302703-9 (v. 3) AMS (MOS) 1970 Subject Classifications: 55F20,57F10,57F15, 57D20,18H25

PRINTED IN THE UNITED STATES OF AMERICA

Respec2fulZy dedicated to the memory of HEINZ HOPF

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Contents xi xiii xxi

Preface

Introduction Contents of Volumes I and 11

Chapter 0 Algebraic Preliminaries

1

PART 1 Chapter I Spectral Sequences 19 25 31 45 48

1. Filtrations 2. Spectral sequences 3. Graded filtered differential spaces 4. Graded filtered differential algebras 5.

Differential couples

Chapter I1 Koszul Complexes of P-Spaces and P-Algebras 1. 2. 3. 4. 5. 6.

53 62 67 70 78 90

P-spaces and P-algebras Isomorphism theorems The Poincar&Koszul series Structure theorems Symmetric P-algebras Essential P-algebras

Chapter I11 Koszul Complexes of &Differential Algebras 1. 2. 3. 4. 5. 6. 7.

P-differential algebras Tensor difference Isomorphism theorems Structure theorems Cohomology diagram of a tensor difference Tensor difference with a symmetric P-algebra Equivalent and c-equivalent (P, 8)-algebras vii

95 103 109 114 126 135 147

...

Contents

Vlll

PART 2 Chapter IV Lie Algebras and Differential Spaces 1. Lie algebras 2. Representation of a Lie algebra in a differential space

157 169

Chapter V Cohomology of Lie Algebras and Lie Groups 1. 2. 3. 4. 5. 6. 7. 8.

Exterior algebra over a Lie algebra Unimodular Lie algebras Reductive Lie algebras The structure theorem for ( o The structure of ( A E*)o= 0 Duality theorems Cohomology with coefficients in a graded Lie module Applications to Lie groups

174 185 188 193 199 206 210 215

Chapter VI The Weil Algebra 1. 2. 3. 4. 5. 6. 7.

The Weil algebra The canonical map p E The distinguished transgression The structure theorem for (VB*)o= 0 The structure theorem for ( V E ) # =o , and duality Cohomology of the classical Lie algebras The compact classical Lie groups

223 231 236 241 249 253 264

Chapter VII Operation of a Lie Algebra in a Graded Differential Algebra 1. Elementary properties of an operation

2. 3. 4. 5.

Examples of operations The structure homomorphism Fibre projection Operation of a graded vector space on a graded algebra 6. Transformation groups

273 278 284 292 300 307

Chapter VIII Algebraic Connections and Principal Bundles 1. Definition and examples The decomposition of R 3, Geometric definition of an operation 4. The Weil homomorphism 5. Principal bundles 2.

314 319 331 340 352

Chapter IX Cohomology of Operations and Principal Bundles 1. The filtration of an operation 2. The fundamental theorem

359 363

Contents

3. 4. 5. 6. 7.

Application&of the fundamental theorem The distinguished transgression The classification theorem Principal bundles Examples

ix

371 3 78 382 390 397

Chapter X Subalgebras 1. 2. 3. 4. 5. 6. 7. 8. 9.

Operation of a subalgebra The cohomology of ( AE*)rF=o. + = a The structure of the algebra H(E/F) Cartan pairs Subalgebras noncohomologous to zero Equal rank pairs Symmetric pairs Relative Poincar6 duality Symplectic metrics

41 1 420 427 431 436 442 447 450 454

Chapter XI Homogeneous Spaces 1. 2. 3. 4. 5.

The cohomology of a homogeneous space The structure of H(GIK) The Weyl group Examples of homogeneous spaces Non-Cartan pairs

457 462 469 474 486

Chapter XI1 Operation of a Lie Algebra Pair 1. 2. 3. 4. 5.

Basic properties The cohomology of BF Isomorphism of the cohomology diagrams Applications of the fundamental theorem Bundles with fibre a homogeneous space

498 509 519 526 540

Appendix A Characteristic Coefficients and the Pfaffian 1. Characteristic and trace coefficients 2. Inner product spaces

547 554

Notes

563

References

574

Bibliography

575

Index

587

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This monograph developed out of the Abendseminar of 1958-1959 at the University of Zurich. It was originally a joint enterprise of the first author and H. H. Keller, who planned a brief treatise on connections in smooth fibre bundles. Then, in 1960, the first author took a position in the United States, and geographic considerations forced the cancellation of this arrangement. The collaboration between the first and third authors began with the former’s move to Toronto in 1962; they were joined by the second author in 1965. During this time the purpose and scope of the book grew to its present form: a three-volume study, ab initio, of the de Rham cohomology of smooth bundles. I n particular, the material in volume I has been used at the University of Toronto as the syllabus for an introductory graduate course on differentiable manifolds. During the long history of this book we have had numerous valuable suggestions from many mathematicians. We are especially grateful to the faculty and graduate students of the institutions below. The proof of Theorem VII in sec. 2.17 is due to J. C. Moore (unpublished), and we thank him for showing it to us. We also thank A. Bore1 for sending us his unpublished example of a homogeneous space not satisfying the Cartan condition, which we have used in sec. 11.15. The late G. S. Rinehart read an early version of the manuscript and made many valuable suggestions. A. E. Fekete, who prepared the subject index, has our special gratitude. We are indebted to the institutions whose facilities were used by one or more of us during the writing. These include the Departments of Mathematics of Cornell University, Flinders University, the University of Fribourg, and the University of Toronto, as well as the Institut fur theoretische Kernphysik and the Hoffmannhaus, both at Bonn, and the Forschungsinstitut fur Mathematik der Eidgenossischen Technischen Hochschule, Zurich. The entire manuscript was typed with unstinting devotion by Frances Mitchell, to whom we express our deep gratitude. A first class job of typesetting was done by the compositors. A. So and xi

xii

Preface

C. Watkiss assisted us with proofreading; however, any mistakes in the text are entirely our own responsibility. Finally, we would like to thank the production and editorial staff at Academic Press for their unfailing helpfulness and cooperation. Their universal patience, while we rewrote the manuscript (ad infinitum), oscillated amongst titles, and ruined production schedules, was in large measure responsible for the completion of this work. Werner G r a b Stephen Halperin Ray Vanstone Toronto, Canada

Introduction T h e purpose of this monograph is to develop the theory of de Rham cohomology for manifolds and fibre bundles. The present (and final) volume is an exposition of the work of H. Cartan, C. Chevalley, J.-L. Koszul, and A. Weil, which provides an effective means of calculating the de Rham cohomology of principal bundles and of homogeneous spaces. In fact, let (P,?T, B, G) be a smooth principal bundle with G a compact connected Lie group, and let E denote the Lie algebra of G. Then H ( G ) is an exterior algebra over the subspace Pc of primitive elements. Let VE" and ( V E"),, respectively, denote the symmetric algebra over E" and the subalgebra of elements invariant under the adjoint representation. Now suppose given the following data: (i) A linear map 7 : Pc +( V E"), . (ii) T h e graded differential algebra (A(B),6) of differential forms on

B. (iii) T h e Chern-Weil homomorphism h: ( V E"), --f H(B). Choose a linear map y : ( V E"), + A(B)such that y(@)is a closed form representing the class h(@). Then a differential algebra (A(B)@ A Pc , V,) is given by v,(y @ XI A ' ' ' A X,) = 6Y , @ XI A ' ' ' A X, B

+(-l)desy

Z(-I)Y-l I=1

YA~(TX,)@X~ A --.3i.j-.-

AX,.

A fundamental theorem of C. Chevalley states that for suitable maps 7 , called transgressions, there is a homomorphism of graded differential algebras ( A ( B )@ A Pc , V,) +( A ( P ) ,6), which induces an isomorphism of cohomology. Next let K be a closed connected subgroup of G with Lie algebra F. An analogous theorem identifies the cohomology of G / K with the cohomology of the graded differential algebra (( V F"), @ A Pc , V), where V is given by

v (y@ X1

A -.*

A X,)

...

XI11

xiv

Introduction

(Here j " : ( V E#),+ ( V F " ) , denotes the homomorphism induced by the inclusion map j : F +E . These two theorems generalize to a single theorem on the cohomology of a fibre bundle whose fibre is a homogeneous space. T h e first part of this volume is chiefly devoted to Koszul complexes. (These are graded differential algebras of the form ( A ( B )0 A P G , V,) described above.) The second part is a careful and complete exposition of the purely algebraic theory (due to H. Cartan) of the operation of a Lie algebra in a graded differential algebra. In particular, the theorems described above are special cases of results about operations. These results reduce problems on operations to problems on Koszul complexes, to which the machinery of Part 1 can then be applied. The applications to manifolds (including the theorems above and a number of concrete examples) are given separately in the last articles of Chapters V to IX, all of Chapter XI, and the last article of Chapter XII. If these articles (which depend heavily on volumes I and 11) are omitted, the rest of this volume is an entirely self-contained unit, accessible to any reader familiar with linear and multilinear algebra. Indeed, Part 2 begins with an account of Lie algebras (Chapter IV) which contains all the necessary definitions and results (but omits some proofs). Moreover, aside from a single quotation in Chapter VI of a theorem proved early in Chapter 11, the results of Chapters I1 and I11 are not used until Chapter IX. Thus the interdependence of chapters is given by

/

I -VI

IX-x-XI-XII

-VIII1 -IH -VIII

IV-V7 A more detailed description of the contents appears below. Unlike volumes I and 11, this volume contains no problems. Much of the material in this volume first appeared in the articles by Cartan, Koszul, and Leray in the proceedings of the Colloque de Topologie (espaces fibrb) held at Brussels in 1950 (cf. [53], [168], and [187]). The first account with complete proofs is [9]. In the notes at the back we shall give more historical details and acknowledge the discoverers of the main theorems, We apologize for errors and omissions.

Part 1 I n this part all vector spaces and algebras are defined over a commutative field I?.

xv

Introduction

Chapter I. Spectral Sequences. This chapter, which has been included only for the sake of completeness, is a self-contained description of the spectral sequence of a filtered differential space. T h e reader already familiar with this material may omit the whole chapter and simply refer back to it as necessary. Almost all the theorems and proofs in the chapter apply verbatim to filtered modules over a commutative ring. Chapter 11. Koszul Complexes of P-Spaces and P-Algebras. Let P = &,. Pk be a finite-dimensional positively graded vector space with Pk= 0 for even k. Let P denote the evenly graded space given by P = P k - l . A P-algebra (S; cr) is a positively graded associative algebra S together with a linear map cr : P+ S, homogeneous of degree 1, such that ~ ( x is ) in the centre of S , x E P. T h e Koszul complex of ( S ; cr) is the graded differential algebra (S @ AP, V,) given by v,(Z

@ XI

A **.

(-

1)deE

-

A X,)

P

2

C ( - l ) i - l ~ .(xi) . 0x1 A ..-$i

A

x,.

i=l

The gradation S @ A P = x k S @ A kPinduces a gradation, H ( S 0 A P ) = & H k ( s @ AP) in cohomology. These and other basic facts are established in article 1. T h e isomorphism theorems in article 2 show, for example, that H + ( S @ r \ P ) = 0 if and only if S Z H o ( S @ A P ) @ V P . Suppose now that S is connected (i.e., So= r).In this case (article 4)the projection S+ I? induces a homomorphism H ( S @ A P ) + A P whose image is the exterior algebra over a graded subspace P c P. Moreover, if p is a complementary subspace, then H ( S @ A P) H ( S @ AP) @ AP. Finally, article 5 deals with symmetric P-algebras ( V Q ; u). The main theorem asserts that if H( V Q @ A P ) has finite dimension, then dim P =dimP+dimQ+k, where k is the greatest integer such that Hk( V Q @ A p ) 0. This is applied to determine the complete structure of H( V Q @ A P ) when dim P = dim P dim Q. T h e results of article 3 are used to obtain the PoincarC polynomial for H( V Q @ A P ) in this case.

+

+

Chapter 111. Koszul Complexes of P-Differential Algebras. A P-differential algebra (or (P, 8)-algebra) is a positively graded associative alternating differential algebra (B, 8,) together with a linear map T : P+ B, homogeneous of degree 1, and satisfying 6 0 T = 0. T h e Koszul complex of (B, 8,; T ) is the graded differential algebra ( B @ AP, V,), where V, = 8, @ L V, . In articles 3 and 4 the isomorphism and structure theorems for Palgebras are generalized to (P, 8)-algebras. Theorem VII in article 4

+

Introduction

xvi

gives six conditions, each equivalent to the surjectivity of the homomorphism H ( B @ A P ) + A P induced by the natural projection. The tensor difference ( B @ S, a,@,, 7 0u) of two (P, a)-algebras (B, as; 7 ) and ( S , 8,; u) is defined by a,@,= 8, @ 1 - W , @ a s and (7 0u)(x) = 7x @ 1 - 1 @ ux. (0, is the degree involution of B.) Its basic properties are established in article 2, including the fact that if S = V P , then H ( B @ V P @ A P ) = H(B). In article 5 the commutativity of the cohomology diagram,

H(S)

-

H ( B @ S @ AP)

-

-H(B)@ AP)

is established. It is also shown that the map H ( B @ S @ AP)+ H( S @ A P)is surjective if and only if the graded spaces H ( B @ S @ A P) and H ( B ) @ H(S @ A P ) are isomorphic. Article 6 gives necessary and sufficient conditions for this to be an algebra isomorphism if S is a symmetric algebra.

Part 2 In this part all vector spaces and algebras are defined over a commutative field I' of characteristic zero.

Chapter IV. Lie Algebras and Differential Spaces. Article 1 starts with the definition of Lie algebras and their representations and then quotes without proof the basic theorems about reductive Lie algebras and semisimple representations. Some material on Cartan subalgebras and root space decompositions is also included. Article 2 deals with representations of Lie algebras in differential spaces. T h e results of this article are fundamental for the following chapters, and complete proofs are given. One such theorem asserts that H((X @ Y ) ,= o) = H ( X , = o) @ H( Y , = o) under suitable hypotheses. (If a Lie algebra is represented in a space X , then X , = ,, denotes the subspace of invariant vectors.) Chapter V. Cohomology of Lie Algebras and Lie Groups. Let E be a finite dimensional Lie algebra. T h e adjoint representation of E determines representations of E in A E and AE*. Moreover, an anti-

Introduction

xvii

derivation of square zero 6, in A E* is determined by (8, x*, x A Y ) = - [x, y ] . Its negative dual 8, is a differential operator in A E , homogeneous of degree - 1. T h e algebra H*(E) = H( A E*, 6,) and the space H,(E) = H(A E, a,) are called, respectively, the cohomology and homology of E. If E is reductive, then ( A E*), = EH*(E) and ( A = H,(E) (article 3). Moreover, in this case there are canonical dual subspaces PE c ( AE*),= and P J E ) c ( A E),= (the primitive subspaces), and the inclusions extend to scalar product preserving isomorphisms of graded algebras ez

A P E S ( AEW),=

and

AP,(E) 5 ( A & =

(articles 4,5 , and 6). Recall from volume I1 (article 4, Chapter IV) that the cohomology of a compact connected Lie group is isomorphic with the cohomology of its Lie algebra. In article 8 this isomorphism is used to translate the results of the chapter into theorems on the cohomology of Lie groups.

Chapter VI. The Weil Algebra. Let E be a finite dimensional Lie algebra. Let E* denote the graded space with E" as underlying vector space and deg x* = 2, x* E E*. In article 1 an antiderivation 6, of square zero is introduced in the algebra W(E)= V E* 0 A E*; the resulting graded differential algebra is called the Weil algebra of E. T h e adjoint representation of E induces a representation of E in (W(E),&). T h e main result of article 1 states that H+(W(E))= 0 and H+(W(E),=o) = 0. Using this result we construct in article 2 a canonical linear map @E : ( V E*)o=o -+ ( A +E")O= 0 , homogeneous of degree -1 (the Cartan map). Theorem I1 in article 4 asserts that if E is reductive, then +

hn @ E = PE

and

ker p E = ( V + E"),=, . ( V E"),=,. +

A transgression in W(E), is a linear map T : P, -+ ( V E*), = homogeneous of degree 1 such that eE 0 T = L . Theorem I of article 4 says that a transgression extends to an algebra isomorphism V P, 5 ( V E *), = o. In particular, ( v E*)o= is the symmetric algebra over an evenly graded space whose dimension is equal to that of P, . I n article 6 these results are applied to determine the algebras ( AE"), = and ( V E"), = o, where E is a classical Lie algebra. Finally, in article 7 we determine H(G) for the classical compact Lie groups. Explicit invariant multilinear functions in E and closed differential forms on G are constructed and shown to yield bases of these spaces. +

Chapter VII. Operation of a Lie Algebra in a Graded Differential Algebra. An operation of a Lie algebra E in a graded differential

xviii

Introduction

algebra (R, 6) consists of a representation 8 of E in (R, 6) together with antiderivations i(x), x E El in R of degree -1 and such that for x, y E E and

i(x)6

+

= qx).

The horizontal and basic subalgebras of R are defined, respectively, by R , = ,= nzEE ker i(x) and R , =o , ,= = (R,=,),= o . R , =, ,= is stable under 6. An action of a Lie group G on a manifold M determines an operation of its Lie algebra on the algebra of differential forms ( A ( M ) ,6) via the Lie derivatives and the substitution operators for the fundamental vector fields (article 6). The Weil algebra of E is a second example of an operation. In articles 3 and 4 the fibre projection eR: H(R,=o ) - +( AE*),= is defined when E is reductive and H(R, = o ) is connected. It reduces to the obvious homomorphism H(M)+H(G) in the example above if G is compact and M and G are connected. 'It is shown that Im eR= AP and H(R,= o ) A 0 A P , where P is a subspace of PEand A is a subalgebra of

H(RB = 0 ) . Chapter VIII.

Algebraic Connections and Principal Bundles. An algebraic connection for an operation of E in a graded differential algebra (R, 6) is an E-linear map x : E*+ R1 such that i(x)x(x*) = (x*, x). I n article 2 it is shown that an algebraic connection determines an isomorphism R , = o0 AE* 4 R. The curvature of an algebraic connection is the linear map 2 : E* +R:= determined by 6(1 0x*) =1 06,~" 2x" 01. It extends to a homomorphism Z V : V E* -+ R, = o , inducing the Weil homomorphism 2" : ( V E *)o= -+ H(R,= , = o ) . Theorem V, article 4, shows that the Weil homomorphism is independent of the algebraic connection. T,B, G ) is a In article 5 we consider the example R = A(P),where (PI principal bundle. Then A(B)g R, = o , ,= o . There is a one-to-one correspondence between algebraic connections and principal connections. Moreover, the corresponding curvatures determine each other, and the Weil homomorphism corresponds to the Chern-Weil homomorphism of the principal bundle as defined in volume 11.

+

,

Chapter M. Cohomology of Operations and Principal Bundles. Suppose an operation of a reductive Lie algebra E in a graded differential algebra (R, 6) admits an algebraic connection X. Let T : PE--f ( V E"), = be a transgression (cf. Chapter VI) and set T~ =%, o T : PE--f R , = o, = o . Then (R,=,,o = o , 6;T ~ is) a ( P E ,6)-algebra (cf. Chapter 111). A funda-

,

Introduction

xix

mental theorem of Chevalley (article 2) gives a homomorphism from the corresponding Koszul complex to (A(P),S), which induces an isomorphism of cohomology. This isomorphism is then used to apply the theorems of Chapter I11 to operations (articles 3 and 4)and to the cohomology of principal bundles (article 6). As examples, the cohomology algebras of the tangent frame bundles of CP" and C P" x CP" are determined.

Chapter X. Subalgebras. A Lie algebra pair (E, F ) is a Lie algebra E together with a subalgebra F. The inclusion map is denoted by j : F +E. The subalgebra F operates in the graded differential algebra ( A E", aE). This chapter deals with cohomology of the basic subalgebra; this is denoted by H(E/F). Let k # :H ( E / F ) - + H ( E )be the homomorphism induced by the inclusion map. In article 1 it is shown that if E is reductive, then I m k # is the exterior algebra over a subspace P c P E , called the Samelson space for the pair ( E , F ) . Suppose (E, F ) is a reductive pair; i.e., E is reductive and F acts semisimply in E. Let T : P E + ( v E')e= be a transgression, and set u =jv 0 T. Then (( V F"),= ; 0) is a symmetric P,-algebra. According to a fundamental theorem of Cartan (article 2), there is a homomorphism from the Koszul complex (( V F"),=o @ APE, -Vo) to (( AE"),F=o, eF=o, 8), inducing an isomorphism in cohomology. I n article 3 this result is applied to translate the structure theorems of Chapter I1 into theorems on H(E/F).Article 4 shows that for a reductive pair dim PE2 dim PF dim P . If equality holds, (E, F ) is called a Cartan pair. For such pairs H ( E / F ) A 0 AP, where A = ( V F')e= o/I and I is the ideal generated by j v ( V E")B = o . Moreover, the PoincarC polynomial of H ( E / F )is given explicitly in this case. A subalgebra F c E is called noncohomologous to zero in E if j " : H*(E)+H*(F) is surjective. These subalgebras are discussed in article 5 . Article 6 deals with equal rank pairs (dim PE= dim PF). I n particular it is shown that if F is a Cartan subalgebra of E, then (E, F ) is an equal rank pair. I n article 7 the results are applied to symmetric pairs, and article 8 gives a relative version of the PoincarC duality theorem.

+

+

ChapterXI. Homogeneous Spaces. Suppose K c G are compact connected Lie groups with Lie algebras F c E. Then there is an isomorphism H(E/F)g H(G/K)which permits us to translate the theorems of Chapter X into theorems on H(G/K)(articles 1 and 2). In particular the Cartan condition is shown to depend only on the topology of GIK. Article 3 is devoted to Leray's theorem, which asserts that if T is a maximal torus in G with Lie algebra H , then j " is an isomorphism from

xx

Introduction

( V E")e= onto the subalgebra of V H " of elements invariant under the action of the Weyl group. Finally, in article 4 the cohomology of the standard homogeneous spaces is determined, while article 5 contains examples of non-Cartan pairs.

ChapterXII. Operation of a Lie Algebra Pair. An operation of E in (R, 8) restricts to an operation of any subalgebra F. We shall say that this is an operation of the pair (E, F)if the inclusion map ReE= +ReF = induces an isomorphism in cohomology. Suppose that the hypotheses of the fundamental theorems of Chapters I X and X are satisfied and let (Ri,= o, BE = @ ( V F ")e = @ A P, , V) be the Koszul complex of the tensor difference of ( R i E= o. BE = o , 6; T ~ and (( V F " ) e = o u) ; (cf. Chapter 111). The main theorem of this chapter (article 2) asserts that I n article 4 the theorems of Chapter I11 on tensor differences are applied to the operation of a pair, and in article 5 these results are used to determine the cohomology of fibre bundles whose fibre is a homogeneous space. I n particular, let (P,7,B, G ) be a principal bundle, where G is semisimple, compact, and connected, and let K be a torus in G. Consider the associated bundle (P/K,e, B, GIK). It is shown that if the graded algebras H(P/K) and H(B) @ H(G/K) are isomorphic, then all the characteristic classes of the principal bundle are zero.

)

Contents of Volumes I and I1 Volume I : De Rham Cohomology of Manifolds and Vector Bundles 0 Algebraic and Analytic Preliminaries I Basic Concepts I1 Vector Bundles I11 Tangent Bundle and Differential Forms IV Calculus of Differential Forms V De Rham Cohomology VI Mapping Degree VIJ Integration over the Fibre VIII Cohomology of Sphere Bundles IX Cohomology of Vector Bundles x The Lefschetz Class of a Manifold Appendix A. The Exponential Map

Volume I1 : Lie Groups, Principal Bundles, and Characteristic

Classes 0 Algebraic and Analytic Preliminaries I Lie Groups I1 Subgroups and Homogeneous Spaces I11 Transformation Groups IV Invariant Cohomology V Bundles with Structure Group VI Principal Connections and the Weil Homomorphism VII Linear Connections VIII Characteristic Homomorphism for Lbundles IX Pontrjagin, PfafFian, and Chern Classes X The Gauss-Bonnet-Chern Theorem Appendix A. Characteristic Coefficients and the Pfaffian

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Chapter 0

Algebraic Preliminaries 0.0. Notation. Throughout this book iX denotes the identity map of a set X. When it is clear which set we mean, we write simply i. The empty set is denoted by 0.The symbol x,, . - .4,j . - .x p means xi is to be deleted. The symbols N,Z, (2, R,and C denote, respectively, the natural numbers, the integers, the rationals, the real and the complex numbers. Throughout the book I' will denote a commutative field, and all vector spaces and algebras are defined over r unless we explicitly state otherwise. Moreover, from Chapter IV o n it is assumed that r has characteristic zero. The group of permutations on n letters is denoted by Sn; if a E Sn, then en = 1 (-1) if a is even (odd). Finally, the proofs of the assertions made in this chapter will be found in [4] and [ 5 ] .

0.1. Linear algebra. We shall assume the fundamentals of linear and multilinear algebra. A pair of vector spaces X",X is called dual with respect to a bilinear function

(,):x"xx+r if (x",

X) = 0

and

(X", x)

=0

imply, respectively, that x" = 0 and x = 0. ( , ) is called the scalar product. If X is finite dimensional, then a pair of dual bases for X" and X is a basis e*i for X" and a basis ei for X such that (eQi,e j ) = di.. 3 If Y c X is a subspace, then the orthogonal complement Y l c X" is defined by Y l = {x" E X" I (x", Y ) = O}. 1

2

0. Algebraic Preliminaries

The direct sum of vector spaces XP is denoted by

~

X

Por

P

OX*. P

The space of linear maps from X to Y ( X and Y arbitrary vector spaces) is denoted by L ( X ; Y), and we often write L ( X ;X ) = L x . If pl E Lx,then its determinant and trace are written det pl and tr pl. Let X", X and Y", Y be pairs of dual vector spaces. Then linear transformations pl E L ( X ; Y) and pl" E L( Y"; X " ) are called dual if

(p"y", x) = ( y " , px),

y"

E

Y",

x.

xE

An inner product space ( X , ( , )) is a finite-dimensional vector space X together with a symmetric scalar product ( , ) between X and itself. ( , ) is called the inner product. If ( X , ( , )) is an inner product space, then the dual p* of a linear transformation p of X is again an element of L,; pl is called skew symmetric if pl" = -e,. A symplectic space ( X , ( , )) is a finite-dimensional vector space X together with a skew symmetric scalar product ( , ) between X and itself. ( , ) is called the symplectic metric. If e, E Lx is skew with respect to ( , ) (i.e., if pl = -p*), then p is called skew symplectic. The tensor product of vector spaces X and Y is denoted by X @ Y. If X", X and Y*, Y are pairs of finite-dimensional dual vector spaces, then X" @ Y" and X @ Y are dual with respect to the scalar product ( , } defined by (x*

Or*,x @ Y > = (x", X'E

X)(Y",Y), X", y"E Y", X E

x,

Y E Y.

It is called the tensol product of the original two scalar products. If X and Yare vector spaces and y E Y, then X @ y denotes the subspace of X @ Y consisting of vectors of the form x @ y , x E X . If X has finite dimension, a canonical isomorphism a:X' @ Y 2% L ( X ; Y ) is defined by a(x* @ y ) ( x ) = (x",

x)y,

x' E X",

xE

x,

y

E

If Y = X and e*i, ei is any pair of dual bases for X", X , then

Y.

0. Algebraic Preliminaries

3

0.2. Gradations. A graded space is a vector space X together with a direct decomposition X = CpszXp. T h e subspace Xp is called the space of homogeneous vectors of degree p . X is called positively graded if Xp = 0 for p < 0. X is called evenly graded if XP = 0, p odd, and oddly graded if Xp = 0, p even. A subspace Y c X is called a graded subspace if

Y

=

2 Y n Xp. P

T h e degree involution wX of a graded space X is defined by wx(x) = (-1)px,

xE

xp.

A graded space X is said to have Jinite type if each Xp has finite dimension. I n this case the formal series

fx

=

C (dim Xp)tp P

is called the Poincare' series for X . If Y is a second graded space of finite type, we write f y 5 f x if dim Yp 5 dim Xp, all p . If dim X is finite, then f-r is a polynomial, called the Poincare' polynomial. I n this case the alternating sum X, = (-1)pdimXP = x r ( - l )

1 P

is called the Euler-Poincare' characteristic of X . Assume that (Xp)* and Xp are dual spaces ( p E 2).Then the scalar products extend to the scalar product between the graded spaces X" = C p (Xp)* and X = C pXp, defined by

X* and X are called dual graded spaces. A bigraded vector space X = Cp,qsz Xp.9 is defined analogously. If X = C Xp~qis a bigraded space, then the gradation X = CrX ( r )given by X(r)=

C

Xp.9

p+q=r

is called the induced total gradation. A linear map v: X + Y between graded spaces is called homogeneous of degree r if it restricts to linear maps from Xp to Y p + r ( p E Z). Homogeneous maps of bidegree ( r , s ) between bigraded spaces are defined

0. Algebraic Preliminaries

4

analogously, and linear maps homogeneous of degree zero (respectively, homogeneous of bidegree zero) are called hmmorphisms of graded spaces (respectively, homomorphisms of bigraded spaces). Let X = & XP be a graded space. Then we write

x+= c XP. P >O

If p: X Y is a homomorphism of graded spaces, its restriction to X + is denoted by --f

p+:

x+

--+

Y+.

0.3. Algebras. An algebra A over F is a vector space, together with a bilinear map A x A --+ A (called the product). A system of generators for A is a subset S c A such that every element of A is a linear combination of products of elements of S. If X and Y are subsets of A, then X Y denotes the subspace spanned by the products xy (x E X , y E Y ) .An ideal I in A is a subspace such that

.

IeAcI

and

A-IcI.

The ideal I . I is denoted by 12. A homomorphism of algebras tp: A B is a product preserving linear map, while a derivation 8 in A is a linear transformation of A satisfying --f

W Y ) = %)Y Given a homomorphism 8,: A + B such that

tp:

A

+

+ x8(Y).

B, a p-derivation is a linear map

Homomorphisms, derivations, and p-derivations are completely determined by their restrictions to any set of generators. In this book we shall consider associative algebras with identity, and Lie algebras (cf. sec. 4.1). In the first case the identity is written 1 and is identified with the subspace r . 1 (so that scalar multiplication coincides with multiplication in A). Homomorphisms between two associative algebras with identity are always assumed to preserve the identity. A graded algebra A is a graded vector space A = & Ap, together with a product, such that Ap . A4 c Apia. A homomorphism pl: A -,B of

r

5

0. Algebraic Preliminaries

graded algebras is an algebra homomorphism, homogeneous of degree zero. A graded algebra A with identity element 1 E Ao is called connected if

AO=

r

and

A p = 0,

p < 0.

An antiderivation in a graded algebra A is a linear map 8: A such that

qXy) = qX)y

+

(-i)PXe(y),

X

E

+A

AP, y E A.

If v: A B is an algebra homomorphism, then a v-antiderivation is a linear map 8,: A + B such that -+

An associative graded algebra A is called anticommutative if

If in addition x2 = 0 for x E Ap, p odd, then A is called alternating. (These notions coincide if I' has characteristic not equal to two.) If A and B are graded algebras, their anticommutative (or skew) tensor product is the graded algebra, A 0 B, defined by

and

B. If A and B are anticommutative (alternating), then so is A In this book the tensor product of graded algebras will always mean the anticommutative tensor product, unless explicitly stated otherwise. There is, however, a second possible multiplication in A 0 B ; it is called the canonical tensor product and is defined by

Assume A , B, and C are graded algebras, and that C is anticommutative. Let p?4: A -+ C and T ~ B: + C be homomorphisms of graded

6

0. Algebraic Preliminaries

algebras. Then a homomorphism

of graded algebras is defined by cp(a @ b) = cpA(a)

cpB(b).

0.4. Exterior algebra. The exterior algebra over a vector space X

is denoted by AX and the multiplication is denoted by A. By assigning ApX the degree p, we make AX into a graded alternating algebra. If el, . . . ,en is a basis for X, we write

AX

= A(el,

. . . ,en).

Let A be any associative algebra, and assume that cp: X -+ A is a linear map such that ( c p ~ )= ~ 0, x E X. Then cp extends to a unique algebra homomorphism cpA: AX

--+

A.

We sometimes denote plA by Acp. (If A is graded and alternating, and cp(X) c A', then ( c p ~ ) = ~ 0, x E X and cpA is a homomorphism of graded algebras.) A linear map p: X -+ ApX(p odd) extends uniquely to a derivation in AX. A linear map y: X --+ ApX (p even) extends uniquely to an antiderivation in AX. Now let X*, X be a pair of dual finite-dimensional vector spaces. Then ApX* and ApX are dual with respect to the scalar product given by (xr

A

---

A

xp", x1 A

- - - A x p ) = det( (xi",

xj)),

xi" E

X*, xi E X .

Thus AX* and AX are dual graded spaces. Moreover, we identify APX* with the space of p-linear skew symmetric functions in X by writing @(XI,

. . . , X p ) = (@,

XI A

---

h X,),

@ E ApX",

Xi

E

x.

Suppose Y*, Y is a second pair of dual finite-dimensional spaces, and let pl E L ( X ; Y ) and y* E L ( Y * ;X") be dual maps. Then the homomorphisms

y,,:AAX-+AY

and

are dual. We will denote (y*), by yA.

(y*),:AX*cAY*

0. Algebraic Preliminaries

7

If x E X , then i ( x ) denotes the unique antiderivation in AX" extending the linear map X" --+ Fgiven by x* H (x*, x). It is called the substitution operator and is homogeneous of degree -1. The substitution operator is dual to the multiplication operator p ( x ) in AX defined by

p(x)b= x

A

b E AX.

b,

More generally, if a E AX, then p ( a ) is the multiplication operator given by p ( a ) b = a A b. The dual operator is denoted by i ( a ) . Clearly,

i(x, A

. - .A x p ) = i ( x p ) . . 0

0

i(xl),

xi E X .

The following result is proved in [ S ; Prop. 11, p. 1381. Proposition I: Let A c AX" be a subalgebra, stable under the operators i ( x ) , x E X . Then

A = A(X* n A ) . Next, suppose X = Y @ Z . Then an isomorphism

AY 0AZ -%A X of graded algebras is defined by a 0b H a A b. If Y", Y and Z", Z are pairs of dual finite-dimensional spaces, then the isomorphisms

AY+@AZ*%AX*

and

AY@AZ--%AX

satisfy (@ @ Y, a @ b ) = (@, a ) ( Y , b ) =

(@A

Y, a h b ) ,

@ P A Y * , YVAZ", a E A Y , b E A Z . Finally, let X = Ck=, X p be an oddly positively graded space. Then

AX=AX'@

* * -

@AXr.

Give AX the gradation defined by

(AX)p =

c

(APlX') 0 * * * @ (A'rX').

P I + ~ P S + . - . +rp,=~

It is called the induced gradation, and makes graded algebra.

AX

into an alternating

8

0. Algebraic Preliminaries

If AX = A ( e , , . . . , e,) and degree ei = g i , then, clearly AX = @ A(er) and so the PoincarC polynomial for AX is given by A(e,) @

-

fAS

=

fi (1 +

tot).

i-1

0.5. Symmetric algebra. The symmetric algebra over a vector space

X is denoted by V X and the multiplication is denoted by v. If A is an associative algebra, and Q): X + A is a linear map, such that Q)X . ~ ) y = ~ ) - yQ)X, x, y E X , then Q) extends to a unique algebra homomorphism q Y :V X

+

A.

A linear map y : X V X extends uniquely to a derivation in V X . Suppose X = Y @ 2. Then an isomorphism of vector spaces --f

VY@ V Z Z V X is defined by U @ b H U V b .

More generally, let X = Ck=, X p be an evenZy positively graded space. Set

(VX). = zp,

+.

c

(VP2XZ)@

* * *

@ (VPrXr);

+rp,-p

then V X becomes a graded anticommutative algebra with respect to this induced gradation. (Note that (VX)p = 0 if p is odd!) If X is the direct sum of graded subspaces Y and 2, then the isomorphism above is an isomorphism of graded algebras. Suppose that A is an evenly graded commutative algebra, and that Q c A is a graded subspace. Then the inclusion extends to a homomorphism of graded algebras

VQ

--f

A.

If this homomorphism is an isomorphism, we will write V Q = A. In particular, if a,, . . . , a, is a homogeneous basis of Q, we write A = V(U,, . . . , u ~ ) . Since then A given by

= V(a,) @

n (1

..

@ V(ar),the PoincarC series for A is

r

=

i-1

-

deg ai = g l .

9

0. Algebraic Preliminaries

Now suppose X", X is a pair of dual finite-dimensional vector spaces, and assume that I' has characteristic zero. Then a scalar product between VPX" and VPX is defined by (x:

v

- - - v x;,

x1 v

. . . v xp) = perm((xr,

c

=

0E

xj>)

Xd1J * *

'

SP

- <x;,

Xdpl).

I n particular we identify VPX" with the space of symmetric p-linear functions in X by writing Y(x,,

. . . ,XP) = (Y, X I v

* * *

Y E VPX",

v xp),

If Y", Y is a second dual pair and 9:X + Y, qP:X" linear maps, then the homomorphisms 9": VX -+V Y

= (x",

t

E

x.

Y" are dual

(v*)~:VX" t V Y*

and

= 9". are dual as well. We write (v,")~ T h e substitution operator is(x) determined by x derivation in VX" satisfying

is(x)x"

xi

xx E

x),

E

X is the unique

X".

Its dual is multiplication by x in VX and is denoted by ,us(x). Finally, assume X* = Y" @ Z" and X = Y @ 2. Then (@v Y, a v b)

=

(@, a)(Y, b)

=

(@

@ E V Y " , YEVZ",

0Y, a 0b), aEVY, bEVZ.

0.6. Poincar6 duality algebras. A Poincare' duality algebra is a finitedimensional positively graded associative algebra A = C&oAP subject to the following conditions:

(1) dimAn = 1. (2) Let ex be a basis vector of (A")". Then the bilinear functions (,):ApxAn-p+r given by ( a , b) = (e", ab),

are nondegenerate.

a E Ap,

b E An-P,

0. Algebraic Preliminaries

10

If A is a PoincarC duality algebra, then isomorphisms D :Ap (A"-P)* are given by ( D a , b ) = (e+, ab).

=

D is called the associated Poincard isomorphism. Note that a PoincarC duality algebra A satisfies dim AP = dim An+,

p

= 0,

. . . , n.

Examples: 1. Exterior algebra: Let E", E be dual n-dimensional vector spaces. Then AE" is a PoincarC duality algebra, with PoincarC = AE given by isomorphism D : AE"

-

D@ = i(@)e, where e is a basis vector of AnE. 2. Tensor products: Let A and B be finite-dimensional graded algebras. Then A @ B is a PoincarC duality algebra if and only if both A and B are. I n fact, write A = AP and B = Cr Bq, where A" # 0 and Bm # 0. Then A @ B = Xttm(A @ B Y , and

x:

dim(A @ B)"+m= dim A n . dim Bm. Thus dim(A @ B)n+m = 1 if and only if dim A n = 1 = dim Bm. Now assume this condition holds. If e" E (A")" and f* E (Bm)" are basis vectors, then e+ @f* is a basis vector for [(A @ B)n+m]s. It follows that the bilinear function in A @ B is the tensor product of the corresponding bilinear functions for A and B (up to sign). In particular, it is nondegenerate if and only if both of them are nondegenerate.

0.7. Differential spaces. A dzyerential space ( X , 6) is a vector space X together with a linear transformation 6 of X such that 62= 0; 6 is called the dzperential operator. We write ker 6 = Z ( X ) ,

Im 6 = B ( X ) ,

Z ( X ) / B ( X )= H ( X , 6 ) (or simply H ( X ) )

and call these spaces, respectively, the cocycle, coboundary, and cohomology spaces of X .

0. Algebraic Preliminaries

11

A homomorphism p : ( X , S ,) + ( Y , 6), of differential spaces is a linear map v: X -,Y such that pdX = 6,~). It restricts to maps between the cocycle and coboundary spaces, and so induces a map, written

p:H ( X ) + H( Y ) , between the cohomology spaces. Assume (X", 6) and ( X , a) are finite-dimensional differential spaces such that X*, X is a pair of dual spaces, and 6 = &a*. Then the scalar product induces a scalar product between H ( X * ) and H ( X ) . A graded differential space (X, 6 ) is a differential space together with a gradation in X , such that 6 is homogeneous of some degree. I n this case Z ( X ) and B ( X ) are graded subspaces

Z ( X ) = 1Zp(X)

Bp(X).

B ( X )=

and

P

P

The space H ( X ) is then graded; we write

H ( X ) = 1HP(X);

H q X ) = ZP(X)/BP(X).

P

Suppose (X, 6,) and (Y, 6,) are graded differential spaces. A homomorphism of graded dz@rential spaces is a homomorphism of graded = Spp. spaces pl: X + Y such that q& Assume ( X , 6) is a finite-dimensional graded differential space. Then the Euler-Poincare' formula asserts that X, = X,, (cf. sec. 0.2); i.e. (-l)P

dim Xp = C ( - l ) p

Let

0

-

dim Hp(X).

P

P

(W, 6,)

I

( X , 6,)

- w

( Y , 6,)

0

be an exact sequence of differential spaces. Then a linear map

a: H ( Y ) + H ( W ) is defined as follows: Let a E H ( Y) and choose x E X so that yx represents a. Then there is a unique cocycle w E W such that pw = d X x . The class p E H ( W ) represented by w is independent of the choice of x, and is given by aa = B. d is called the connecting homomorphism.

a

12

0. Algebraic Preliminaries

Elementary algebra shows that the triangle

H ( W )’p”H ( X )

is exact. If the differential spaces are graded differential spaces, and ip and y are homogeneous of degrees k and I, then d is homogeneous and deg d = deg dX - k - 1. I n this case we obtain a long exact sequence (m = deg a)

-

HP( Y )

a

4

HP+m(

W )f +

v*

Hp+m+k(X)

-

HP+m+k+l( Y )

*

Let ( X , 6,) and ( Y , 6,) be graded differential spaces where 6, and dP are homogeneous of the same odd degree k. Then their tensor product is the graded differential space ( X 0 Y, B X B y ) given by

We will often write

Consider the inclusion

Z ( X ) 0Z ( Y ) + Z ( X 0 Y ) . It induces the (algebraic) Kiinneth isomorphism

zH ( X @ Y ) .

H ( X ) 0H ( Y )

0.8. Differential algebras. A graded dzzerential algebra (R, 6,) is a positively graded associative algebra R with identity, together with an antiderivation 6,, homogeneous of degree 1 and satisfying 6; = 0. If (R, 6,) is a graded differential algebra, then Z ( R ) is a graded subalgebra and B ( R ) is a graded ideal in Z ( R ) . Thus H ( R ) becomes a graded algebra.

0. Algebraic Preliminaries

13

A homomorphism p: (R, 6,) + (S,6s) of graded diferential algebras is a homomorphism of graded algebras which satisfies pdR = Ssp. The induced map p# is then a homomorphism of graded algebras. A positively graded differential algebra (R, 6,) is called c-connected if (cf. sec. 0.3) the algebra H ( R ) is connected. Finally, assume (R, 6,) and (S, 6,) are graded differential algebras. Then so is (R, 6,) @ (S, dS). Moreover, inclusions jR:

(R, 6,)

-+

(R, 6,)

8(s, 68)

and

jS:

(s,6,)

+

(R, 6R)

8(s, 6,)

are given by j R ( x ) = x 81 andjs(y) = 1 B y . In this case the Kunneth isomorphism is the isomorphism of graded algebras given by a

8B -jR#(a)

*

j,"(B),

a E H(R),

B E H(S).

Remark: In the literature it is usual to regard cohomology as a contravariant functor and homology as a covariant functor ; thus "the cohomology of a topological space is the homology of its cochain complex." Whatever the aesthetic advantages of this convention, it would, in this book, lead to a great deal of artificial lowering and raising of indices. For example (cf. Chapter V), for a Lie algebra E w e would have

while for a manifold M we would have

and for a smooth map p: M -+N inducing p": A ( M )+ A ( N ) we would have (@)# = p#: H ( M ) + H ( N ) . For this reason we have arbitrarily declared all differential spaces (with a single exception in sec. 5.5) to have cocycles, coboundaries, and cohomology, and used the notation p#: Hp(R) -+ H p ( S )for the map induced by a map p: R + S. 0.9. n-regularity. A homomorphism p: X + Y of positively graded spaces is called n-regular if the restrictions pp: X p .+ Yp satisfy: pp is an isomorphism, 0 I p 5 n, and pn+l is injective.

14

0. Algebraic Preliminaries

Proposition 11: Let v:(X, 6,) -+ (Y, 6,) be an n-regular homomorphism of positively graded differential spaces. Assume 6, and By are homogeneous of degree 1. Then $9# is n-regular. Proof: It follows at once from the hypotheses that $9:

zqx)4 ZP(Y ) , 0 5 p 4 n

and lp: B q X ) 5 Bp( Y ) , 0 5

Since

v:Zfl+l(X)-+

p 4 n + 1.

Zn+l(Y) is injective, the proposition follows. Q.E.D.

0.10. c-equivalent differential algebras. Let (R, 6,) and (S, 6,) be graded alternating differential algebras. Then a cohomological relation (c-relation) from (R, 6,) to (S, 6,) is a homomorphism

of graded differential algebras, such that $9# is an isomorphism. If such a c-relation exists, we say that (R, 6,) is c-related to (S, d,), and write

(R, 6,)

7(8, &I.

Note that this relation is not an equivalence. However, it generates an equivalence relation in the following way: Definition: Two alternating graded differential algebras (R, 6,) and (S, 6,) are called cohomologically equivalent (c-equivalent) if there is a sequence (Xi, di), i = 1, ,n of alternating graded differential algebras, satisfying the following properties :

. ..

(1) (Xl , 61) = (R, 8,) and (Xn , 8,) = (S,

(2) Either (Xi 3 Si) (i = 1, . . . ,n).

7(Xi+l,di+1)

w

or (Xi+lpdi+l) 7(Xi 9

I n this case we write

(R, 6R) 7 (s,6S)*

Si)

0. Algebraic Preliminaries

15

A specific choice of the (Xi, di), together with a specific choice of crelations pi between them will be called a c-equivalence between (R,6,) and (S, 8s). T h e isomorphisms determine an isomorphism

pr

H ( R )5 H ( S ) , which will be called the isomorphism induced by the given c-equivalence. An alternating graded differential algebra (R, 6,) is called split if there exists a homomorphism of graded algebras p: H ( R ) + Z(R)which splits the exact sequence

0 + B ( R ) -+Z(R)+ H ( R )

-

0.

Thus, if (R,6,) is split, then ( H ( R ) ,0) 7( R , 6,). More generally, (R,6,) will be called cohomologically split (c-split), if

I n this case the c-equivalence can always be chosen so that the induced isomorphism of H ( R ) is the identity. Such a c-equivalence will be called a c-splitting.

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PART 1 In this part r denotes a commutative field of arbitrary characteristic. All vector spaces and algebras are defined over r.

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Chapter I

Spectral Sequences Most of the results (and proofs) in this chapter continue to hold if F, instead of being a field, is allowed to be an arbitrary commutative ring with identity.

51. Filtrations 1.1. Filtered spaces. A decreasingfiltration of a vector space M is a family of subspaces Fp(M),indexed by the integers p E Z, and satisfying the conditions

Fp(M) 3 Fp+'(M)

and

M

=

U Fp(M). P

(In this book we consider only decreasing filtrations.) The definition of Fp(M) is extended to p = fco by writing

F-"(M)

=M

and

F"(M)

= 0.

With a filtered vector space M is associated its associated graded space AM= A&, defined by

0,

A& = F p ( M ) / F p + l ( M ) ,

--OO

< p < 00.

The canonical projection F p ( M )-+ A& is denoted by & or simply by ep. Suppose M and A? are filtered vector spaces. A linear map 9:M + A?i is called filtration preserving (or a homomorphism. of filtered spaces) if it restricts to linear maps p: Fp(M) + Fp(A?i). In this case 9 induces a 19

I. Spectral Sequences

20

unique linear map vA:A M

-+

Aa of graded spaces such that the diagrams

FP(M) LFp(rn)

A&-A&,

-O0

< p < 00,

PA

commute.

If M = C pMp is a graded vector space, then by setting Fp(M) = CpspMp we obtain a filtration of M. This filtration is said to be induced

r

by the gradation. In this case e p restricts to an isomorphism M p A&, and these isomorphisms define an isomorphism M AM of graded spaces.

-f.

1.2. Filtered differential spaces. A jltered dz$erential space is a differential space ( M , 6), together with a filtration (Fp(M)},,, of M such that the subspaces Fp(M) are stable under 6. If ( M , 6) is a filtered differential space, then a filtration of H ( M ) is defined by

Fp(H(M))= n(FP(M) n Z ( M ) ) , where 7c: Z ( M ) + H ( M ) denotes the canonical projection. Thus we can form the associated graded space A H ( M ) . Consider the associated graded space /Iizi. Since the Fp(M) are stable under 6, an operator in A M is induced by 6 . Clearly ( A M , 6,) is a differential space, and dL1is homogeneous of degree zero. 1.3. The differential spaces (Ei, di). Let ( M , 6) be a filtered differential space. Define subspaces Zf c M by Zf = F p ( M )n 6-'(FP+'((M)),

I t follows from the definition that

-m

< p < 00,

0 5i

< 00,

1. Filtrations

21

Next, define subspaces Dip by

and

D& = P ( M ) n 6(M). It follows that Df c DLl c Dip-'. Since 0% c Z g , we have the inclusion relations

Now form the factor spaces

Ef

= Zf/(Zf!$l

+ D,P_l),

1 5 i 5 00,

and extend the definition to i = 0 by setting

The canonical projections of Zf onto Ef will be denoted by It is easy to verify that

6: Zf

+

Zp'

and

6: ker $'

-

Hence, if i is finite, a linear map dip: Ef mutative diagram 2;

8

ker

qp,

+ Ef+" is

rf: Zp i --+ Ef .

0 5 i < 00.

defined by the com-

.q+i

It is clear that d p ' o dip = 0. Now consider the direct sums Ei = E f . If 0 5 i < 00, the operators d,P define a differential operator d i , homogeneous of degree i in the vector space Ei.Hence we can form the vector spaces

Bp

H(E,, di) = ker di/Im di. Since each vector space Eli is graded by the subspaces E f , an induced

I. Spectral Sequences

22

gradation in H ( E i , di) is given by

H P ( E i ,d i ) = ker df/Im dfPi. The projections ker df

--+

H p ( E i , di) will be denoted by nr

1.4. The spaces E, and E m . Proposition I: (1) The graded differential space ( E o , do) coincides with ( A M ,dA). (2) The graded space Em is isomorphic to the associated graded space of H ( M ) with respect to the filtration induced by the filtration of M.

Prwfi (1) follows immediately from the definition. T o prove (2) observe that ker 7% = zp," D%.

+

On the other hand (cf. sec. 1 2 ) , the filtration of H ( M ) is given by

FP(H(M))= n(Z%). Since (clearly) n(D%)= 0, we have

FP+'(H(M))= n(Z&+' Hence a surjective linear map commutative diagram

+ DL).

ug:EL + Ag(M)is

d i n e d by the

zz n FP(H(M))

But, evidently

It follows that ~JLis an isomorphism. Q.E.D.

23

1. Filtrations

1.5. Homomorphisms. Let v: ( M , 6) + (A,8) be a homomorphism of filtered differential spaces. Then the subspaces Zf, Dip are mapped into the subspaces Dp . Hence linear maps p?': EF -+ &' are given by the commutative diagrams

zf,

ei.

A simple computation shows The pp define linear maps pi: Ei + that pidi = dipi; i.e., pi is a homomorphism of graded differential spaces. Thus vi induces a linear map

homogeneous of degree zero. Proposition 11: If p: M + @j is a homomorphism of filtered differential spaces, then the linear map v#: H ( M ) + H ( m ) preserves the filtration (the filtration being defined in sec. 1.2). Moreover, if (T*).~: A H ( M+ ) AH(@)is the induced homomorphism of the associated graded spaces, then the diagram

commutes. Proof: Since

@(FP(H(M)))= fp"(n(Z&))= ji(q(ZL)) c q* is filtration preserving.

q.2)= F P ( H ( B ) ) ,

24

I. Spectral Sequences

Now consider the diagram

It follows from the definition of cp# and (P#)-~that the back face commutes. T h e definition of & (sec. 1.4) implies that the top and bottom faces commute. T h e definition of & (above) shows that the left-hand face commutes. Since &(hf, 0 n is surjective so is $, , and the right-hand face must commute.

Q.E.D.

$2. Spectral sequences 1.6. The spectral sequence of a filtered differentialspace. Definition: A spectral sequence is a sequence (Ei , d i , ai), m 5 i < 00, where ( E i , d i ) is a differential space, and ai is an isomorphism of Ei+, onto H ( E , , d i ) . We often omit the cri from the notation, and refer simply to the spectral sequence (Ei, di). A homomorphism of spectral sequences a : (Ei , d , , ai)

-

(I&,J i , Si)

is a system of homomorphisms of differential spaces ai: ( E i , d i ) -+ (&, di),such that the isomorphisms ai and Bi identify with ar :

A spectral sequence ( E i , d i , ai) is said to collapse at the kth term if di = 0, i 2 k. I n this case H ( E i , d i ) = E i , i 2 k, and so ai is an isomorphism from Ei+l to Ei. Let ( M , 6) be a filtered differential space, and consider the sequence ( E i , di)i20 of differential spaces constructed in sec. 1.3. We shall now construct isomorphisms ai: Ei+l 5 H ( E i , di),

i 2 0,

of graded spaces. T h e resulting spectral sequence ( E i , d i , ~ 6 will ) be called the spectral sequence of the jltered digwential space ( M , 6 ) . T h e ai are defined as follows: Consider the projection qf: 2: -+ E f (cf. sec. 1.3). I n Lemma I below we show that

qf(Z5,) = ker df'. Thus composing qf (restricted to Zf+l) with the projection nf:ker d,P + Hp(Ei, d i ) yields a surjective linear map

25

I. Spectral Sequences

26

Evidently, ker yf = Zf+l n (($)-l(Im d f i ) ) . I n Lemma 11, below, we show that this space coincides with Q+' Of. It follows that yf induces an isomorphism

+

1?

a:

E:+l G Hp(Ei, di).

The af define the desired isomorphism ai. Remark: In view of Proposition I, (l), sec. 1.4, a. is an isomorphism:

ao: El 5 H(&

, dA).

Lemma I: T $ ( Z ~ = + ~ker ) df.

Proofi We show first that ($)-l(ker d f ) Fix z E Zf'. Since df$ = $+'6,

= Z&l

it follows that dfqfz

But this occurs if and only if x E Zf+l the definitions. Thus (1.1) is proved. Now apply T$ to (1.1) to obtain

Lemma 11: 2:"

+ Zf!'.

(1.1) =0

if and only if

+ Z?
+ Dr = Zc1 n [($)-l(Im

dp)].

Proof: It follows at once from the definitions that

27

2. Spectral sequences

On the other hand, an easy calculation yields Zf-$I n Zf+, = Zf+I

Df'

and

c

Zf+l.

Thus intersecting (1.2) with Zf'+, yields the lemma. Q.E.D. Proposition III: Let q : (M, 6) + ($3, 8) be a homomorphism of filtered differential spaces. Then the maps qi: (Ei 8 di)

--+

(ei 4 ) 9

of sec. 1.5 form a homomorphism of the corresponding spectral sequences. Proof: It has to be shown that Biqi+l = qi#ai, i 2 0. Consider the diagram .If

ET+l

+

N -

ker dip

HP(Ei).

04

The outside and inside squares commute, as follows from the definition of ai and 6,.The left, right, and upper faces commute, as follows from the definitions of qi+l,qr, and qi. Since rB1 is surjective the lower face must commute. Q.E.D. 1.7. Filtrations induced by a gradation. Let M = C p e zMp be a differential space, and consider the induced filtration Fp(M)=

C Mp. PZP

I. Spectral Sequences

28

Assume that for some k 2 0,

p

6: Mp -+ Fp+k(M), Then 6 can be written uniquely as 6 = D

D(Mp) c Mpfk

and

E

Z.

+ 8, where;

8(Mp) c Fpfk+l(M), p E Z.

(1.3)

It is easy to see that D2 = 0. As an immediate consequence of formula (1.3), we have the relations

Zf

= Fp(M),

i 5 k,

and

Of

= 6(Fp-'((IM)) c

Fp-'tk(M) c F*'(M),

i 5 k - 1.

This shows that ker q2p = Fp+'(M), i 5 k, whence

Zf

= M p@

ker

rf,

i 5 k.

I t follows that for i 5 k the inclusion map j : Mp + Fp(M) can be composed with qf to yield an isomorphism tf : M p <Ef . T h e isomorphisms Ef define isomorphisms of graded spaces N

It is immediate from the definitions that di = 0, 0 5 i < k . Thus for 0 5 i < k we can regard ti as an isomorphism

t i :( M , 0) 5 (4,di),

(1.4)

of graded differential spaces. Next we show that t k D = dktk,so that

tk:( M , D ) -% (Ek, dk) is an isomorphism of graded differential spaces. In fact, fix z E Mp, Then formula (1.3) yields

(1.5)

2. Spectral sequences

29

and so (1.5) is proved. Formula (1.5) yields an isomorphism ti : H ( M , D ) 5 H ( E k , d k ) . Combining t$ with oil, we obtain an isomorphism 01

= '0 ;

0

[k#

: H ( M , D ) 5 Ek+'.

(1.6)

As a straightforward consequence of Lemmas I and I1 (sec. 1.6) or by direct computation we now obtain the formulae

Zl+l = Z p ( M ,D ) @ Fp+'(M) and Zf+'

(1.7)

+ D i = D ( & F k ) @ Fp+'(M).

Next, assume that &? = Cpsli@p is a second differential space and let it have the induced filtration. Moreover, assume that 8 ( i @ P ) c Fpfk(i@), p E Z, and let b:i @ p -+ f i p + k be defined as above. Assume that q ~ :M -+ i@ is a homomorphism of graded spaces which satisfies qd = 8cp. Then

vD = Dv, and so 9 determines a linear map I $ :

H ( M , D ) -+ H ( m , b).

On the other hand, v preserves the filtrations and so it induces a homomorphism of spectral sequences 'pi:

i 2 0.

(Ed,di) + (&, Ji),

It follows from the definitions that

.'if Ri is the induced isomorphism. where Fi: Passing to cohomology and using Proposition I11 we obtain the relations p7k#

0

tk#= {f Vg 0

and

9:

0

ok =

sk

0

P)k+l.

I. Spectral Sequences

30

Thus the diagram

commutes. 1.8. The homogeneous case. Suppose now that ( M , 6) is as above but that 6 is homogeneous of degree k. Then D = 6. I t follows that the spectral sequence collapses at the (k 1)th term. In fact, the homogeneity of 6 implies that 2: = 2% Pi-k, i 2 k 1, and hence for z E Z,P (i 2 k l),

+

+

+

+

di$z = ~YGZ = 0.

+

Since 17: is surjective, this implies that di = 0, i 2 k 1. Since D = 6 it follows from formula (1.5), sec. 1.7, that there are natural isomorphisms

H ( M , 6 ) LZH(Ek, dk)

Ek+l

Ek+2

*

+

*

;

i.e.,

Ei E'H ( M , 6 ) ,

k < i < 00.

It is simple to show that this relation holds for i =

00

as well.

$3. Graded filtered differential spaces 1.9. Graded filtered spaces. Let M = CrZn M r be a graded space which is filtered by subspaces Fp(M),p E Z.The filtration of M induces the filtration of the subspaces M r given by

Fp(Mr)= Fp(M) n Mr. We will call the filtration of M compatible with the gradation if the Fp(M) are graded subspaces; i.e., if

It is standard practice to call such spaces graded filtered spaces. However, in this book a graded filtered space will mean a graded space M = Cr2,,M r filtered by subspaces Fp(M) such that (1.9) holds, and in addition Fp(Mr)= 0 if p > Y. (1.10) Let M be a graded filtered space, and (cf. sec. 1.1) consider the associated graded space AAf. Since P ( M ) and Fp+'(M) are graded subspaces of M , a gradation is naturally induced in each A&:

and so A&,is a bigraded space. If x E AGq,we say it has base or filtration degree p , jibre degree q and total degree p q. Observe that condition (1.10) implies that Agq = 0 if q < 0, and that Ago = F p ( M p ) .

+

31

32

I. Spectral Sequences

The total gradation of AAliis given by

A,,

=

C A.:;',

AL\i)=

r

C

A.%q.

P+Pr

Note that A!$ is the associated graded space of the filtered space Mr. Suppose fi is a second graded filtered space and let y: M + i@ be a linear map, homogeneous of degree zero, which preserves the filtration. Then p is called a homomorphism of graded filtered spaces. Every such homomorphism induces a linear map T - ~ : -+ A@ (cf. sec. 1.1) which preserves the bigradation. 1.10. Graded filtered differential spaces. Let ( M , 6) be a graded differential space and assume that 6 is homogeneous of degree 1. Assume further that a filtration is given in M which makes M into a graded filtered space and into a filtered differential space. Then M is called a graded filtered differential space. Clearly the induced filtration and gradation on H ( M ) make H ( M ) into a graded filtered space. Now consider the subspaces Zf and Df defined in sec. 1.3. It follows from the homogeneity of 6 and formula (1.9) that 2: and Df are graded subspaces of M :

Then the relations above become

From sec. 1.3 and the fact the 6 is homogeneous of degree 1 we obtain the relations Qsq = FP(MP+P) n d-I(Fp+i(Mp+q+I))

(1.11)

and Df.Q = g(Zf-i.P+i-l),

The relation

O(i<m.

3. Graded filtered differential spaces

33

follows at once from the definitions. I t shows that the gradations of Zf and Df induce the gradation in Ef given by

Thus Ei becomes a bigraded space

Ei = C EP"'. P.9

+

As above, p , q, and p q are respectively called base degree, Jibre degree, and total degree. The total gradation of Ei induced by the above bigradation is given by

~ . The restriction of $' to Zfsg will be denoted by ~ f ?Thus

is a surjective map, and

Now consider the operators df: Ef -+ Ef+'. I t is immediate from the definitions that d f maps Efsq into Ef+i*q+l-i;i.e., di is homogeneous of bidegree (i, 1 - i) and total degree 1. Thus di restricts to operators

df.9; Ef.9

--+

Ep+i.P+l-i

In particular, dOp.9; E P . 4 0

--+

EP,P+l 0

dp9; EP.9 1

9

-+

EP+l,9 1

and dP.9: E P , 9 2

2

--+

EPt2.9-1

T h e bigradation of Ei determines a bigradation of H(Ei , d i ) , written

H(Ei, di) = C HP*g(Ei,di). P.9

I. Spectral Sequences

34

The corresponding total gradation, H ( E i , d i ) = C,.H(,.)(Et,d j ) , is induced by the total gradation of E i :

Finally consider the isomorphisms N

0 :

E;P,l =L.H p ( E i ,d J ,

defined in sec. 1.6. Evidently restricts to isomorphisms

G!

maps Ef;f into HPgq(Ei,di). Hence it

E ? $ Z HPSq(Ei, di). In the same way it follows that the isomorphism a&: E& 5 A$(M) (cf. sec. 1.4) is homogeneous of bidegree zero, and hence restricts to isomorphisms

(~2': E?'

= N

A:&).

We close this section with a condition that forces the collapse of a spectral sequence. Proposition IV: Let (M, 6 ) be a graded filtered differential space with spectral sequence ( E i , di). Assume that, for some m, Em is evenly graded with respect to the total gradation :

Ek) = 0,

r odd.

Then the spectral sequence collapses at the mth term. Proof: Since E, is evenly graded, and d, is homogeneous of total degree 1, it follows that d, = 0. Hence Em = H(E,, d,). Since the isomorphism 0,: Em+,S H(Em,d,) is homogeneous of degree zero, it follows that Em,, is evenly graded. Now an induction argument shows that di = 0 for i 2 m. CS

Q.E.D. 1.11. Bigraded differential spaces. Let M = C,,, MPJ be a biq 2 0 and q 2 0, graded vector space such that 1MP.q = 0 unless p

+

3. Graded filtered differential spaces

35

and consider the induced total gradation

Then the subspaces

make M into a graded filtered space. Now assume that 6 is a differential operator in M homogeneous of degree 1 with respect to the total degree, and such that for some fixed k 2 0, 6 : Mp-g FPtk(M), all p , q. ---f

Let D be the differential operator defined as in sec. 1.7. Then D is homogeneous of bidegree (K, 1 - k). It follows from the results of sec. 1.7 that Ef.9 M p * q , i 5k and

El;!

Hp*q(M, D).

1.12. Convergent spectral sequences. Let ( M , 6) be a graded filtered space, with spectral sequence ( E j ,d J . Since Fp(M‘) = 0, p > r, formula ( l . l l ) , sec. 1.10, implies that

Thus the first equality induces a surjective linear map yT.9:

Ef.9

-

E Pm. 9,

iPq+2.

Definition: T h e spectral sequence ( E i , di)is said to converge if for each ( p , q ) there is some i 2 q 2 such that yPP is an isomorphism.

+

Remarks: 1. T h e linear map yrsq is an isomorphism if and only if

36

I. Spectral Sequences

and yfiq is the identity map. 2. Let (Ei , di) be a convergent spectral sequence for a graded filtered differential space (M, 6) which collapses at the nzth term. Then

Proposition V: Let (M, 6) be a graded filtered space. Assume that for each Y there is a finite integer K(Y) (possibly negative) such that

FP(M")= M",

pI k(y).

Then there is a finite integer i(r) with the following property: yP"-" is defined, and an isomorphism, for all p and all i 2 i ( ~ ) I, n particular

Eir) = E(r) (f) m = A H ~ M , i 2 i(y), N

9

and the spectral sequence is convergent. Proof: It follows from our hypothesis that

+

2 Hence yr,r-p is defined, and an isomorphism, whenever both i 2 Y - p and i 2. p - K(Y- 1). In particular if we set i(v) = max(r - k(r) 2, Y - K(Y- l)), then yf.'" is an isomorphism for i 2 i ( ~ )and k(r) 5

+

p

I y .

On the other hand, our hypothesis shows that

whenever p < K(Y) or p > Y. In particular, yfSr+ is always an isomorphism for p < k(;) or p > Y.

Q.E.D. 1.13. The base space. Given a graded filtered differential space (M, 6) consider the graded subspace B = CPBP defined by

BP = ZP.0 1 ,

PEZ.

3. Graded filtered differential spaces

37

Thus Bp consists of the elements of Fp(Mp) which are mapped into FP+l(Mp+')under 6. B is called the base of (M, 6 ) . Since (by definition) 8(BP) c P+l(Mp+l),it follows that the base is stable under 6. Hence it is a graded differential space. A homomorphism e ) : M + A? of graded filtered differential spaces restricts to a linear map between the respective base spaces. Now let B have the filtration induced by the gradation; i.e., set @ ( B ) Bp. Since 6 is homogeneous of degree 1 we may apply the results of sec. 1.8 to obtain the spectral sequence for this filtration; it is given by

- xPzp

(Rip 4)

=

i = 0, i = 1, i 2 2.

1

(B, 61, , ; :(: ; O),

Moreover, in view of sec. 1.11, the bigradation off& is given by

Ep

BP,

@=o,

q>o

@so

H*(B),

Bq= 0,

q

(i=O,l),

and

> 0 (i 2 2).

Next, consider the inclusion e : B -+ M . It induces a linear map e*: H(B) --t H ( M ) . Moreover, e is filtration preserving, and so it de-

termines a homomorphism ei: ($

,di)

+

( E d ,di)

of spectral sequences. Proposition VI: The maps

are isomorphisms. I n particular,

mo

BP

and

EPo

Hp(B).

Proof: We show first that each el"*' is an isomorphism. In fact, by definition Gso= BP, while pkL-1 0

= 0 = DP.0 0

38

I. Spectral Sequences

(cf. formula ( l . l l ) , sec. 1.10). I t follows that @" is an isomorphism from BP onto Ef.'. Now the commutative diagram

implies that e ? * O is an isomorphism. Next note that the differential operator d, is homogeneous of bidegree (1, 0). Thus, for fixed q, the direct sums C pEf,' are stable under d,, and so

WE1

9

c

4 )= H ( C E?*, 4 ) r

Since e l : El Z

C pET,',

l

(1.12)

P

it follows that

e:: H ( E , , d,)

H(

c

d,)

~ r - 0 ,

.

P

In view of formula (1.12), and Proposition 111, sec. 1.6, this implies that eP,o: E P . 0 2

2

5 Ei.0. Q.E.D.

Corollary: The maps

eiso: H p ( B )+ El,' are surjective for K 2 2. 1.14. Homomorphisms of graded filtered differential spaces. Assume that Q : M + h? is a homomorphism of graded filtered differential spaces. Then the induced maps v i : Ei+ (cf. sec. 1.5) preserve the bigradation, and hence restrict to linear maps

ei

Since the isomorphisms ai: Ei+l 5 H ( E , , di) and am:E ,

s

AfI(.ll)

3. Graded filtered differential spaces

39

also preserve the bigradation, we have the commutative diagrams

(1.13)

and

(1.14)

Recall from sec. 0.9 that a homomorphism p: Cpz0A p -+ Cp20B p of graded spaces is called n-regular, if p p : A p B p is an isomorphism for p 5 n and injective for p = n 1. --f

+

Theorem I (Comparison theorem): Suppose p : M + I@ is a homomorphism of graded filtered differential spaces whose spectral sequences are convergent. Assume that for some i the induced homomorphism pi: Ei -+ is n-regular (with respect to the total gradation). Then

ei

pi: Ei

+

ej,

j 2 i,

and p#: H ( M ) -+ H ( B )

are n-regular. Proof: Since pi is n-regular, so is pf (cf. Proposition 11, sec. 0.9). But the isomorphisms oifl and B i + l identify pi+lwith pr (cf. Proposition 111, sec. 1.6) and so pi+l is n-regular. I t follows by induction that pi+j is n-regular for all j 2 0. But for any p , q and sufficiently large k we have by hypothesis

Hence ym is n-regular. Since

T (,

identifies pMwith (v*)-~,this implies

I. Spectral Sequences

40

that ( T * ) ~is n-regular. Now the theorem follows from Proposition VII, below.

Q.E.D. Corollary: If

vt:Ed Ed is

an isomorphism for a certain i, then

-+

and

are isomorphisms.

-+a

Proposition VII: Let v: M be a homomorphism of graded filtered spaces and consider the induced linear maps

is Suppose ~ 2is) injective (respectively, surjective). Then vr:Mr -+ injective (respectively, surjective). In particular, if ~ ( a ' )is an isomorphism, so is vr. Proof: (1) Assume that pl(a') is injective, and fix x E ker vr. Then, since M = Fp(M), it follows that x E Fp(Mr) for some p . But if x E P ( M r ) , then

up

0 = QPVX

= q$QPX.

Since cpz) is injective, this implies that

x

E

QPX

= 0; i.e.,

Fp+'(Mr).

Now any easy induction argument yields

x

E

nF P ( M ~ ) . P

Hence x = 0 and pr is injective.

ar.

(2) Assume that ~ 2is)surjective, and fix x E We show by induction that x E Fp(A?k) p ( M 7 )for all p . As above, we have x E Fa(@) for some s; then x E F S ( B r ) v ( M r ) . Now assume

+

+

+

x E FP(ar)v ( M r ) .

3. Graded filtered differential spaces

This relation yields (for some y

E

41

Mr)

x - py E FP(Mr).

Since p)$) is surjective, it follows that for some z E Fp(Mr),

pqx

- py) = p..@(z) = p p p z .

Hence x - p)y - pz E ker

pp;

i.e.

x - py - p?zE

FP+l(@).

+

I t follows that x E F p + l ( i @ r ) p(Mr). Proceeding in this way yields

In particular, since Fr+'(A?)

= 0, x E

p ( M r ) .Thus pr is surjective. Q.E.D.

Corollary I: Suppose p: M -+A? is a homomorphism of graded filtered differential spaces, and assume that

-

(I): E(r) Pm m

E(r) m

is injective (surjective) for some r . Then the mapping

is injective (surjective). In particular, if pMis injective (surjective), then p)# is injective (surjective). Proof: I t follows from the diagram (1.14) that pa')is injective (surjective) if pk) is. Now the corollary follows from Proposition VII, applied to the graded filtered spaces H ( M ) and H ( f i ) .

Q.E.D.

-

Corollary 11: Let v: M i@ be a homomorphism of graded filtered differential spaces whose spectral sequences are convergent and collapse at the Kth term. Then (p#y:H r ( M ) Iir(@) is injective (surjective) whenever &): EF) + EF) is injective (surjective). ---f

I. Spectral Sequences

42

Corollary In: Let M be a graded filtered differential space with base B. Assume that the spectral sequence of M collapses at the second term and is convergent. Then e # : H ( B ) -+ H ( M )

is injective. Proof: Since e,: 8, + E, is injective (cf. Proposition VI, sec. 1.13) this is an immediate consequence of Corollary 11.

Q.E.D. 1.15. Poincar6 series. I n this section M will denote a graded filtered differential space with a convergent spectral sequence. Proposition VIII: Let ( E i ,di)be the spectral sequence for M and assume that for some i, the spaces El') all have finite dimension. Then the spaces Ej'" (i C j 5 co) and H r ( M )all have finite dimension. Moreover (1.15) f i 4fi+l2 - . * >fm= f H ( M ) ,

wheref, denotes the PoincarC series of Ej with respect to the total gradation. Finally, the relation

fj = f n w holds if and only if the spectral sequence collapses at thejth term.

B?

T h e isomorphism oj restricts to an isomorphism H(')(Ej).Hence, if Eir)has finite dimension, so has Eiii and by induction we obtain this for every j >_ i ( j finite). Since the spectral sequence is convergent, it follows that Egj has finite dimension. The isomorphism E z ) E A&) (cf. sec. 1.4) shows that A&af)has finite dimension. Finally, it is a straightforward exercise in linear algebra (via Proposition VII, sec. 1.14) to construct a linear isomorphism Proof:

(1.16)

homogeneous of degree zero, Hence Hr(M) has finite dimension. This proves the first part of the proposition. For the proof of the second part we observe that, since Ej:; E H(r)(Ej), dim Ej'" 2 dim Ej':;,

i 5 j < 00,

3. Graded filtered differential spaces

43

while, because the spectral sequence converges, for sufficiently large j , dim Ej’” = dim EL)

It follows that fi2

-

. 2 fW. On the other hand, the existence of the isomorphism (1.16) and the isomorphisms EZ) A&) show that

I t remains to prove the last statement. If the sequence collapses at the jth term, it follows from Remark 2, sec. 1.12, that

whence fi = fW =fH(M).Conversely, assume fj = fH(M).Then formula (1.15) yields fj = f j + 1

=

. .*

=fH(M).

In particular, dim H‘r’(Ej+k) = dim EjiL+l= dim Ej’&

r 2 0, It >_ 0.

This implies that dj+, = 0 (k 2 0), and so the spectral sequence collapses at the jth term.

Q.E.D. Corollary: Suppose dim Ei < 00 for some i. Then dim H ( M ) < 00, and the spectral sequence collapses at the kth term if and only if dim Ek = dim H ( M ) .

1.16. Euler characteristic. Proposition IX: Assume that M is as in sec. 1.15. Suppose that dim Ei < 00 for some i. Then the EulerPoincarC characteristics X E , satisfy X E , = XEt+, =

* * *

= XE- = X H ( M ) -

Proof: Since the isomorphism Ei+* 2 H(E,) preserves the total gradation, it follows from the Euler-PoincarC formula (cf. sec. 0.7) that XEI+l = X H ( E t ) = XEt.

44

1. Spectral Sequences

Since dim Ei < 00, El"' = 0 for all but finitely many Y. Thus because the spectral sequence converges, it follows that for some fixed finite n,

EZ) = Eg),

all

Y.

Q.E.D.

$4. Graded filtered differential algebras 1.17. Filtered algebras. Let R be an algebra (over T).Afiltration OJ the algebra R is a filtration of the vector space R by subspaces Fp(R) of R which satisfy Fp(R) P ( R ) c Fp+q(R). (1.17) Given a filtration of R, consider the associated graded space

Relation (1.17) allows us to define a multiplication in An by setting @PX

py = p + q x

- y),

xE

Fp(R), y

E

P(R).

This multiplication makes A R into a graded algebra, called the associated graded algebra of the filtered algebra R . Remark: If Fo(R) = R then P ( R ) is an ideal, and so multiplication in FP(R)induces a product in A$. But this product is trivial for p 2 1. I n fact, if x E P ( R ) and y E Fp(R),then (if p 2 1) xy E F2p(R) c Fp+l(R),whence ,p(xy) = 0. On the other hand, A: equipped with this multiplication is a subalgebra of the associated graded algebra.

1.18. Graded filtered differential algebras. Let (R, 6) be a graded differential algebra. Suppose R is filtered by subspaces Fp(R) so that the filtration makes R into a filtered algebra (cf. sec. 1.17) and also into a graded filtered differential space (cf. sec. 1.9 and sec. 1.10). Then (R, 8 ) is called a graded jiltered dijjkential algebra. Assume that ( R , 6) is a graded filtered differential algebra. Then we have the relations

45

I . Spectral Sequences

46

In fact, if u E Zj'sg and v E Z:.', then

Moreover,

whence (1.18). Formula (1.18) implies in particular that for p 2 0

zp i

Zf c Zf

and so Zj' is an algebra if p 2 0. To prove (1.19) observe first that this relation is trivial for i = 00. Hence we may assume that i < 00. Let u E Zf,gand v E Z ~ : , ~be ' ~any ' elements. Write u

*

Sv =

-(--l)P+%4

*

w

+ (-1)p++

*

v).

It is easy to see that

- v E Z?T+l

Su

and

-

S(u w ) E O f ! .

Formula (1.19) follows. In view of relations (1.18) and (1.19) a multiplication is defined in Ei, by ($24)

*

(VfV)

= $+"u

*

v),

11

E

Zf ,

V

E

zi.

In this way the space Ei becomes a bigraded algebra. T h e operator di is an antiderivation with respect to the total gradation of E i . Thus (Ei, d i ) is a graded differential algebra. Next observe that the isomorphisms ui: Ei+lLH ( E i , di) N

and

u,:

E,

-

= AH(M)

are algebra isomorphisms, as follows directly from the definitions. Moreover, the algebra structures of E, and AR coincide (recall from sec. 1.4 that E, and An are equal as spaces). Note also that the basic subspace B is a subalgebra of R, as follows from formula (1.18). Finally, let R and be graded filtered differential algebras. A homomorphism of graded differential algebras rp: (R, 6) (R, 6) which

-

4. Graded filtered differential algebras

47

preserves the filtrations will be called a homomorphism of graded filtered dzflerential algebras. The induced mappings (Pi:

(Ei, 4 )

-j

(4,Ji)

are homomorphism of graded differential algebras.

§5. Differential couples 1.19.

Let M = C p p zMp be a graded space and let

C M@

P(M)=

/L2P

be the corresponding filtration of M. Let 6, , 6, be differential operators in M such that for some k 2 0

6,:

Mp

--*

6,:

and

Mp+k

Pik+, (M).

(1.20)

Mp -+

Assume further that 8,6,

+ 6,6, = 0.

(1.21)

Then (M, 6,, 8,) will be called a difJerentia1 couple of degree k. It follows from (1.21) that 6 = 6, 6,

+

is again a differential operator. 6 is called the total diflerential operator of the couple ( M , d,, 6,). Relations (1.20) imply that (M, 6) is a filtered differential space satisfying the conditions of sec. 1.7. Moreover, the corresponding homogeneous differential operator D is precisely 6,. Next consider the differential space (M, d2). It is also a filtered differential space and satisfies the conditions of sec. 1.7. Hence 6, determines a differential operator D, in M, homogeneous of degree k 1, and such that

+

6,

- D,:

Mp + Pik+' (W.

It follows from (1.21) and an argument on degrees that

6,D2+ D,dl

= 0.

Hence D, induces a differential operator D: in H ( M , d,), homogeneous 1. of degree k

+

48

5. Differential couples

49

Theorem II: Let ( M , S,, 6,) be a differential couple of degree k. Then the first terms of the spectral sequence for the filtered differential space ( M , 6) are given by:

(1) ( E i , di) (2)

(Ek,4 )

( M ,O ) ,

= ( M , 61); =( W M ,

0 5 i < k;

(3) (E,,, , 4 + 1 ) 611, X ) ; (4) Ek+, H ( H ( M , 41, @).

=

Remark: The actual isomorphisms are important, and appear explicitly in the proof. Proof: The isomorphisms (1) and (2) are constructed in sec. 1.7 (formulae (1.4) and (1S ) ) . Moreover, in sec. 1.7 (formula (1.6)) we constructed an isomorphism Ektl. I t is immediate from the definition of a that the a:H ( M , 6,) diagram

. q M , 6,)

inclusion

Z+l

HP(M,4)7 E + 1 N

commutes. T o establish (3) we show that

(12 2 )

GSD;= dk+,a. Fix z E Zp(M, dl). Then

4(n,z)

+k+lD

= r&1

2z

and df+,a(n,z)= r]g:y6z = qg:ps,z.

Subtract the second relation from the first to obtain

But clearly

(D, - B,)(x) and so (1.22) follows.

E

p+k+l FP+k+2(M) n Z ( M , 6,) c ker l;lk+l

50

I. Spectral Sequences

It follows from (1.22) that a induces an isomorphism oc*:

H ( H ( M , 61)1D t ) 4 H(E,+,, dk+l),

Composing a# with the isomorphism phism (4).

oill

we obtain the isomor-

Q.E.D. A differential couple ( M , dl, 6,) of degree k is called homogeneous if S, is homogeneous of degree K 1,

+

If (MI 6,, 6,) is a homogeneous differential couple, it follows that D,= 6,. Hence Theorem I1 reads

(El.+,, 4 + 1 )

=( H ( M

61)l

83

and

H(H(M, di), a,#).

&+a

(a,

2.20. Homomorphisms. Let ( M , S,, 6,) and &, 6,) be differential couples of degree K. Then a linear map y : M +A?, homogeneous of degree zero, is called a homomorphism of di'erential couples if

p6, = &p In particular, setting 6 = 6,

p6, = 8,y.

and

+ 6,

and

8 = 8, + 8,,

we see that

ya = 8q.l.

Moreover, pD, = 23,~. Let pz): H(Ml 8,) -+ H ( a , &) be the induced linear map. Then since 0

2

=

D,?,

ycl,D: = ab;,. #

be the induced map.

5. Differential couples

Proposition X:

51

Let p be as above. Then the diagrams

( 1.23)

and

(1.24)

commute, the isomorphisms being defined in Theorem 11. Proof: Diagram (1.23) follows directly from diagram (1.8) in sec. commute with the appropriate differential 1.7. Since G I , pg), a, and operators, we have

P$P#

=

c#(PL)"

*

This, together with (1.23) and the relation p$+lak+l= sec. 1.6), yields (1.24).

C ? ~ + , P ) ~ +(cf. ~

Q.E.D. 1.21. Graded differential couples. Let ( M , 6,) 6,) be a differential couple and assume that the spaces Mp are graded, Mp = CqMPvq. Suppose that M p s q = 0 unless p q 2 0 and q 2 0. Then M is a bigraded space M = C,,, MP,q. As usual we set

+

zr20

and observe that M = T h e couple ( M , 6,, 6,) is called a graded dzfferential couple, if 6, and 6, are homogeneous of degree 1 with respect to the total gradation of M . In this case ( M , 6) (6 = 6, 6,) becomes a graded filtered differential space (cf. sec. 1.10).

+

I. Spectral Sequences

52

Remark: If 6, and 6, are homogeneous of bidegrees (0, 1) and ( 1 , O ) respectively, ( M , 6) is a double complex in the notation of [2; p. 601.

T h e operators 6, and D, of a graded differential couple are bi-homogeneous. Hence a bigradation is induced in the spaces H ( M , 8,) and H ( H ( M , d1), D:). The isomorphisms of Theorem 11, sec. 1.19, are homogeneous of bidegree zero :

s MP*Q, i l k Ef;; s Hpsq(M,d,), Ef-9

and

Ef;;

Hpsq(H(M,8,), DE).

A homomorphism of graded dzfferential couples is a homomorphism of differential couples, homogeneous of bidegree zero. Proposition XI: Let p: ( M , S,, 6,) + (A?, 8,, 8,) be a homomorphism of graded differential couples of degree k. Suppose the spectral sequences are convergent. Assume that the homomorphism p k : H ( M , 4) H(@, +

81)

is n-regular. Then so is the homomorphism cp*: H ( M ) + H(A?). Proof: I t follows from the hypothesis and diagram (1.23) in Proposition X that the homomorphism pk+,:Ek+,+ Ek+,is n-regular. Now Theorem I of sec. 1.14 implies that p* is n-regular.

Q.E.D. Corollary:

If p;) is an isomorphism, then so is p*.

Chapter I1

Koszul Complexes of P-Spaces and P-Algebras

xk

I n this chapter P = Pk denotes a finite-dimensional positively graded vector space which satisfies

Pk= 0

if k is even.

P denotes the evenly graded vector space defined by Pk= Pk-'. Note that P and P are equal as vector spaces. T h e gradations of P and P determine gradations

in the algebras

AP

x1 A and

and V P ; these are defined by the relations

. . . xp E ,j

x, v - .

v

(l\p)il+**.+iv

xq E (VP)il+...+i*

if x, E Pi. if x, E Pi..

$1. P-spaces and P-algebras 2.1. P-spaces. A P-space is a positively graded vector space

S=CSk k2O

together with a bilinear map SXP + S (written ( z ,x ) ++ z 0 x) which satisfies the conditions

(zox)oy=(zoy)ox, and 53

zES, x,yEP

(2.1)

54

11. Koszul Complexes of P-Spaces and P-Algebras

A P-linear map (or a homomorphism of P-spaces) between P-spaces S and T is a linear map 9:S T such that --f

9 ( z ox) = (pz) ox,

Z E

s,

X f

P.

If p is homogeneous of degree zero, it is called a homomorphism of graded P-spaces. An exact sequence of P-spaces is an exact sequence

in which S , , S , , and S, are P-spaces, and the arrows are P-linear maps. A P-subspace of a P-space S is a graded subspace S , c S such that z 0 x E S, whenever z E S , and x E P. A P-subspace S, is itself a P-space, and the inclusion S , -+ S is a P-linear map. I n particular the subspace of S spanned by the vectors of the form z o x ( z E S, x E P ) is a P-subspace; it is denoted by S 0 P. If S, is any P-subspace of S , then the quotient space S / S , admits a unique P-space structure for which the projection n: S + S / S , is a P-linear map. The P-space SlS, is called the quotient or factor space of S with respect to S,. If p: S -+ T is a homomorphism of graded P-spaces, then ker 9 and Im ip are respectively P-subspaces of S and T. Moreover, q induces an isomorphism

of graded P-spaces. Given a P-space S , a graded VP-module structure is defined in S by z.(x*v...vxp)=zox,o

* " O X P ,

ZES,

X,EP,

(cf. formulae (2.1) and (2.2)). This establishes a 1-1 correspondence between P-spaces and graded VP-modules. Evidently, P-subspaces, P-factor spaces, and P-linear maps correspond respectively to submodules, factor modules, and module homomorphisms. 2.2. Koszul complexes. With each P-space S is associated the following differential space: In the tensor product S @ A P define a linear operator 0, by setting

qy(z @ 1) = 0,

zE

s,

1. P-spaces and P-algebras

55

and

The relation z o xio x j = z 0 x j o x i (cf. sec. 2.1) implies that V'. = 0. Thus (S @ AP, Ts) is a differential space; it is called the Koszul complex associated with the P-space S. T h e corresponding cohomology space H ( S @ AP, &) is called the cohomology space associated with the P-space S. The gradations of S and P induce a gradation in S @ AP. It is written

S @ AP = C ( S @AP)7, r

and is uniquely determined by the following condition: If z E xi E P I ,then z @ x1 A

*

.-

A X,

Sq

and

( S 0AP)Q+Pl+.*.+pm.

E

It follows from formula (2.2) that Ts is homogeneous of degree 1. On the other hand, a second gradation is defined in S @ A P by S@AP=C(S@AP),,

where

(SOAP),= S@AkP.

k

To distinguish it from the first gradation (called, simply, the gradation) we call it the lower gradation. Evidently 6 is homogeneous of degree - 1 with respect to the lower gradation. The two gradations of S 0AP define the bigradation given by S @ AP

=

1( S @ AP);,

where

( S @ AP); = ( S @ AkP)'.

k,T

The elements of ( S @A")' are called homogeneous of degree r , lower degree k, and bidegree (r, k). Since qs is homogeneous of bidegree (1, -- 1), a bigradation is induced in H ( S @ A P ) ; it is denoted by

H ( S @ AP) =

C Hk(S @ AP). I; ,T

Elements of HL(S @ AP) are called homogeneous of degree r, lower degree k, and bidegree ( r , k). We shall write

H'(S @ A P ) = C Hk(S @ AP) and k

Hk(S@ AP) =

c Hi(S @ AP). r

56

11. Koszul Complexes of P-Spaces and P-Algebras

Next consider the inclusion map

I,: S

+

S @ AP

given by & ( x ) = x @ 1. We may regard

,I as an isomorphism

S 5 Zo(S @ AP)

(Z(S @ AP) = ker G). It restricts to an isomorphism

S~PZB,(S@AP) ( B ( S @ AP) = I m Vs). Thus Is induces a commutative diagram S

in which 18 is surjective and is is an isomorphism (both homogeneous of degree zero). We may write simply l, I*, and for I, Is#, and &. Finally, recall from sec. 0.4 that each x* E P* determines a unique antiderivation i ( x * ) (substitution operator) in AP, homogeneous of degree -1, such that i ( x * ) ( x ) = (x*, x),

x* E

P", x E P.

We will also denote by i ( x * ) the operator in S ~ ( x * ) ( z0@) = (-1)"~ @ ;(A+)@,

0AP given by z E SP, @ E

AP.

A simple calculation shows that

where e, that

, e+v is any pair of dual bases for P and P". This relation implies i(x*)17,

+ V,i(x*) = 0,

x*

E

P".

(2.31

In particular, i ( x * ) induces a linear operator i(x")* in H ( S 0AP).

57

1. P-spaces and P-algebras

Examples. 1. Direct sums: Given P-spaces S and T make S @ T into a P-space by setting

(z,W ) 0 x = (Z 0 X, w 0 x),

w E T , x E P.

z E S,

Then

and I,,,

=

1s @ I,.

I n particular,

H((S @ T )@AP) = H ( S @AP) @ H ( T @AP). 2. Tensor products: Let S be a P-space and let F be a graded vector space. Make the graded space F 0S into a P-space by setting

U E F , ZES,XEP.

(U@X)OX=U@(ZOX),

Then the corresponding Koszul complex is ( F @ S @ AP, W F @ O,), where oFis the degree involution in F. It follows that H ( F @ S 0AP) = F @ H ( S @ AP). Moreover, lFOs= i p @ I, and

I:,s

=

iF

@ Is#.

2.3. Homomorphisms. Let 9:S T be a homomorphism of Pspaces. Then q~ @ i: S @ AP + T @ A P is a homomorphism of differential spaces, homogeneous of lower degree zero. Thus it induces a linear map, homogeneous of lower degree zero, ---f

(9@ i)*: H ( S @ AP)

+H (T

@ AP).

We denote the restrictions of (p 0L ) * by (p @ I):: H p ( S 0AP)

+H,(T

@ AP).

T h e P-linear map p determines commutative diagrams

I

SP

It

I

H(S @AP) (~101)'

T y H ( T @ AP) 1T

@I

iS

HO(S @ A P ) Sl(S O p >y and

T / (T o P )

1

(Po'):

Ho(T @ AP).

11. Koszul Complexes of P-Spaces and P-Algebras

58

Moreover, if p is homogeneous of degree zero, then

If a, is homogeneous of degree k, then so are p @ i and (p @ I ) " . If y : T + W is a second homomorphism of P-spaces, then y 0 9" is P-linear, and (Y

O

v 01)"

= (Y

0I ) *

O

(p'

0I ) # *

The identity map of S induces the identity in S @ AP and H ( S @ AP). Next, consider a short exact sequence

O - S L T L

W-0

of P-spaces. The induced sequence of differential spaces is again short exact, and so it determines an exact triangle of cohomology spaces. Since the differential operators have lower degree -1, while p @ I and y @ t have lower degree zero, this triangle yields the long exact sequence -----+

Hk(S @ A P )

(pol)"

Hk( T

@ AP)

(@I)*

Hk(

W @ AP)

I

Hk-,(S @ AP)

1

(POCI*

Hk-i(

T @ AP)

--+

Now suppose that p and y are homogeneous of degree zero. Then the connecting homomorphism is homogeneous of degree 1, and we have the system of long exact sequences

2.4. P-algebras. A P-algebra is a pair (S;a), where:

(1) S is a positively graded associative algebra with identity, and

59

1. P-spaces and P-algebras

(2) a: P satisfies

--+

S is a linear map, homogeneous of degree 1, which .(x)

*

x

=

z

- a@),

xE

P, z

E

s.

(2.5 1

a is called the structure map of the P-algebra (S; a).

A homomorphism of P-algebras p: (S;a) -+ ( T ;t) is an algebra homomorphism p: S -+ T which satisfies p 0 a = t and p(1) = 1. If p is homogeneous of degree zero, it is called a homomorphism of graded Palgebras. With each P-algebra (S; a) is associated the P-space structure of S given by z E s, x E P. x 0x =x . I+),

(Observe that a homomorphism of P-algebras is simply a P-linear algebra homomorphism. ) On the other hand, a extends to a homomorphism u,: V P +

s

of graded algebras (cf. formula (2.5)). This formula also shows that the image of u, is in the centre of S. Thus a" makes S into a graded VPalgebra, In this way P-algebras are put in 1-1 correspondence with graded VP-algebras. Finally, observe that each VP-algebra is, in particular, a VP-module. Evidently, x * fY"(Y) = z 0 Y, zE YE VP,

s,

and so the diagram P-algebras

I graded VP-algebras

-

P-spaces

I:

graded VP-modules

commutes. 2.5. The Koszul complex of a P-algebra. To the Koszul complex of a P-algebra (S; a) we assign a multiplication in the following way:

60

11. Koszul Complexes of P-Spaces and P-Algebras

(skew tensor product of algebras, cf. sec. 0.3). This makes S @ A P into a bigraded associative algebra with identity. If S is anticommutative, then so is S @ AP. The Koszul complex for a P-algebra (S;CT) will be denoted by (S@AP, Vo)(i.e., we use R, rather than G). A straightforward computation (using formula (2.5), sec. 2.4, and the fact that Pk= 0 for even k) shows that R, is an antiderivation with respect to the gradation of S @ AP. Thus ( S @ AP, R,) is a graded differential algebra, and so H ( S @ AP) becomes a bigraded algebra. I n particular,

H ( S @ AP) = H o ( S @ AP) @ H+(S @ A P ) decomposes H ( S @ AP) as a direct sum of a graded subalgebra and a graded ideal. Finally observe that the maps ls, Is#, and ls (cf. sec. 2.2) are all homomorphisms of graded algebras. Moreover, the operators, i(x") (x" E P"), are antiderivations in S @ A P (with respect to the gradation). Hence the operators i(x3)#are antiderivations in H ( S @ AP). 2.6. The algebra V P T h e identity map P -+ P makes V P into a P-algebra ; the corresponding P-space structure is given by

zox=zvx,

ZEVP,

X E P .

We shall show that

In fact, recall that the differential operator V (=.Vvp) is an antiderivation in the graded algebra V P @ AP. Define a second antiderivation k in this algebra by setting

k(1 @ @)

=0

and Q

k ( y , V . * * ~ y , @ @ ). =~ y l ~ . - * j Vi Y** OYi * * A @, 2-1 yi E P, @ E AP.

+

Set A = Vk KV. Then d is a derivation which reduces to the identity map in (P @ 1 ) @ ( 1 @ P). I t follows that A reduces to ( p q)t in V p P@ AqP. This implies formula (2.6) if r has characteristic zero.

+

1. P-spaces and P-algebras

61

If has characteristic different from zero, we obtain the result as follows: Fix a homogeneous basis xl, . . . , x, of P (and hence of P) and let Pi and Pi denote the 1-dimensional subspaces of P and P spanned by xi. Then the Kunneth formula yields H ( V P @ AP) = H(VP, @ API) @ where the differential operator

* * *

@ H(VP, @ APr),

c in V P i @ APi is given by

K(z @Xi + w @ 1 ) = z v x i @ 1 ,

z, w

E

VPi.

A trivial verification shows that

and so the same formula must hold for H ( V P @ AP). 2.7.* Interpretation of H(S @ A P ) as Tor. Make by setting jlox=o,

r

into a P-space

A E ~ ,~ E P .

Let S be any P-space. Then (cf. [2; p. 1061 for the definition of Tor)

H ( S @ AP) = TorVP(S,r). T o see this, consider the Koszul complex ( V P @ AP, V ) of sec. 2.6, and let E : V P + I' denote the canonical projection with kernel V+P. Then the sequence

+ V P @ A ~ P ~ V P @ A ~ - -... ' PL~v n , @ i L r - o is exact, as was shown in sec. 2.6. Evidently, V P @ AkP is a free V P module, and the maps V and E are linear over V P . Thus this sequence is a free resolution of the VP-module r. Hence, by definition, TorVP(S,r)= H ( S O V p ( V P@ AP), L @ V ) . But clearly,

S @vp(VP @ A P ) = S @ A P It follows that TorV"(S,

and

r)= H ( S @ AP).

1

@V

= 0,.

$2. Isomorphism theorems 2.8. The first isomorphism theorem. I n this section we establish Theorem I: Let p: S + T b,e a homomorphism of graded P-spaces. Then the following conditions are equivalent:

(1) p is an isomorphism. (2) (p @ t ) # is an isomorphism. ( 3 ) (9@ t ) $ is an isomorphism and (p @ L)? is surjective. The proof is preceded by some preliminary results. Lemma I: Let F be a graded subspace of the vector space S such that S = F S 0 P. Then

+

S=FoVP. ( F o VP denotes the VP-module generated by F.) Proof: A simple induction from the hypotheses yields the relations I.

S=

2 F0ViP+ S O V ~ + ~ Pk, =

0,1,2,

i=O

... .

Since Sj = 0 and Pj = 0 for j 5 0, it follows that

( S 0 Vbf'P)"

= 0.

Hence b

Sk =

C (F i=O

o

ViP)",

k

= 0, 1,

. .. .

Lemma 11: Let S be a P-space such that H,(S @ AP)

s= 0.

Q.E.D. = 0.

Then

Proof: Choose a graded subspace F of S so that S = F @ (S 0 P). Then (cf. sec. 2.2) F H,(S @ AP), and so F = 0. Hence Lemma I yields = F 0 V P = 0. Q.E.D.

s

62

63

2. Isomorphism theorems

Proposition I: Let q : S + T be a homomorphism of graded P-spaces. Then p is surjective if and only if (p @ L): is surjective. Proof: According to the commutative diagram in sec. 2.3, (p @ L)O# is surjective if and only if the induced map q : SlS 0 P TIT 0 P is surjective. If p is surjective, then, clearly so is p. Conversely, assume that p is surjective. Then T = p(S) T o P. ---f

+

Applying Lemma I (with F

T

= p(S)) we

= q ( S )0 V

find

P = q(S 0 V P ) = p(S),

and so p is surjective.

Q.E.D. Proof of Theorem I: Clearly, ( 1 ) 3 (2) 3 (3). Now assume that (3) holds. Then Proposition I implies that we have a short exact sequence

0 - K - S - Ti - 0

'p

of graded P-spaces, where K = ker p. This yields the long exact sequence

-

H,(S @ A P )

(p161):

H I (T @ A P )

la

Ho(K@ AP)

H o ( S@ AP)

(p61):

H,,( T 0AP)

(cf. sec. 2.3). Since (p @ L): is surjective, it follows that 8 = 0. Since (p 0L)$ is injective, (i @ t ) ! = 0. These relations imply that H o ( K @ A P ) = 0. Now Lemma I1 shows that K = 0 and so p is injective; i x . , ( 3 ) (1). Q.E.D.

-

2.9. The second isomorphism theorem. Let S be a P-space. Recall that the map I#: S + H o ( S @ AP) is surjective (cf. sec. 2.2) and choose a linear injection y : Ho(S O A F ) -+ S,

64

11. Koszul Complexes of P-Spaces and P-Algebras

homogeneous of degree zero, so that I #

0

y =

Then

1.

S = Im y @ S o P. Now make Ho(S @ AP) @ V P into a P-space by setting (a@@)ox=a@@vx,

@EVP,

a~ H,(S@AP),

XE

P,

(cf. Example 2, see. 2.2, and see. 2.6). Then a homomorphism

g: Ho(S@ AP) @ V P --* S of graded P-spaces is defined by g(a @ @) = y ( a ) 0 @.

Theorem 11: Let S, y , and g be as above. Then the following conditions are equivalent: (1) g is an isomorphism. (2) (g @ L ) * : H(Ho(S @ AP) @ V P @ AP) morphism. (3) I*: S -+ H ( S @ AP) is surjective. (4) H + ( S @ AP) = 0. (5) H,(S @AP) = 0.

-+

H ( S @ AP) is an iso-

Proof: Theorem I, sec. 2.8, shows that (1) o (2). Since I m I* @ AP), it follows that (3) e (4). Next, assume (2) holds. Using Example 2, sec. 2.2, and sec. 2.6, observe that = H,(S

--

H+(Ho(S@ AP) @ V P @ AP) = Ho(S@ AP) @ H+(VP @ AP) = 0. Hence (2) +-(4). Clearly, (4) (5). I t remains to be shown that (5) (2). Assume that ( 5 ) holds. Then (obviously) (g @ 1): is surjective. Moreover, the diagram

//

HdS 0AP)

(Ho(S @ AP) @ V P ) / ( H o ( S@ AP) @ V+P)

I

S / ( S0 P )

commutes; hence, 2 is an isomorphism. Thus (g @ L); is an isomorphism (cf. the commutative diagram of sec. 2.3).

2. Isomorphism theorems

65

Since (g @ L): is surjective and (g @ L): is an isomorphism, Theorem I, sec. 2.8, shows that (g @ L ) # is an isomorphism. Thus (5) => (2). Q.E.D. Theorem 111: Let S , y , and g be as in Theorem 11, and assume that S is evenly graded. Then conditions (1)-(5) in Theorem I1 are equivalent to

(6) H ( S 0AP) is evenly graded.

--

Proof: We show that (3) (6) (5). Suppose (3) holds. Then, since S is evenly graded and I# is surjective, H ( S 0AP) is evenly graded. Thus (3) 2 (6). Now assume that (6) holds. Then, in particular H:(S @ A P ) = 0, Y odd. On the other hand, since Pk= 0 for even k, while Sk = 0 for odd K , it follows that

( S @ P)'

= 0,

This shows that H:(S @ AP) = 0, i.e., (6) e-( 5 ) .

Y

Y

even.

even. Thus H,(S

0AP) = 0; Q.E.D.

2.10. P-algebras. Let ( S ;u) be a P-algebra, and let y:

H,(S @ A P )

---f

S

be a linear map, homogeneous of degree zero such that E#oy=

and

i

y(1)

=

1,

(cf. sec. 2.9). Let

g: Ho(S @ AP) @ VP

--f

S

be the corresponding P-linear map, and observe that' the diagram V P A H o ( S 0AP) @ V P

S commutes, where q(Y)= 1 @Y,

YEVP

66

11. Koszul Complexes of P-Spaces and P-Algebras

Proposition 11: Let (S; u) be a connected P-algebra. Then

(1) uv is surjective if and only if (Z#)+ = 0; i.e., if and only if H,(S 0AP) = H;(S 0AP) = r. (2) If H , ( S @ AP) = 0, then uv is injective. (3) ov is an isomorphism if and only if H ( S @ AP) = r. Proof: (1) If uv is surjective, then

and so H,f(S O A P ) = 0. Conversely, if H , f ( S @ A P ) = 0, then S = r @ S 0 P. Hence, by Lemma I, sec. 2.5,

S = 1 0 VP = Im uv. (2) Assume that H,(S @ AP) = 0. Then, according to Theorem 11, sec. 2.9, g is an isomorphism. Now commutative diagram (2.7) implies that uVis injective. (3) This follows at once from (1) and (2) and sec. 2.6.

Q.E.D.

§3. The PoincarC-Koszul series 2.11. Definition: Let E = C,,, EpP be a bigraded vector space such that E,P < GO, all r. dim

C

p+q=r

A simple gradation of E is defined by E = C pEp, where

E P = C EpP. 9

The Poincark-Kosxul series of E is the formal series

where c, =

C

( - l ) q dim EZ.

p+q=r

If E has finite dimension, then UE is a polynomial. I n particular, in this case U E ( - l ) = C ( - l ) ~dim Ep; P

i.e., U E ( - l ) is the Euler-PoincarC characteristic X E of the graded space E. Example: Suppose that E = C pEp is a graded vector space of finite type. Define a bigradation in E by setting

Et

=

EP

Then

E:

and

=

0,

q > 0.

UE = C dim Eptp = fE, P

where f E denotes the Poincark series for E . T h e following lemma is easy to check: Lemma 111: Let E = CP,, E4p and F = C p , qF4p be bigraded vector spaces satisfying the conditions above. Then

UE,, = UE

+ Up

and 67

U,y,,

=

UE

-

Up

68

11. Koszul Complexes of P-Spaces and P-Algebras

(Set ( E @ F): = C Er @ F," where the sum is over those p , q, m, and n such that p m = r, and q n = s.)

+

+

2.12. The Poincar&Koszul series of a P-space. Let S be a P-space of finite type. Then S @ A P and H ( S AP) are bigraded spaces satisfying the condition of see. 2.1 1. Thus we can form the corresponding PoincarC-Koszul series. Proposition III: The PoincarC-Koszul series of S @AP and H ( S @AP) satisfy

us,,,

U H ( S @ A P )=

= fs

UAP= fs

*

+ (fvP)-l,

where AP is given the bigradation (AP); = (AiP)i and f v p and fs denote the PoincarC series of V P and S. Proof: Let N k be the subspace of S @ A P given by

Nk=

C

(SOAP):.

itj-k

Define a gradation in N k by

N k = C NF,

NF

i

=

( S @ AP)f-i.

Then N k is stable under & , and 0, maps N t into NE1. Applying the Euler-PoincarC formula to the differential space ( N k ,Vs) we obtain

C (-l)i

dim Hi(Nk)= C (-l)idim N f . i

i

But Hi(Nk)= Ht-i(S @AP), and so

This shows that UH,s,Ap,= Us,Ap. On the other hand, Lemma 111, sec. 2.11, and the example of sec. 2.11 yield USWP =

US

*

UAP =fS

'

uAP*

3. T h e PoincarC-Koszul series

69

Finally, recall from sec. 2.6 that H(VP 0AP) = I', It follows that 1= Hence

U A p = (fVp)-l,

U H W P Q A P ) =fvp

*

UAP,

and so

Corollary I: Suppose that the PoincarC polynomial of P is given by fp

=

+

+ tor.

Then

Corollary 11: Assume that H+(S 0AP) = 0. Then fIf(S@AP) = UH(SEOAP)9

and so

Corollary 111: Let 0 -+ S of graded P-spaces. Then u H ( ? * Q A p )=

Corollary IV: T h e series

of

s.

-+

T -+ W + 0 be a short exact sequence

+

UH(SBAP) UHWOAP) U H ( S O A p )is

*

independent of the P-structure

Finally, suppose H ( S 0A P ) has finite dimension. Then its EulerPoincarC characteristic is given by X N ( S O A P ) = uH(SOAI')(-l)

(2.8)

(cf. sec. 2.11). If, in addition, S is evenly graded, then Proposition I11 shows that

UH(SOAP)(~) = UH(SWP)(-~).

s4. Structure theorems In this article (S; a) and (2'; T ) denote connected P-algebras. We identify So and T o with T via the isomorphisms A++ A . 1, A E I'. Recall that the Koszul complexes are written (S @ AP, 0,) and ( T 0AP, q). 2.13. The Samelson projection. Define a linear map pkq;S @ A P + A P

by setting ps(1 @ Y

+ z @ @) = Y ,

@, Y E AP,

x

E

S'.

Then, since S 0 P c Sf, we have es 0 0, = 0. Hence ps induces a homomorphism QB : H ( S @AP) --LAP of graded algebras. Definition: The homomorphism ps is called the Samelson projection for (S;a), and the graded space

Ps = P n Im pQ is called the Samelson subs3ace of P. If p: S + T is a homomorphism of graded P-algebras, then p T = ps; thus the diagram

H ( S @AP)

H ( T @ AP) commutes. I n particular the Samelson spaces satisfy 70

Ps c P,.

0

(p'

@ 1)

4. Structure theorems

71

Theorem IV: Let (S; G) be a connected P-algebra. Then

Im

Proof: Evidently i(x") 0 i(x") 0

@;

e$

= APs.

es = ps

=

@$

0

o

i(x"), xx E P",whence

x" € P".

i(x")#,

This relation shows that the subalgebra Im es is stable under the operators i(x"). Now Proposition I, sec. 0.4, implies that I m @$ = APs. Q.E.D. Definition: A Samelson complement for ( S ;a) is a graded subspace

Ps of P such that P = ps @ Ps. Proposition IV: Let ( S ;a) be a connected P-algebra. Then an element x of P is in ps if and only if

.(x) Moreover, if

Sf a(P).

E

ps is a Samelson complement, .(Ps)

then

c Sf * a(Ps).

Proof: ( 1 ) Fix x E P. It is immediate from the definitions that

PS = ps(ker K n (S @ P)). Ps

Therefore x E such that

if and only if there are elements xi E P and ziE S+

E(l i.e., x

E

Ps if

ox+

and only if a(.)

l Z i @ X i i

has the form

But this condition is equivalent to a(.) E S+* o(P). (2) Observe via (1) that

.(Pi)

c

s+ 1i k .(Pi) *

11. Koszul Complexes of P-Spaces and P-Algebras

72

Now induction on k yields the relation Q.E.D. 2.14. The cohomology sequence. The sequence

V P -+s5 H ( S 0AP) 1'

0"

-5AP

is called the cohomology sequence of the P-algebra (S;0). A homomorphism q ~ :S -+ T of graded P-algebras induces the commutative diagram

s ts"H ( S 0A P )

H ( T 0AP)

T1%

of cohomology sequences. On the other hand, let P, be a second finite-dimensional positively graded vector space such that P: = 0 for even k. Assume that a: P, P is a linear map, homogeneous of degree zero. Then, setting CT, = u 0 a, we obtain a P,-algebra (S;a,). Evidently

-

(L

@ a A ) :S @ APl

-+

S @ AP

is a homomorphism of graded differential algebras, preserving the bigradation. Thus we obtain a commutative diagram of cohomology sequences

VP,

___+

a"]

-

Ii

S

H(S 0APl) I(saa,,)*

N ( S @ AP)

VP-S0v

-

API I S A

AP.

Note as well that

i(x")#

0

(1

@ah)# = ( L @a,)#

o

i(a"(~"))*,

X"

E

P".

4. Structure theorems

73

Finally we have Proposition V:

I n the cohomology sequence for ( S ;a): (1) Is# 0 a+ = 0, and ker Is# coincides with the ideal in S generated by I m . : a I n particular, (Is#)+ = 0 holds if and only if a, is surjective. (2) ,& 0 (Is#)+ = 0, and so ker es" contains the ideal in H ( S @ AP) generated by Im(Zs#)+. Proof

Cf. sec. 2.2, and Proposition 11, sec. 2.10.

Q.E.D.

2.15. The reduction theorem. Let P and P denote the Samelson subspace and a Samelson complement for the P-algebra ( S ;u). Then multiplication defines an isomorphism

g: AP 0AP 5 AP of graded algebras. Next, let ( S ;5 ) be the P-algebra obtained by restricting u to P, and denote its Koszul complex by ( S @ AP, E). Then V6 is the restriction of I?,to S @ AP (cf. sec. 2.14). Moreover,

( S @ A P @AP,

v6 0

1)

is a graded differential algebra. Theorem V (reduction theorem): Suppose that ( S ; u) is an alternating connected P-algebra with Samelson space p, and let P be a Samelson complement. Then there is an isomorphism

-

-

f:(S@APOAP,~O1)---,(SOAP,~',) of graded differential algebras, such that the diagram

S 8 AP @ AP $,@I

I

-

AP @ AP ---

f N

N

S @ AP

lea

AP

commutes. ( ,Il and 1, are the obvious inclusion maps.)

11. Koszul Complexes of P-Spaces and P-Algebras

74

Proof (1) Construction off:

/I: P

-+

Choose a linear map

ker 0, n S @ P,

homogeneous of degree zero, such that es o B = 1. Since S is alternating, so is S @ AP. Thus, since Pk = 0 for even k, we have x

@(x)' = 0,

Hence

P.

E

/I extends to a homomorphism

PA:Af3 + ker 0,. Now define f by setting f(2

@ @ @ Y) =

(2 @

@)

x

@,(Y),

E

S, @ E AP, Y E AP.

(2) f is a homomorphism of graded dzfferential algebras: Clearly f is a homomorphism of graded algebras. Moreover, using the relations oBA=

0

0 IsoAp= 0,

and

we find that for z E S , @ E AP, and Y E AP,

@ @ @ y , = Pb((' @ @)

(% Of)('

= 0(' =

*

(f 0 (0, @ L ) ) ( Z @ @ @ Y).

Thus 0,O.f = f 0 ( & @ L ) . (3') f is an isomorphism: Since B(x) = 1 @ x

@ @)

es

+ w,

0

/3 = w

1,

E

it follows that for x

E

P

s+@ P.

This implies that

f-t@g:Sk@AP@AP-+C

Sj@AP.

(2.10)

3 >k

Now set

Then the algebras S @ AP @ AP and S @ AP are filtered respectively by the ideals I k @ AP @ AP and I k @ AP. Moreover, i @g and i @g-l are filtration preserving isomorphisms; hence i @g induces

4. Structure theorems

75

an isomorphism AtOgbetween the associated graded algebras (cf. sec. 1.1). Since L @g is filtration preserving, formula (2.10) shows that so is f a n d that A, == A , @ g . Thus A, is an isomorphism; hence so is f (cf. Proposition VII, sec. 1.14). (4) The diagram commutes: It is trivial that the upper triangle commutes. Moreover, formula (2.10) yields

Corollary I: f induces an isomorphism of graded algebras

f*: H ( S @AP) @ A P Z H ( S @ A P ) for which the diagram

H ( S @ AP)

commutes.

Corollary 11: Th e diagram

H ( S @ AP) commutes. In particular, (&$)+= 0.

11. Koszul Complexes of P-Spaces and P-Algebras

76

Proof: T h e Samelson theorem (sec. 2.13) shows that I m ,p$ = AP, It follows that (cf. Corollary I )

(6s")' = 0. Now Corollary I1 follows from Corollary I.

Q.E.D. Corollary 111:

f# restricts to an isomorphism j":Im ;7 25 I m IS#.

Corollary lV:

f* restricts to an isomorphism

f":H + ( S @ A P )@ A P zker,pz 2.16. Simplification theorem. The theorem of this section is, in some sense, a generalization of Theorem V. Let ( S ; u ) be an alternating graded connected P-algebra. Assume a:P -,S+ u(P) is a linear map, homogeneous of degree 1, and define a second P-algebra ( S ; r ) by setting z = u a.

-

+

Theorem VI (simplification theorem): With the hypotheses and notation above, there is an isomorphism

f: ( S @ AP, F) 5 ( S @ AP, Vo) of bigraded differential algebras, such that the diagram

SOAP

SOAP commutes.

-

Proof: Choose a linear map y : P -+ S+ 0P, homogeneous of degree zero, and such that 0 y = a. Define @: P S @ P by

8(.)

=1

0x +

x

p,

77

4. Structure theorems

and set

f (0 ~ @)

= (Z

z E S, @ E AP. 01 ) . /?I,,(@),

Then it follows, exactly as in the proof of the reduction theorem, that f is an isomorphism making the diagram above commute. Moreover

01 = 4%) 01 + Kl(Y(.)) xEP = (K o f ) ( l 0x),

(f

K)(1 0x)

(fo

F)(z @ 1) = 0 = (K of)(.

= .(X)

and

0l),

x E S.

Since f o 0, and 0, of aref-antiderivations, this implies that f

0

0,= 0,of. Q.E.D.

Corollary I: There is an isomorphism c=

g: (S B A P , E) which satisfies g 0 ls = 1, and ps og

=

( S @AP, &) ps.

Corollary II: f induces an isomorphism

f*: H ( S 0AP, 0,)5 H ( S 0AP, 0,) of graded algebras, which makes the diagram

commute. (Note that we have used homomorphisms! )

Is# (and

p d ) to denote two different

s5. Symmetric P-algebras 2.17. The main theorem. Let Q be an evenly graded finite-dimensional vector space with Qk = 0 for k 5 0. Let VQ have the induced gradation; i.e.,

vy,)

deg(y,v

=

+ + degy,.

degy,

Then if a: P + VQ is a linear map homogeneous of degree 1, the pair (VQ; a) will be called a symmetric P-algebra. Given a symmetric P-algebra (VQ; a) with Samelson space P, define an integer k(VQ; u) by k(VQ; u) = dim P - dim P - dim Q.

A main purpose of this article is to establish Theorem VII: Let (VQ; a) be a symmetric P-algebra such that

H(VQ @ A P ) has finite dimension. Then k(VQ; a) 2 0 ; i.e.,

8 dim P 2 dim i Moreover, if

f

+ dim Q.

is a Samelson complement, then

Hj(VQ 0Af) # 0,

j

= k(VQ; a),

Hj(VQ @AP)= 0,

j

> k(VQ; a).

while

In particular, the following conditions are equivalent :

+

dim Q. (1) dim P = dim P ( 2 ) H+(VQ @ A P ) = 0. The first statement is established in Proposition VI, below. The rest of the theorem is proved in Proposition VII, sec. 2.18. Proposition VI: If dim H(VQ @ A P ) 78

< 00, then k(VQ; u) 2 0.

5. Symmetric P-algebras

79

Proof: Let p be a Samelson complement. According to Corollary I of the reduction theorem (sec. 2.15),

H(VQ @ A P )

H(VQ @ Ap) @ AP.

Thus H(VQ @ AP) has finite dimension; in particular, its PoincarCKoszul series U H ( v Q o # ) is a polynomial. We show that k(V&; o) is the multiplicity of 1 as a root of the polynomial U H ( V Q B A F ) . Write the PoincarC polynomials of p and Q in the form ii

ffi

=

2

tgi

(gi odd),

ri: = dim P,

1 t4

(lj even),

m

i=l

and m

j Q= Then

n (1

j=1

= dim

m

A

fvp =

Q.

- pi+')-'

and

f v Q=

i=l

fl (1 - t4-1,

j=1

Thus Corollary I to Proposition 111, sec. 2.12, yields

Now let r (r 2 0) be the multiplicity of 1 as a root of U H ( v Q o ~ . ) . Then the equation above implies that m r = ri:, whence

+

k(VQ; c ) = ri:

-

m = r.

Q.E.D. Corollary: The relation

dim P

= dim

P

+ dim Q

holds if and only if H(VQ @ Ap) has nonzero Euler-PoincarC characteristic. Proof:

Since VQ is evenly graded,

(cf. sec. 2.12). Now apply the proposition.

Q.E.D.

11. Koszul Complexes of P-Spaces and P-Algebras

80

2.18. Proposition VII: With the hypotheses of Theorem VII,

H,(VQ @,AP)If 0, j = k(VQ; u) and

Hj(VQ

@AP) = 0, j > k(VQ; 0).

Proof: Since dim P = dim P - dim P, and since the Samelson space for the P-algebra (VQ; 3) is zero (cf. Corollary I1 to Theorem V, sec. 2.15), it is clearly sufficient to consider the case P = 0, P = P. We do so from now on. Define a graded space T by setting Tp = p+'( p = 1, 3, 5, . . .). Let dim T = dim Q = m and dim P = n ; then k(VQ; u) = n - m. Let k denote the greatest integer such that

H,(VQ @ AP) # 0. We must show that k = n - m. Consider the algebra R = VQ @ AP 0AT (skew tensor product). Define a bigradation in R by setting

dmote the corresponding total gradation by

Set Rp,* = CqRP.q and R*,q = C pRp~q. Let (VQ @ AP; t) be the T-algebra defined by

t ( Y )" Y

01,

Y

E

T,

and denote the corresponding differential operator in VQ @ AP @ AT by F. Zt is homogeneous of bidegree (0, 1). On the other hand, if K, denotes the differential operator in VQ @ AP, then c7, @ L is homogeneous of bidegree (1,O). Set

D- Vo@~+V'r. Then D 2= 0, and so ( R , D ) is a graded differentia1 space (with respect to the total gradation defined above). The proposition is now an immediate consequence of the following two lemmas.

81

5. Symmetric P-algebras

Lemma IV: H("(R, D ) = 0, 0 5 r

< m, and W r n ' ( R0) , f 0.

Proof: Define a linear map a:P + V Q @ 1 @ T such that Then the linear map 8: P -+ R given by xE

@(x) = 1 @ x @ 1 - cr(x),

0,o

(x

= LT.

P,

extends to an algebra homomorphism PA:AP + R. Now consider the algebra homomorphism p: R + R given by

It is easy to verify that p is an isomorphism of graded algebras ( R has the gradation defined just above). A straightforward computation shows that Llq - yK is zero in V Q @ 1 @ 1, 1 @ P @ 1, and 1 @ 1 @ T . Since these spaces generate the algebra R and since Dy - p0, is a p-antiderivation, it follows that

Dp

= pK.

This imp,lies that p induces an isomorphism,

Since p is homogeneous of degree zero, and since H(VQ @ A T ) = (cf. sec. 2.6) we obtain isomorphisms

r

The lemma follows. Q.E.D. Lemma V:

Let k be the greatest integer such that Hk(VQ 0AP)

f 0 . Then

H(')(R,D ) = 0, r < n Proof:

- k,

Filter R by the subspaces

and

H(n-k)(R,D ) f 0.

11. Koszul Complexes of P-Spaces and P-Algebras

82

Then the E,-term of the corresponding spectral sequence is given by

= H ( H ( V Q 0AP) @ AT, 0:)

E,

(cf. Theorem 11, sec. 1.19, and sec. 1.21). Here

(2.11)

F#is defined by

F#(a @ 1) = 0

and o,"(U

=

A

(-1)'

* *

.AYP) P

(-1)"Cr

*

i&(yi) @)ylA

- * -

jt* * .

hYp,

1-1 (I

E

H'(VQ @ AP), yi E T.

In particular, the spaces Hp(VQ @ A P ) @ AT are stable under This implies that

F'.

H(H(VQ @ AP) @ A T , 0:) = 2 H(Hp(VQ@ AP) @ A T ) P

=

1 Hq(Hp(VQ@ AP) @ A T ) .

P,*

Composing this with (2.11) we obtain

@" = Hm-q(H,-p(VQ@ AP) O A T ) .

(2.12)

Next we establish the relations

p < n - k, i 2 2,

= 0,

p n 2 - k

(2.13)

f 0,

(2.14)

and q3-k)

i 2 2.

# 0,

(2.15)

Formula (2.13) (for i = 2) is an immediate consequence of the definition of k, and the formula for i > 2 follows at once from the definition of the l2iQ.p.

T o prove (2.14) choose a non-zero element a E AmT and a non-zero element a E H,(VQ @ A P ) with maximum degree (with respect to the gradation defined in sec. 2.2). Then CL & ( y ) = 0, y E T , and so

-

K#(a @ a ) = 0. Thus a @ a is a nonzero cocycle in Hk(VQ 0AP)

AmT.

5. Symmetric P-algebras

83

Since T has dimension m,

hence a @ a represents a nonzero cohomology class. This implies that Hm(Hk(VQ0AP)

0A T ) # 0.

Combining this relation with formula (2.12) we obtain formula (2.14). T o prove (2.15) observe that, in view of (2.13), EPp = 0 if p < n - k and i 2 2. Since di;EpP EQ+i,P-i+l a ,

-

a straightforward induction, starting with (2.14), establishes (2.15). In particular, relations (2.13) and (2.15) imply that

H " ( R , D)= 0,

Y

< n - k,

and

H(n-k)(R, D)

EEPk)f 0. Q.E.D.

Remark: With the proof of Lemma V, the proofs of Proposition VII and Theorem VII are completed.

2.19. The decomposition theorem. In Theorem VIII below we give a number of conditions on a symmetric P-algebra (VQ; u) which are equivalent to conditions (1) and (2) of Theorem VII. Part of Theorem VIII remains true for any P-algebra; however, that part is true in even greater generality (P-differential algebras) and will be established in Chapter I11 (cf. sec. 3.16). Theorem VIII: Let (VQ; u) be a symmetric P-algebra with Samelson space P and a Samelson complement P. Assume that H(VQ @ A P ) has finite dimension. Then the following conditions are equivalent :

+

dim Q. (1) dim P = dim P (2) The kernel of &e coincides with the ideal generated by Im(Z&)+ VQ -+ H(VQ 0AP) is surjective. (3) The map (4) There is an isomorphism

r&:

[VQlVQ 0 PI @ AP ----+ H(VQ 0AP), N

84

11. Koszul Complexes of P-Spaces and P-Algebras

of graded algebras which makes the diagram

CvQlvQ

0

PI 0Ap

HWQ 0AP) commute. ( 5 ) There is an isomorphism [VQlVQ 0 PI

0VP 5 VQ

of graded p-spaces which makes the diagram

commute. ( 6 ) H,(VQ @ A P ) = 0. (7) H+(VQ OM’)= 0. ( 8 ) H(VQ @AfJ) is evenly graded. (9) H(VQ @ AP) has nonzero Euler-PoincarC characteristic. (10) T h e map 5”: VP + VQ is injective. Proof: We show that

( 1 ) e (7): This is Theorem VII, sec. 2.17. (2) o (3): In view of the Corollaries 111 and IV to the reduction theorem (sec. 2.15), condition (2) is equivalent to the following condition:

85

5. Symmetric P-algebras

The ideal in H+(VQ @ AP) generated by Im l#is all of H+(VQ @ AP). But (by the same argument as the one used in Lemma I, sec. 2.8), this happens if and only if Im

= H+(VQ @ AP).

(3) o (4): This is an immediate consequence of the reduction theorem and the fact that VQ 0 P = VQ 0 P (cf. Proposition IV, sec. 2.13). (5) o (7): In view of the relation VQ 0 P = VQ 0 P, this follows from Theorem 11, sec. 2.9. (6) o (7) e (3): This corresponds to the equivalent conditions (5), (4), and (3) in Theorem 11, sec. 2.9. (6) o (8): This follows from Theorem 111, sec. 2.9. (7) o (9): In view of the corollary to Proposition VI, sec. 2.17, (9) is equivalent to the condition dim P = dim P dim Q. By Theorem VII, sec. 2.17, this is equivalent to (7). (7) e (10): In view of the results above, (7) =+-(5) 3 (10). Conversely, if (10) holds, Lemma VI, below, implies that dim P < dim Q. On the other hand, Theorem VII, sec. 2.17, shows that dim P 2 dim Q. Thus dim P = dim Q; hence, again by Theorem VII, H+(VQ 0AP) = 0; i.e., (10) (7). Q.E.D.

+

-

Corollary: Assume (VQ; u) is a symmetric P-algebra such that H(VQ @ AP) has finite dimension. Then the following conditions are equivalent :

(1) dim P = dim Q. (2) ZTQ is surjective. ( 3 ) H(VQ 0AP) VQlVQ 0 P. (4) There is an isomorphism Y

g : H,(VQ @ AP) @ V P

of graded P-spaces such that g(1)

(5) (6) (7) (8)

=

VQ

1.

H+(VQ 0AP) = 0. H(VQ @ AP) is evenly graded. The Euler-PoincarC characteristic of H(VQ @ AP) is nonzero. o, is injective.

86

11. Koszul Complexes of P-Spaces and P-Algebras

Lemma VI: Let Q1 and Q2 be strictly positively graded, finitedimensional vector spaces. Give VQ1 and VQ2 the induced gradations and assume that v: VQ1 VQz is a linear injection, homogeneous of degree zero. Then dim Q1 5 dim Q2. --f

Proof: Let Y

fi =

C

8

tki

and

=

f2

C tzf

j=l

i=l

be the PoincarC polynomials of Q1 and Q2. Then the PoincarC series of VQ1 and VQz are given by

These series are absolutely convergent for 0 5 t < 1. Moreover, by hypothesis, dim(VQ1)P 5 dim(VQ,)P,

p

= 0,

1, . . .

Hence, for 0 5 t < 1,

ThusfVQl(t)
n (1 7

fVQ,(t)/fVQ,(t)=

i=l

- t")

1"7

j=1

(1 - t

y

=

(1 - t)'-"g(t),

where g is a continuous function in the interval 0 5 t < 00. Thus (1 - t)'-"(t) L 1 if 0 5 t < 1. Since g is continuous at 1, it follows that r 5 s; i.e. dim Q1 5 dim Q2. Q.E.D. 2.20. Poincar6 series. Let (VQ; 0) by a symmetric P-algebra such that H(VQ @ A P ) has finite dimension and dim P = dim P dim Q. Let P be a Samelson complement. In view of Theorem VIII, sec. 2.19, and Corollary I11 to the reduction theorem, sec. 2.15, we have isomorphisms

+

H(VQ @ AP)

= Ho(VQ @ AP)

I m I&

87

5. Symmetric P-algebras

and

I m ITQ @ AP.

H(VQ @ AP)

Thus Corollary I1 to Proposition 111, sec. 2.12, gives f I m 1 ; ~==

f V Q fTh *

9

whence fH(VQOAP) = fIm l;,fAP

Since f v p

= fvpf v p ,

= fVQ

*

fij ' fAa.

these equations yield frm

= fva

*

fvp

*

fG

(2.16)

and f H ( VQO A P )

f V QfVPf AP

(2.17)

f V P

I n particular write the PoincarC polynomials of P, P, and Q in the form

(giodd, k i even, r equations read

=

dim P, s = dim Q, r

-

s

=

dim P ) . Then these (2.18)

and (2.19) Now consider the Euler-PoincarC characteristics of I m 1;Q and H(VQ @ AP). Since Im ZFQ is evenly graded, we obtain from formula (2.18) (2.20a) Moreover, if

p # 0,

then XW(VQOAP) = 0; while if

P = 0,

then

H(VQ @ AP) = I m 1;Q and XH(VQBAp)=

dim H(VQ @ AP) =

JX=l(gi + 1 ) .

nL ki

(2.20b)

88

11. Koszul Complexes of P-Spaces and P-Algebras

2.21. A third structure theorem. Theorem IX: Let (VQ; a) be a symmetric P-algebra with Samelson space P and a Samelson complement P. Then the following conditions are equivalent:

(1) (I&)+ = 0. ( 2 ) uv:V P + VQ is surjective. ( 3 ) 5,: VP 3 VQ is an isomorphism. (4) &*: H(VQ 0AP) 3 AP is an isomorphism. (5) T h e algebra H(VQ @ AP) is generated by 1 together with elements of odd degree. If these conditions hold, then dim P

=

dim

+ dim Q.

Proof: According to Proposition V, (l), sec. 2.14, (1) * (2). Now we show that

(1)

=>

(3)

=+

-

(4) (5) * (1).

I n fact, suppose (1) holds. Then Corollary I11 of the reduction theorem (sec. 2.15) shows that (21.,)+ = 0. Moreover, the Samelson space for (VQ; 3 ) is zero, as follows from Corollary I of the reduction theorem. Hence we can apply Lemma VII below (with P replacing P) to show that 3, is an isomorphism. Thus (1) * (3). Suppose (3) holds. Then by sec. 2.6, H(VQ 0AP) = r. Now Corollary I to the reduction theorem implies that

f *: Af' --%H(VQ @ AP). Moreover, it is an easy consequence of the commutative diagram in that corollary that e & = (f#)-'.Thus ( 3 ) * (4). (4)* (5) is obvious. Suppose that (5) holds. Since V Q is evenly graded, so is H,(VQ 0A P ) ; thus the elements of odd degree are contained in the ideal H+(VQ @ AP). It follows that H+(VQ @ AP),together with 1, generates H(VQ @AP). Hence H,+(VQ 0AP) = 0, and so (Z,#,)+ = 0. This shows that (5) (1). Finally, if these conditions hold, then (3) implies that dim P = dim Q, whence dim P ) = dim P dim Q. dim P = dim p

+

+

Q.E.D.

5. Symmetric P-algebras

Lemma VII:

89

Let (VQ; a) be a symmetric P-algebra such that

(IfQ)+ = 0

P

and

=

0.

Then a,: V P + VQ is an isomorphism. Proof: By Proposition V, ( l ) , sec. 2.14, the condition (ITQ)+ = 0 implies that 0, is surjective. Hence there is a linear injection p : VQ + VP, homogeneous of degree zero, such that 0" 0 9 = 1. Now Lemma VI, sec. 2.19, yields dim Q 5 dim P.

On the other hand, let 7 t : V+Q + Q be the projection with kernel (V+Q) . (V+Q). We show that 3c 0 a: P + Q is injective. In fact, if n(a(x)) = 0 for some x E P, then U(X) E

(V'Q)

(V+Q) = a,(V+P)

*

av(V+P).

Now Proposition IV, sec. 2.13, shows that x E P. Hence, x = 0 and so 0 0 is injective. Since dim Q 5 dim P,it follows that 7t 0 0 : P + Q is an isomorphism of graded vector spaces. Hence V P and VQ have the same PoincarC series. Thus a, restricts to linear surjections

3c

0,:

(VP)k + (VQ)k,

k

= 0, 1,

. ..

between spaces of the same dimension. Hence u, is an isomorphism. Q.E.D.

$6. Essential P-algebras 2.22. Essential P-algebras. Let (VQ; a) be a symmetric P-algebra. Consider the ideal V+Q . V+Q in VQ. Define a graded subspace PI of P by P , = a-'(V+Q . V+Q).

I t is called the essential subspace for the P-algebra (VQ; a). Note that the Samelson space ?' of the P-algebra (VQ; a) is contained in the essential subspace P,: P c P I . In fact, if x E P, then, by Proposition IV, sec. 2.13, C(X) E V+Q a(P). 3

In particular U(X)E V+Q V+Q, and so x E P I . A symmetric P-algebra is called essential if P, = P ; i.e., if a ( P ) c V+Q * V+Q. 2.23. The associated essential PI-algebra. Given a symmetric Palgebra (VQ; o) with essential subspace P, ,we shall construct an essential PI-algebra (VQ,; a,) (with Q, a graded subspace of Q) such that

H(VQ @ AP)

= H(VQ, @ AP,).

Choose a graded subspace P, c P so that

P = P, @ Pz. Let n: V+Q -+ Q denote the projection with kernel V+Q . V+Q. Then the map n 0 a: P + Q restricts to a linear isomorphism N

n

0

a: P, LL Im(n

0

a).

Now choose a graded subspace Q1 c Q so that

Q = Q1 @ Im(n o a). 90

6. Essential P-algebras

91

Then a homomorphism of graded algebras q : VP, @ VQ1 + VQ is given by

YEVP,,

q(Y@O)=o,(Y)v@,

@€VQ1.

Lemma VIII: q is an isomorphism of graded algebras.

Then q = y y (here V ( P , @ Ql) is identified with VP, @ VQ1). Now filter the algebras V(P, @ Ql) and VQ by the ideals

Since 17 is a homomorphism, it is filtration preserving. Denote the induced homomorphism of associated graded algebras by A,. Next, observe that the canonical linear isomorphism

Q 5 V'QlV'Q

*

V+Q

N

extends to an algebra isomorphism VQ 5A,, (A,, the associated graded algebra; cf. sec. 1.17). Similarly we obtain an algebra isomorphism

0$21) =AvcP&oQ,, N -

V(P2

*

A simple computation shows that the diagram V(pz

-

O QI) A Avcqog,,

it is a linear isomorphism. Hence (n o y), is an isomorphism and so the

11. Koszul Complexes of P-Spaces and P-Algebras

92

commutative diagram implies that A, is an isomorphism. It follows now from Proposition VII, sec. 1.14, that 7 is an isomorphism. Q.E.D. T h e lemma shows that VQ admits the direct decomposition

VQ = VQ1@ VQ

*

u(P~).

This decomposition determines a projection

with kernel VQ a(P,); y is a homomorphism of graded algebras. Next define a linear map u, : P, -+ VQ by u1 = y 0 u. Then the P,algebra (VQ,; ul) is essential. In fact, since

-

u(Pl) c V+Q V+Q

y(V+Q) c V+Q1,

and

it follows that ~i(Pic )

r(v+Q

*

V+Q) c

r(v+Q) r(v+Q)c V+Qi *

*

V+Qi.

The P,-algebra (VQ1; u,) is called the associated essential P,-algebra for (VQ; 0)Note that the associated essential P,-algebra depends only on the choice of the graded subspaces P, of P and Q1 of Q. I n particular, if (VP; u ) is itself essential, then P , = P, Q1 = Ql and ul = a. Thus (VQl; u,) = (VQ; u ) in this case. T o construct an isomorphism between the cohomology algebras of (VQ; u ) and (VQ,; ul)l let /?:P-+ P, denote the projection induced by the direct decomposition P = P, @ P2. Extend B to a homomorphism PA:A P AP, . Then

-

Y0

(VQ

/?A:

0AP, K)

(VQi

0APi

9

Val)

is a homomorphism of graded differential algebras and the diagram

VB @dl

VP,

- ov

1.

VQ

-

VQi

W

commutes.

I

V

IvQ

@JQ

V Q O A P -AP ]@PA

lvQi

VQ1 0AP,

]PA

@JQl

AP,

(2.21)

6 . Essential P-algebras

93

Theorem X: (1) y @ p1 induces an isomorphism of graded algebras

(Y 0

PA)#:

H(VQ

0AP)

N

H(VQi

0APi).

(2) The diagram

commutes. (3) The Samelson spaces of the P-algebras (VQ; CT) and (VQ1; a,) coincide.

Proof: (1) Identify VQ with V P 2 @ VQ1 via 7,and write VQ@AP=VP2@VQl@APl@AP2. Then the subspaces F-P

=

C VP, @VQ1 @ApPl @ A P 2 ,

p

=

0,

. . . , dim P,

/L S P

define a filtration of VQ @ AP. The El-term of the corresponding spectral sequence is given by

El

VQ1 @ AP, @ H(VP2@ APz)

(cf. Theorem 11, sec. 1.19). Since H(VP2 @AP2) = T (cf. sec. 2.6), it follows that El z V Q 1 @APl.

On the other hand, filter VQ1 @ A P , by the subspaces P-P =

C VQ1 @ A'P,. P SP

Then El = VQ1 @ APl . Moreover, the map y @ PAis filtration preserving and the induced homomorphism of El-terms is simply the identity map of VQ, @ A P l . Thus, by the comparison theorem (sec. 1.14), (y @ PA)# is an isomorphism.

94

11. Koszul Complexes of P-Spaces and P-Algebras

(2) This follows immediately from the commiitative diagram (2.21). (3) I n view of (2), p,, restricts to a surjective linear map between the Samelson spaces P and P , . But since c P, (cf. sec. 2.22) p is the identity map in P . This shows that P = PI. Q.E.D.

Chapter I11

Koszul Complexes of P-Differential Algebras I n this chapter P L- CkPk denotes a finite-dimensional positively graded vector space satisfying Pk= 0 for even k. P is the evenly graded space given by Pk = Pk-l, and AP and V P are the graded algebras described at the start of Chapter 11.

$1. P-differential algebras 3.1. Definition: A P-diaerential algebra ( ( P , 8)-algebra) is a triple

(B, 6,; r ) where (1) ( B , 6,) is an associative, alternating, positively graded, differential algebra with unit element 1. (2) t:P ker 6, is a linear map, homogeneous of degree 1. --f

T h e differential algebra ( B , 6,) is called the base of the (P, 6)-algebra ( B , 6,; t). Note that if ( B , 8,; t) is a ( P , 8)-algebra, then ( B ; t) is a P-algebra. A homomorphism of ( P , 6)-algebras

is a homomorphism 31: ( B , 6,) -+ (B,8,) of graded differential algebras that satisfies (p(1) = (l), and 31 0 t = 4. A semimorphism of (P, 6)-algebras is a homomorphism of graded differential spaces y: ( B , 8,) ( P , 68) that satisfies y(1) = 1, and

-

y(b * z(x))

= yp(b)

- %(x),

b E B, x E P.

Thus every homomorphism is a semimorphism. 95

96

111. Koszul Complexes of P-Differential Algebras

3.2. The Koszul complex. With each (P, 8)-algebra (B, BB; z) is associated a graded differential algebra ( B @ AP, VB) as follows: B @ AP denotes the skew tensor product of the graded algebras B and AP, with the gradation given by (B@AP)’=

1

BP@(AP)Q.

p+q=r

Define operators SB and 0, in B @ A P by dB = SB @ L, and

0,(b @ 1) = 0,

Then (B @ AP, 0,) is the Koszul complex of the underlying P-algebra (B; T); in particular, 0, is an antiderivation of square zero, homogeneous of degree 1. Moreover, since 8’ 0 t = 0, it follows that

Then VB is also an antiderivation homogeneous of degree 1 which satisfies Vi = 0. Thus (B @ A P , 0,) is a graded differential algebra. I t is called the Koszul complex of the (P, 8)-aZgebra ( B , 8B; z), and VB is called the differential operator in B @ AP. T h e graded algebra H(B @ AP, V’) is called the cohomology algebra of the (P, 8)-algebra (B, BB; T). We sometimes denote it simply by H ( B @ A P ) . Next observe that the inclusion map b H b @ 1 ( b E B ) defines a homomorphism IB:

(BY8B)

--f

(B @ Ap,

VB),

of graded differential algebras; lB is called the base inclusion. It induces a homomorphism

Zg: H(B) -+ of graded algebras.

H ( B @ AP)

1. P-differential algebras

97

-

-

Remark: Clearly 1: restricts to an isomorphism H o ( B )G H o ( B @A€'). Thus H ( B ) is connected if and only if H ( B @ AP) is connected. I n this case the (P, 6)-algebra ( B , 6,; t) is called c-connected. On the other hand, recall from sec. 2.2 that each element x+ E P" determines the linear operator i ( x " ) in B @ A P given by

i(x")(b @ @)

=

(-1)Pb @ i ( x + ) @ ,

b E BF, @ E AP

Evidently i(x") is an antiderivation and satisfies

i ( x * )o 6,

+ an

o

i(x*) = 0

and

i(x")c7,

+ c7,i(x") = 0

(cf. sec. 2.2). These relations imply that

i(x")V,

+ V.i(x") = 0.

Hence i(x+) induces an antiderivation i(x*)# in H ( B @ AP). Thus we obtain operators i ( u ) in B @ AP and i(a)# in H ( B 0AP) (u E AP*) by setting i(xF

A

..

A

xp")= i(xp") o . . . o i(xF),

xr

Furthermore, since B is alternating and z(P)c the relations

.(x). Thus

t

z ( y ) = .(y) . .(x),

x,y

E

E

&even

P". Bp, we have

p.

extends to a homomorphism t,:V P + B

of graded algebras. Since 6, 0 t = 0 it follows that 6' 0 z, = 0. Hence, composing t, with the projection ker 6, + H ( B ) we obtain a homomorphism -cf: V P -+ H ( B ) of graded algebras. Now suppose e;: (B, B,; t) (8, 6 ~ f); is a semimorphism. Then the map p @ 1 : B @ AP + B @ AP commutes with the differential operators V' and V ~ Jand , so it induces a linear map --f

(p'

@ L ) * : H ( B 0AP) + H ( B 0AP),

98

111. Koszul Complexes of P-Differential Algebras

homogeneous of degree zero. Moreover, the commutative diagram 1B

B-BOAP VP

/ \

plQi

B-B@AP lg

induces the commutative diagram

H(B)

H(B)

I;;

H ( B @ AP)

-

H ( B @ AP)

1;

in cohomology. The relations i(x") 0

(fp

@

1)

== (fp

@ 1)

0

i(x"), x"

E

P", imply that

If p is a homomorphism, then so are p @ i and (fp @ 1)". Finally observe that if dB = 0 (so that ( B , dB; T ) is a P-algebra), then the definitions and notation above reduce to those in Chapter 11. 3.3. The associated Falgebra. Let ( B , BB; T ) be a (P, 6)-algebra. Since dB 0 T = 0, T induces a linear map T # : P H ( B ) which makes H ( B ) into an alternating P-algebra. It is called the associated P-algebra. Its Koszul complex is ( H ( B )@ AP, I+),and I& is the operator induced in H ( B ) @ A P by 0,; i.e., = (0,)#. If fp: ( B , dB; T ) (8, 8n; f) is a semimorphism of (P, 8)-algebras, then fp# is a homomorphism of graded P-spaces. Hence p # @ i commutes with the differential operators cl,, and K,. It induces a linear map ---f

-

(p* @

1)":

H ( H ( B ) @ AP, V+)

-

H ( H ( B ) @ AP, K.).

1. P-differential algebras

99

If p is a homomorphism of (P, 8)-algebras, then y # is a homomorphism of P-algebras, q ~ #@ L is a homomorphism of graded differential algebras, and (pl# @ 1 ) " is a homomorphism of graded algebras. 3.4. The spectral sequence. Let ( B , dB; T) be a (P, 8)-algebra. Filter B @ A P by the ideals

1 Bp @ AP.

FP(B @ AP) =

(3.1)

PZP

Then ( B @ AP, VB) becomes a graded filtered differential algebra (cf. sec. 1.18). This filtration leads to a convergent spectral sequence of graded differential algebras; it is called the spectraZ sequence of the (P,8)-aZgebra ( B , 8s;.). Observe that the filtration (3.1) arises out of the bigradation

B @AP

=

C BP @ (AP)q. P>q

With respect to this bigradation ( B @ AP, SH , 0,) is a graded differential couple of degree 1 (not in general homogeneous) (cf. sec. 1.21). Thus the spectral sequence of ( B , 6,; T ) is the spectral sequence of this couple; in particular,

E,

=

El

B @AP

and

E,

H(B)@AP,

as follows from Theorem 11, sec. 1.19. T h e basic subalgebra (cf. sec. 1.13) for this filtration is simply B @ 1. It follows that ZB induces surjective maps

H p ( B )+ Ef,',

i 2 2,

homogeneous of degree zero (cf. Proposition VI, sec. 1.13). Moreover, the operators i ( x * ) are filtration preserving, and hence induce antiderivations in each E j . Finally, let q ~ B : + & be a semimorphism of (P, 8)-algebras. Then (9@ L ) : B @ A P + B @ A P is filtration preserving, and so it induces a homomorphism of spectral sequences. T h e induced homomorphisms of the E,, , E l , and E, terms correspond under the identifications above to ip @ L, 9 @ L , and i p # @ L respectively. 3.5. The lower spectral sequence. Consider the bigradation of B @ AP given by ( B @ Ap)(-k,Z)z ( B @ AkP)pk+l (3.2)

100

111. Koszul Complexes of P-Differential Algebras

It gives rise to the filtration of B @ A P by the subspaces

k k ( B@ A P ) =

k

C B @/lip,

R

= 0,

. . . ,n

(n = dim P).

i=O

Then L-k . L-‘ c L-(k+ll,and

In this way ( B @ AP, 0’) becomes a graded filtered differential algebra. The corresponding spectral sequence is denoted by (E:k,z),d,) and satisfies

Eik,l)= 0

(any r 2 0)

+

unless -n 5 R 5 0 (n = dim P)and K I2 0. Hence it is convergent (cf. Proposition V, sec. 1.12). It is called the lower spectral sequence of the (P, d)-algebra (B, 6,; T). Remarks: 1. If one used the convention that X-P = X , to raise and lower the indices of graded spaces, then the bigradation above is determined by the bigradation (B @ AP),” = ( B @ A‘P)’ used for P-spaces in Chapter I1 (cf. sec. 2.2). This explains the terminology “lower spectral sequence.” 2.

Note that with the notation of this section, the PoincarC-Koszul series for B @ A P is given by

U,,,,

=

(- l)kdim(B @ AP)(kp9J

whenever the right-hand side is defined. With respect to the bigradation (3.2) the operators 6, and are homogeneous of bidegrees (0, 1 ) and (1,O). Thus the lower spectral sequence is the spectral sequence of the graded homogeneous differential couple (or double complex) ( B @ AP, d B , q). In view of Theorem 11, sec. 1.19, the first terms of the lower spectral sequence are given by

and

ES

cz H(H(B) @ AP, F:.).

1. P-differential algebras

101

I n particular, the &-term is exactly the Koszul complex of the associated P-algebra ( H ( B ) ;.#). Moreover, the isomorphisms above restrict to isomorphisms

EIBsZ)

( H ( B )@ k k P ) k + l and Eik,l) g H!iz(H(B)@ AP, K:.).

Finally, let 9:B

+

(3.4)

& be a semimorphism of (P, 6)-algebras. Then

p @ 1 preserves filtrations and so it induces a homomorphism of lower

spectral sequences. T h e induced homomorphisms of the E,, , E l , and E, terms correspond under the isomorphisms (3.3) to q @ I, tp* @ L, and (p* 0t ) # , respectively. 3.6. Example: Let ( B , 6,; T) be a (P, 6)-algebra where P is a 1dimensional space homogeneous of degree g. Let x* be a basis vector of P". Then

0 +B

ZB

B @ AP

i(z*)

B --+ 0

is an exact sequence of differential spaces (up to sign). Hence we have the exact triangle

H ( B 0AP)

H(B)

called the Gysin triangle. The corresponding long exact sequence

is called the Gysin sequence. T o compute d, let x E P satisfy (x*, x) = 1. Then, if a E H p ( B ) is represented by a cocycle b, da is represented by the cocycle

v B ( ( - l ) p b @ x)

=b

.

.(x).

Hence, 301 = a

T#(x),

01

E

H(B ).

(3.5)

The Gysin sequence yields the important short exact sequence

0 --+ coker

i;

a +H ( B @ AP)

- iW)*

ker d

0.

(3.6)

111. Koszul Complexes of P-Differential Algebras

102

Now consider the associated P-algebra and its Koszul complex ( H ( B )@ AP, E#).Then

H ( H ( B )@ AP, c7,*) = Ho(H(B) @ A P ) @ H,(H(B) @ AP), Evidently,

H o ( H ( B )@ AP) = H ( B ) / ( H ( B )0 P ) = coker d, and

H , ( H ( B ) 0AP) -- ( H ( B )@ A'P) n ker O,,

=

(ker d) @ x.

These equations show that there is a linear isomorphism

H ( H ( B ) @ AP, V'+) zz H ( B 0AP, V B ) of graded vector spaces ; however, in general this isomorphism cannot be chosen to preserve products.

$2. Tensor difference (S, 8,; a) be (P, 8)-algebras, and consider the skew tensor product of the graded algebras B and S. Set 3.7. Definition. Let ( B , 8,;

6BO.S

=

t) and

dB

@ L - W B @ 8,S,

where 01, denotes the degree invoiution of B. Then ( B @ S , ),8 is a graded, alternating, differential algebra. T h e multiplication in B 0S induces an isomorphism

H ( B 0S,

H ( B )@ H(S).

8Ba.q)

Now define a linear map t

by (t

0a)(.)

0a: P-t =t ( . )

B@S

@ 1 - 1 @ a(.).

Then ( B @ S , d,,,; t 0a) becomes a P-differential algebra. It is called the tensor dzfference of ( B , 8,; t) and (S, 8,; a). The differential operator of the Koszul complex ( B 0S @ AP, V'',) is the sum of four anticommuting antiderivations : b 3 S

(dB

+ K)

-

+ %)*

Here 6, and 6, are the obvious extensions to B @ S @ AP, while K and Vu denote the operators corresponding to the P-algebras ( B @ S; r ) and ( B @ S; a). In particular, and 0, restrict to operators in B @ A P and S 0AP, respectively. T h e Koszul complexes for ( B , dB; T) and (S, 6,; a) are given by

( B @ AP, 6,

+ K)

and

( S 0AP, 6,

s

+ Vu).

Next, suppose that y: B + B and y : S -+ are semimorphisms (respectively, homomorphisms) of (P, 8)-algebras. Then y B y :B

S+ 103

0s

104

111. Koszul Complexes of P-Differential Algebras

is a semimorphism (respectively, homomorphism) between the tensor differences. Thus it induces a linear map (respectively, a homomorphism)

(v oY@ p :H ( B 3 s @AP) + H ( B @ 5 @ A P ) . Example: If 6, = 0 (i.e., if (S, 6,; cr) is simply an alternating Palgebra), then the tensor difference of ( B , 6,; t) and (S;o) is simply ( B @ S , B B ; t 0cr). I n this case VBOSis given by 8B

%@S

+ 0,

- 0,.

(Note that we consistently write d B = 6s @ i =

BB

@i@

1.)

3.8. Tensor difference with VP. Recall the P-algebra ( V P ; cr) given ) x, x E P (cf. sec. 2.6). Suppose ( B , 6s; 7) is any (P, 6)by ~ ( x = algebra and form the tensor difference ( B @ VP, 6,; t 0cr). Then a homomorphism mB:

(BY6,)

--f

( B @ vP @ AP,

of graded differential algebras is given by m,(b)

VBBBOVP)

=b

@ 1 @ 1.

Proposition I: The homomorphism mB induces an isomorphism mB#:H ( B ) 5 H ( B @ V P @ AP),

of graded algebras. Proof: A linear map a : P --+ B , @ V P homogeneous of degree zero, is given by x E P. .(X) = .(X) @ 1 1 @ x,

+

Since B @ V P is alternating and V P is evenly graded, we have a ( x ) a ( y ) = a ( y ) a ( x ) ,x,y, E P. Thus a extends to a homomorphism a,: V P + B @ VP.

Now define a homomorphism of graded algebras y: B @ V P -+B @ V P by setting y(b @ Y )= ( b @ 1 )

*

a,(!P),

Filter B @ V P by the subspaces

!P E VP, b

CpZpB* @ VP.

E

B.

Then y is filtration

2. Tensor difference

105

preserving and induces the identity in the associated graded algebra. Hence, in view of Proposition VII, sec. 1.14, y is an isomorphism. Next observe that the relation dn 0 t = 0 implies that y

@

6u

= SI,

@

y.

Moreover, (Y

O

(.

0.I)(.)

01 - 1 0x)

= y(.(.) -

-1 @ x = -a(%),

x E P.

Thus y : ( B @VP, 6s;

T

0CT)

c

( B @ V P , 6,; -a)

is an isomorphism of (P , 6)-algebras. Hence we have an induced isomorphism ( y o i)#: H ( B @ V P @ A P , Ss+Vr-

Finally, since y

0

mB = m u ,

V a ) zH ( B @ V P @ A P , SB - 0,).

we have the commutative diagram

H ( B 0V P 0AP, SB But H ( V P @ AP, l(,) == yields

H ( B @ V P @ AP,

6B

r (cf. -

-

On).

sec. 2.6), and so the Kunneth theorem

Vo)= H ( B ) @ H(VP 0AP, l(,) =: H ( B ) .

This shows that the lower m g in the diagram above is an isomorphism. Hence so is the upper mg. Q.E.D.

-

Now we shall construct an isomorphism inverse to rng. Define a homomorphism of graded algebras 9:B 0V P @ AP B by V ( b @ !P @ 1) = b . T , ( ! P ) and

q(b@Y@O)=O,

b e B , !PEEP, O E A \ + P .

111. Koszul Complexes of P-Differential Algebras

106

g

Proposition 11: (SB 0, - 0,)

+

0

Q

=

is a homomorphism of graded differential algebras: SB 0 9 , and satisfies e; 0 mB = 1. I n particular, y~ =

Proof: Since S R 0

t, =

(rn%)-l.

0 it folloivs that dB

=

813

9.

Moreover, q ~ ( ~ - V ~ ) ( b @ ! € ' @ l ) = O , ~ E B Y, E V P ,

and p0

(E - V,)(l

@ 1 8x) = p ( t ( x ) @ 1 @ 1 - 1 @ x @ 1) = 0,

x

€

P.

Since p 0 (0, - 0,) is a g-antiderivation and since B @ V P @ AP is generated by the elements of the form b @ Y @ 1 and 1 @ 1 @ x, it follows that p 0 (0,- 0,)= 0. Hence, (6,

+ 0,

-

0,)= 40 6,

= 8B

v.

Finally, it is obvious that p o mB = 1. Thus e;# 0 nzj$ = 1. By Proposition I, m% is an isomorphism. Thus the above relation shows that q++ is the inverse isomorphism. Q.E.D. 3.9. Spectral sequences of a tensor difference. Consider the tensor difference of two (P, d)-algebras ( B , d,; t) and ( S , d,; u). Bigrade the Koszul complex by setting

( B @ S @ AP)Psg = BP @ ( S @ AP)g. The corresponding filtration of the algebra B @ S @ AP is given by the ideals FP(B 0S @ A P ) = C Bp @ ( S @ A P ) P 2P

and leads to a convergent spectral sequence of graded differential algebras. It is called the B-sequence of the tensor difference and does not coincide with the spectral sequence of the (P, 8)-algebra ( B @ S, B B B S ; z 00).

2. Tensor difference

107

The B-sequence is the spectral sequence of the graded differential K), dn E). Since couple (B @ S 0AP, -( ds

+

6,:

BP

@S

+

0A P -

BP+l@ S @ A P

and Vr:BP@S@AP+

C Bp+p@S@AP, Qr2

the first terms of this sequence are given by

E:.q BP 0( S 0AP)" Ergq zz BP 0HQ(S@ AP, Vs) and

E,P."

(3-7)

H p ( B )0Hq(S0AP)

(cf. sec. 1.19 and sec. 1.21). All the above isomorphisms are algebra isomorphisms. s" be semimorphisms of (P, 8)-algebras. Now let pl: B B and y : S Then the map qj 0y 0i preserves the filtrations. The induced homomorphisms of spectral sequences correspond under (3.7) to the linear maps --f

P 0Y

04

97

0(Y 01 ) * ,

respectively. Next, define a bigradation in B

(B @ S 0

=

and

0S 0A P

C

B p

pl*

0(Y 0L ) # ,

by setting

@ S' @ (AP)..

/ltv=l

The induced filtration is given by the ideals

T h e corresponding spectral sequence is called the S-sequence of the tensor difference. A canonical isomorphism B 0S @ AP % S @ B 0AP is given by b 0s 0@ (-I)% 0b @ @ ( b E Bp, s E Sq, @ E AP). It induces isomorphisms of bigraded algebras

-

Ei'9.l'

S' @ ( B 0AP)'

El'." E Sk0H ' ( B @ AP, 0,) and

Eik-''

H k ( S )@ H L ( B@ AP)

(3.8)

111. Koszul Complexes of P-Differential Algebras

108

for the first terms of the S-sequence. Under these isomorphisms the homomorphisms of spectral sequences induced by semimorphisms and y : S --+ correspond to the linear maps Q]: B + y

0(9'011,

respectively.

Y

0(9'0

L)#,

and

Y*

0(v 0L ) # ,

$3. Isomorphism theorems T h e purpose of this article is to generalize the theorems of article 2, Chapter 11, to (P, 8)-algebras. 3.10. n-regularity. Recall that a linear map p': E -+ F between graded FP is an isomorphism for vector spaces is called n-regular if p p : Ep p 5 n and injective for p = n 1. --j

+

Theorem I: Let p: ( B , dB; r ) + ( B , 8'; ?) be a semimorphism. Then the following conditions are equivalent :

H ( B ) + H ( B ) is n-regular. (2) (p' @ L ) # : H ( B @ AP, 0,)+ H ( B @ AP, V') is n-regular. (3) For all (P, 8)-algebras (S, 8,; o ) the linear maps

(1)

p'#:

(p 0L

0L ) # :

H ( B @ S @ AP, V'Bs)

+

H ( B @ S @ AP, V'Bs)

are n-regular (cohomology of tensor differences).

-

Proof: (1) (2): Apply the comparison theorem of sec. 1.14 to the spectral sequence of the (P, 8)-algebras (cf. sec. 3.4). (2) (3): Apply the comparison theorem to the S-sequence of the tensor differences (cf. sec. 3.9). (3) (1): Consider the commutative diagram

-

H ( B @ V P @ AP) mf

I

(p6101)"

H ( B @ V P @ AP)

g

H(B) (cf. Proposition I, sec. 3.8). If shows that p# is also n-regular.

N

c"

(p'

___c

1

1.5

H@)

@ L @ 1 ) " is n-regular, the diagram

Q.E.D. 109

110

111. Koszul Complexes of P-Differential Algebras

Corollary: The following conditions on the semimorphism g, are equivalent :

(1) y e : H ( B ) + H ( B ) is an isomorphism. (2) (g, @ L ) # : H ( B @ AP, V') + H ( B @ AP, 0') is an isomorphism. (3) (p* @ 6)s: H ( H ( B ) @ AP, I&) + H ( H ( B ) 0AP, I&) is an isomorphism.

(4) ( y * @ i)$ is an isomorphism and ( y # @ L)? is injective. Proof: Theorem I above implies that (1) tj (2). Theorem I, sec. 2.8, applied to the P-linear map p # shows that (1) o ( 3 ) o (4).

Q.E.D. Remark: The corollary generalizes Theorem I, sec. 2.8.

3.11. A second isomorphism theorem. In this section we generalize Theorem 11, sec. 2.9. Let ( B , 8,; T ) be a (P, 8)-algebra. Recall from sec. 3.2 that the inclusion : ,Z B + B @ AP induces a homomorphism Z j : H ( B ) + H ( B @I AP). Form the (P, 8)-algebra (Im Z$ @ VP, 0; o), where x E P. .(X) = 1 @XI

Its Koszul complex is simply (Im Z i @ V P @ AP, Vo). Next choose a linear map, homogeneous of degree zero, y : Im 1:

+ ker

BB,

which satisfies Z;ozoy==

i

y(1) = 1.

and

(Here E : ker 8, + H ( B ) denotes the projection.) Then define a map g: I m Zg @ V P B ---f

by g(u @ @)

= y ( a ) . T,(@),

a

E

Im

Is#, 0 E VP.

Evidentlyg is a semimorphism of ( P , 8)-algebras; in particular it satisfies 8s o g = 0. Theorem 11: Let ( B , 8,; T ) be a ( P , 8)-algebra and let y , g be as above. Then the following conditions are equivalent:

111

3. Isomorphism theorems

(1) g#: I m 1; @ V P + H ( B ) is an isomorphism. (2) (g @ L ) # : H(1m 1; 0V P @ AP) + H(B @ AP) is an isomorphism. (3) Ze#: H ( B )+ H ( B 0AP) is surjective. (4) H + ( H ( B )@ AP, K*) = 0. (5) H , ( H ( B ) @ AP, K*) = 0. Proof: We show that (1) o (2) o (3), (4)o (5), (1)

- -

(4),(4) (3).

(1) o (2): Apply the corollary to Theorem I, sec. 3.10. (2) o (3): I t follows from the definitions of g and y that the diagram H(Im ZB#

0V P 0AP)

H ( B 0AP) commutes, where i and j are the obvious inclusion maps. Moreover, it follows from the example in sec. 2.2 and from sec. 2.6 that i* is an isomorphism. Thus (g @ t ) # is an isomorphism if and only if j is; i.e., if and only if j is surjective. But this is equivalent to 1: being surjective. (4)o ( 5 ) : This is proved in Theorem 11, sec. 2.9. (1) (4): This follows from Theorem 11, sec. 2.9 ((1) (4)). (4)=> (3): Recall from sec. 2.2 the linear map

-

-

and denote its image by F. If (4)holds, Theorem 11, sec. 2.9, yields a P-linear isomorphism y:F@VP%H(B)

which satisfies p(1) = 1. Next, choose a linear map 7 : F zero, such that ~ ( 1 = ) 1, and

+ ker

q*(a) = y ( a @ l),

dB , homogeneous of degree a

E

F.

111. Koszul Complexes of P-Differential Algebras

112

Then a semimorphism y : ( F @ VP, 0;

U)+

( B , Sg; T )

is defined by ~ ( C Z@

Y)

= T / ( u )*

E

T,(Y),

F, Y E VP.

The induced map y # : F @ V P -+ H ( B ) is given by

-

a

y#(a @ Y ) = q*(c) t$(!P)= cp(a @ Y ) ,

E

F, Y E V P ;

i.e., y # = cp. I n particular, y* is an isomorphism. Hence, by the corollary of Theorem I, sec. 3.10, (y @ L ) * is an isomorphism. Since the diagram

^I

H ( F @ V P @ AP)

F

(V@LP

N

H ( B @ AP) 12;

7P

+

H(B)

commutes, it follows that Is# is surjective.

Q.E.D. 3.12."

In this section we generalize Theorem 111, sec. 2.9, to

Theorem 111: Let (B, dB; T) be a (P, 8)-algebra. Then the following conditions are equivalent :

(1) H ( B ) is evenly graded, and the conditions of Theorem 11, sec. 3.1 1 hold. (2) H ( B @ A P ) is evenly graded. Proof: ( 1 ) * (2): If (1) holds, then Z jis surjective (cf. Theorem 11, (3), sec. 3.11). Since by hypothesis H ( B ) is evenly graded, so is H ( B @ AP). (2) (1): Recall from Proposition I, sec. 3.8, that

H ( B @ V P @ AP) On the other hand, the &term

H(B).

of the VP-spectral sequence for the

113

3. Isomorphism theorems

tensor difference ( B B V P , 8,;

t

0u)

is given by

E, EH ( B @ A P ) @ V P

(cf. sec. 3.9). Thus El is evenly graded. I t follows that El = Em (cf. Proposition IV, sec. 1.10 and Proposition V, sec. 1.12) and so we have isomorphisms of graded vector spaces

H(B)z H ( B @ V P @ A P ) E E E , r E , . In particular, H ( B ) is evenly graded. It remains to show that I;: H ( B ) + H ( B 0AP) is surjective. Assume first that dim P = 1. Then we have the exact sequence

0

-

coker 8

if

H(B @ AP)

- i(x*)#

ker

a

0

of sec. 3.6. All three spaces in this sequence are evenly graded, but i(x")# is homogeneous of odd degree. It follows that i(x")* = 0 and so, by exactness, ; Z is surjective. In the general case we argue by induction on dim P. Write P = P, @ P , where P , and P, are graded subspaces. Then t restricts to a linear map t,:P , + B and thus yields a ( P , , 8)-algebra ( B , 8,; tl) whose Koszul complex will be written ( B 0AP, , V,). Next define a map t2:P, + B 0AP, by setting t2(.)

= .(X)

01,

x E P,.

This yields a ( P z , 6)-algebra ( B 0AP,, F; t,). Its Koszul complex is given by ( B 0AP, 0AP,, V,) L= ( B 0AP, V'). Finally, since H((B 0AP,) 0AP,, V,) = H ( B 0AP, V '), we have that H ( ( B O A P , ) @AP,, 6 ) is evenly graded. It follows, as above, that H ( B @ AP,, V,) is evenly graded. Thus, by induction, the maps

H ( B ) --+ H ( B @ A P l )

and

H ( B 0APl)

---f

H ( B @ AP,

0AP,)

are surjective. Hence so is their composite, I;. This closes the induction. Q.E.D.

$4. Structure theorems In this article all (PIS)-algebras will be assumed to be c-connected. The purpose of this article is to generalize the theorems of article 4, Chapter 11, to (PI 6)-algebras. 3.13. The Samelson theorem. Let ( B , 8,; T ) be a (P, 8)-algebra and consider the projection B + Bo with kernel B+. I t gives rise to a projection

@B:B@AP+B'@AP. Lemma I:

eB satisfies the conditions el,o 0' = 0 and QB(kerV')

c 1 @ AP.

Proof: The first relation is obvious. T o prove the second, fix

x E ker 0 ' and write z = pBx

+ z1,

x1 E B' @ AP.

Then while 8 , ~ ~E zB' 0AP. Since 6&# f qpBz 6seBz = 0. Thus p'z E (ker

+ vBx1 = VBx = 0,

6B)O

@ AP

these relations imply that

= Ho(B)@ AP =

1 @ AP. Q.E.D.

In view of Lemma I there is a unique homomorphism

&: H ( B @ AP) + AP, which makes the diagram ker 0,

H ( B @ AP)

-

114

et

AP

4. Structure theorems

115

commute. & is called the Samelson projection for ( B , aB; T ) and the space P, = P n (Im eB) is called the Samelson subspace. A graded complement of p , in P will be called a Samelson complement for (B, 6,; T ) . Note that these definitions reduce to the definitions of sec. 2.13 if dB = 0. Exactly the same argument as given in Theorem IV, sec. 2.13, establishes Theorem IV: Let ( B , B B ;

T)

Im

be a (P, d)-algebra. Then

& = AP,.

Finally, observe that if p: (B, B B ; then

T) +

( P , 68; i) is a semimorphism,

eB (e, 04*= e:: O

In particular,

P,

c

PE.

3.14. The cohomology sequence. T h e cohomology sequence of a (c-connected) (P, d)-aZgebra ( B , S B ; T ) is the sequence

VP

5 H ( B ) 5 H ( B @ AP) 5 AP.

'A semimorphism q ~ :(B, aB; T tive diagram

H(B)

~+ )

(B,68;

i) determines the commuta-

It

H ( B @ AP)

Next, suppose P, is a second graded space satisfying the same conditions as P (cf. the beginning of this chapter) and let GI:P, -+ P be a linear map homogeneous of degree zero. Define a map t,:PI B by

-

111. Koszul Complexes of P-Differential Algebras

116

Then ( B , 6,; diagram VP,

VP

tl)is

a (PI, 6)-algebra, and we have the commutative

- - (71):

4

H(B)

H ( B @ AP,)

H ( B @ AP)

H(B)

-

AP, (3.9)

AP.

Note as well that for x* E P",

i(x*) 0 ( L @ a,)*

= (1

@ a,)#

0

i(a*(x*)).

Proposition 111: Let ( B , 6,; t) be a c-connected (P, 6)-algebra. Then (1) Zg o (t$)+= 0 and so ker 1; contains the ideal generated by Im( t$)+. (2) e; o (@)+= 0 and so ker e$ contains the ideal generated by Im( I;)+. Proof: (1) It is sufficient to show that Z$ Z,t(x)

= .(x) @

1

= V,(1

@ x),

0

z# = 0. But

xE

(2) This is obvious.

P. Q.E.D.

3.15. The reduction theorem. I n this section we generalize the results of sec. 2.15. Let pB be the Samelson subspace of a c-connected (P,6)-aIgebra (B, d B ; t). Choose a Samelson complement P. Then multiplication defines an isomorphism

g: AP @ AP -5AP of graded algebras. Next, let ( B , 6,; f) be the (P, 6)-algebra determined by restricting t to P, and denote it8 Koszul complex by ( B @ AP, (Then vB- is the restriction of V, to B @AP.) In particular, ( B @AP @Ap,0, @ 1 ) is a graded differential algebra.

.)'v

Theorem V (reduction theorem): Suppose that ( B , B B ; t) is a cconnected (P,6)-algebra. Let p be the Samelson subspace and let P be

4. Structure theorems

117

a Samelson complement. Then there is an isomorphism

f: ( B @ AB @ AP, VB @ L)

Z ( B @ AP, VB),

of graded differential algebras, such that the diagram

commutes ( A , and 1, are the obvious inclusions). Proof: Choose a linear map @:

P + ker VB,

homogeneous of degree zero, and such that ( @ B 0 @)(.)

=

1 @ x,

x E P.

Then the proof of Theorem V, sec. 2.15, with trivial modifications (using this map, 8) establishes this theorem as well. Q.E.D. Corollary I: f induces an isomorphism of graded algebras

f #: H ( B @ Afr) @ AP -% H ( B @ A P ) which makes the diagram

H ( B @ AP)

commute.

118

111. Koszul Complexes of P-Differential Algebras

Corollary 11: The diagram

f"

\

AP

H(B @ AP) commutes. I n particular, (@j$)+= 0. Proof: Apply the Samelson theorem (sec. 3.13) and Corollary I.

Q.E.D. Corollary 111: f" restricts to an isomorphism

Corollary IV: f # restricts to an isomorphism

3.16. The decomposition theorem. T h e results of this section generalize much of Theorem VIII, sec. 2.19. Theorem VI: Let (B, B B ; T ) be a c-connected (P, 8)-algebra. Then the kernel of e j contains the ideal generated by Im(l:)+. Moreover, if P and P denote respectively the Samelson subspace and a Samelson complement, then the following conditions are equivalent :

(1) T h e kernel of Im(&)+.

& coincides with the ideal generated by

(2) The map &: H ( B ) + H ( B @AP) is surjective. (3) There is an isomorphism of graded algebras

4. Structure theorems

119

making the diagram

H ( B @ AP) commute.

.' (4) There is an isomorphism Im 13 @ VP

H ( B ) of graded

p-

spaces, which makes the diagram

H(B) commute. Remark: For further equivalent conditions see Theorem 11, sec. 3.11, and Theorem 111, sec. 3.12.

-

Proof: (1) (2): In view of Corollaries I1 and 111, sec. 3.15, we may at once reduce to the case i? = 0. In this case we have to prove that l$ is surjective if and only if

-

Im(Zi)+ H ( B @ AP) = H+(B 0AP). This follows by an elementary degree argument (in the same way as Lemma I, sec. 2.8). ( 2 ) => (3): This follows from Corollary 11, sec. 3.15. (3) => (1): This is obvious. ( 2 ) - (4): This is proved in Theorem 11, sec. 3.11.

Q.E.D.

Corollary I: If the conditions of the theorem hold, then ker 1; coincides with the ideal generated by Im(tf)+.

120

111. Koszul Complexes of P-Differential Algebras

Proof: In view of condition (2) of the theorem, we may apply Theorem 11, sec. 3.11, to the (P, 8)-algebra (B, BB; f) where 5 denotes the restriction of t to P. This yields a semimorphism I

g: I m 'l of

(P, 8)-algebras

0VP + B

which induces a commutative diagram Im 1:

H(Im 1;

0V P

'#I

8'

1

' H(B)

,.,

-

ii

H ( B 0AP).

0VP @ AP)

Now recall that if S is a P-space, then the kernel of I,# is the space S o P (cf. sec. 2.2). Thus in the diagram above ker I#

=

(Im @ @ Vp)

0

p.

It follows that ker

1:: = g#(ker I # )

= H(B) 0 P = H(B)

- t"(P).

Finally, Corollary I1 of the reduction theorem, sec. 3.15, shows that ker I:: = ker 9. Thus we may combine the relation above with Proposition 111, sec. 3.14, to obtain ker

13 = H ( B ) . t"(P) c H(B) - t:(V+P)

c ker I;.

Q.E.D. Corollary 11: If the conditions of the theorem hold, then there is a commutative diagram

(H(B)/H(B) . Im

0AP

t#)

H ( B 0AP) in which the vertical arrow is an isomorphism of graded algebras.

4. Structure theorems

121

Proof: Note that by Corollary I

Im Zjj

H(B)/ker 1;

= H ( B ) / H ( B ) Im t*. 1

Now apply part (3) of the theorem.

Q.E.D. 3.17. (P,6)-algebras with AP noncohomologous to zero. Let ( B , 6,; T) be a c-connected (P, 6)-algebra. Then AP is said to be noncohomologous to zero in B 0AP (n.c.z.) if the homomorphism

&: H ( B @ AP) -+ AP is surjective. Theorem VII: Let ( B , 6,; T ) be a c-connected (P , 8)-algebra. Then the following conditions are equivalent :

(1)

e3 is

surjective. (2) There is a linear isomorphism of graded vector spaces

H ( B ) @ AP 22 H ( B @ AP, V,) which makes the diagram

H(B)@AP

(3.10)

H ( B @ AP) commute. (3) 1:: is ‘injective. (4) T* = 0. (5) The spectral sequence for ( B , S B ; T ) (cf. sec. 3.4) collapses at the &-term. (6) There is an isomorphism of graded differential algebras

f:( B 0AP, 6,) -3( B 0AP, V,)

I1 I. Koszul Complexes of P-Differential Algebras

122

such that

f(b

f(1 0X)

@ 1) = b @ 1,

-

1 0x

b E B, x

B+ @ 1,

E

E

P,

and i ( x " ) of = f 0 i ( X u ' ) ,

Xu'

E

P".

( 7 ) There is an isomorphism of graded algebras g: H ( B ) @ AP 25. H ( B @ AP, 0,) which makes the diagram (3.10) commute, and satisfies i(x")" og

-

=g

xu' E P".

i(X*)#,

0

Proof: We show that

(1) and

(2) =;> ( 3 )

(6)

=+

(4) =+- (6)

-(5)

-(7)

(1)

(3).

(1) => (2): If (1) holds, then the Samelson subspace is all of P . In this case (2) follows from Corollary I1 of the reduction theorem (cf. sec. 3.15). (2) * (3): This is obvious. (3) * (4): This follows from the relation 1s 0 (t$)+ = 0 (cf. Proposition 111, sec. 3.14). (4) (6): If t # = 0, then there is a linear map a : P B f homogeneous of degree zero, and such that

-

-

t =

Define

8: P + B @ AP

-&

OC!.

by

B(x) = .(X)

01 + 1 @ x,

Then, clearly, 0 ' o ,!I= 0. Extend to a homomorphism morphism of graded algebras

x E P.

PA:AP -+ B

@ AP and define a homo-

f: B @ A P + B @ A P by f ( b @ @)

=

(b @ 1)

*

(PA@),

b E B , @ E AP.

123

4. Structure theorems

Exactly as in the proof of Theorem V, sec. 2.15, it follows that f is an isomorphism. Finally, since 0, 0 /I= 0, we have

0,

o f = f o

6,.

The relations of (6) are trivial consequences of the definition o f f . (6) => (7) * (1): This is obvious.

-

(6) ( 5 ) : The ideals FP(B @ A P ) which determine the spectral sequence are given by

Thus if (6) holds, then f is an isomorphism of filtered graded differential algebras; i.e., f andf-' preserve the filtration. Hence f induces an isomorphism of spectral sequences. But the spectral sequence for ( B 0 AP, 6,) collapses at the second term (cf. sec. 1.8). (5) * (3): Recall from sec. 3.4 that B 01 is the basic subalgebra of B @ AP with respect to the given filtration. Thus, if the spectral sequence collapses at the &-term, Zg is injective (cf. Corollary I11 to Proposition VII, sec. 1.14). Q.E.D. Corollary: If AP is n.c.z. in B @ AP, then H ( B @ AP) is generated by the classes which can be represented by cocycles of the form

z@1

or

w@lfl@x,

Z,WEB,~ E P .

Finally, assume AP is n.c.z. in B @ AP, and let q : AP -+ H ( B @ A P ) be a linear map, homogeneous of degree zero, and such that @ ;

0

q

= 1.

Define

h : H ( B ) @ AP

+H ( B

0A P )

by setting

h(. @ @)

= @(a)*

q(@),

CY E

H ( B ) , @ E AP.

Proposition IV: T h e map h is an isomorphism of graded spaces. Moreover, if is a homomorphism, then h is an isomorphism of graded algebras.

124

111. Koszul Complexes of FDifferential Algebras

Proof: Let g: H ( B ) @hp% H ( B @ A P ) be the isomorphism of part 7 of Theorem VII. Then g-' 0 h is an endomorphism of H ( B ) @ A P satisfying (g-1 o h - L ) : Hp(B) @ AP

Hi(B) @ AP.

+ i,p

Filter H ( B ) by the subspaces CjrpHj(B) @ A P ; observe that g-I 0 h induces the identity' in the associated graded space; conclude via Proposition VII, sec. 1.14, that g-' o h is an isomorphism. Since g is an isomorphism, so is h. Q.E.D. 3.18. Poincark series. I n this section the PoincarC series of a graded

vector space X will be written f x . The relation f x 5 fy will mean that dim Xp 5 dim Y p ,

each

p.

Proposition V: Let ( B , 8,; r ) be a c-connected (P, 8)-algebra. Then H ( B ) is of finite type if and only if H ( B @ AP) is of finite type. In this

case the PoincarC series satisfy the relations

(3.11) and

(3.12) Equality holds in (3.11) if and only if A P is n.c.2. in B @ A P

Proof: Suppose H ( B ) is of finite type. T h e &-term of the spectral sequence for B @ A P is given by

E,

H ( B ) @ AP

(cf. sec. 3.4). Hence E, is of finite type and

Now Proposition VIII, sec. 1.15, implies that: (1) H ( B @ A P ) has finite type, (2) relation (3.11) holds, and (3) equality holds in (3.11) if and only if the spectral sequence collapses at the &-term. I n view of Theorem VII, sec. 3.17, this last is equivalent to A P being n.c.2. in

B @ AP.

125

4. Structure theorems

Conversely, assume that H ( B @ AP) is of finite type. Recall from sec. 3.9 that the E,-term of the VP-spectral sequence of B @ V P @ AP is given by E, V P @ H ( B @ AP), It follows that E, has finite type and that

But according to Proposition I, sec. 3.8, H ( B ) Hence, H ( B ) has finite type and (3.12) holds.

H ( B @ V P @ AP). Q.E.D.

Corollary: If H ( B ) has finite dimension, then so does H ( B @ A P ) . In this case the Euler-PoincarC characteristic of H ( B @ AP) is zero. Moreover, dim H ( B 0AP) 5 dim H ( B ) dim AP

and equality holds if and only if AP is n.c.z. in B @ AP. Proof: Apply Proposition IX, sec. 1.16, to the spectral sequence of sec. 3.4 to obtain X H ( B B A p ) = 0. The rest follows at once from the proposition. Q.E.D.

$5. Cohomology diagram of a tensor difference In this article ( B , 6,; T ) and ( S , as; a) denote c-connected ( P , 8)algebras. ( B @ S , Gees; T @ o ) denotes their tensor difference (cf. sec. 3.7) and ( B @ S @ AP, VBBs) denotes the corresponding Koszul complex. 3.19. The homomorphisms p j and p i . cohomology sequences

Recall from sec. 3.14 the

V P --+H ( B ) -% H(B @ AP) r:

5AP

and

VP

5 H ( S ) 5 H ( S @ AP) 5 AP.

-

In this section we construct homomorphisms

p#,:H ( B @ S

@ AP)

H ( S @ AP)

and

pg : H ( B @ S @ AP) Extend the projection B PB:

Then, clearly, P B homomorphism 7B:

0

+

--+

H (B @ AP).

Bo to the projection

B @ S @ A P + Bo @ S @ AP

G'BBos = - ( L

@ V8)0 p B , and so we have an induced

H(B @ S @ A P ) -+ Bo @ H(S @ AP).

The image of this homomorphism is contained in 1 @ H ( S 0AP). In fact, let a E H ( B @ S @ AP). Lemma 11, below, shows that a can be represented by a cocycle d, such that @E

(1 @S@AP)@(B+@S@AP).

Thus p B ( @ )E 1 @ S @ AP, and represents qB(a). It follows that qB(a)E 1 @ H ( S @ AP). 126

5. Cohomology diagram of a tensor difference

127

Since Im rB c 1 @ H ( S @ AP), there is a unique homomorphism -+ H ( S @ AP) such that

p g : H ( B @ S @ AP) VB((L)

=

1 @pg((L),

W

Similarly, extend the projection S

ps: B

@ S @ AP

E

H ( B @ s@ A p ) .

+ Soto

+B

a homomorphism

@So@AP.

Exactly as above we obtain a homomorphism

pg : H(B @ S 0AP)

---f

H ( B @ AP).

Thus each a E H ( B @ S @ A P ) can be represented by a cocycle @ in ( B @ 1 @ AP) 0( B @ Sf @ AP), and ps(@) represents pg(a). Lemma 11: Let Q E B @ S @ AP be any cocycle. Then there is an element Y E B @ S @ AP such that

9 - G'BaSY

E

Proof: Write 9 = Q0

(1 @ S @ A P ) 0(B+@ S @ AP).

+ 9, +

Q2,

where

Q 0 cB O O S O A P , Q l e B l @ S @ A P , and 9 , 1~B j @ S @ A P . 32 2

Denote coB @ Vs by

VsQ0

qq. Then =0

an argument on degrees shows that

and

dBQ0

-

VsQl

= 0.

(3.13)

Now choose a subspace C c Bo so that Bo = 1 @ C , and let n:Bo -+ be the corresponding projection. Since H ( B ) is connected, there is a linear map h: B1 Bo such that

r

-+

n-L

h 0 6s.

(3.14)

Since n @ t @ L and h @ L @ t commute up to sign with 0, in B @ S @ AP, relations (3.13) and (3.14) yield Qo-

( V s o ( h @ t @ t ) ) Q I E1 @ S O A P .

Hence Q - (Vs@s0 ( h @ L @ L ) ) QE~(1 @ S

@ A P ) @ (B+@ S @AP). Q.E.D.

111. Koszul Complexes of P-Differential Algebras

128

3.20. The cohomology diagram. Consider the inclusion maps

m B :B

--f

B @ S @ AP

and

ms: S

-+

B @ S @ AP

given by mB(b)= b @ 1 @ 1 and m s ( z ) = 1 @ x @ 1. They are homomorphisms of graded differential algebras and hence induce homomorphisms

mj: H ( B ) + H ( B @ S @ AP)

and

mg: H ( S ) -+ H(B @ S @ A P )

of graded algebras. Obviously, p j o (m$)+ = 0 and pg 0 (mg)+ = 0. Combining these homomorphisms with the cohomology sequences for ( B , dB; r ) and ( S , as; 0) we obtain the diagram

H ( S @ AP)

+

e.3

AP.

It is called the cohomology diagram for the tensor dzzerence. Proposition VI: T h e cohomology diagram commutes. Proof: It is immediate from the definitions that

In view of these relations the two triangles and the lower square of the diagram commute. It remains to show that m j 0 T$ = m# 0 o$. Since these maps are homomorphisms it is sufficient to verify that they agree in P. But for x E P we have (mBr, - mso,)(x) = ~ ( x @ ) 1@ 1 = V&S(l

-

1 0~ ( x 0 ) 1

@ 1 0x).

Q.E.D.

5. Cohomology diagram of a tensor difference

129

Example: Suppose that ( S , 8,; G) is given by S = VP, 6, = 0, and P, (cf. sec. 3.8). Then the cohomology diagram yields the commutative diagram ~ ( x= ) x, x E

H(B)

H ( B @ V P @ AP) where m$ and cp* are the inverse isomorphisms of Propositions I and 11, sec. 3.8. 3.21. Tensor difference with S 0AP noncohomologous to zero. Suppose (B, 6,; r ) and ( S , ds: G) are c-connected (P, d)-algebras. Then S @ AP is called noncohomologous to zero in B @ S @ AP if the projection p$: H ( B @ S @ AP) ---+ H ( S @ AP)

(cf. sec. 3.19) is surjective. Remark: This is not the same as saying that AP is n.c.z. in B @ S @ AP (cf. sec. 3.17). Example: If the map l#: H ( S ) H ( S @ A P ) is surjective, then S @ AP is n.c.z. in B @ S 0AP. I n fact, since I# = pg 0 mg,pg is surjective. --f

Theorem W I : Suppose (B, 6,; 7 ) and ( S , ds; CT) are c-connected (P, 6)-algebras. Then the following conditions are equivalent:

(1) p$ is surjective.

(2) There is a linear isomorphism of graded vector spaces

f:H ( B ) @ H ( S @ AP) -5H ( B @ S @ A P )

130

111. Koszul Complexes of P-Differential Algebras

which satisfiesf(a @ j 3 )

=

mjj(a) f(1 @ j3) and makes the diagram

H ( B) @ H ( S @ AP)

H(B @ S @ A P ) commute. (3) The B-spectral sequence for the tensor difference collapses at the &term.

-

Proof: We show that

(1)

(1) Since zero,

(2) * ( I ) ,

(21,

(2): Let

TZ:

(1)

Z(S @ AP, &)

-+

-

(31,

(3)

=+-

(1).

H ( S @ AP) be the projection.

pjj is surjective, there is a linear map, homogeneous of degree 8: H ( S @ AP)

--j

ker OB,, ,

such that (i) p , o 8: H(S @ AP) -+ 1 @ Z ( S 0AP). (ii) n 0 p B 0 6 = 1. (iii) e(1) = 1. Define a linear map q: B @ H ( S @ A P ) -+ B @ S @ A P

by setting y(b @ j3) = rn,(b) 0 0 = 0, we have

O(b), b E B, 0 E H ( S @ AP). Since

VB,,

v

@ L,

= VB@S

Thus p induces a linear map

q#:H(B)OH(S@AP)~H(BOS@AP). I t follows from (ii), (iii), and the definition that p#(a

0B ) = mB(.)

*

v#(1 0

j 3 ) 9

5. Cohomology diagram of a tensor difference

131

and that the diagram H(B) @ H ( S @ A P )

\

H(B@ S @AP) commutes. T o show that p # is an isomorphism, filter B @ H ( S @ A P ) by the subspaces

Then QI is filtration preserving with respect to this filtration and the filtration of B @ S @ AP giving rise to the B-spectral sequence (cf. sec. 3.9). Thus p induces homomorphisms p i : (pi, d i ) 4 ( E ,, di)of the spectral sequences. I n particular, po is the homomorphism po:B @ H ( S @ A P ) + B @ S @ A P

given by Po(b

0P ) = b @ M ( B ) ,

(To see this observe that p , Z ( S @ AP) and that

o

b

E

B,

P E H ( S 0w.

8 is a linear map from H ( S @ AP) to

e(p) - 1 g p , e ( p )

E

B+ 0 s ~ A P . )

Since cpl = @, it follows from the relation above that v1 = 1. Hence, by the comparison theorem (cf. sec. 1.14), p* is an isomorphism. (2) * (1): This is obvious. (1) 3 (3): Let q~ be the linear map constructed above and consider the induced maps p i :(&, di) -+ ( E i ,d i ) . Since y1 is an isomorphism so is each pL, i 2 1. Now the spectral sequence (pi,dj) for B @ H ( S @ A P ) collapses at the &-term. Hence so does the B-sequence (Ei , dJ. Recall from sec. 3.9 that the &-term of the B-sequence (3) 2 (1): is given by EZp*q H p ( B )@ H q ( S @ AP).

111. Koszul Complexes of P-Differential Algebras

132

Let fi E Hq(S @ AP) be arbitrary, and choose a representing cocycle z E S @ AP. Then, in the notation of sec. 1.10,

1@x E

z!y,

Since by hypothesis E, G Em, it follows that

+ x,, where

Thus we can write 1 @ z = z1

z1E Zzq

( c Z4(B @ S @ AP))

In particular, p B ( z l )= x. Thus if a E H ( B @ S @ AP) is the element represented by zl,then pf:(a)= B. This shows that p j is surjective.

Q.E.D. Corollary I: Assume that S @ AP is n.c.2. in B @ S @ AP and let q: H ( S @ AP) H ( B @ S @ AP) be a linear map, homogeneous of degree zero, and such that pf:0 11 = 1. Then an isomorphism of graded spaces h : H ( B ) @ H ( S @ A P ) -%H ( B 0s @ A P ) --f

is defined by h(a @ j3) = m j ( a ) . .I(,!?).

Proof: This follows in exactly the same way as Proposition IV, sec. 3.17.

Q.E.D. Corollary 11: Assume that Is#: H ( S ) -+ H ( S @ AP) is surjective and let y : H ( S @ AP) H ( S ) be a linear map, homogeneous of degree zero, such that 18 0 y = 1. Then an isomorphism of graded spaces -+

h: H ( B ) @ H ( S 0AP)

H ( B @ S @ AP)

is given by

h(a 0B ) = m l b ) * m#(r(B)),

a E WB),

B E H ( S 0w.

5. Cohomology diagram of a tensor difference

Proof:

133

Observe that

and apply Corollary I. Q.E.D. Finally, consider the homomorphism y : H ( B ) @ H ( S ) + H ( B @ S @ AP)

given by y ( a @ B ) = mg(a) m$(p). It follows from the commutativity of the cohomology diagram that a

y(."(x) @ 1 - 1 @ u " ( x ) ) = 0,

xE

P.

Let I denote the ideal in H ( B ) 0H ( S ) generated by elements of the form ~ " ( x )@ 1 - 1 @ cr#(x), x E P. Then y factors to yield an algebra homomorphism

Corollary III: Assume that Is#: H ( S ) H ( S @ AZ') Then p is an isomorphism of graded algebras. ---f

Proof: Consider the (P, 6)-algebra ( B @ S, SBes;

t

is surjective.

00).Evidently,

Y = l&S.

By Corollary 11, is surjective. Now Corollary I to Theorem VI, sec. 3.16, applies, and shows that

-

ker ZgOs= [ H ( B )@ H ( S ) ] (x 0.)*(P) = I. Thus

is injective. Q.E.D.

Corollary IV: Assume that H ( B ) and H ( S @ A P ) are of finite type. Then so is H ( B @ S @ AP), and the PoincarC series satisfy fH(LlQS0AP)

5fHcB)

*

fH(S@,AP)

Equality holds if and only if S @ AP is n.c.2. in B @ S @ AP.

134

111. Koszul Complexes of P-Differential Algebras

Corollary V: If H ( B ) and H ( S @ AP) are finite dimensional, then so is H ( B @ S @ A P ) , and

dim H ( B @ S @ A P ) 5 dim H ( B ) . dim H ( S @ AP). Equality holds if and only if S @ AP is n.c.2. in B @ S @ AP. Corollary VI: If H ( B ) and H ( S @ A P ) are finite dimensional, then the Euler-PoincarC characteristic of H ( B @ S @ A P ) is given by XH(BOSOAP)

= xH(B ) X H ( S @ A P )*

$6. Tensor diEerence with a symmetric P-algebra In this article ( B , 6,; T ) denotes a c-connected (P, 8)-algebra. ( V Q ; u) is a symmetric P-algebra and ( B @ V Q @ AP, V,7BovQ) denotes the Koszul complex of their tensor difference. We shall carry over all the notation of article 6, Chapter 11, unchanged. In particular, PI c P denotes the essential subspace for ( V Q ; u) and we write (cf. Lemma VIII, sec. 2.23)

P

= P, @

P,

V Q = VP,@VQ1.

and

P

will denote the Samelson space for (VQ; 0). Recall that Theorem VIII, sec. 3.21, gives a necessary and sufficient condition for a linear isomorphism

H ( B ) @ H(VQ @ AP)

H(B

V Q @ AP)

which makes the appropriate diagram commute. A main result of this article (Theorem IX) gives necessary and sufficient conditions for this to be an algebra isomorphism. Note that this contrasts with the situation for ( P , 8)-algebras where the existence of a linear isomorphism H ( B ) 0A P H ( B @ AP) implies the existence of an algebra isomorphism (cf. sec. 3.17). 3.22. Cohomology of the tensor difference. Recall from Theorem X, sec. 2.23, that H(VQ @ AP) H(VQl @ APl). I n this section that theorem will be generalized to yield an isomorphism

H ( B @ V Q @ A P ) E H ( B @ V Q 1@ AP, , G). Unfortunately, 0, is more complicated than the differential operator corresponding to the tensor difference of (B, 8,; z Ir,) and (VQ1;ol). First define a homomorphism p: B @ V Q

+

135

B @VQl

136

111. Koszul Complexes .of P-Differential Algebras

It satisfies ip 0 dB = dB 0 9. Next, define a linear map t,:

P, ---* B @VQ1

by tl(x) = t(x) @ 1 - v(l @ a(x)), x E P,. Then ( B @ VQ,, dB; t,) is a (P,, d)-algebra. Denote its Koszul complex by ( B @ VQ, @ AP, , V,). The inclusion map m,:B B 0VQ, @ AP, is a homomorphism of graded differential algebras. Moreover, since H ( B ) is connected, the projection p , : B @ VQ, 0AP, -+ Bo 0VQ, @ AP, induces a homomorphism --f

p?: H ( B 0VQi 0APi)

+

H(VQi 0A P i GI) 9

in exactly the way described in sec. 3.19. Finally, let 8: P + P, be the projection with kernel Pz, and extend it to homomorphisms

BA:AP+APl

and

/lV:VP+VPl,.

Proposition VII: With the hypotheses and notation above :

(1) i p @ B A : ( B O V Q O A P , VB&"@) ( ~ O V Q I O A P , 0,) , is a homomorphism of graded differential algebras. (2) (v @FA)#is an isomorphism. (3) The diagram +

H ( B @ VQ 0AP)

-

H(VQ @ AP)

commutes, where y : VQ + VQ, denotes the projection defined in sec. 2.23.

6. Tensor difference with a symmetric P-algebra

Proof: The proof 2.23, that are left to

137

( 1 ) and (3) are straightforward consequences of the definitions. of (2) is essentially the same as the proof in Theorem X, sec. (y 0PA)* is an isomorphism. T h e necessary modifications the reader. Q.E.D.

3.23. The algebra isomorphism theorem. Theorem IX: Let ( B , S B ; T) be a c-connected (P, 8)-algebra, and let (VQ; u) be a symmetric P-algebra. Then the following conditions are equivalent:

( 1 ) There is a homomorphism y: VQ graded algebras, which makes the diagram

VP

7:

---f

H ( B ) @ VQ/VQ 0 P of

’H ( B )

commute ( I is the projection). (2) There is an isomorphism

f:( B @ VQ 0AP, V B B V Q )

N

--+

( B 0V Q 0AP, S B - V u )

of graded differential algebras, which makes the diagram

B@VQ@AP

B@VQ@AP commute. (3) There is an isomorphism

g: H ( B 0VQ 0AP, VBBOVQ) 5 H ( B ) @ H(VQ @ AP)

111. Koszul Complexes of P-Differential Algebras

138

of graded algebras, which makes the diagram

H ( B ) 0H(VQ 0AP>

t

\

commute. Remark: Consider the tensor difference as a generalization of (P, 6)algebras with V Q 0AP replacing AP. Then Theorem VIII, sec. 3.21, and Theorem IX together generalize Theorem VII, sec. 3.17. I n fact, if one sets Q = S = 0 in Theorems VIII and IX, one finds that the conditions Theorem VIII: (l), (2), (3) and Theorem IX: (l), (2), (3) reduce respectively to the conditions Theorem VII: (l), (2), (5), (4), (61, (7). However, in general the conditions in Theorem VIII are not equivalent to those in Theorem I X (cf. the corollary to Proposition VIII, sec. 3.25, and sec. 12.31).

3.24. Proof of Theorem IX: (1) * (2): tative diagram of (l), we may write y ( @ )=

where

+: Q

--+

P(@)

In view of the commu-

+ 1 0I(@),

@E

Q,

H + ( B )@ V Q / V Q 0 P. Choose a linear map

q: Q

--+

Z + ( B )@ V Q ,

homogeneous of degree zero, so that n 0 + H ( B ) @ VQ/VQ 0 P is the projection.) Define 7 : Q -+ Z ( B ) @ V Q by

v(@)

= ij(@)

+ 10

@,

+ = +. (Here z: Z ( B ) 0V Q @E

Q,

and extend 77 to a homomorphism 7 " : V Q + Z ( B ) @ V Q . Then n

o

=y

and

vV(@)- 1 0@ E Z + ( B )@ V Q ,

@ E VQ.

(3.15)

6. Tensor difference with a symmetric P-algebra

139

On the other hand, the commutative diagram of (1) shows that

01 = yo(.),

."(A?)

Next observe that since

T(X)

x E P.

and qVo(x)have even degree we can write

Thus we obtain from formula (3.15) that

~(x)01 - (rvg(~) - 1 0a (%))

E

J,(B+) 0VQ

+ Z+(B)0(VQ

0

P),

P.

X E

I t follows that there are linear maps

O,:P-B+@VQ

and

d,:P-Z+(B)@VQ@P,

homogeneous of degree zero, such that

Extend f3 to a homomorphism 8,: AP --+ B define a homomorphism

f:B B V Q by f(b 0

0@)

=

0V Q 0AP.

Finally,

0AP + B @ V Q @ AP

(b 01 01) * (7°F 01) - (O,@), bEB, Y E V Q , OEAP.

Since (cf. formula (3.15)) qv(Y)01 - 1 B Y 0 1 E Bf @VQ B A P ,

Y EVQ,

111. Koszul Complexes of P-Differential Algebras

140

and (by definition)

it follows that

(f-1):

C Bp@VQ@AP+ C P

aJ

... .

Bfi@VQ@AP, p=O,1,

/I 2 P + l

(3.17)

This implies (as in the proof of Theorem V, sec. 2.15) that f is an isomorphism. The commutativity of the diagram of ( 2 ) is an immediate consequence of formula (3.17) and the definition off. Finally, a simple computation, using formula (3.16) and the relation SB 0 qv = 0 shows that

f

VB@VQ = ( d B

- V) O f .

( 2 )=-( 3 ) : This is obvious. ( 3 ) =- ( 1 ) : Recall from sec. 2.2 that H,(VQ @ A P ) = VQ/VQ 0 P. Thus we have a commutative diagram of algebra homomorphisms

H ( B ) @ H(VQ @ AP)

H(VQ @ AP)

"@

H ( B ) @ VQlVQ 0 P

VQlVQoP

where e denotes the projection with kernel H.k(VQ @ AP). Now set Y = (lOe)ogOm$Q.

Then, combining the cohomology diagram for the tensor difference (sec. 3.20) with the commutative diagram of ( 3 ) and the commutative diagram above, we obtain the commutative diagram of ( 1 ) . Q.E.D. Corollary: If z # = 0, then the conditions of Theorem IX hold.

6. Tensor difference with a symmetric P-algebra

141

Proof: Set y ( Y ) = 1 0I(!#'), Y E VQ, and observe that the diagram of (1) commutes. Q.E.D.

+.

In this section we study the com3.25. The homomorphism mutative diagram in Theorem IX, (1). Proposition VIII: Let (VQ; G) be an essential symmetric P-algebra such that VQ/VQ 0 P has finite dimension. Let A be a connected graded anticommutative algebra. Suppose

VP

9

+A

li

VQ A A @VQ/VQ 0 P

is a commutative diagram of homomorphisms of graded algebras, where i and e are the obvious inclusion and projection. Then, if r has characteristic zero, (1) I m y+ c A @ (VQIVQ 0 P)+ and (2) p+ = 0.

-

Proof: In view of the commutative diagram, (2) is a direct conseA be the (unique) homoquence of (1). T o establish (l), let q : VQ morphism satisfying y ( Y ) - q(Y) @ 1 E A @ (VQlVQ

0

P)+,

Y E VQ.

Then ( 1 ) is equivalent to the relation

F(V+Q)

= 0.

(3.18)

T o prove formula (3.18) we show by induction on k that

q(QP)

= 0,

1 < p 5 k.

(3.18)k

Formula (3.18), is trivially correct. Now assume that formula (3.18)k-1 holds.

142

111. Koszul Complexes of P-Differential Algebras

Let I be the ideal in VQ generated by .(P) n:VQ/VQ 0 P

+

+ Cj
VQ/I

be the corresponding projection. Further, set R = A/&s Aj, and let A + R be the projection. Set

jcR:

YIC = (nf<

@ n) y ,

z=noi.

F,

i ? ~= Z n

PR =ZR

Then the diagram (of algebra homomorphisms)

1

1

VQLR~VQII

VQlI commutes. Since (VQ/I)P = 0, 1 5 p < k, it follows that Y R ( ~=) PR(Y)@ 1

+ 1 @ Z(Y),

YE

Qk-

Fix Y E Qk and let n (n 2 1) denote the least integer such that [@')lo = 0. (Recall that VQ/VQ o P has finite dimension.) Since Rp = 0, p > k, [ijR(!?')I2 = 0. Thus we obtain yR(Yn)

= nFR(yI)

+

@ [z(Y)]"~'

= ? q R ( Y l )@

8 [z(Y)l"

[Z(Y)]"-l.

(3.19)

On the other hand, since

Z(!P)

= [Z(Y)]" = 0,

it follows that Y nE I . Hence, Lemma 111, below, implies that YR(Y") = 0.

(3.20)

Combining formulae (3.19) and (3.20) yields ijR(Y) = 0 (since I' has characteristic zero).

6. Tensor difference with a symmetric P-algebra

143

Finally, observe that ~ c R :Ak Rk. Since 7~,9(Y) = 0, it follows that y(Y) = 0. Thus formula (3.18), is established, and the induction is closed. Q.E.D. Lemma 111: Assume formula (3.18)k-1 holds. Then (with the notation established in the proof of Proposition VIII)

I Proof: Since R p

= 0,

c

keryR.

p > k, we have

vR(Qk)TR(Qk) = 0

vR(Qp) = 0,

and

*

p > h.

Moreover, formula (3.18)Ii-1 implies that vB(QP) = 0, p < k. These relations show that TR(V+Q) * TR(V+Q) = 0. Now the commutative diagram of the proposition implies that

and so yR(c(P))= q R ( c ( P )@ ) 1. Since (VQ; a) is essential, we have a(P) c V+Q V+Q; hence

-

Y d m ) = vn(V+Q)

*

v)n(V+Q) 01

= 0.

(3.21)

On the other hand, since I 2 Qp, p < k, we have (VQ/I)p = 0, 0 < p < k. I t follows that YR(Q')

c

R p 01,

P < k.

This, together with formula (3.18)n.-1yields

Relations (3.21) and (3.22) show that

But the space on the right generates the ideal I ; thus since keryR is an ideal we have ker y R 2 I. Q.E.D.

144

111. Koszul Complexes of P-Differential Algebras

I n view of Theorem IX, sec. 3.23, and its corollary, we obtain the following Corollary: Let (VQ; CT)be as in the proposition, and let (B, dB; z) be a c-connected (P, 8)-algebra. Assume that I' has characteristic zero. Then there is an algebra isomorphism

g: H ( B @ VQ @ AP) % H ( B ) @ H ( V Q @ AP), making the diagram of Theorem IX,(3) commute, if and only if

t* = 0.

3.26. Theorem X: Let (VQ; c) be a symmetric P-algebra with essential subspace P, such that H ( V Q @ AP) has finite dimension. Let (B, 6,; z) be a c-connected (P, 8)-algebra, and assume that the tensor difference satisfies the conditions of Theorem IX, sec. 3.23. Then P, is contained in the Samelson space for ( B , 8,; z):

P, c Im ef. Lemma IV: With the hypotheses of Theorem X

t * ( P , ) c H + ( B )* t*(P,) (where P = Pl 0Pz). Proof: I n view of Theorem IX, (1) we have the commutative diagram

VP

_1 2 VQ

H ( B ) @ VQlVQ 0 P

(3.23)

Moreover, the projection VQ + VQ, induced by the decomposition VQ = VP, @ VQ, induces an isomorphism

VQlVQ 0 P -5VQ,lVQ,

*

CT,(P,).

6. Tensor difference with a symmetric P-algebra

145

We identify these algebras under this isomorphism. Observe that crl: PI+ VQ1 is the linear map of sec. 2.23. Denote by A the factor algebra

A

=~

-

( ~ ) / ."(PZ) ~ ( ~ )

and let nA:H ( B ) -+ A be the projection. Define

VQi lVQi ai(Pi) commutes. T h e commutativity of the triangle is obvious. T o prove that the square commutes, recall that VQ = VP, 0VQ1 and that the restriction of ov to VP2 is given by @ E VPZ. OV(@) = @ @ 1, Moreover, ~ ( x E) V t Q Thus we can write

- VtQ,

.(x) = a(x) @ 1

+c i

xE

Pi.

0!Pi + 1 @ q(x),

@i

x E PI,

where a(x) E (V+Pz) * (V+P2),

Qi E

V+P,,

and

!Pi E V+Q,.

111. Koszul Complexes of P-Differential Algebras

146

Hence

This shows that the square commutes. According to sec. 2.23, (VQ1, crl) is an essential P-algebra. Thus Proposition VIII, sec. 3.25, implies that yi(V+Qi) c A

0(VQilVQi

*

ci(Pi))+.

It follows that

Since y ( P z ) c t*(Pz)@ 1, this implies that Y(V+Q) c H ( B ) 0(VQlVQ 0 PI+

+ ( f f ( B ) ~ " ( P z0 ) )V Q P Q *

0

P.

(3.24)

v:

Now let VQ -+ H ( B ) be the homomorphism obtained by composing y with the projection H ( B ) 0VQlVQ 0 P + H ( B ) . Then formula (3.24) yields V(V+Q) c H ( B ) * t#(P,), whence

@(V+Q V+Q) c H+(B) - t#(P,).

-

But for x E P , , ~(x) E V+Q V+Q, and so .*(X)

= fj3(o(x))E

H+(B) t * ( P 2 ) .

Q.E.D.

Proof of Theorem X: In view of Lemma IV, there is a linear map 8,: P,

such that O'Ol(x) P'(1

0x + e,(x))

-,z + ( B ) 8P, + B+ 01,

= -t(x)

=0

This shows that P, c Im

01, x E P,. and

Hence

eB(l 0x + O,(x)) = x,

x E PI.

&. Q.E.D.

$7. Equivalent and c-equivalent (P, 6)-algebras 3.27. Equivalent (P,6)-algebras. Two (P, 6)-algebras ( B , 6’; T ) and (8, 6 f i ; f ) are called equivalent if B = B, 6~ = a, and f # = T*. Proposition IX: Let ( B , 6,; T ) and ( B , 6,; a ) be equivalent (P, 6)algebras. Denote their Koszul complexes by ( B @ AP, 0’) and ( B @ AP, 0 ’). Then there is an isomorphism of graded differential algebras

f:( B @ A P , V n ) z ( B @ A P , such that f o i ( x * ) = i ( x * ) of, x*

E

vn)

P*,and the diagram

B @AP

B @AP commutes. In particular, f# is an isomorphism. Proof: Since T # = f # ,there is a linear map a : P of degree zero, such that

B: P + B @ AP I?(.)

=

and define f:B @ AP

f(b @ @)

1

by

0x + .(X)

.+B

=

B+,homogeneous

f = 6,oa.

z-

Define

--f

@ 1,

x E P,

@ AP by

(b @ 1 )

*

PA(@), 147

b E B, @ E AP.

111. Koszul Complexes of P-Differential Algebras

148

It follows exactly as in the proof of Theorem V, sec. 2.15, that f is an isomorphism. The remaining properties are straightforward consequences of the definition off. Q.E.D. Remark: Theorem VII, sec. 3.17 ((4)3 (6)) is a special case of the proposition above. Proposition X: Let (B, S B ; t) and (B,Sg; 5 ) be ( P , S)-algebras. Assume that v: B B is a homomorphism of graded differential algebras such that v* 0 r* = i*. Then there is a homomorphism of graded differential algebras ---f

y : ( B 0AP, V')

such that y

0

+

(B0AP, V')

i(x") = i(x") o y , x" E P", and the diagram

B-

lB

B

0A P BQ0 L AP

commutes. Proof: Consider the map 4:P + B given by 4 = 9 0 t. Then q ~ :( B , S B ; r ) -+ (I?, Sg; 4) is a homomorphism of (P,S)-algebras; hence

0&:

vB) (B~

(B~ A P ,

+

A PdB,

+6)

is a homomorphism of the Koszul complexes. 68; f); thus ProposiOn the other hand (B,8'; f) is equivalent to (8, tion IX yields an isomorphism

f: (

B ~ A P , s ~ + + ) ~ ( B ~ A P , ~ ~ )

of Koszul complexes. Now set y = f 0 (901 ) .

Q.E.D.

3.28. c-equivalent (P,Shalgebras. A ( P , 8)-algebra (B, 6; t)will be called cohomologically related (c-related) to a ( P , 8)-algebra (B,8; f) if there is a homomorphism of (P,S)-algebras 91: ( B , 6; t) (B,8; 5 )

-

7. Equivalent and c-equivalent (P,@-algebras

149

such that p#: H ( B ) + H ( 8 ) is an isomorphism. I n this case we write

(8,8; f)

(B, 6; t)

and call q~ a c-relation. (Note that then ( B , 6 ) (8, ~8); cf. sec. 0.10). Two (P, 6)-algebras ( B , 6; t) and (8, 8; f) will be called cohomologically equivalent (c-equivalent) if there is a sequence of (P, 6)-algebras, (B,, 6,; T ~ )(i = 1, . . . , n) such that

8, 2 ) . (1) ( B , , 6,; 2,) = ( R , 6; T ) and (B,, 6,; t,) = (8, ti+l), (2) For each i (1 5 i 5 n - l), either ( B ,, 6,; ti)-p (Bi+,,c?,+~; t i + ,7 ) ( B , , 6,; T,). This is an equivalence relation; it or (Bi+l, is denoted by ( B , 6; t) 7 (8, 8; 5). A specific choice of the (Bi, 6,; t i ) ,together with a specific choice of the c-relations between them, is called a c-equivalence between ( B , 6; T ) and (8, 8; f). Let {(Bi, 6,; ti), qi} be a fixed c-equivalence. Then { ( B , , dL), v,} is a c-equivalence between the graded differential algebras ( B , 6) and (8,8) (cf. sec. 0.10). On the other hand, Theorem I, sec. 3.10, implies that ( ( B ,@ A P , VBB1), pl, @ L } is a c-equivalence between the Koszul complexes ( B @ A P , 0') and (8@ A P , 0 ~ ) . T h e resulting isomorphisms

H(B) and

= H( 8)

H ( B @ A P ) E H ( 8 0AP),

(3.25) (3.26)

will be called the isomorphisms induced by the c-equivalence { ( B ,, 6,; ti),v,}. It is immediate from the definitions that the isomorphisms (3.25) and (3.26) make the diagram

H(B)

1:

H ( B @ AP)

H ( 8 ) 4H ( 8 0AP) la

150

I1 I . Koszul Complexes of P-Differential Algebras

commute. (The right-hand triangle is to be omitted unless H ( B ) and H ( 8 ) are connected.) Moreover, the c-equivalence between ( B @ AP, VB) and ( 80AP, G'n) determines an isomorphism of their spectral sequences, and of their lower spectral sequences (cf. sec. 3.4 and sec. 3.5). Finally, let (S,dS; c) be any (P, d)-algebra. Then

is a c-equivalence between the Koszul complexes ( B @ S @ AP, G'',,) and (S 0S @ AP, V'80s) for the tensor differences. The induced isomorphism (3.27) H ( B @ S @ A P ) H ( 8 @ S 0AP), together with the isomorphisms (3.25) and (3.26), defines an isomorphism of cohomology diagrams. Moreover, there is an induced isomorphism of B- and S-spectral sequences (cf. sec. 3.9). 3.29. Let ( B , 6 ; T ) and (8,8; 4) be (P, 8)-algebras. Suppose there is a c-equivalence between the differential algebras ( B , d) and (8,8) with induced isomorphism y : H ( B )5 H ( 8 ) .

Proposition XI: Assume y 0 T # = f#. Then there is a c-equivalence of (P, d)-algebras ( B , 8; .) 7 (8,8; f),

such that the induced isomorphism H ( B ) -% H ( 8 ) coincides with y. Proof: It is easy to reduce to the case that there is a homomorphism of graded differential algebras

9:( B , 6)

--+

(8,8)

such that q ~ #= y. Let y : ( B @ A P , VB) + (S @ A P , 0 ) be the homomorphism con' structed from ? in Proposition X, sec. 3.27. Extend y in the obvious way to a homomorphism

B @ V P @ A P + S @ V P @A P ,

7. Equivalent and c-equivalent (P,&)-algebras

151

!Po

such that 7j(l 0 1) = 1 @ Y @I 1. Then the relations y = i(x*) o y, xx E P x , (cf. Proposition X) imply that

3 O VRRoVP = V&VP

O

0

i(x*)

3.

(Here VBovp and V E denote ~ the ~ differential ~ operators in the tensor differences-cf. sec. 3.8.) Now consider the (P, 6)-algebras, ( B @ V P 0AP, VBsvp;C T ) and ( B 0V P 0AP, V B ~6),~ where ~ ; CT(X)=

Then

1 @I x 01

and

6(x) = 1 0x

3 is a homomorphism of (P, 6)-algebras. 3 o mB = mg q ~ ,

01,

x E P.

Moreover

0

as follows from the construction of y . Since (cf. Proposition I, sec. 3.8) mg and hi% are isomorphisms, and since q ~ * is an isomorphism by hypothesis, it follows that 7j* is an isomorphism. Thus

( B 0V P 0AP,VBOVI.;c ) 7(B0v p 0AP,VBBVP;5). Finally, observe that Proposition 11, sec. 3.8, implies that

( B 0V P 0AP, VBovP; ). and

7( B , 6 ; T)

(P @ V P 0AP, V',@Vp; 6) 7(a,8; f).

These three c-relations define a c-equivalence between ( B , 6; T) and (8,8; f). Moreover (cf. Proposition 11, sec. 3.8), the induced isomorphism j3 between H ( B ) and H ( & is given by

B = (mjj-1 o 3'

o

m:

== p' = y.

Q.E.D. Corollary: Equivalent (P , 6)-algebras are c-equivalent.

Example: Suppose ( B , 6; T ) is a c-connected (P, 6)-algebra such that the differential algebra ( B , 6) is c-split. Then we can apply Proposition XI to a c-splitting

(86) 7 ( H ( B ) ,01, (cf. sec. 0.10).

152

111. Koszul Complexes of P-Differential Algebras

This yields a c-equivalence

inducing the identity in H ( B ) . Thus (B, 8 ; T) is c-equivalent to its associated P-algebra ( H ( B ) ;r*) (cf. sec. 3.3). The commutative diagram of sec. 3.28 reads

(The vertical arrow is the isomorphism induced by the c-equivalence.) In particular, it follows that ker

= ker

&.

3.30. Symmetric &algebras. Theorem XI: Let (VQ; 0)be a symmetric P-algebra with Samelson space p. Then the graded differential algebra (VQ 0AP, Vu) is c-split if and only if

dim P = dim Q

+ dim p.

(3.28)

Proof: First assume that (3.28) holds. Let P be a Samelson complement. The reduction theorem (sec. 2.15) shows that (VQ

0AP, 0), 7 (VQ 0Afs, K) 0(Ap, 0).

Define a homomorphism of graded differential algebras y : (VQ

-

0Afs, Vu)

(VQlVQ

0

fs, 0)

by setting

y(!P@l)=@P)

and

y(!P@@)=O,

YEVQ,

@EA\+P.

Since dim P = dim Q, Theorem VIII, sec. 2.19, implies that H+(VQ @ Afs) = 0, and it follows easily that y # is an isomorphism.

7. Equivalent and c-equivalent (P, d)-algebras

153

Thus (VQ 0Ap, Vu)7 (VQlVQ 0 p , 01, and so (VQ @ AP, K) is c-split. Hence so is (VQ @ AP, Vo). Conversely, assume that (VQ B A P , 0,) is c-split. Define an oddly graded space T = Cs Tk by

Tk= Q k f l ,

k

=

1, 3,

,

.. ,

and consider the (T, 8)-algebra (VQ @ AP, 0,; z), where =x

.(X)

x

@ 1,

E

T.

Since the base of this (T, 8)-algebra is c-split, the example of sec. 3.29 (with B = VQ @ A P , S = K) yields the commutative diagram H(H(VQ

H(VQ @ AP)

0AP) 0AT, VZ*)

-

H(VQ @ AP @ A T , V').

%@AP

Next, let 'p: VQ @ AP 0AT -+AP be the obvious projection. It satisfies 'p 0 V ' = 0 and an easy spectral sequence argument shows that it induces an isomorphism

v#: H(VQ @ AP @ A T , VB) 5 AP. Moreover, if evQ: VQ

0AP

---f

AP denotes the projection, we have lVQ@hP

=

eVQ

9

whence v* 0 l$QOhp = e$Q. Combining this with the commutative diagram above yields the commutative diagram H(H(VQ @ AP) 0A T )

AP.

154

111. Koszul Complexes of P-Differential Algebras

It follows that

c#)

Finally, observe that (H(VQ @ At') @ AT, is the Koszul complex of the T-algebra (H(VQ @ A P ) , t"). Thus, in view of Proposition V, (11, sec. 2.14, the kernel of lj$(vQoAp, coincides with the ideal generated by Im(z$)+. Since t, = ZvQ: VQ

4VQ

0AP

(as follows from the definition of t),we have tf = ItQ.Thus ker Zj$(vQoAr, is the ideal generated by Im(ZqQ)+. Now relation (3.29) shows that ker edQ coincides with the ideal generated by Im(ldQ)+. Thus, applying Theorem VIII, sec. 2.19, ((2) 3 (l)), we obtain (3.28).

Q.E.D.

PART 2 I n this part I' denotes a commutative field of characteristic zero, and all vector spaces and algebras are defined over I'.

This Page Intentionally Left Blank

Chapter IV

Lie Algebras and Differential Spaces $1. Lie algebras 4.1. Basic concepts. A Lie algebra is a vector space E together with a bilinear map [ , 1: E XE-+ E

which satisfies the conditions

[x, X] and

[x, [ y ,213

+

A homomorphism

[z, [x,

pl:

E

+

= 0,

x E E,

Yl1 + [Y, [z,X I 1 = 0,

x, Y , 2 E E (Jacobi identity).

F of Lie algebras is a linear map

PIX, Yl =

f!?w,dY)l,

p?

such that

x, Y E E.

A subalgebra of E is a subspace El c E, which satisfies [x,y] E E l , x,y E E l . A subspace I of a Lie algebra E is called an ideal if [x, y ] E I , x E E , y EI. If I is an ideal in E , there is a unique multiplication in E / I making E/I into a Lie algebra, such that 7c: E + E / I is a homomorphism of Lie algebras. The Lie algebra E/I is called the factor algebra o f E with respect to the ideal I. The direct sum of two Lie algebras E and F is the vector space E @ F with Lie product given by [xl O r l ,x, @ y,] = [xl,x2] 0[yl ,Y2], xi E E, yi E F. E and F are ideals in E @ F. If v: E + F is a homomorphism of Lie algebras, then the kernel of rp is an ideal in E. The elements x E E which satisfy [x, X]

= 0, 157

x E E,

158

IV. Lie Algebras and Differential Spaces

form an ideal Z E in E , as follows from the Jacobi identity. Z E is called the centre of E. A Lie algebra E is called abelian, if it coincides with its centre; i.e. if [x,y ] = 0, x, y E E. The derived algebra E' of a Lie algebra is the subspace of E spanned by the products [x,y ] , x,y E E. E' is an ideal in E . If { I z }is a family of ideals in E, then the spaces nDIIol and C a I aare again ideals. If El and E , are subspaces, we denote by [ E l , E,] the subspace of E spanned by the products [x, y ] , x E El, y E E , . If I , and I , are ideals, then [I,,12]is an ideal and [ I I ,12]c I , n I,. 4.2. Representations. Let V be a vector space and consider the space L , of linear transformations Q ^ : V + V . Then the bilinear map Lvx L, -+ L, given by (v,y ) t-+ y 0 y - y 0 ps makes Lv into a Lie algebra, A representation of a Lie algebra E in a vector space V is a homomorphism of Lie algebras

t):E+LV. Given two representations 0,: E + L , and err.:E + Ltr. of a Lie algebra E, a linear map p: V + W is called E-linear if

e,(x)

=

or+)

0

w,

x E E.

Let 8: E + L , be a representation of E in V. Then a subspace W c V is called stable under 0 or simply E-stable if

W,

XE

E,

E

W.

In this case t) induces a representation of E in W. I t is called the restriction of 8 to W. If W c V is a stable subspace, there is a unique representation of E in V / W such that the projection V + V / W is E-linear. On the other hand, if F is a subalgebra of E , then 8 restricts to a homomorphism OF: F L , . This representation is called the restriction of e to F. A vector v E V is called invariant under 8 if --f

The invariant vectors form a stable subspace of V , denoted by Voe0, and called the invariant subspace.

1. Lie algebras

159

A second stable subspace of V is the vector space generated by the vectors of the form 8 ( x ) v , x E E, ZI E I/. It is denoted by O( V ) . A representation 8 of E in V is called faithful if the homomorphism 0: E Lv is injective. If 8, O v , and eM..are representations of E in U , V , and W , we obtain representations of E in V @ W, V @ W, U*, AU, V U given respectively by 8vorv(x) = e v ( x ) 08rdx)

-

~",,(X)

=8dX)

eyx)

=

= e",x)

x )

-e(x)*

T h e representations 8 and o",,,(x)

01 + 1 0M

Oh

o @+),

( 0 , ) ~= (Oh),

are called contagredient. Evidently, e",,,,(x)

and

= (O,)h

e",~)

o+ 1

1

0M x ) ,

= (Oh),.

We shall write

A-representation 8 of E in a graded space is a representation such that the operators O(x) are homogeneous of degree zero. A representation in an algebra is a representation such that each 8(x) is a derivation. A representation in a graded algebra is a representation such that each O(x) is a derivation, homogeneous of degree zero. Let 8 be a representation of E in a finite-dimensional vector space V. Then the trace form of 8 is the bilinear function T,: E XE + r given by

It satisfies

T h e aa'joint representation of a Lie algebra E is the representation L, , given by ad: E --f

(ad x)y = [x,y ] ,

x, y E

E.

IV. Lie Algebras and Differential Spaces

160

The Jacobi identity implies that this is indeed a representation of E. Observe that each map ad x is a derivation in E ;i.e., ad x( [ y , .I) = [ad ~ ( y )XI,

+ [y, ad x(z)l,

x, y, x E

E.

The trace form of the adjoint representation of a finite dimensional Lie algebra E is called the Killing form of E, and is denoted by (x, y ) H K(x, Y >. 4.3. Semisimple representations. A linear transformation p of a vector space V is called semisimple if whenever W is a subspace stable under v, then there is a second p-stable subspace W , such that V =

W @ w,. A representation 8 of E in V is called semisimple if, for every E-stable subspace W c V, there is a second E-stable subspace W, c V such that V = W @ W,. As an immediate consequence of the definitions we obtain

Lemma I: Let 8 be a semisimple representation of E in V and assume that W is a stable subspace. Then

V = v ~0 =e(v). ~ (2) If dim V is finite, then 8b is a semisimple representation in V". (3) The restriction of 8 to W and the induced representation in V / W are semisimple. (4) W0,, = W n Vo=, and 8(W)= W n 8 ( V ) . ( 5 ) (V/W)e=o= Ve=o/We=o and 8 ( V / W )= e(V)/O(W). ( 6 ) If 2 is another vector space, then the representation x H 8(x) @ L in V 02 is semisimple. (1)

A representation 8 of E in V is called quasi-semisimple if the restriction of 8 to every finite-dimensional stable subspace is semisimple. Proposition I: Let BV and Ow be representations of a Lie algebra E in vector spaces V and W, and assume that Ow is quasi-semisimple. Let 8 denote the representation in V 0W induced by OV and Ow, and let Y E ( V 0W)e=o. Then there are finite-dimensional subspaces Y c V , 2 c W with the following properties :

(1) Y (respectively, 2 ) is stable under the operators 8,(x) tively, O,.(x)), x E E.

(respec-

1. Lie algebras

161

(2) The induced representations O y and O2 of E in Y and Z are semisimple. (3) YE (Y@Z),=,. Proof: Write

where the vi and the wi are linearly independent. Let Y denote the space spanned by the vectors v i and let Z denote the space spanned by the vectors w i . Then (3) is obvious. We show first that the spaces Y and Z are E-stable. I n fact, since O ( x ) Y = 0, x E E, we have

Thus

C O v ( X ) v i O w i E ( ~ @ Z ) n ( Y O W ) =Y @ Z . i

Since the wuiform a basis for 2, it follows that Ov(x)v,E Y,

XE

E, i = 1, . . . , m.

Thus Y is E-stable. Similarly, Z is E-stable. Let OY and 0, denote the restrictions of Ov and 8,. to Y and 2. Now the canonical isomorphism a : Y 8 Z 5L( Y";2 ) satisfies N

.(qx)di)

=

e,(x)

-

e",X),

di E Y 0 z,x E E.

I t follows that a(Y) is an E-linear map from Y" to 2. Elementary linear algebra shows that a(Y) is an isomorphism. But since O,br is quasi-semisimple, and Z is finite dimensional, Oz is semisimple. Since a(Y) is an E-linear isomorphism, it follows that 0; is semisimple. Hence so is O Y . Q.E.D. 4.4. Semisimple Lie algebras. A Lie algebra is called simple if it is nonabelian and contains no proper nontrivial ideals. Theorem I: Let E be a finite-dimensional Lie algebra. Then the following conditions are equivalent :

(1) T h e Killing form of E is nondegenerate.

162

IV. Lie Algebras and Differential Spaces

(2) E is the direct sum of simple ideals. (3) Every representation of E in a finite-dimensional vector space is semisimple. Proof:

Cf. [ l ; Theorem I, p. 71, Prop. 2, p. 74, Theorem 2, p. 741.

A Lie algebra that satisfies the (equivalent) conditions above is called semisimple. Theorem I, (1) shows that if E is a semisimple Lie algebra (over r)and 12 is an extension field of F, then E Q is a semisimple Lie algebra (over Q). If E is semisimple, then 2, = 0 and E' = E. A finite-dimensional Lie algebra E is called reductive if E=Z,q@E' and E' is semisimple. It follows from Theorem I that E is reductive if and only if the adjoint representation of E is semisimple. A finite-dimensional Lie algebra E over R is called compact if it admits a negative definite inner product ( , ) which satisfies

([.,rl, 2) +- 0, [x, X I > = 0,

x, y, 2 E

E.

It follows from Proposition XVI, sec. 1.17, volume 11, that the Lie algebra of a compact Lie group is compact. Evidently every compact Lie algebra is reductive. Theorem 11: Let E be a finite-dimensional Lie algebra. T h e following conditions are equivalent : (1) E is reductive. (2) E admits a faithful, finite-dimensional representation with nondegenerate trace form. (3) E admits a faithful, finite-dimensional, semisimple representation. Proof:

Cf. [ l ; Proposition 5 , p. 781.

Theorem 111: A representation 8 of a reductive Lie algebra E in a finite-dimensional vector space V is semisimple if and only if each transformation e(x), x E 2, , is semisimple. Proof:

Cf. [ l ; Theorem 4, p. 811.

In the rest of this article all Lie algebras are assumed to beJinite dimensional.

1. Lic algebras

163

Let F be a subalgebra of a Lie algebra E. Restricting the adjoint representation of E to F yields a representation of F in E ; it is called the adjoint representation of F in E and denoted by adE,F. If this representation is semisimple, then F is called reductive in E. If F is reductive in E, then clearly F is reductive. Moreover, every E (y E Z F ) is semisimple. Conversely, if F transformation a d y : E is a reductive subalgebra of a Lie algebra E and each a d y ( y E 2,) is semisimple, then Theorem I11 shows that F is reductive in E. -j

Example: Let F be reductive in E and assume that F is abelian and that r is algebraically closed. Fix a E F" and set

Ea=

{XE

E l ( a d h ) x = a ( h ) x , h E F}.

In particular,

Eo

= {X E

E I [h,X]

= 0,

h E F},

and so F c Eo. Moreover,

E=EoOCEa, U

where the sum is extended over all nonzero a. Finally, observe that

[Em > Ep]

c

Ea +p

a>B E

F".

4.5. Cartan subalgebras. Given a Lie algebra E, define ideals E(k)c E inductively by E(0) = E and E ( k ) = [E, E(k-1'1.

E is called nilpotent, if E ( k )= 0 for some K. A Cartan subalgebra of a Lie algebra E is a nilpotent subalgebra H such that

H = { y E E I [Y,Hl = H I . I t follows from the definition that every Cartan subalgebra of E contains the centre 2,. According to [6; Theorem I, p. 591, every Lie algebra contains a Cartan subalgebra. Now assume that E is reductive and let H be a Cartan subalgebra. Then H has the following properties (cf. [6; $1, $2, Chap. IV]): (1) H is abelian. (2) H is reductive in E. (3) E = H 0 [H, El.

164

IV. Lie Algebras and Differential Spaces

Moreover, if the coefficient field the direct decomposition

r is algebraically closed, then we have

E=H@

C E,

af O

and every nonzero subspace E, has dimension 1. If E, f 0, 01 is called a root, and E, is called the corresponding root space. (4.1) is called the root space decomposition of the reductive Lie algebra E with respect to the Cartan subalgebra H. Lemma 11: Let E be a Lie algebra and let F be an abelian subalgebra which is reductive in E. Let E, be the subalgebra of E given by

E,

=

{ y E E I [u, F ] = O } .

Then F is contained in every Cartan subalgebra of E, , and every Cartan subalgebra of E, is a Cartan subalgebra of E. Proof: Evidently F c Zz0 and so F is contained in every Cartan subalgebra of Eo. Now let H be a Cartan subalgebra of E,. We must show that H is a Cartan subalgebra of E. Clearly H is nilpotent and so it is sufficient to prove that if x E E satisfies [x, HI c H , then x E H. First observe that (since F is reductive in E )

Now assume that [x, HI c H. Then [x, F ] c [x, HI c H c E,.

On the other hand, [x, FI

= [F, El.

It follows that [x, F ] = 0 and so x E E,. But H is a Cartan subalgebra of E, and so the relations [x,

HI c H

and

XE

E,

imply that x E H. Q.E.D.

165

1. Lie algebras

Proposition 11: Let E be a reductive Lie algebra and let F be an abelian subalgebra which is reductive in E. Let E, be the subalgebra of E given by E, = (X E E I [x, F ] = 01.

Then E, is reductive in E. Proof: First, set E , = [F, E l ; then E = E, @ El. Now we show that E, and El are orthogonal with respect to the Killing form K of E. For this we may assume that T is algebraically closed, and write (cf. the example of sec. 4.4)

El

=

C

E,.

,EF* U f O

The relations [E,, ED]c Eor+D imply that ad x o ad y : EB+

XE

Ear,

Y E

E,.

Hence K ( x , y ) = 0. Now observe that E' = (E, n E ' ) @ E l . Since E' is semisimple, K restricts to a nondegenerate bilinear form in E', and so, by what we have just proved, the restriction of K to E, n E' is again nondegenerate. But this is the trace form of the faithful representation of E, n E' in E'; thus Theorem 11, sec. 4.4,implies that E, n E' is reductive. Now the decomposition E, = Z , @ (E, n E ' ) shows that E, is reductive. Finally, let H , be a Cartan subalgebra of E,. According to Lemma 11, H , is then a Cartan subalgebra of E ; in particular, H , acts semisimply in E. But since ZE0 c H , , ZE0acts semisimply in E. Now Theorem 111, sec. 4.4,implies that E, is reductive in E.

Q.E.D. 4.6. Examples. 1. Consider the Lie algebra L,, where V is a finitedimensional vector space. Then ZLv= r . I and (15,)' is the Lie algebra Lo(V )of transformations with trace zero. It follows that L , = Z L , @ L; . Moreover, the Killing form of L,(V) is given by

K(a, B )

= tr

a

0

B,

a, B

E

L,(V),

and so Lo(V ) is semisimple. It follows that Lv is reductive. 2. Let V be a finite-dimensional Euclidean space. Then the linear transformations u which satisfy d = - u (5 denotes the adjoint trans-

IV. Lie Algebras and Differential Spaces

166

formation with respect to the inner product) form a Lie algebra, which will be denoted by Sk,. The Killing form of SkV is negative definite, and so SkV is semisimple. 4.7. Semisimple representations. Proposition 111. Let 8 be a semisimple representation of a Lie algebra E in a finite-dimensional vector space V . Let F be a subalgebra which is reductive in E. Then the restriction 0, of 8 to F is semisimple. Proof: Cf. [l; Corollary 1 to Proposition 7, p. 841. Corollary: Assume that H is reductive in F and that F is reductive

in E. Then H is reductive in E. Proposition IV: Let 8 be a faithful representation of E in a finite-

dimensional vector space V. Let F c E be a subalgebra, and assume that the restriction 8, of 8 to F is semisimple. Then F is reductive in E. Proof: Without loss of generality we may consider F and E as subalgebras of L,. T o show that F is reductive in E, it is clearly sufficient to show that F is reductive in Lv. Since F admits a faithful semisimple representation, it is reductive (cf. Theorem 11, sec. 4.4). Thus we need only show that each transformation ad y : Lv-+ L , ( y E 2,) is semisimple. But the canonical isomorphism V s 0 V z Lv identifies the transformation -8$(y) 01 1 0OE,(y)with ad y . Since 8, is semisimple and y E ZF, the transformation OF(y) is semisimple (cf. Theorem 111, sec. 4.4). Hence so is ad y. Q.E.D.

+

4.8. Involutions. Proposition V: Let w be an involutive isomorphism of a reductive Lie algebra E. Then the subalgebra F given by F = {x E E I W(X) = x} is reductive in E. Proof:

Since w preserves 2, and E', we have the direct decomposition

F

=F

n 2, @ F n E'.

Thus it is sufficient to consider the case that E is semisimple. Write E- = {x E E I w ( x ) = -x}. Then the relations

E = E - @ F,

[E-, F ] c E - ,

[ E- , E-] c F

1. Lie algebras

167

imply that F and E- are orthogonal with respect to the Killing form of E. Hence the restriction of this Killing form to F is nondegenerate, and so F is reductive (cf. Theorem 11). Thus we have only to show that each transformation ad y: E + E ( y E 2,) is semisimple (cf. Theorem 111, sec. 4.4). Without loss of generality, we may assume that is algebraically closed.

r

I : F is nonabelian. Then F' # 0. Since F is reductive, F' is semisimple. Hence, by Theorem I, F' is reductive in E. Now let H be a Cartan subalgebra of F'. Then H is reductive in F'. Hence, by the corollary to Proposition 111, sec. 4.7, H is reductive in E. Now let E, c E be the subalgebra given by Case

E,={x~EI(adx)H=0}. Then Proposition 11, sec. 4.5, shows that E, is reductive in E. Moreover, E, # E. In fact, since E is semisimple, 2, = 0, while ZEo2 H f 0. Next observe that since H c F , w restricts to an involution of E, with fixed point subalgebra F, = F n E,. Since E, is a proper reductive subalgebra of E, we may assume by induction on dim E that F, is reductive in E, and hence reductive in E (cf. Proposition 111, sec. 4.7). But evidently 2, c ZFoand so it follows that F is reductive in E. Case 11: F is abelian. Define subspaces EA c E (1 E F " ) by x if and only if

(ady

--

Then

E = E,

, ? ( y ) ~ ) ~ (= x ) 0,

0 1 E,

and

E

EA

y E F, n = dim E.

[ E , , E,]

c

E,+,.

I f 0

Hence the restriction of the Killing form of E to E, is nondegenerate, and so 3,is reductive. Now we show that F c ZE,. (4.2) Since F c E, and Eo is stable under co, we have

E,

=;

F @ (E, n E - ) .

Now assume that (4.2) fails. Then there is a least integer p ( p 2 2) such that (ady)p(x) == 0, y E F , x E E,.

IV. Lie Algebras and Differential Spaces

168

Let Z c E, be the subspace given by

Z

= {x E

E, I (ady)p-'x

= 0,y E

F).

Then clearly

[F, E, n E-] c Z

and

[E, n E - , E, n E-] c F c 2.

Hence (E,)' c 2, and so Z is an ideal in E,. Since E,, is reductive, we obtain the direct decomposition

Eo = Z @ Z,,

[Z,Z,]

= 0.

Since F c 2, it follows that (ady)p-'(x) = 0, y E F, x E E,. This contradiction proves (4.2). Finally, choose a Cartan subalgebra H of E,. Then (4.2) implies that F c H. Thus, if z E E satisfies [z, HI c H , then certainly [z, F ] c E,. This implies that z E E,, and so, since H is a Cartan subalgebra of E,, z E H. It follows that H is a Cartan subalgebra of E. I n particular, F c H is reductive in E.

Q.E.D. Example: Let ( , ) be a nondegenerate bilinear form in V which is either symmetric or skew symmetric. Then an involution qj H -qj* in the Lie algebra L , is defined by

The fixed point subalgebra consists of the transformations which are skew with respect to ( , ) and, by the proposition, this subalgebra is reductive in L, .

$2. Representation of a Lie algebra in a differential space 4.9. Let E be a Lie algebra and let ( M , dA1) be a differential space. A representation of E in the dzfferential space ( M , d M ) is a representation O,, of E in M such that O N ( X ) b = ~ M ~ '(x), ,,

xE

E.

I n particular the cocycle space Z ( M ) and the coboundary space B ( M ) are stable under such a representation. Hence each map Ohi(x) induces a linear map OM@)#: H ( M ) H(M). ---f

Observe that the correspondence x H Oh1(x) defines a representation 8% of E in H ( M ) . It is called the induced representation in the cohomology , , be simply space. The corresponding invariant subspace ( H ( M ) ) o ~ =will denoted by H(M),,=, . A representation of E in a graded differential space ( M , SAW) is a representation O,, in the differential space ( M , 8,) such that each map O,(x) is homogeneous of degree zero. In this case t?$(x) is also homogeneous of degree zero. A representation of E in agraded differential algebra ( R , S f ( ) is a representation of E in the graded differential space ( R , 6,:) such that each O,(x) is a derivation in R. I n this case each t?z(x) is a derivation in H ( R ) and so 0: is a representation in the graded algebra H ( R ) . Let OM be a representation of E in a differential space ( M , &). Then, evidently, the invariant subspace and the subspace O(M)are stable under 6. Hence we can form the cohomology spaces H(Mo=o)and H(O(M)).T h e inclusion map M,=, M induces a linear map H(M,=,) + H ( M ) . Clearly we may regard this as a map into H(IM),,=,. If O,, is a representation of E in a graded differential space (respectively, in a graded differential algebra), then these maps are homomorphisms of graded vector spaces (respectively, of graded algebras). ---+

4.10. Semisimple representations. Let 0, be a semisimple representation of a Lie algebra E in a differential space ( M , &). Since Z ( M ) and B ( M ) are E-stable subspaces and 0, is semisimple, we can 169

IV. Lie Algebras and Differential Spaces

170

find E-stable subspaces CM c M

Z(M)

and

A,W c

and

Z ( M ) = A, @ B ( M ) .

such that

M

= Z,, @

C,,

In particular, the projection Z ( M ) + H ( M ) restricts to an E-linear isomorphism A,w H ( M ) . I t follows that 0; is a semisimple representation.

-

Theorem IV:

Let OM be a semisimple representation of a Lie algebra + M induces

E in a differential space ( M , &). Then the inclusion an isomorphism,

H(M,=,) 5H(M)()*=,. Proof: Since O,

is semisimple, we have

z ( o ( M ) )= Z ( M ) n e ( M )= o ( z ( M ) ) . It follows that the inclusion O(M) + M induces a map H ( O ( M ) )+ O(H(M)). Moreover, because Oh, is semisimple, M = M,=, @ B(M). Since 19; must also be semisimple, it follows that

H(M,=,)

o H ( e ( M ) )= H ( M ) = H(M),*=, 0 e ( H ( M ) ) .

But H(M,=,) c H(M),,=, and H(O(M)) c B(H(M)). The theorem follows.

Q.E.D.

Corollary I:

If ; 0 = 0, then H(M,=,) 5 H ( M ) .

Corollary 11: If 6 ;

= 0,

then H(O(M))= 0.

Proposition VI: Let O,v be a semisimple representation of a Lie algebra E in a graded differential space ( M , SM). Then there are E-linear maps AM: H ( M ) + M , nM: M + H ( M ) , h M : M +M ,

respectively homogeneous of degrees 0, 0, and -1 lowing properties: (i) nfi,SfiI = 0, SMAM = 0, nfi1&= SMh,, . (ii) 1 - A M n M = h M S f i I

+

L.

and having the fol-

171

2. A Lie algebra in a differential space

Proof: Choose E-stable subspaces C, c M and An* c Z ( M ) such that and M = Z ( M )@ C M Z ( M )= A M @ B(M).

Then the projection Z ( M ) -+ H ( M ) restricts to an E-linear isomorphism a : A M -5H ( M ) , while S, restricts to an E-linear isomorphism

-

8:

c,

-5B ( M ) .

Now set

Now suppose that ON is a semisimple representation of E in a second graded differential space N . Let

q):M-+N

and

y:M-+N

be homomorphisms of graded differential spaces such that E-linear.

q) - y

is

Proposition VII: Assume that q ~ #= y*. Then there is an E-linear map k : M N , homogeneous of degree - 1, such that --f

Proof: Write q) - y = X . Using the notation of Proposition VI, define a linear map XA:

AM - - + A N

by x,(a) = ANnNX(a),a E A M . Then the diagram %A

AM-AN

commutes. Since X* = 0, it follows that X A = 0.

IV. Lie Algebras and Differential Spaces

172

On the other hand, the linear map AM 0 nM : M so that (A N n N ) 0 X 0 (A f i f n M ) X A 0 (AMn,)

-+ M ,

has image A,,

= 0.

1

It follows that

Q.E.D. Corollary I: If H (. M.) = 0, then there exists an E-linear map M , homogeneous of degree -1, and satisfying

k: M

--f

t =

6Mk

+ k6,.

Corollary 11: If 8; = 0, then there is an E-linear map k: 8 ( M ) -+ 8(M) homogeneous of degree -1 such that I

= 6Mk

+ k&.

Proof By Corollary I1 to Theorem I, we have H(B(M))= 0. Now apply Corollary I above to the differential space ( 8 ( M ) ,6,). Q.E.D.

4.11. Representations in a tensor product. Let E be a Lie algebra, a graded differential space with 6, homogeneous of degree 1, and let Ox be a semisimple representation of E in ( X , 6,) such that 8; = 0. Let O y be a representation of E in a second graded differential space ( Y , 6,) and consider the differential space ( X 0 Y , S), where

( X , 6,)

6(u @ v ) = 6x24 @ v

+ (-1)”u

@ Syv,

u

€

Finally, let 8 denote the induced representation in X Theorem V:

X”,

2)

€

Y.

0Y.

With the notation and hypotheses above, the inclusion

map

Xe-0 @ Ye=,

--+

( X @ Y)o=o,

2. A Lie algebra in a differential space

173

induces an isomorphism

X

Proof

Since Ox is semisimple, we have the direct decomposition

= X,=,

@ O(X), whence

( X 0 Y)e=o= (X6-o 0 Y w ) 0 ( e ( X )0 Y)o=o. Since S commutes with Ox and OI. (and hence with 0 ) all terms in this relation are stable under 6. This implies (via the Kunneth formula) that

H ( ( X 0 Y)e=o)= H(Xo-0)

0H ( Yo=,)0H ( ( O ( X )0 Y)o=o)*

Hence it has to be shown that

q(e(x)o Y ) ~ ==~0.)

(4.31

Corollary 11 to Theorem IV, sec. 4.10, implies that H(O(X))= 0. Hence, by Corollary I1 to Proposition VII, sec. 4.10, there is an E-linear operator k : O(X) --+ O(X), homogeneous of degree -1, such that Sxk

Now set

k = k @ 1.

Then

+ k6x =

k is E-linear, Sk + kS

=

1.

and satisfies

1.

Hence k restricts to an operator k,=o in (O(X) implies that &o ks=oS = 1,

(4.4) Y)e=o.Relation (4.4)

+

and so (4.3) follows.

Q.E.D.

Chapter V

Cohomology of Lie Algebras and Lie Groups In this chapter E denotes a finite-dimensional Lie algebra. T h e multiplication operators determined by a E AE and @ E AE* will be denoted by A a ) and 4@): p(a)(b)= a A b

and

p ( @ ) ( Y )=

@ A

Y.

The (dual) substitution operators are denoted either by i E ( a ) :AE" AE* and iE(@): AE + AE, or simply by i(a) and i(@).

-+

sl. Exterior algebra over a Lie algebra 5.1. The operators O,(x) and OE(x). Given a Lie algebra E consider the dual space E". The linear maps a d x ( X E E) extend to unique derivations 6 E ( ~in) the algebra AE. Similary, -(adz)* extends to a derivation O,(x) in AE". BE and e E are contragredient representations of E in the graded algebras AE and AE*, extending the contragredient representations ad and adQ. The derivation property of these operators is expressed by the formulae

eyX),+)

- pu(a)eE(x)

= pu(eE(.+),

a E AE,

xE

E,

and @ E ( X ) p ( @ ) - P ( @ ) @ E ( x )= p(@E(x)@),

174

@

AE", x f E.

175

1. Exterior algebra over a Lie algebra

In particular, O E ( X ) ~ E ( Y ) - ~ E ( Y ) ~ E ( x= ) i ~ ([x, y ] ) ,

X,

y E E.

(5.1)

T h e representations OE and 6 E determine the invariant subalgebras (AE),=, and (AE")o,, , as well as the stable subspaces O(AE) and O(AE") (cf. sec. 4.2). If a and @ are invariant, the relations above reduce to simple commutation formulae. I n particular, the invariant subalgebras and the subspaces O(,IE) and O(AE") are stable under multiplication and substitution by invariant elements. Since BE and OE are contragredient, we have the relations

(AE)&,

=

O(AEX)

and

(AE*),I,, = O(AE).

Moreover, since the restriction of BE to E is the adjoint representation, it follows that

(A'E),=, Finally, let e,, Then

e*V

= 2,

and

O(AIE)= E'.

(v = 1, . . . , n) be a pair of dual bases for E and E".

OE(x) = - C ,u(e"V)i([x, e V ] )

and

OE(x) =

C ,u([x, e V ] ) i ( e X V ) . (5.2) V

Y

(Since both sides are derivations, these formulae have only to be verified in E and E", where they are immediate consequences of the definitions.) 5.2. The operators 6, and b,. given by

Consider the linear map v: A2E

V(X A Y ) = [X, y ] ,

x, Y E E.

Extend the negative dual, -v":

A2E" +- E",

to an antiderivation dE:

AE"

+-

AE",

homogeneous of degree 1 (cf. [ S ; p. 1111). We shall establish the fundamental relations

and

-

E

176

V. Cohomology of Lie Algebras and Lie Groups

Since

the first relation holds when applied to elements in E". Since both sides are derivations in AE", it must be true in general. T o prove the second relation observe that the first relation yields

Since Si is a derivation, it follows that Si = 0. Finally, the last relation is established by applying S, to both sides of the first, and using the fact that 8; = 0. It follows from the relations (5.3) that (AE*, S,) is a graded differential algebra, that 0, represents E in (AE", S,), and that 0; = 0. Next, let a,: AE + AE be the linear map, homogeneous of degree - 1, given by dB = - S z . Then a,(x~y)=[x,y],, dEx=O,

and

aE(A)=O,

Dualizing formulae (5.3) yields the formulae Ax)aE

+ a,/+)

= 0"(x),

8%= 0

and eE(X)a,

=

a,eyx),

xE

E.

x , y ~ E AET. ,

1. Exterior algebra over a Lie algebra

177

In particular, BE represents E in the graded differential space (AE, 8,). Note that d E is not in general an antiderivation in the algebra AE. 5.3. The Koszul formula.

In this section we shall establish the

Koszul formula SE

4 2 p(e*V)eE(e~),

=

(5.4)

where e v , e*v (v = 1, . . . , n ) is a pair of dual bases for E and E*. Since SE and the operator on the right-hand side are both antiderivations, it is sufficient to verify this formula for elements in E*. Let x* E E* and x E E be arbitrary. Then, using relations (5.3) and (5.1), we find that iE(x)SEx* = 6 E ( x ) x * ,

while iE(x)

4 C p(e*Y)~E(e,Jx*= SeE(x)x* - 4 C ~ ( e * ~ > i ~ ( x ) e ~ ( e , > x * V

V

=

SeE(x)x* -

4 C p(esY)iE([x, e,])x*. V

Now formula (5.2) yields

which completes the proof. Dualizing the Koszul formula we obtain the equation dE =

+ C fP(ev)iE(e*v)

(5.5)

V

(contravariant Koszul formula). As an immediate consequence we obtain dE(x0h

- . -AX,)

C

=

V

CP

(-I)Y+~+I[x,, x],

AX,, A

g,,

-.

4,

-

- '

X ~ E. E

AX^,

(5.6)

Thus, if APE* is interpreted as the space of p-linear skew functions in E, then B E is given by (6E@)(x0

=

9

-

,

c ( - l ) ' + " @ ( [ x V , x , ] , x o , ..., d v , ..., d,, ...,

Y
X,)'

(5.7)

V. Cohomology of Lie Algebras and Lie Groups

178

5.4. The product formula for relations &(a

A

+ (-1)Pa b + (-1)pa

b ) = dEa A b

dE(a A b ) = aEa A

A

a,.

I n this section we derive the

+ C iE(exv)a O"(e,)b (5.8) aEb + ( - l ) ~C OE(eY)a i E ( e f V ) b

dEb

A

Y

A

A

V

and &(a

A

b ) = -dEa

A

b

+ (-1)Pa

A

dEb

+ C eE(eY)(iE(e*V)a b), A

V

aE

APE, b E AE,

(5.9)

where e , , e + V ( v = 1, . . . , a ) are dual bases for E and E". To prove these, use the contravariant Koszul formula to obtain dE(aA b) =

& C P ( e , ) i E ( e f u ) ( aA b ) V

C ${efi(ey)iE(e+u)aA b + iE(e*u)aA BE(e,)b +(- l)@E(e,)a A iE(e"l)b + (- 1)pa A OE(eV)iE(efv)b} = a,n A b + (- 1 ) ~ Aa aEb + & C iE(e+u)aA OE(e,)b =

V

Y

+ $ C (-

l ) ~ € P ( e , )AaiE(e*v)b.

V

The two last terms in this relation are equal. I n fact for xi, yj E E, we have

Thus the relations (5.8) follow. T o verify (5.9) observe that, in view of the derivation property of

OE(ev),

C OE(e,)(iE(e+~)a A b ) = 2dEa b + C iE(ewu)aBE(e,)b. A

V

Subtracting this from (5.8) yields (5.9).

A

Y

179

1. Exterior algebra over a Lie algebra

5.5. Cohomology and homology of a Lie algebra. Let E be a Lie algebra of dimension a,and recall the graded differential algebra (AE", 6,) introduced in sec. 5.2. T h e cocycle (respectively, coboundary) algebras of this differential algebra will be denoted by

Z * ( E ) = C ZP(E)

B"(E) = C Bp(E).

and

P

P

The corresponding cohomology algebra

H"(E) =

cW ( E ) P

is called the cohomology algebra of the Lie algebra E. Notice that H O ( E )= r. Next consider the graded differential space (AE, aE).T h e cycle (respectively boundary) subspaces will be denoted by

Z,(E) = C Zp(E)

B,(E)

and

=

C Bp(E). P

P

The corresponding homology space

H"(E) =

c HP(E) P

is called the homology space of the Lie algebra E. For example, Z,(E) = E, B , ( E ) = aE(A2E)= E', and so

H , ( E ) = E/E'. Since d, = -6%, it follows that

B,(E)

= Z*(E)l

Z,(E)

and

= B*(E)l,

and so a natural duality is determined between the spaces H * ( E ) and H , ( E ) . This duality restricts to a duality between each pair Hp(E), H,(E) ( p = 0, , . . , n). In particular, dim Hp(E) = dim H p ( E ) , p = O , . . . , n. The integers

p

bp = dim Hp(E),

= 0,

. . . , n,

are called the pth Betti numbers of E and the polynomial n

fH(E)=

C bptp p=o

V. Cohomology of Lie Algebras and Lie Groups

180

is called the Poincare' polynomial of H # ( E ) . Since Z p ( E ) c APE", it follows that ( n = dim E ) . 5.6. Homomorphisms. Let rp: F + E be a homomorphism of Lie algebras, and consider the dual map rp": F" t E". Extend rp and rp" to (dual) homomorphisms

vA:AF -+ AE

and

rpA: AF"

+

AE".

The relations

y E F,

T o a d y = adrpyorp, imply that

vh eF(y) = eE(py)

q A 9

y

F*

(since both sides are p,,-derivations). Dualizing we obtain

In particular, rpA restricts to a homomorphism

Moreover, if v is surjective, then rpA maps invariant elements into invariant elements. Finally, use the expressions (5.6) and (5.7) for dE and BE (at the end of sec. 5.3) to obtain

Thus rpA (respectively, rpA) is a homomorphism of graded differential algebras (respectively, graded differential spaces). The induced maps in cohomology and homology are denoted by

v*: H # ( F ) c H " ( E )

and

rp*: H,(F)

-

H#(E).

They are dual, and homogeneous of degree zero. Moreover, algebra homomorphism.

v # is

an

1. Exterior algebra over a Lie algebra

181

5.7. The space (A3E*),=o. Let E be a Lie algebra. Extend the representation of E in E" to the representation Bs of E in WEE"given by O,(x)(x" v y " ) = -(ad x)*x* v y "

+ x* v (-ad x E E,

x>*y*, x*, y" E E".

We shall construct a canonical linear map

e: (V2E")e=0

---+

(A3E*)e=0.

In fact, let Y E (V2E")e=0.Then a 3-linear function @ is defined in E by @(x,y , z ) = Y( [x,y ] , z ) ,

x,y , z E E.

Since Y is invariant, it follows that

This relation shows that @ is skew symmetric. Moreover, the Jacobi identity yields ( b ? ( w ) @ ) ( xY, ,

Yl,4 X , y, Z,w E E.

= (e,(w)w[x, =

0,

Thus @ is invariant. T h e correspondence Y ~--f@ defines a linear map

-

e: (V2E")e=o

(A3E"),,o.

Proposition I: Let E be a Lie algebra such that H,(E) = 0. Then e is a linear isomorphism.

=0

and

H,(E)

Proof: T o show that

e

is injective assume that Y E ker e. Then

This implies that Y ( u , z ) = 0, u E E', z E E. But since H , ( E ) = 0, E = E' (cf. sec. 5.5). Thus Y = 0. Now we show that e is surjective. Let @ E (A3E")e=o. Then (@, a ) = 0, u E O(A3E). T h e Koszul formula (5.5), sec. 5.3, shows that I m dE c O(AE). On the other hand, the relations at the end of sec. 5.2 yield y

A

dBw = 8"(y)w - dE(yA w ) ,

y

E

E, w E A3E.

182

V. Cohomology of Lie Algebras and Lie Groups

It follows that d E ( yA w ) E O(AE) and y and

(@, d E ( y A w ) ) = 0

A

dEw E O(AE), whence (@, y

A

d E w ) = 0.

Now since H , ( E ) = 0, d E maps A2E onto E. Set

where u is an element such that dEu = x. T o show that Y is well-defined, suppose dEu = dEv. Then dE(u- v ) = 0. Since H 2 ( E )= 0, there is an element w E A3E such that dEw = u - v. Thus

It follows that Y is a well-defined bilinear function in E. Next, use formula (5.9), sec. 5.4, to obtain the relation

Since 8,: A2E --+ E is surjective, it follows that Y is symmetric. T h e invariance of Y is immediate from the definition. Finally, it follows directly from the definitions that $P . = @. Hence e is surjective.

Q.E.D. 5.8. Abelian Lie algebras. Let E be an abelian Lie algebra of dimension n. Then the operators e E ( x ) , OE(x), S, and dE are all zero. Thus we have

(AE"),=, = H * ( E ) = AE"

and

(AE),=, = H , ( E )

= AE.

In particular b,(E) = (;), 0 5 p 5 n, and so the PoincarC polynomial of E is given by f E = (1 t)".

+

5.9. Direct sums. Assume E and F are Lie algebras and consider the direct sum E @ F. Consider the isomorphisms (of graded algebras)

AE @ AF 5 A ( E @ F )

and

AE* @ AF* 5 A(E @ F)",

given by a 0b I-+ a A b and @ @ Y H @ A Y. These algebras will be

183

1. Exterior algebra over a Lie algebra

identified via these isomorphisms. T h e scalar products are given by (@

0y,a 0b ) = <@, a>(U: b ) , @ E AE",

Y E AF",

aE

AE, b E AF.

With these identifications we have

It follows that

The analogous formulae hold in A(E @ F ) . Next we establish the relations

(oEdenotes the degree involution in AE" and in AE.)

I n fact, it follows at once from the definition of the Lie product in wE 0d F in A2(E @ F ) . Dualize and conclude that BEeF and BE 0i w E 0SF agree in ( E @ F ) " ; since both operators are antiderivations, they coincide. Dualize again to obtain that dE@F= dE @ L w E @ dF. I n view of these relations and the Kunneth formula it follows that H"(E @ F ) H X ( E )@ H * ( F ) and H,(E @ F ) H,(E) @ H,(F). Moreover, the first isomorphism is the algebra isomorphism given by

E @ F that dE@Fagrees with dE @ i

+ +

+

-

-

F are the projections. where q :E @ F -+E and x 2 :E 0F Now consider the case F = E. Then the diagonal map A : E E @E given by x E E, A ( x ) = x @ X, is a Lie algebra homomorphism. T h e induced homomorphism of graded differential algebras A": AE" + AE" @ AE" is given by

AA(@0!P) = @ A Y,

@, Y E AE".

184

V. Cohomology of Lie Algebras and Lie Groups

It follows that the induced homomorphisms

A:=,,: (AE+)B=O+- (AE")e=o @ (AE")e-o and d #: H " ( E ) +- H"(E) @ H " ( E )

are given, respectively, by

$2. Unimodular Lie algebras 5.10. Unimodular Lie algebras. Consider the vector t" fined by x E E. (t", x ) = tr ad x,

E

E" de-

The vector t" is invariant, as follows from the relation

(8E(x)t",y ) = -tr ad[x, y ] = -tr(ad x o ad y

- ad y o

ad x) = 0, X , ~ E. E

In terms of dual bases we have t" =

C Be(eY)e*v.

(5.10)

Y

I n fact, for x E E,

Relation (5.10) implies, in view of the formulae in sec, 5.1, that i(t")

=

C 8E(e,)i(e*v) - 1 i ( e + ~ ) V ( e , ) .

(5.11)

V

Y

A Lie algebra is called unimodular if tr ad x this is equivalent to t*

=

= 0,

xE E;

0. If E is unimodular, formula (5.11) reduces to

Hence the contravariant Koszul formula (cf. formula ( 5 . 5 ) , sec. 5 . 3 ) can be written in the form

dE =

4 C i(e*p)BE(e,). Y

185

(5.12a)

186

V. Cohomology of Lie Algebras and Lie Groups

Dualizing we obtain (5.12b) Proposition II: Let E be a Lie algebra. Then

(AE”),,,

c

Z*(E)

B,(E) c O(AE).

and

If E is unimodular, then also

(s4E)O-o c Z,(E)

and

B*(E) c B(AE*).

Proof: The first statement follows from the Koszul formulae of sec. 5.3. The second follows from formulae (5.12a) and (5.12b) above. Q.E.D.

Every reductive Lie algebra is unimodular. I n fact, if E is reductive, we have the direct decomposition E = 2, @ E‘. Since tr ad[x,y] = tr(ad x 0 a d y - a d y 0 ad x) = 0,

x , y E E,

this decomposition implies that E is unimodular. 5.11. Poincark duality. Let E be a unimodular Lie algebra of dimension n, and let e E AnE. Then BE(x)e= tr(ad x)e = 0,

x E E,

and so e is an invariant element. Now choose a fixed nonzero vector e E AnE. Then the Poincare‘ innH product in AE* determined by e is the nondegenerate bilinear function ( , ) defined by (@, Y ) =

(@A

@, Y E AE*.

Y , e),

The corresponding PoincarC isomorphism D : AE* E.+AE is given by

D(@)= i(@)e (cf. sec. 0.6).

Since e is invariant, the formulae of sec. 5.1 imply that

D 0 e,+)

=eyx)

Hence D restricts to an isomorphism

D,

x E E.

187

2. Unimodular Lie algebras

On the other hand, since a E ( f ? ) = 0 (cf. Proposition 11, sec. 5.10), the antiderivation property of dE yields a,o

D

=D

0

8E

WE.

0

( w E denotes the degree involution in AE".)

Hence, D induces an isomorphism

D*:H*(E) % H,(E). Now let definition,

E E

H,(E) be the class represented by e. Then, in view of the (a

*

B, &)

=


a, B E H"(E).

Hence a nondegenerate bilinear function, denoted by PE, is determined in H * ( E ) by 9E(a,

B ) = ( a B, &),

a, B

H"(E).

Thus H"(E) is a PoincarC algebra (cf. sec. 0.6). Finally, observe that D restricts to isomorphisms

APE" -5

(ApE')e-o

N

(An9E)e-o,

while

D # : Hn-p(E) -5H p ( E ) . In particular, b,(E) = bnJE) (0 5 p 5 a).

$3. Reductive Lie algebras 5.12. Let E be a reductive Lie algebra. Since ad x = 0, x E Z E , it follows that eE(x) =

0

and

OE(x) = 0,

XE

2,.

Hence, by Theorem 111, sec. 4.4, the representations 8 E and 8" are semisimple. I n particular, we have the direct decompositions

AE"

=

(AE"),=, @ O(AE")

and

On the other hand, the duality of

O(AE")l = (AE),=,

8E

and

AE

=

(AE),=, @ O(AE).

and OE implies that

8 ( A E ) l = (AE"),=,.

(5.13)

Thus, in the decomposition above, the scalar product between AE" and AE restricts to a scalar product between (AE#),=, and (AE),=, as well as between O(AE*) and O(AE). Lemma I: Let E be a reductive Lie algebra. Then

( 1 ) Z"(E) = (AE")B=o @ B"(E). (2) Z#(E) = (AE),=o 0B # ( E ) . (3) B"(E)'= 8(Z"(E))= Z"(E) n O(AE").

(4) B,(E)

=

O(Z,(E)) = z,(E)n O(AE).

Proof: First we establish (3). I n fact, the relations

On the other hand, in view of Proposition 11, sec. 5.10,

B X ( E )c Z*(E) n B(AE*). 188

189

3. Reductive Lie algebras

Since O E is semisimple, Z*(E) has an E-stable complementary subspace in AE*. It follows that

B(Z*(E)) = Z*(E) n B(AE"), and these three relations prove ( 3 ) . T o establish (l), observe from the Koszul formula (formula (5.4), sec. 5 . 3 ) that (AE*),=, c Z*(E). Thus (AE"),=, = (Z*(E)),=,. But, since 6 E is semisimple, so is its restriction to Z*(E). Thus

Z*(E) = (Z"(E)),=o@ O(Z"(E))== (AE*)O=O @ O(Z"(E)), and so (1) is a consequence of ( 3 ) . Relations (2) and (4)are proved in the same way. Q.E.D. Theorem I:

Let E be a reductive Lie algebra. Then the projections

Z"(E) -+ H * ( E )

and

Z,(E)+ H , ( E )

restrict to linear isomorphisms

Moreover, nE is an algebra isomorphism. Proof: Apply Lemma I.

Q.E.D. Remarks:

1. T h e projections zE and z* satisfy

(TcE@, Z * U )

= (@, a ) ,

@ E (AE*)O=O, a E (AE),=,.

2. If v: F + E is a homomorphism of reductive Lie algebras, then the diagram

(AF"),=,

commutes.

@Lo

(AE"),=,

V. Cohomology of Lie Algebras and Lie Groups

190

3. The diagram De=a

(AEX)e=oy (AE),,,

H"(E)

+H,(E)

commutes, where D,=,and D* are the isomorphisms of sec. 5.11. 5.13. The Pontrjagin algebra. Let E be reductive. Then there is a unique multiplication in H , ( E ) which makes the linear isomorphism z* : (AE)o=,, H , ( E ) into an isomorphism of graded algebras. With this multiplication H , ( E ) becomes a graded anticommutative algebra, called the Pontrjagin algebra of E. N

Remark: Note that in general H,(E) does not have a natural algebra structure since d E is not an antiderivation.

Let Q : F + E be a homomorphism between reductive Lie algebras. The map q,,:AF + AE (in general) does not restrict to a map from (AF),+o to (AE)e=o. I n this section we shall, nonetheless, construct a homomorphism

v# : (AF),,o

-

(AE)o=o.

In fact, consider the projections

with kernels O(AF) and O(AE). Set

Propositian 111: With the notation and hypotheses above:

(1) (2)

'p+

v+

is an algebra homomorphism. is dual to the. homomorphism,

191

3. Reductive Lie algebras

(3) The diagram

commutes. Lemma 11: Let E be reductive. Then

Proof: Lemma I, (4), sec. 5.12, implies that qr: o dE = 0. Thus applying vE to formula (5.9), sec. 5.4, yields 7 E ( d E a A b) = (-l)PqE(a

A

dEb),

a E APE, b

E

AE.

In particular, ~ ; I E ( UA

3,')

=

u E Z,(E),

0,

ZI

E

AE.

On the other hand, in view of Lemma I, (2), sec. 5.12, we can write (for 2 1 7 z2 E 2, ( E ) )

zi= V E X i

+ dEUi,

i = 1,2.

Now the relation above gives

Proof of Proposition 111: ( I ) Let u, 21 E ( I I F ) ~ -Then ~ . by Lemma I, sec. 5.12, u and ZI are dy cycles. Since p is a homomorphism of Lie algebras, it follows that =

~ E P A ( ~ )

P , ~ P ( U )= 0.

192

V. Cohomology of Lie Algebras and Lie Groups

(3) This follows immediately from the definitions, and Lemma I, (Z), sec. 5.12. Q.E.D. Corollary: The induced map q # :H,(F) phism between the Pontrjagh algebras.

-

H , ( E ) is a homomor-

$4. The structure theorem for (AE),=,, In this article E denotes a reductive Lie algebra. 5.14. The primitive subspace. Consider the diagonal map

A:E+E@E (cf. sec. 5.9). Since

(A(E 0E))o=o= (AE)e=o0(AE)e=o, the induced homomorphism A , (cf. Proposition 111, sec. 5.13) is a homomorphism

(AE),,o Lemma 111: Let a

E

+

(AE)0=00(AE)e=n.

(A+E),,,.

Then

d,(a) = a @ 1 + b

+ 1 @a,

Then by Proposition 111, (2), sec. 5.13, and sec. 5.9,

(0, a,> = (001,

=

=

(@, a>,

@ E (AE")e=o.

NOWthe duality between (AE*),=, and (AE),,, (cf. sec. 5.12) implies that a, = a. Similarly, a2 = a. Q.E.D. Definition: An element a E (A+E),=, is called primitive if

d,(a)= a 0 1 193

+ 1 @a.

V. Cohomology of Lie Algebras and Lie Groups

194

T h e primitive elements form a graded subspace

P,(E)

=

f P,(E)

(n = dim E )

j-1

of (A+E),,,. It is called the primitive subspace. T h e dimension of P,(E) is called the rank of E. Lemma IV: (1) Every homogeneous primitive element has odd degree.

(2) If a,, . . . , up are linearly independent homogeneous primitive elements, then U1A

- * *

AUpfO.

Proof: (1) Let a be a homogeneous primitive element of even degree. Since A , is a homomorphism (cf. Proposition 111, sec. 5.13),

A,(&) = (A,(a))k = ( a @ 1

+ 1@

a)k,

k

= 1,2,

. .. .

Since a has even degree, the elements a @ 1 and 1 @ a commute. Thus the binomial theorem yields

Now choose k to be the least integer such that ah = 0. (Since E has finite dimension, this integer exists.) We show that k = 1. I n fact, assume that k > 1. Then (for degree reasons) the elements a, . . . , uk-l are linearly independent, which contradicts the formula above. Thus k = 1. It follows that a = 0 and so (1) is established. 5 k,. Since (2) Set deg ai = k, and number the ai so that k, 5 A , is a homomorphism and the ai are primitive,

--

In particular, the component w of d,(a, is given by

where a,,

. . . ,up are the elements of

A

.-

A

u p )in

degree k,.

@ (AE),,,

4. The structure theorem for (AE)o=,

195

By induction on p we may assume that

Since the ai are linearly' independent, it follows that w # 0. Thus A a,) # 0, and so a, A . - . A up # 0. A,(a, A Q.E.D. a

5.15. The ideal D"(E). I n this section we shall obtain a different

description of the primitive subspace P*(E).Let D"(E) = C j Dj(E) denote the graded ideal in (AE")e,o given by

D"(E)

=

(A+E"),=o * (A+E"),=,.

If F is a second reductive Lie algebra, we have the relation

D"(E @ F) = (D*(E) 01) @ [(A+E"),=o 0(A+F"),=o] @ (1 @ D"(F));

(5.14)

this follows by squaring the formula

(cf. sec. 5.9). Lemma V: T h e primitive subspace P"(E) is the orthogonal complement of D*(E) with respect to the duality between (A+EB)o=O and (A+E)&O P*(E)= D " ( E ) L 9

Proof: Let a E P#(E). Then the duality between the maps and A , (cf. Proposition 111, (2), sec. 5.13) shows that for @, Y E (A+E*),=,, (@A

Y, a )

0Y), a ) = (@ 0Y, d , ( a ) )

=

(A;=,(@

=

( @ @ Y , a @ l + 1 @a)=O.

It follows that a E D"(E)I. Conversely, suppose that a E D*(E)I. Since A;=o is a homomorphism, it restricts to a homomorphism

V. Cohomology of Lie Algebras and Lie Groups

196

Hence the dual map d, restricts to a linear map

Thus d , ( a ) E D*(E @ E ) I . But in view of formula (5.14) (applied with F

D"(E

= E),

0E)* = D # ( E ) l @ 1 + 1 @ D"(E)l.

This, together with Lemma 111, implies that d , ( a ) and so a is primitive.

=a

@1

+ 1 @ a, Q.E.D.

Corollary I:

If E and F are reductive Lie algebras, then

Corollary 11: If Q:: E -+ F is a homomorphism of reductive Lie algebras, then v# restricts to a linear map

I n view of Lemma Iv,

5.16. The structure theorem for (l), sec. 5.14, we have aA

Thus the inclusion map of algebras

U ==

0,

U E

P#(E).

P*(E) (AE),=, extends to a homomorphism

x,:

--f

A(P#(E)) (AE),=O. -+

If AP,(E) is given the gradation induced by that of P#(E), then x# is homogeneous of degree zero. Theorem 11: Let E be a reductive Lie algebra. Then x, :

W*(Jq)( W , = O +

is an isomorphism of graded algebras. Thus the invariant subalgebra of AE (and hence the Pontrjagin algebra of E ) are exterior algebras over graded vector spaces with odd gradations.

197

4. The structure theorem for (AE)e=,,

Proof: To avoid confusion, the product in A(P,(E))will be denoted by U A V .

(1) x, is injective: Let ul, . . . , a, be a homogeneous basis of P, ( E ) . Then, by Lemma IV, (2), sec. 5.14,

Now let u be a nonzero element in A(P,(E)). Then, for some v E A(P* (E))l U A V = U, A

* * *

(cf. Example 1, sec. 0.6). Hence x * ( u ) This shows that x+ is injective.

(2)

x, is surjective:

A

A U,

x,(v)

# 0 and so

x,(u)

f0.

Since x, is injective it is sufficient to show that

dim A(P, ( E ) )2 dim(AE),=,

.

But dim A(P*(E))= 2'(r = dim P,(E)),and dim(AE),=, = dim(AE*),=,.

Thus we have only to show that

Choose a graded subspace U c (A+E"),=, so that

Then, by Lemma V, sec. 5.15, U is dual to P,(E). Thus, all the homogeneous elements of U have odd degree. It follows that @ A @ = o ,

c

@EU.

Hence the inclusion map i: U + (A+E"),=,, extends to a homomorphism i,: AU -+ (AE"),,,. It satisfies the relation

+ (A+E"),=, . (A+E*),,, = (A+E*)B,o.

Im(i,+)

V. Cohomology of Lie Algebras and Lie Groups

198

Squaring both sides yields Im(it) . Im(i:) whence Im(i,+)

+ ((A+E*),=,)3 = ((A+E*),,,)2,

+ ((A+E*),=,)3= (A+E*),=,.

Repeating this argument shows that Im(i,+)

+ ((A+E*),=,)p = (A+E*),=,,

p

=

2, 3,

. .. .

Since ((A+E*),=,)'1+1= 0 ( a = dim E ) , it follows that Im(iz) = (A+E*)B=O; i.e., iAis surjective. Finally, since U and P,(E) are dual, dim U

= dim

P#(E)= r .

Since i, is surjective, dim AU 2 dim(AE*),=,; i.e.

2' >_ dim(AE*),=,.

Q.E.D. Corollary: Let y : F + E be a homomorphism of reductive Lie algebras, and let ( Q ? ~ P,(F) )~: P , ( E ) be the restriction of rpx to P*(F)(cf. Corollary 11, to Lemma V, sec. 5.15). Then the diagram --f

A(P*(F))

A(v,)P

___*

commutes.

A(P*(E))

§5. The structure of (AE*)B,o In this article, E again denotes a reductive Lie algebra. 5.17. The comultiplication in (A\E*)B=O. Let p : E @ E linear map given by P(X,

y) =x

+ Y,

x,y

E

-+

E be the

E.

Then the homomorphism p,:

AE @ AE

--f

AE

is simply multiplication. Hence it restricts to the multiplication map

The linear map dual to (p,),+o will be written

It is called the cornultiplication map for E. Let q : A ( E @ E)" (AE"),=o @ (AE")e=o +

denote the projection with kernel B(A(E @ E)"), and let p ~A:( E @ E)"

+ AE"

be the homomorphism extending p". Then YE(@) =

Lemma VI:

(q PA)(@),

@ E (AEb)o=o.

: + E be a homomorphism of reductive (1) Let q ~ F 199

200

V. Cohomology of Lie Algebras and Lie Groups

Lie algebras. Then the diagram

-I

I

commutes.

(2) y E is an algebra homomorphism.

Proof: (1) I n view of Proposition 111, (l), sec. 5.13,

is an aIgebra homomorphism. Hence, since ( P , ) @ = ~is multiplication,

Dualizing this relation yields (l), as follows from Proposition 111, (2), sec. 5.13.

(2) An automorphism Q of (AE),=, 0(AE),=, @ (AE),,, @ (AE),=, is given by

The dual automorphism of (AE"),=, @ (AE"),=, @ (A\EX)o=O@ (AE"),,, is given by

Now the multiplication maps for (AE),=, @ (AE),=, and for (AE")o-o @ (AE"),,, are given by EOE (PA

10-0

= ((~Jo=o

0(PA)e=o) Q 0

and by

(diaz)tj=o = (A$=, @ A$=,)

0

Q".

5. The structure of (AE*)e=,,

201

Dualizing the first relation yields

Hence

and so y E is a homomorphism.

Q.E.D. 5.18. The primitive subspace of (AE*)e,o. Exactly as in Lemma 111, sec. 5.14, it follows that

where Y E (A+E*),=, @ (A+E*),=,. The invariant elements @ in (A+E*),,, which satisfy YE(@)= @ @ 1

+ 1@ @

are called primitive. They form a graded subspace

P,=CPi j

of (A+E*),=,, called the primitive subspace. If p: F -+ E is a homomorphism of reductive Lie algebras, then by Lemma VI, sec. 5.17, p;=o restricts to a linear map pp: Pp

+P E .

T h e following lemma is proved in exactly the same way as Lemma IV, sec. 5.14,and Lemma V, sec. 5.15. Lemma VII: (1) The homogeneous primitive elements of (AE*),=, have odd degree.

(2) If 01,. . . , @ p are linearly independent homogeneous primitive elements, then A . A # 0.

- -

202

V. Cohomology of Lie Algebras and Lie Groups

(3) If D,(E) denotes the ideal ((A+E)o-o)zin (AE)+o, then PE = D*(E)I (with respect to the duality between (A+E")o,o and (A+E),=,). Lemma VII implies that @EJ?E.

@A@==,

Thus the inclusion map FE xE:

--f

(A+E")e-o extends to a homomorphism

APE

-+

(AE*)o=o.

If APE is given the gradation induced from that of PE , then x E is homogeneous of degree zero. Theorem 111: Let E be a reductive Lie algebra. Then

is an isomorphism of graded algebras. Thus (AE"),,, and H * ( E ) are exterior algebras over graded subspaces with odd gradation. Proof: The theorem is established with the aid of Lemma VII in exactly the same way as Theorem 11,sec. 5.16, was proved from Lemmas IV and V.

Q.E.D. 5.19. Corollary I: If p: F algebras, then the diagram

-+

E is a homomorphism of reductive Lie

commutes. Proof: Apply Theorem 111, and the second remark after Theorem I, sec. 5.12, noting that all maps are homomorphisms.

Q.E.D.

203

5. T h e structure of (AE*)e=o

Corollary 11: The PoincarC polynomial of (AE"),=o (and hence the PoincarC polynomial of H " ( E ) ) has the form

f = (1 + tot) - - . (1 + t"). Here the exponents are odd and satisfy (n = dim E ) .

g, = n

Proof: Let tQi be the PoincarC polynomial of PE. Then the gi are odd and the product above is the PoincarC polynomial of APE. T o prove the second statement, observe the elements of top degree in APE have degree g, - .. g,, while the elements of top degree in (AE"),=o have degree n.

+

+

Q.E.D. Corollary 111: T h e Retti numbers of a reductive Lie algebra satisfy n

n

(-l)pbp = 0

and

p=o

C bp = 2' p=o

(n = dim E,

r

=

dim PE).

Moreover, n = r (mod 2). Proof:

Apply Corollary 11, noting that gi = 1 (mod 2).

Q.E.D. Finally, let y p : PE -+ PE @ PE denote the diagonal map yp(@) = @

@ @,

@E

PE.

Extend it to a homomorphism Ay,: APE + APE @ APE. Proposition IV: Suppose E is a reductive Lie algebra. Then the diagram

XE

(AE"),,, commutes.

AYP

1- I

APE

APE @ APE N

YE

XEOXE

(AE*),=o @ (AE")e-o

V. Cohomology of Lie, Algebras and Lie Groups

204

Proof: I n fact, for @ E PE we have (by the definition of P E ) (YE

xE)(@)

=@

8 f 1 @ @ ==

(%J3

@ xE)(yP@)*

Since all the maps are homomorphisms, the proposition follows. Q.E.D. 5.20. The invariant subspaces of low dimensions. Let E be a semisimple Lie algebra with PoincarC polynomial

(cf. Corollary 11, sec. 5.19). Since E’ = E, it follows that b,(E) = 0. Thus, because the gi are odd, gi2 3 (i = 1, . . . , Y). It follows that

b,(E) = 0. These equations in turn imply that (A3E*)0=0= P i Moreover, they show that the hypotheses of Proposition I, sec. 5.7, are satisfied. Hence the linear map e defined in that section is an isomorphism e : (V2E*)0,0 P i . N

CJ

I n particular, a nonzero primitive element @ = e ( K ) ( K , the Killing form); i.e., @(x,y, z ) = tr(ad[x, y ]

Furthermore it follows that b,(E)

0

ad z ) ,

=

E

P i is given by

x, y , z E

E.

(5.15)

dim(V2E*)0,0.

Proposition V: Let E = El @ . @ Em be the decomposition of a semisimple Lie algebra E into simple ideals. Then:

(1) b l ( E ) = b,(E) = 0. (2) b,(E) 2 m. (3) If either r is algebraically closed, or I’ = W and the Killing form is negative definite, then b,(E) = m. Proof: ( 1 ) is proved above. In view of (l), the Kunneth formula (cf. sec. 5.9) implies that

H3(E) =

2 H3(Ei).

i-1

205

5. The structure of ( A E * ) O = ~

I t is thus sufficient to consider the case that E is simple; i.e., m = 1. T o prove (2), observe that formula (5.15) determines a nonzero element in (A3E"),=,. Thus b3(E)2 1. It remains to establish (3). Assume that r is algebraically closed, and let Y E (V2E"),=,. Since the Killing form is nondegenerate, Y determines a linear transformation y : E + E such that x , y E E.

Y ( x , y )= K(yx,y),

In view of the invariance of Y we have y

0

(ad x) = (ad x)

0

y,

xE

E.

I t follows that if 1 is an eigenvalue of y , then ker(y - izi) is a nonzero ideal in E. Since E is simple, this implies that ker(y - h )= E ; i.e., y = 11.This shows that Y = AK, whence dim(V2EX),=,= 1. Finally, suppose r = R and K is negative definite. Proceed as above, observing that y is self-adjoint with respect to I(,and so has real eigenvalues. Q.E.D. Corollary I: Let E be a reductive Lie algebra over an algebraically closed field. Let I = dim 2, and denote by m the number of simple ideals in E. Then

b,(E)

= I,

b,(E) =

( 1 ) , b ( E ) ( I ) + m, and M E ) =

=

Corollary 11: Let E be a simple Lie algebra of rank 2 over an algebraically closed field. Then the PoincarC polynomial of H ( E ) is

f = (1

+ t3)(l + tn-3)

(n = dim E ) .

$6. Duality theorems In this article, E denotes a reductive Lie algebra. 5.21. The duality between primitive spaces. Proposition VI:

Let

E be a reductive Lie algebra. Then the scalar product, ( , ), between (AE*)s,o and (AE),=, restricts to a scalar product between the primitive subspaces PE and P*(E). Moreover, it satisfies

Proof: The isomorphism x E (cf. sec. 5.18) restricts to an isomorphism XE:

(A+PE). ( A f P E )-5(A+E*),_o (A+E*),=o.

Thus (cf. sec. 5.15)

(A+EX),,o = PE @ D*(E). On the other hand, by Lemma V, sec. 5.15,

P*(E) = D"(E)L. This shows that the spaces PE and P * ( E ) are dual with respect to the restriction of ( , ). Next observe that formula (5.16) holds if @ and a are homogeneous of the same degree. Thus we need only show that iE(U)@ =

0,

a

E

P,(E), @ E P i , I? > j .

But, by Lemma VII, sec. 5.18, PE = D*(E)L. This implies that for b E (Ak-jE)o,O,

(iE(a)@,b ) = (@, Hence iE(a)@

a /'Ib ) = 0.

= 0.

Q.E.D. 206

207

6. Duality theorems

5.22. Duality theorems. As we observed in the preceding section, the scalar product between (AE*),=, and (AE)e=orestricts to a scalar product between PE and P,(E). This scalar product in turn induces a scalar product between APE and AP,(E).

Theorem IV: Let E be a reductive Lie algebra. Then the isomorphisms x E : APE -5(AE")e=o

and

x,:

AP,(E) -2, (AE)e=o,

preserve the scaIar product ; i.e., (x,@,

%,a> = (@, a),

@ E APE, a

E

AP,(E).

Proof: We observe first that this relation holds for @ E a E P,(E) by definition. Now let

PE

and

x%: AP,(E) + (AE),=,

be the linear map dual to x E with respect to the given scalar products. It has to be shown that xj$ = x ~ l . Observe first that ( x E @ , %,a) = (@, a ) if a E P * ( E ) . Indeed, if @ is primitive this is true by definition, while if @ is a product of primitives both sides are zero (cf. Lemma V, sec. 5.15). I t follows that a

.%(a) = x;'(a),

E

P,(E).

Thus to prove xf = x;* we have only to show that xj$ is a homomorphism. For this purpose we establish two lemmas. Lemma VIII:

Let a E P*(E).Then the operator ~ E ( u . ) : (AE')o=o

-

(AE"),+,

is an antiderivation. Proof: Since the map A , : (AE),=, (AE),=, 0(AE),=, is morphism (cf. Proposition 111, (l), sec. 5.13), we have --f

A d * ( a ) ) O 4= 4O Aa). Dualizing this relation we obtain

a

homo-

V. Cohomology of Lie Algebras and Lie Groups

208

Thus

But, since a is primitive, it follows that d , ( a ) Hence

=a

@1

+ 1 @ a.

Lemma IX: Denote by i p ( a )the substitution operator in the algebra APE induced by a E P#(E).Then xE

ip(a) = i E ( a )

xE.

Proof: Since ip is an antiderivation in A P E , x E o ip(a) is a xE-antiderivation. On the other hand, Lemma VIII shows that iE(a) is an antiderivation in the algebra (AE"),=,. Thus iE(a)0 x E is a xE-antiderivation. Hence it is sufficient to prove that %E

if'(a)(@)= i E ( a )

@E

xE(@),

PE.

But in PE,x E reduces to the identity. Thus in view of the definitions of the scalar products and Proposition VI, sec. 5.21, XE

ip(a)(@)= (@, a ) = i E ( a ) @ = iE(a)

xE(@),

@E

PE. Q.E.D.

5.23. Proof of Theorem IV: Dualizing the formula in Lemma IX gives Llp(a) O

=

OLl(a),

where p p ( a ) denotes the multiplication operator (induced by a E P* (E)) in the algebra AP,(E).Since a = xga, this can be rewritten as x B ( a ) A xB(b) = x s ( @A

b),

aE

P'(E),

b E (AE)~=O.

209

6. Duality theorems

Clearly this yields

Since Pu; ( E ) generates (AE),=, , X % must be a homomorphism. I n view of sec. 5.22, Theorem IV is now proved. Q.E.D. Corollary: For any b E AP*(E), XE

0

&(b) = i,(X*b)

0

XE.

$7. Cohomology with coefficients in a graded Lie module

In this article E denotes an arbitrary finite-dimensional Lie algebra, and O,, denotes a representation of E in a graded vector space M = C pMp.

0AEX. Consider the graded vector space = 1 ( M 0AE")', where ( M @ AE")' = C Mp @ APE".

5.24. The space M

M @ AE"

r

p+q=r

Extend the multiplication operators p(@) (@ E APE",q = 0, 1 , . . .) to operators p(@) in M 0AE" by setting

,u(@)(z0Y) = ( - 1 ) P q ~

0@ A

@ E A'E",

Y',

Y E AE",

z E Mp.

Extend the substitution operators iE(a) ( a E A E ) to the operators iE(u) in M 0AE" defined by

i E ( a ) ( z@ Y) = (-

1)pqx

0iE(a)Y,

aE

AqE, Y E AE",

z E Mp.

In particular, iE(x) (x E E ) is homogeneous of degree -1. Similarly denote the operators O,,(x) @ i and i 0O,(x) simply by O,,(x) and O,(x). Then O,, and OE are representations of E in the graded space M 0AE". They determine the representation 6 given by

O(x)

= e,,(x)

+ O&),

x E E.

More explicitly,

e(x)(Z 0Y) = e,,+lZ 0Y + 0eE(x)y', The symbols O(M @ AE*) and ( M 0AE"),,,

E

E,

E

M , Y E AE*.

will refer to this represen-

tation (cf. sec. 4.2). In view of sec. 5.1, q x ) i E ( a )- iE(a)O(x)= iE(OE(x)a),

x E E,

In particuIar, if a is invariant,

O(x)i,(a) = iE(a)O(x), 210

x E E.

aE

AE.

7. Coefficients in a graded Lie module

211

Next, extend S, to the operator (again denoted by 6,) in M @ AE" given by QE(z

z E Mp, !?' E AE".

@ Y ) = ( - 1 ) p ~ @ dE!?',

Then the relations (5.3) of sec. 5.2 yield the formulae &(x)Q,

s;

+

=

QEiE(.)

O,(x),

= 0,

(5.3)

and Q,O,(x)

= O&)QE

for the corresponding operators in M @ AE". Clearly, d E o M ( x ) = O,(x)S,, and so Q,O(x) = e(x)dE , x E E. and 8. Define an operator So in M @ AE" by

5.25. The operators

C OM(ev)x @ e+y A !€I,

QO(x@ Y ) = ( - 1 ) ~

x

E

Mp, Y E AE",

V

where e"", eu are dual bases for E" and E. Evidently, So is independent of the choice of the dual bases. Moreover, Qo =

c

P(egv)eM(ev)=

V

c

~M(ev)Lc(e**).

V

Next, define an operator 6 : M @ AE"

+

by 6 = dB

M @ AE",

+ do.

Observe that dE , S o , Q are all homogeneous of degree 1. If M @ APE" is identified with the space L ( A P E ;M ) (0 5 p 5 n) by the equation

9 E M @ APE", a E APE,

9 ( a ) = iE(a)9, then 6 is given by the formula

+ c
,Xj], xo .. . &, . . . ,gj . . . ,x p ) , xi E E , i = 0, . . . ,p , Q E L(APE;M ) ,

1)i+iQ(

[X(

(cf. formula (5.7), sec. 5.3).

y

y

y

V. Cohomology of Lie Algebras and Lie Groups

212

It will now be shown that 6 satisfies the relations (5.17) In fact, it follows from the definition that

This, together with (5.3), yields the first relation. To establish the second, let e+v, e, be a pair of dual bases for E* and E. Observe that for x, y E E,

C

(e*Y

A

e+@,x A y ) [ e , , e,] =

V
(edV A V

e+p,

x A Y ) 8 E ( e V A e,)


=

AY),

and

It follows that, in E @ A2E*,

Now an easy computation gives

Since 6%= 0, it follows that

Finally, applying 6 on the left and right of the first relation of (5.17) yields the third relation. As an immediate consequence of (5.3) and (5.17) we have the relation S,O(x) = O(x)Se,

xE

E.

7. Coefficients in a graded Lie module

213

5.26. The cohomology and the invariant cohomology. It follows from the relations of sec. 5.25 that ( M @ AE*, 6) is a graded differential space. T h e corresponding cohomology space H ( M @ AE", 6) is called the cohomology of E with coeficients in M , and is denoted by

H"(E; M ) . If 0, is the trivial representation, then 6,

=

0 and so 6 = S E . I n this

case we have

H " ( E ; M ) = M @ H"(E). On the other hand, relation (5.17) of sec. 5.25 implies that 0 represents E in the graded differential space ( M @ AE", S). Thus the invariant subspace is stable under S. T h e corresponding cohomology is called the invariant cohomology of E with coeficients in M and is denoted by H ( ( M @ AE"),=o, 6). Finally, the Koszul formula of sec. 5.3, together with the definition of d o , gives 263

+

p(e"")O(e~)-

68 ==

(5.18)

V

It follows that in the invariant subspace ( M @ AE"),=,, S, reduces to - 2 6 E , while 6 reduces to - B E . Thus

5.27. Representation in graded algebras. Assume that 0, is a representation of E in a graded algebra M . Then the operators 0,,(x) (in M ) are derivations, homogeneous of degree zero. Give M @ AE" the algebra structure defined by the anticommutative tensor product. Then a simple verification shows that :

( 1 ) iG(x) (x E E ) is an antiderivation, homogeneous of degree -1. (2) BE(x), 0,(x), and 0(x) ( X E E ) are derivations, homogeneous of degree zero. (3) B E , S,, and S are antiderivations, homogeneous of degree 1. Thus M @ AE" and ( M @ AE*),=, become graded differential algebras, and so the corresponding cohomology spaces become graded algebras. 5.28. Reductive Lie algebras. Proposition VII: Let E be a reductive Lie algebra, and let 0, be a representation of E in a graded vector space M . Then

214

V. Cohomology of Lie Algebras and Lie Groups

(2) If Ow is a semisimple representation, then the inclusion ( M @ AE*),=, -+M @ AE* induces an isomorphism

H ( ( M @ AE*),=, , 6) % H + ( E ;M ) . Proof: (1) Recall from sec. 5.26 that in ( M @ AE+)o,o, 6 = - B E . Thus apply Theorem V, sec. 4.11 with

( X , 6,)

=

(AE*, 6,)

and

( Y , 6,)

=

( M , 0).

(2) It follows from formula (5.17), sec. 5.25, that the representation fl# of E in H ( M @ AE") is trivial. Thus (2) is a consequence of Theorem IV, sec. 4.10.

Q.E.D.

$8. Applications to Lie groups Throughout this article, E denotes a Lie algebra of a Lie group G. 5.29. The algebras H + ( E ) and H*(G). Recall from sec. 1.2, volume 11, that each vector h E E determines a unique left invariant vector field x h on G such that X,(e) = h. Moreover, we have the operators i(X,), 8 ( x h ) , and S (substitution operator, Lie derivative, and exterior derivative) in the algebra A ( G ) of differential forms on G. These operators restrict to operators iL(h),BL(h) and SL in the algebra A,(G) of left invariant forms (cf. sec. 4.5, volume 11). Moreover, an isomorphism

5 AE'

rI,:A,(G)

is defined by tL(@) = @ ( e ) (cf. Proposition 11, sec. 4.5, volume 11). Under this isomorphism, the operators iL(h), B,(h), and SL correspond, respectively, to the operators iE(h), O,(h), and 6, defined in sec. 5.1 and sec. 5.2. (To see this, apply Proposition 111, sec. 4.6, volume 11, and formula (5.7), sec. 5.3.) In particular, zL induces an isomorphism tL#:H(A,(G),

S,) 5 H * ( E )

-

(which, in volume 11, is denoted by (tL)#). Composing t z l : AE' 5 A L ( G ) with the inclusion AL(G) we obtain a homomorphism &G:

(AE", 6,)

of graded differential algebras. Let 8;:

H*(E)

-

A(G),

( A ( G ) ,6)

H(G)

denote the induced homomorphism. On the other hand, if G is connected, then tL restricts to an isomorphism r z :A,(G) 5 (AE"),=, , 21 5

216

V. Cohomology of Lie Algebras and Lie Groups

where A,(G) is the algebra of bi-invariant forms (cf. sec. 4.9, volume 11). In view of sec. 4.10, volume 11, we have the commutative diagram

AdG)

- HdG)

H(G) (5.19)

Here zEdenotes the homomorphism induced by the inclusion (AE*),=, -+Z"(E); if E is reductive, it coincides with the isomorphism zE in Theorem I, sec. 5.12. T h e composite 0 nE will be denoted by a G , GIG:

(AEQ),=o-+ H(G).

Note that, for @ E (AE")B=O,aG(@)is represented by the (closed) biinvariant form t~'(@). I t follows from Theorem 111, sec. 4.10, volume 11, that if G is compact and connected, then all the maps in diagram (5.19) are isomorphisms of graded algebras. Next, let p: K G be a homomorphism of connected Lie groups, and let p': F E be the induced homomorphism of Lie algebras (cf. sec. 1.3, volume 11). From p we obtain homomorphisms -+

.--f

q?": A ( K ) +- A(G),

-

P,L": A@)

AdGI,

F?I": A m

-

-

AdGI,

as well as homomorphisms v#: H ( K ) +- H ( G ) and q#L:H L ( K ) H,(G), induced by p* and p,L".On the other hand, p' induces homomorphisms (p')", (p')i=o,and (p')", as described in sec. 5.6. It follows from the results of sec. 4.7, volume 11, that the diagram

(AE"),,,

nl3

H"(E)

H(G)

commutes. This shows that (5.20)

217

8. Applications to Lie groups

Example: Products: Let K and G be connected Lie groups with Lie algebras F and E. Let nl:K x G + K and n2:K X G -+ G be the projections. Then 7 ~ ;and 7 ~ ;are just the projections from F @ E to F and E. Now write (A(F @ E)")o=O= (/IF*),=, @ (AE"),,,, as in sec. 5.9. If 0 E (AF"),+o, Y E (AE")B=o, then

On the other hand, recall that the Kunneth homomorphism x#:

H ( K ) 0H ( G ) + H ( K x G )

) sec. 5.17, volume I). is given by %#(a0B ) = nf(a) $(,!I(cf.

Proposition VIII:

If K and G are connected, then the diagram

commutes.

Now apply formula (5.20), sec. 5.29, to the homomorphisms n1 and n 2 , to obtain aKxc(@

0Y) = ni+(ad@)) n f ( 4 W ) = (ClK 0aQ)(@ @ Y). *

%# 0

Q.E.D. 5.30. The map cp. Let G be a connected Lie group with reductive Lie algebra E. Define a smooth map p: G X G G by -+

y ( x , y ) = xy-',

x, Y E G.

It induces a homomorphism y # : H ( G x G ) + H ( G ) .

218

V. Cohomology of Lie Algebras and Lie Groups

Since p(e, e ) = e, the derivative dp restricts to a linear map p': E @ E E. Evidently p'(h @ k) = h - k, h, k E E. Now extend (p')" to a homomorphism (p')^:AE" + A ( E E)". ---f

Let 7EeE:

A ( E O E)"

be the projection with kernel

+

( A ( E 0E)")e=o

oEeE(A(E

@ E)"), and define a linear map,

Proposition IX: With the hypotheses above, the diagram

(AE")e-o

qJh

( A ( E O E)")e=o

commutes. Proof: Observe that

It follows that the map p*: A ( G x G) +- A ( G ) restricts to a homomorphism q ? : A,r,(Gx G) + A,(G). Now fix @ E (AE"),=, and define Y E A,(G) by Y = T?'(@) (cf. sec. 5.29). Then (again cf. sec. 5.29) Y is closed, and represents a G ( @ ) . Thus q?(Y) represents p*n(:(@). On the other hand, since yj+'(Y)is left invariant, and

219

8. Applications to Lie groups

Now Theorem I, sec. 5.12, (applied to E Q E A ( E 0E)*, (p')4 = @@

Finally, apply

t~' to

+

0E)

shows that for some

BEeEQ.

this equation to obtain

Thus, ~y-l(ph@)also represents y#ac(@);i.e.,

5.31. The comultiplication. Let G be a connected Lie group with reductive Lie algebra E. Then the multiplication map pG: G X G + G (given by p G ( x , y ) = xy) determines a homomorphism of graded differential algebras pz: H ( G x G) +-- H(G). On the other hand, we have the homomorphism

defined in sec. 5.17. Identify (AE*),=, 0(AE*),=, with ( A ( E @ E)*),=, as in sec. 5.29 and sec. 5.9. Proposition X: With the hypotheses above, the diagram

commutes. Proof: Define a diffeomorphismf: G X G -+ G X G by

V. Cohomology of Lie Algebras and Lie Groups

220

These equations imply that the induced isomorphismf" restricts to an isomorphism

f:: A,(Gx G)

A,(Gx G).

Next observe that f ( e , e ) = e and that the derivative df restricts to the linear isomorphism f' of E @ E given by f ' ( h , K)

=

(h, -12).

I t follows that the automorphism (f')"of A ( E @ E)" commutes with the operators 8 E @ E ( h , 12). Hence it restricts to an automorphism (f'):30 of the invariant subalgebra. It is immediate from the definition that the diagram 'I

A(Gx G) c-A,(Gx G) N ( A ( E 0E)")e=o f+

1

=

commutes. It follows that

Finally, observe that the maps pG and 9 are connected by PUa = Y of,

where rp is the map defined in sec. 5.30. Thus,

(cf. Proposition IX, sec. 5.30). On the other hand, since (f')"commutes with the operators eE,$,E(h, k ) , it follows that (f')"commutes with the projection v E ~ Edefined in sec. 5.30. This implies that for @ E (AE"),,,,

8. Applications to Lie groups

Since ,&(h, k) = h

221

+ k, it follows from sec. 5.17 that (5.22)

The proposition follows from relations (5.21) and (5.22).

Q.E.D. 5.32. The primitive subspace. Now assume that G is compact and connected. Then the Lie algebra E is reductive (cf. sec. 4.4). Use the Kunneth isomorphism to identify H ( G x G) with H(G) @ H ( G ) (cf. Theorem VI, sec. 5.20, volume I). Combining the example of sec. 5.29 with Proposition X, sec. 5.31, and observing that a. is an isomorphism (cf. Theorem 111, sec. 4.10, volume 11) we obtain the commutative diagram

-

(AEX),=, aG

1

YE

(AEf)O=O

i

aG@aG

G

H(G)

0(AE"),=o

+

PZ

H ( G ) 0H ( G )

In sec. 4.12, volume 11, we defined the primitive subspace PG c H+(G) to be the subspace of classes a E H+(G) satisfying p#c(a)= CI @ 1

+ 1 @ a.

It follows that aG restricts to an isomorphism

Thus there is a commutative diagram

of algebra isomorphisms, where xE is the isomorphism defined in sec.

222

V. Cohomology of Lie Algebras and Lie Groups

5.18 while AG is defined in sec. 4.12, volume 11. This diagram shows that in the compact case Theorem 111, sec. 5.18, is equivalent to Theorem IV, sec. 4.12, volume 11. Moreover, Theorem IV, sec. 4.12, volume 11, asserts that dim PG = dim T , where T is a maximal torus in G. It follows that the rank of the Lie algebra of a compact Lie group is equal to the dimension of a maximal torus. An algebraic version of this result will be established in Chapter X (sec. 10.23).

Chapter VI

The Weil Algebra I n this chapter E denotes a finite-dimensional Lie algebra.

$1. The Weil algebra 6.1. The algebras VE" and VE. Consider the graded vector space E" which is defined as follows: E" is equal to E" as a vector space and the gradation of E" is given by

deg x"

= 2,

x" E

E",

The induced gradation of the symmetric algebra VE" is given by

(VE*)'q

= VqE",

(VE")Zrl+l = 0,

420.

Thus VH" is evenly graded, and so it is a graded anticommutative algebra. T h e representation x ++ -(ad x)" of E in E" gives rise, evidently, to a representation of E in 177%.Extending the linear transformations -(ad x)* to derivations O,(x), we obtain a representation of E in VE". I n a similar way we form the graded space E ( E = E as a vector space; deg x = 2, x E E ) and the symmetric algebra V E . Then V E is a graded anticommutative algebra and the adjoint representation of E in E extends uniquely to a representation 8s of E in the graded algebra V E . T h e representations 0, and Os are contragredient with respect to the induced scalar product between VE" and V B . Thus we have

(VB"),,,

=

O(VE)'-

and 223

O(VE*) = (VE)i=,.

224

VI. The Weil Algebra

Next, let p: F E be a homomorphism of Lie algebras. Then p and its dual rp" extend to homomorphisms of graded algebras -+

vv: V F

-+

VE

pv:VF"

and

+-

VE".

Clearly we have

and so

pv

restricts to a homomorphism

Now suppose E is reductive. Since

it follows that the representations 0, and 8, are semisimple (cf. Theorem 111, sec. 4.4). Hence in particular

VEX= (VE*)o=O0O(VE*)

and

VB = (VE),=,

0 O(VS).

It follows from these decompositions that the scalar product between VE" and VE restricts to a scalar product between (VE*)o=oand (VE)e=o. 6.2. The algebra W(E). Consider the anticommutative graded algebra W ( E )= Crz0W'(E) given by

W ( E ) = VE" @ AE", W r ( E )=

C

(VE")' @ AqE* =

ptq=r

1

VPE" @ APE".

2p+q=r

Thus WO(E) = r, W ( E )= 1 @ E", and W 2 ( E )= E" @ 1 @ 1 @ A2E#. We shall now translate some of the results of article 7 of the preceding chapter, with VE* = M and 0, = 0., I n fact, as in sec. 5.24, extend the operators p ( @ ) (@ E AE"), iE(a) ( a E AE), e E ( x ) , and O,(x) to W ( E ) by setting p(@)= 6 @ p ( @ ) ,

iE(a) =

6

@ iE(a),

&(x) = &(x) @ 6 ,

eE(x) = 6

@ eE(x).

(Note that VE" is evenly graded.)

1. The Weil algebra

Then

8E

225

and OS are representations of E in W(E).Next set b ( x ) = OE(x)

+ eS(x),

x E E*

Then Ow is also a representation of E in the graded algebra W ( E ) (it corresponds to the representation 8 of sec. 5.24). T h e invariant subalgebra and the "8"-subspace for this representation will be written W(E),=, and 8( W ( E ) )respectively. Assume E is reductive. Since 8 , ( x ) = 0, x E Z E , it follows from Theorem 111, sec. 4.4,that the representation BIv is semisimple. I n particular,

W ( E )= w(E),=,

oe ( W ( q .

Next, extend d E to the operator SE = L @ B E in W ( E ) , and let 68 be the operator determined by the representation 8, (cf. sec. 5.25). .Then 68

1

==

p(e"")eS(ev),

Y

where e"", e, is a pair of dual bases for E" and E. T h e operators BE and 8 8 are antiderivations, homogeneous of degree 1. In view of formula ( 5 . 1 7 ) , sec. 5.25, we have the relations

f (6,

z'E(x)(dE f

+ 0, (6, + 6B)elY(x)

(6, and

+ dO)iE(x)

=

eW(x),

(6-1)

=

=

8W(x)(6B

+

E*

On the other hand, formula ( 5 . 1 8 ) , sec. 5.26, yields

6.3. The antiderivations h and k. Define operators h and k in W ( E) by

h ( Y @ 1) = 0,

Y E VB",

xXi E E X ,

K ( 1 @ @)

@ E AE",

~ " Ei

and by

= 0,

E".

226

VI. The Weil Algebra

Then h and k are antiderivations in W ( E ) ,homogeneous of degrees 1 and - 1 respectively. I n terms of dual bases eQu,e, we can write

h = C ,us(e"u)0iE(e,)

and

k = C is(e,) Y

0,u(e"'),

where ,us(xf) and is(%) are the multiplication and substitution operators in VB". The operators h and k satisfy

In fact, since

h(l

0x")

= X"

k(l

0x")

= 0,

01,

h(x" @ 1) = 0

and

k(x"

+

01 ) = 1 0x",

it follows that hk kh reduces to the identity in E" But hk kh is a derivation and so (6.3) follows. Similar arguments show that

+

6.4. The antiderivation 6,.

X"

E

EL,

01 and in 1 @ E".

Set

Then dIt. is an antiderivation in W ( E ) ,homogeneous of degree 1 . If E is abelian, then dE = dB = 0 and so 6 , reduces to h. We shall now establish the relations

1. The Weil algebra

227

I n fact, the first and third relation follow at once from formula (6.1), sec. 6.2, and formula (6.4), sec. 6.3. Moreover, since (6, 88)2= 0 and h2 = 0, the second relation is equivalent to

+

+

( 6 ~ 8e)h X"

+ h(6E + So)

= 0.

T o prove this, we may restrict ourselves to elements of the form @ 1 and 1 @ x*, xu; E E", because both sides are derivations. Now

+ 6e)h + ~ ( S +B Se)](x"

[(d~

@ 1) = h68(~"0I ) =

1 Os(eV)x* v e*V. Y

But

On the other hand, [(SE

+ 6e)h + ~ ( S +E &)](l @ x") C (8,(eV)x* = C (8,(e,)xf

=

V

+ e", @ i,(eV)dEx") @ e*v + eeu @ 8,(eV)x*).

@ e*v

V

This is a vector in E* @ E". Its scalar product with x B y ( E E @ E ) is given by (X", - [Y, X I ) (x", - [x, yl) = 0. Hence [ ( 6 E f 6O)h f h(6E 60)](1 0x") = 0-

+

+

This completes the proof of (6.5). T h e graded differential algebra ( W ( E ) ,6,) is called the Weil algebra of the Lie algebra E. Formulae (6.5) show that 8, is a representation of E in the graded differential algebra ( W ( E ) ,6,) and that 6% = 0. I n particular, the invariant subalgebra W(E),,, is stable under 6,. Moreover, the relation (6.2), sec. 6.2, shows that the restriction of 6, to W(E),=, is given by

6, = h - 6 E .

228

VI. The Weil Algebra

6.5. Homomorphisms. Let p: F + E be a homomorphism of Lie algebras. Then p induces a homomorphism pw = pv 0pA:W ( F )+- W(E).

I t follows from sec. 5.6 and sec. 6.1 that

and so pw restricts to a homomorphism,

On the other hand, pw is a homomorphism of differential algebras: pwdw= dww.I n fact, we know from sec. 5.6 that pwdE= dww. It is immediate from the definitions that pwh = hp,. Finally, observe that pwd,(l @ x") = 0 = d/jpw(l

0x")

while

Since p w b and d e w are pw-antiderivations, it follows that pwdo= Bepw , and so V w b = dwm. 6.6. The cohomology of the Weil algebra. T h e purpose of this section is to establish the following Proposition I: Let E be a Lie aIgebra. Then the cohomology of W(E)and of W(E),=ois trivial,

1. The Weil algebra

229

Recall the definition of the antiderivation k in sec. 6.3. Define a derivation A in W(E),homogeneous of degree zero, by d

=

S,k

+ k6,.

Lemma I: Let 9 E W7(E),r 2 1. Then

where c,, = 1 and

In particular, the restriction of A to W+(E)is an isomorphism. Proof:

Set A1

=

(6,

+ Se)k + k ( 6 , + SO) = A - (hk + Ah).

Then

A , : VQE"0AE"

-+

Vq-lE" @ AE",

q = 0,1,

.. . .

Next, recall from sec. 6.3 that

(hk

+ kh)D = ( p + q)D,

D E VQE"@ APE".

Define operators Tg (q = 0, 1, . . .) in W ( E ) by

D E W'(E).

T g (Q )= (r - q)D, Then

Tg(D) = (hk

+ kh)Q,

D E VQE"@ AE",

and so it follows that

A

-

Tq:VQE"0AE"

-+

Vq-lE" @ AE".

Iterating this process we obtain

( A - To) 0 . * *

0

( A - T,)(Q) = 0,

9 E VQE"@ AE".

(6.6)

230

VI. T h e Weil Algebra

On the other hand, since T , reduces to scalar multiplication in each Wr(E),it commutes with every operator homogeneous of degree zero. Thus AT, = T,A and TpTq= TqTp. In particular, the relation (6.6) continues to hold for

9E

C VjE* @ AE". j sq

Now let SZ

E

W'(E),Y 2 1. Then Q E ( A - To)o

*

-

0

xjcr-lVjE" 0AE", and hence

( A - T,.-l)Q= 0.

Expanding this relation and using the fact that

Ti, *

* *

Ti,(Q) = ( Y

-

il)

*

-

*

(Y

- ip)Q,

we obtain the formula of the lemma. This formula shows that A rest:icts to surjective maps

W'(E) + W ( E ) ,

Y

2 1.

Since Wy(E)has finite dimension, d must be an isomorphism in each

W'(E). Q.E.D. Proof of the proposition: Since dP =

(6,k)P

+

db = 0 and k2 = 0, we have

(kd,)P,

p 2 1.

Thus if 9 E W'(E), Y 2 1, and dwQ = 0, Lemma I yields with

D = dwQl

It follows that H'(W(E)) = 0, Y 2 1. On the other hand, we have, trivially, Ho(W ( E ) )= r. Finally, suppose 9 E Wr(E)o=o, I 2 1, and 6,s = 0. Since K and dw commute with the operators e,(x), x E E, it follows that Ql is invariant. Hence, H+(W(E),=,)= 0. Q.E.D.

$2. The canonical map p E I n this article we shall construct a canonical linear map eE:

(V+E"),=o

-

(A+E"),=o

for an arbitrary Lie algebra E. 6.7. Definition: Let nE:W ( E )+ AE" denote the projection defined

by nE(1

@ @) = @

@ E AE*,

nE(Y' @ @) = 0,

and

Y E V+H*.

Then nE is a homomorphism of graded algebras. It satisfies the relations n,&

= 6EnE

and

nE&V(x) = OE(x)nE,

x E E.

I n particular, nE restricts to a homomorphism

Since (cf. formula (6.2), sec. 6.2) 6, reduces to i d , and n E h = n E 6 , = 0, it follows that

6w

(nE)&O

Moreover, if

+ h in

W(E)e=o,

(6.7)

= 0.

v : F + E is a homomorphism of Lie algebras, then

Lemma 11: Let Y E (V+E*),,o. Then there is a unique element @ E (A+E*),=o such that for some D E W+(E),,o, nEQ

=@

and

6&'=

Y @ 1.

Proof: Evidently, aW(P@ 1) = 0. Hence, by Proposition I, sec. 6.6, there exists an element f2 E W+(E),,, such that 6WQ =

Y @ 1.

Set @ = n E ( f 2 ) . 231

232

VI. The Weil Algebra

If 9, E W+(E),,, is another element such that

d&2

=Y@

fiWQl

1, then

- 9,)= 0.

Hence, again by Proposition I,

9 - 9, = sWQ,

QE

for some

w+(E),,,.

(cf. formula (6.7) above). Thus @ is independent of the choice of 9.

Q.E.D. The correspondence Y H @ defines a linear map @E:

(V+E")o,o

+

(A+E")o=o,

homogeneous of degree -1. I t will be called the Cartan map for E. In view of the definition we have

I n sec. 6.14 it will be shown that if E is reductive, then ker

@E

=

(V+E*)i=,

and

Im

@E =

PE.

Example: Let E be an abelian Lie algebra. Then the Cartan map is given by @ E y = Y,

!P E E";

@E!i?

= 0,

Y E (V'B")

*

(V'E").

I n fact, in this case dE = do = 0. Thus, if Y E W E " , then

Y @ l = - h1k ( Y @ l ) =

P

dw - k ( Y @ l ) ) .

(;

Hence @ E y=

1 - Z E k ( Y @ 1).

P

If p 2 2, then k ( Y @ 1 ) E V+E" @ AE* and so then k ( Y @ 1) = 1 @ Y and so @BY = ng(1 @ Y) = Y.

@ E y = 0.

If p

=

1,

2. The canonical map

-

eE

233

Proposition 11: If y : F E is a homomorphism of Lie algebras, then ye^=, @ E = @F '?$=O* Proof: Let Y E (V+E*)o,O and let Q E W+(E),,, satisfy &7Q

Then yvY @ 1 = yWS&

=

=

Y @ 1.

dlvy,mQ,and hence

6.8. Explicit formula for p E . I n this section we give an explicit expression for eE. Proposition 111: T h e Cartan map p E for a Lie algebra E is given by (cf. sec. 6.3 for k)

Lemma 111: Let Y E (V*EX)e=o.Then

where a, = 1 and

Proof: We adopt the notation of sec. 6.6, and show first that (d - Tl)

* * .

(A

-

Tp)(Y@ 1) = 0.

(6.8)

As in Lemma I, sec. 6.6, observe that

(d

-

TI)*

* '

( A - T*)(Y 01) = 1 0@,

where @ E (A2qE*),=,. Also note that the space W2g(E)o=o is stable under d and that each Ti restricts to scalar multiplication in W*g(E),=,. It follows that there are constants a, such that 1 @ @ = 2 (Yvd*(Y @ 1 ) V

=

c a,( d,+.k),(Y 01). V

VI. The Weil Algebra

234

Hence

1 @ @ = a0(Y@ 1 )

+ 6,Q,

some Q E W(E)o=o.

It follows that (cf. formula (6.7), sec. 6.7) @ = n E ( 1 @ @) =

(nE)o=oSW(Q) =

0.

This proves formula (6.8). Finally, obtain the lemma by expanding formula (6.8) in the same way as in the proof of Lemma I, sec. 6.6.

Q.E.D. Proof of Proposition 111: Recall from sec. 6.4 that in W(E)o=o, 6, = h - 6,. Further note that

These formulae imply that for !P E (VqE"),=,

n,k(S,k).(!P@

1) = 0,

,

p
and

n~k(G~~k)q-'@ ( ! P1 ) = (-1)q-17CEk(6&)q-1(p @ 1) (-l)~-'k(GEk)Q-'(!P @ 1).

1

Using these relations and Lemma I11 we obtain the proposition.

Q.E.D. Next we shall derive a second explicit formula for e E , considering VE" and AE* as the spaces of symmetric and skew symmetric multilinear functions in E. Proposition IV: The Cartan map for a Lie algebra E is given by

2. The canonical map

Proof: Define a linear map ~ 1 W : E"

dY)=

-+

eE

235

A2q-lEICbY

k(d,k)q-l(Y @ l ) , (2q- I)!

Y E WE".

Then, in view of Proposition 111 above, p restricts to eE in (VE*)o-O. On the other hand, since k is an antiderivation, a simple computation yields

Hence for Y E W E " ,

Corollary: The linear map e : (V2E")o-o3 (A3E")o,o of sec. 5.7 satisfies @

=

--2e3.

§3, The distinguished transgression I n this article E denotes a reductive Lie algebra with primitive space PE c (A+E")e=o. We shall construct a linear map tE:

PE

+

(V+E")e,o,

homogeneous of degree 1 , such that

6.9. The space W(E)i,,o. Fix an element a E (APE)@,,-,,p 2 1 , and consider the operator iE(u) in W ( E ) (cf. sec. 6.2). Since a is invariant, the relations of sec. 5 . 1 yield xE

iE(u)r3,(x) = B w ( x ) i E ( a ) ,

E.

Moreover, dualizing formula (5.8), sec. 5.4,and observing that a is invariant we find that

Since E is reductive and hence unimodular, dEu = 0 (sec. 5.10) and so the formula above becomes

iE(a)6E = (-l)PdEiE(a). Clearly, iE(a)o h = ( 6 @ i E ( a ) )o C pAY(e+")iE(e,) = (-1)Ph

0

iE(a).

Y

Now define a graded subspace W(E)+,, of W ( E ) by

W(E)+, = (52 E W ( E )I iE(a)Q = 0, a E (A+E),=o}. The relations above imply that Orv restricts to a representation of E in W(E)+,, and that the space W(E),,=, is stable under dE and h. Hence the invariant subspace W(E)ir=o,e=o= W(E)i,-o 236

n W(E)e=o

237

3. The distinguished transgression

is stable under S E and h. Since Slv coincides with h - BE in W(E),=, (cf. sec. 6.4) it follows that W(E)i,=o,e=o is stable under SIP. Proposition V: Let E be a reductive Lie algebra. Then the inclusion map j : (VE*),=, -+ W(E)ir=o,o=o induces an isomorphism

j ' : (VE")o=o

N

H ( W(E)i,,o,o=o,a,,-).

Lemma IV: The inclusion map (VE")o=o3 W(E)+,,,=, induces an isomorphism (vE")O=o Proof:

e

H ( W(E);,=O,O=OS E ) *

Let (AEe)iI=obe the subspace of AE" given by

(AE")i,=, =

0

ker

~E(u).

a E ( AfE)O=,,

Since OE(x)iE(u)= iE(U)OE(x), x E E, the representation of E in AE" restricts to a representation 8 in (AE")+, . Moreover, since the operators i E ( x ) , O E ( x ) ,and BE commute (up to sign) with iE(u) for an invariant element a, the relations 8 E ( x ) = Z ' E ( x ) S E S E i E ( x ) restrict to (AE")i,=o and imply that 8# = 0. Hence, applying Theorem V, sec. 4.11, and observing that the restriction of BE to ( A E ~ ) i r = o , ois= ozero, we find that the inclusion map induces an isomorphism

+

(VE")o=o

0(AE")iI=o,o=o5 H ( W(E)i,=o,o=o,6,).

But, in view of Lemma IX, sec. 5.22,

and so the lemma follows.

Q.E.D. Proof of the proposition: Set

and filter W(E)ir=o,o=o by the subspaces Fp

B E : Mp -+ Mp

and

=

Cj2p Mj. Then

h : Mp --+ MP+l,

p >_ 0.

VI. The Wed Algebra

238

Hence, in view of formula (1.6), sec. 1.7, the &-term of the corresponding spectral sequence is given by

H ( W(E)i,=O,O=o 8,).

El

9

Next we filter (VE#),=, by the subspaces f 3 p = Cj2p(VE+)je,o. Then, giving (VE#),=o the zero differential operator, we obtain a spectral sequence with

El = (VE'*),,o. Finally, observe that the inclusion map (VE")8,0 + W(E)iI=o,s-o, is filtration preserving. The induced map I?, + El is precisely the isomorphism of Lemma IV (cf. sec. 1.7). Thus the proposition follows from Theorem I, sec. 1.14.

Q.E.D. 6.10. The map T ~ .Lemma dIi4

V: Let @ E P E . Then

0@) E

W(E)i,-o,e=o.

Proof: Since 1 @ @ is invariant, we have for a E (APE),=,, p >_ 1, that

On the other hand, since P*(E) generates (AE),=o (cf. Theorem 11, see. 5.16) Proposition VI, sec. 5.21 implies that iE(a)@E Thus

r.

dTV(1@ i&'(a)@) = 0.

Q.E.D. I n view of Lemma V, a linear map

p: PE

-

z(W(E)ipO,8-13 b)

is given by

p(@) = dF(1 @ @), Let

B": Pz

+

@ E P,lj.

H(W(E)+o,e=-o, 6,) be the induced map, and define

3. The distinguished transgression

239

N

wherej#: (VB+)o=o sition V, sec. 6.9. Definition: Lie algebra E.

t Eis

H ( W(E)iI,o,e=o, dJv) is the isomorphism of Propo-

called the distinguished transgression for the reductive

Proposition VI: Let E be a reductive Lie algebra. Then the distinguished transgression has the following properties:

(1) It is homogeneous of degree 1. such that (2) For each @ E P E , there exists an SZ E W+(E)+o,e=o dW(1 @ @

(3)

@Eo Z E =

+ Q) = t E ( @ )

@ 1.

L.

Moreover, zE is uniquely determined by these properties. Proof: (1) and (2) are immediate from the definitions. T o prove ( 3 ) , recall that W(E)iI=o = VE+ 0(AE*)iI,O, and so

(because (A+EC)iI,O,e,O = 0-cf. the proof of Lemma IV, sec. 6.9). NOW let @ E PE and write tE@

01 = &(l

@@

+ SZ),

SZ E W+(E)+o,e=o.

Our calculation above shows that nESZ = 0. Thus

Finally, suppose t:PE --+ (VE+)e,O is any linear map satisfying properties ( 1 ) - ( 3 ) . Then, for @ E P E ,

[.(a)

- tE(@)]

@ 1 = SWQ,

E W+(E)g=o,e=o.

Now Proposition V, sec. 6.9, implies that r ( @ )= t~(@).

Q.E.D. 6.11. Homomorphisms. Let q ~ :F --+ E be a homomorphism of reductive Lie algebras and consider the induced maps

pi=o:P p +- PE

and

~ ) j = ~(VF*),,o :

+

(VE*)e,o.

VI. The Weil Algebra

240

Then, in general, the diagram

9%-0

1

does not commute (cf. Example 3, sec. 6.16). Proposition VII: If 9:F + E is a surjective homomorphism of reductive Lie algebras, then the diagram (6.9) commutes. Proof: We have the relations

b ( Y ) vrv = PW

h ( 9 Y ) and

iF(b)

9 l v = %'

;h'(V'hb)

y

F, E AF.

Since 9 is surjective, q,,maps (AF),,, into (AE)e=o. It follows that the map pw: W ( F )c W ( E ) restricts to a homomorphism (qW)ir=o,e=o:

4 - 0

W(F)il=o,e=o

+

w(E)iI=o,e-o-

1

clearly commutes, and the proposition follows.

Q.E.D.

$4. The structure theorem for (VH")e,o I n this article E denotes a reductive Lie algebra. 6.12. The filtration of W(E)e=o. Consider the ideals

They make W(E),=,into a graded filtered differential algebra (cf. sec. 1.18). Since in W(E),=,, 6, = h - S, and since

it follows that the spectral sequence associated with the filtration begins with (6.10) (EO, do) = (W(E),=o9 - 6,) (cf. sec. 1.7). Hence

E,

= H ( W(E),,, , -S,).

As an immediate consequence of Theorem V, sec. 4.11, we have Lemma VI: The inclusion map (VE")o=o@ (AE"),=, -+ W(E),,, induces an isomorphism

(VE"),=o @ (AE"),+o

3

H ( W(E)o=o - 8),.

+

3

6.13. Transgression. A transgression in the filtered differential algebra W(E),,, is a linear map z:

PE

-

(VE"),,,

with the following properties : 241

242

VI. The Weil Algebra

(1) t is homogeneous of degree 1. (2) For every @ E P E , there is an element Q E W+(E),=osuch that 6,&

= t@

01

1 0@ - Q E F'(W(E),=,).

and

Lemma VII: (1) A linear map t:PE (VE+)e,o, homogeneous of degree 1, is a transgression if and only if it satisfies @E0 t = 1. (2) The distinguished transgression t E defined in sec. 6.10 is a transgression. I n particular, a transgression always exists. --f

Proof:

(1) follows from the relation ker(ne),=o = F'( W(E),=o)

(cf. sec. 6.7). (2) is a consequence of Proposition VI, sec. 6.10.

Q.E.D. Theorem I: Let E be reductive and let z: PE-,(VE*)e=o be a transgression. Let PE be the evenly graded space defined by Pi = Pk-'. Then the induced homomorphism zv: VPE

--f

(VB"),=o

is an isomorphism of graded algebras. In particular, (VE"),,o is a symmetric algebra over an evenly graded vector space whose dimension is the rank of E. Proof: Consider the PE-algebra ((VE"),=,; sec. 2.8, it is sufficient to show that (tv @ L)':

H(VPE 0APE)

--+

t).I n

H((VE"),=o

view of Theorem I,

0A P E , 0;)

is an isomorphism. But H(VP8 0APE) = H O ( V P0 ~ APE) = r (cf. sec. 2.6), and so we are reduced to proving that H+((VE"),j,o

0A P E , Or)= 0.

Since t is a transgression there is a linear map a : P E homogeneous of degree zero, such that d,,-u(@)= T(@) @

1

and

(6.11) ---+

W(E),=o,

a ( @ )- 1 0@ E F1(W(E),+o). (6.12)

4. The structure theorem for (i~'E*)5=~

243

Extend a to a homomorphism (xA; APE -+ W(E),=,, and define a homomorphism 0: (VE*)o=O @ APE W(E),=o, by setting o(Y 0@) = (YO 1) * a,(@). +

A straightforward computation, using (6.12), shows that u

0

0, = d,,.

0

u.

Thus u induces a homomorphism u':

H((VE*),=o @ APE, Vz)

+

H ( W(E),,o, fiiv).

Now filter (VE*)o=o@ APE by the ideals E P

C (VE*)i=o 0APE.

=

J2P

The corresponding spectral sequence starts off with

Moreover, u is filtration preserving with respect to the filtration of sec. 6.12 and the filtration above. T o compute the map U " : I--+? ~ Eo, first write it in the form ((VE*)o=o @ APE, 0 )

+

(W(E)O=O,- f i E )

(cf. formula (6.10), sec. 6.12). Use the isomorphism X E of Theorem 111, sec. 5.18, to identify APE with (AE*)o,o.Then, in view of formula (6.12), a0(Y'

0@) = \v @ @,

Y/

E

(VE*)o=O, @

E

(AE*)e=O.

Thus uo is simply the inclusion map

and so, by Lemma VI, sec. 6.12, u$ is an isomorphism. Hence, by the comparison theorem (sec. 1.14), u # is an isomorphism. Finally, observe that, since u # is an isomorphism,

(cf. Proposition I, sec. 6.6), and so (6.11) is proved.

Q.E.D.

244

VI. The Weil Algebra

n;=, +

Corollary: If J;J(E)= (1 P) is the PoincarC polynomial of (AE"),=,, then the PoincarC series of (VLE*),=~ is given by

6.14. The image and kernel of p E , Theorem 11: Let E be reductive. Then the image and the kernel of the Cartan map are given by ker

@E =

(v+E*)o=o * (v+L??*),,, and

I m @g

= PE.

Proof: We first show that

(V+E*),=,

*

(V+B*)e=o c ker

Let Yl ,YzE (V+@*),=o. Choose Ql E W+(E),,, so that 6&2, Then since Sw(Yz@ 1) = 0, MQl

- (Yz 01)) = Yl v

Y 2

(6.13)

QE.

=

Y1 @ 1.

01,

whence v Yz) = nEQ1

AnE(Yz

@ 1) = 0.

This proves (6.13). Next let t be a transgression in W(E),=,,. Then Lemma VII, sec. 6.13, and Theorem I, sec. 6.13, yield respectively the relations eEOZ=L

and

(V+E")s=o = t(PE) @ (V+B"),=o * (V+E"),=o.

Theorem I1 is an immediate consequence of these relations and formula (6.13). Q.E.D. Corollary: A linear map if and only if

t homogeneous

t - tE: P E

where

tE is

+

(V+E"),=,

of degree 1 is a transgression

*

(V+B*)e=o,

the distinguished transgression (cf. sec. 6.10).

Proof: Since ker p E = (V+B*),=, corollary is equivalent to

- (V+B")e,O,

the condition of the

e ~ t Q E ~ E= L. z=

Now apply Lemma VII, sec. 6.13. Q.E.D.

4. T h e structure theorem for (VE*)o=,

245

6.15. Homomorphisms. Proposition VIII: Let p: F E be a homomorphism of reductive Lie algebras. Then the following conditions are equivalent: ---f

(1) p#: H " ( F ) +- H * ( E ) is surjective. (2) (AF*)O=O + (AE*),=, is surjective. pp: PF + PE is surjective. (3) (4) p:,,: (VF*)o,o t is surjective. ( 5 ) There are transgressions u and t in W(F),=,and W(E),,, such that q?;=o 0 t = CT 0 (p;,o. Proof: ( 1 ) 0 (2)u (3): This follows from Corollary I to Theorem 111, sec. 5.19. (3) * (4): Choose a linear map a : PF + PE, homogeneous of degree zero, so that pp 0 a = 1.

Let t be any transgression in W(E),,, and define a linear map U: PF W(F),,o by u = p:=, 0 z 0 u. --+

Then u is homogeneous of degree 1, and by Proposition 11, sec. 6.7,

Thus, by Lemma VII, sec. 6.13, u is a transgression in W(F)O=o. I t follows from Theorem I that u(PF)generates (VF"),,,. Since

Im pg=O Y this shows that p;;=ois surjective. (4) 3 (3): Observe that, by Proposition 11, sec. 6.7, @=O

@ E = @F

Since pi=, is surjective and Im sec. 6.14), it follows that

Thus pp is surjective.

q?i=O.

eE = PE,I m eF = PF (cf. Theorem 11,

VI. T h e Weil Algebra

246

(4) (5): Let

tl

be any transgression in W(E)o=o.Then, for

@ E ker q p ,

@&i=otl(@)

= QO^=OQETl(@) = @=O(@)

= 0.

Hence, in view of Theorem 11, sec. 6.14,

Since

Q;=~

is surjective, it follows that there is a linear map

homogeneous of degree 1, and such that

Now choose a graded subspace P c PE so that

PE = P @ ker pp. Define

t:PE ---*

(V+E*)O=O by

Then, in view of the corollary to Theorem 11, sec. 6.14, gression in W(E),=,. Moreover, by the definition of /I, t : ker

t

is a trans-

qp -+ ker qiEO.

On the other hand, since (4) (3), pr is surjective. Thus pp restricts to an isomorphism P-% P F . Let a : PF 32 P denote the inverse isomorphism. Define u : PF + (VF*)o=oby

Then eF 0 u = L (as in the proof that (3) e-(4))and so u is a transgression in W(F)e=o.

4. The structure theorem for (VE*),g_o

247

Finally, since

-

it follows that

o t = CT o

( 5 ) (4): Suppose t and G are transgressions satisfying ( 5 ) . Then Im cr c I m But by Theorem 11, sec. 6.14, I m CT generates (VF"),,,. Hence p&, is surjective. Q.E.D. 6.16. Examples. 1. Let E be the Lie algebra of linear transformations with trace zero in a 2-dimensional vector space. Then, with respect to a suitable basis, h, e, f,of E,

[ h , f ] = -2f,

[h, el = 2e,

[ e , f ] = h.

Since E is semisimple, it follows from Proposition V, sec. 5.20, that PE is a 1-dimensional subspace of degree 3. Moreover, if K denotes the Killing form of E, then @E(K)= - & ( K ) # 0 (cf. sec. 6.8 and Proposition I, sec. 5.7). Thus, by Theorem I, sec. 6.13, (VB"),,, consists of the polynomials in K ; i.e., 1, K , K 2 , , . . , K*, . . . is a basis for (V"B)o,o. 2. Let F be the abelian subalgebra of E (cf. Example 1) spanned by h. Then F is reductive in E. Moreover,

(AF")o,o = AF"

=

A(h")

(VF"),g,o = VF"

and

= V(h").

Thus (VF"),g-o consists of the polynomials in h". Now consider the inclusion map p: F -P E. Then, since PE = P i and PF = Pi, it follows that the restriction of &o to PE is zero, pp = 0. On the other hand, let e", f",h* be the basis for E" dual to e, f,h. Then K = 4(hY v h* e* v f " ) ,

+

and hence

&l(K) = 4 ( q 2 . Note that in this case none of the conditions of Proposition VIII, sec. 6.15, can hold.

VI. The Weil Algebra

248

3. Consider the Lie algebra L = E 0F, where E and F are the Lie algebras of Examples 1 and 2. Define y : F + L and x: L -+ F by

Y(~)=PY@Y, y € F

n(xOy)=y, X E E , ~ E F .

and

Then n o y = L and so y;=, o n&, = L. In particular, y;=, is surjective. Hence, Proposition VI 11, sec. 6.15, shows that, for appropriate transgressions z and u in W(L),=, and W(F),=o9 y;=, 0 = u 0 yP-0. Nonetheless yi=o 0 t L # t F 0 y i X 0 , where guished transgressions. Indeed, write

PL = PE 0P F

and

t L

and

t Fare

the distin-

(VE")s=o = (VE*)o=o@ (VF")o,o.

Then it follows immediately from the definitions that tL(@1

@ @Z)

=

zE(@l)

@

+ 1@

tF(@2),

@I

PE,

On the other hand,

Thus, if

# 0, it follows from Example 2 that (yi=o

zL)(@i

0@ z ) f

(zF

Y,e"=o)(@i @ @z).

@z E

PF.

$5. The structure theorem for (VE)o=o, and duality In this article E denotes a reductive Lie algebra. 6.17. The structure of (VE),=, tion Os of E in V E .

.

Recall from sec. 6.1 the representa-

Theorem 111: Let E be a reductive Lie algebra. Then the graded algebras (VE),,, and (VP"),=,, are isomorphic. I n particular, (VE),=, is a symmetric algebra over an evenly graded vector space, whose dimension is the rank of E. Proof: By Theorem 11, sec. 4.4,there is an inner product ( , ) in E such that X,Y, 2 E E. (Cx,yI, 2) ( Y , 1x9 23) = 0,

+

Denote the corresponding linear isomorphism by a: E 5 E*,

Then a o ad x = -(ad x)"

o

a, whence

N

Thus a, restricts to an isomorphism (VE),=, 5(VE*),=,. Now the theorem follows from Theorem I, sec. 6.13. Q.E.D. 6.18. Duality. I n this section it will be shown that, in contrast with the results on exterior algebra (cf. article 6, Chapter V), it is in general impossible to choose dual generating subspaces U* c (VE*)o=o and U c (VE),=O such that the isomorphisms

VU* -3(VEB)O=O

and

preserve the scalar products. 249

V U 25 (VIE),=,

VI. The Wed Algebra

250

First recall from sec. 5.9 that the diagonal map d: E + E @ E is a homomorphism of Lie algebras. Hence A induces a homomorphism A:=,: (VE'")e=o + (VE")e=o @ (VE")e=o.

I n view of the duality between (VE"),,, and (VE)e=othere is a dual map (A:-0)":

-

(VE)e-o

(VE)e-o @ (VE)e-o.

Proposition IX: Let E be a reductive Lie algebra. Assume that there are subspaces U" c (VE'")~,oand U c (VE),=, such that the inclusion maps induce isomorphisms y : VU"

5 (VIE") e=o

and

9:V U

(VE)e=o,

which preserve the scalar products. Then the map (A:,,)" morphism. Proof: The diagonal map D:U to dual homomorphisms

D,:V U ---* VU @ V U

and

---f

is a homo-

U @ U and its dual D* extend

Dv:VU* +- VU* @ VU".

Moreover, since Dv(u* @ v") = u* v v*, u*, v* E VU", it follows that y(u") v y ( v " ) = yD'(u" @ v"). On the other hand, for @, Y E (VE"),=o,

0Y).

@v Y = Hence we have yD"(u" @ v " )

=

d;=,(yu" @ yv'"),

and thus yD" = A;=, 0 ( y @ y ) . Dualizing this formula and observing that y* = p-l, we obtain

( A ~ - o ) += (q @ q ) 0 Dv0 9-l. This shows that (d~=,)*preserves products.

Q.E.D. Proposition X: homomorphism.

If E is semisimple, then the map

is not a

5. The structure theorem for (VE)e-,,, and duality

251

Proof: First observe that

-

where n:V E @ V E (VE)e-o @ (VE),=o denotes the projection with kernel eE@E(VE@ V E ) . Next, since the Killing form of E is nondegenerate, it induces, as in the previous section, an E-linear isomorphism a,: V E 5 VE". N

I n particular, a, restricts to an isomorphism (VE),j=o(VE"),,o. ; Let u E (V2E)e-obe the element given by u = a;'(K), K the Killing form. A straightforward computation shows that

where w E E @ E c V E @ V E . Since E is semisimple, E = 8(E). Hence E @ E = eE&,E(E @ )?!i C kern, and so n(w)= 0. It follows that (A;=,)"(u)= u @ 1

+ 1 @ u.

On the other hand, since A , is an algebra homomorphism

A,(u v u ) = (u @ 1 + w = (u @

+ 1 @ u)Z

+ 2(u @ 1 + 1 @ u ) v w + w2.

1 + 1 @u)2

Applying n to this equation we obtain (&,)"(u

v u ) = (u @ 1

+ 1 @ u)Z +

n(w2).

Hence it remains to be shown that n ( w 2 )f 0; i.e., that w2 is not orthogonal to the space (VLE)~=, @ (VE),,, with respect to the scalar product ( , ) induced by the Killing form. We shall prove that (u @ u, w2) f

0.

I n fact, let en, ea be a pair of dual bases of E (with respect to the Killing form). Then u = 4 C e, v elL, U

252

VI. The Weil Algebra

and so

It follows that w2

=

e, v ea @ el'v e , .

Finally, it follows from the definitions that (u, x v y ) = (x,y ) . Hence (u @ u, w 2 ) =

C (e,,

ea)(ep, e,) = dim

E f 0.

PJ

Thus n(w2)# 0, and so

is not a homomorphism.

Q.E.D. Corollary: If E is a semisimple Lie algebra, it is not possible to and U c (VE),=, such that the choose dual subspaces U" c (VEQ)B=O isomorphisms VU" -5(VE*)8,0 and V U 5 (VE),=, preserve the scalar products.

§6. Cohomology of the classical Lie algebras 6.19. The Lie algebra L(n). Let X be an n-dimensional vector space and consider the Lie algebra L ( n ) of linear transformations of X . According to Example 1, sec. 4.6, L(n) is reductive. In sec. A.2 and sec. A.3 the invariant, symmetric p-linear functions C , , Tr, E (VpL(n)*),+o ( p = 1, . . . , n), are introduced. On the other hand, consider the invariant skew symmetric functions

(AzP-lL(n)x)o=o( p = 1, . . . , a)

@zp-l

E

,a 2 p - 1 )

=

defined by @2p--1(G

9

--

*

1

,&P-'

8,

tr ad]) O

* * *

O

aU(2p-l)

ai E L(n).

,

I t follows directly from Proposition IV, sec. 6.8, and the definition of

is the Cartan map for L(n). In particular (cf. Theorem 11, where eL(n) sec. 6.14) the @2p--1 are primitive. Using that same theorem, and Proposition 111, sec. A.3, we see that

c, + OP p Tr,

E

(V+L(n)*),,,

(V+L(n)x)e,o

=

ker

eL(n).

It follows that (6.15) We shall show that the @ 2 p - 1 are a basis for the primitive space of L(n) (cf. Theorem IV below). First we need Lemma VIII:

QZp-]

# 0, 1 s p 5 n.

Proof: Let Pi,ei be dual bases for X*and X . Define a, pi, y i E L(n) by a ( . )

=

(@I,

x)el, Pi(x) = (eei-l, x ) e i , yi(x) = { F , 253

25iI p.

VI. T h e Weil Algebra

254

Then the only nonzero products of these transformations are given by

Now consider a nonzero product of these 2p - 1 transformations (each occurring once) with u the pth factor. I n view of (6.16) it must be ppopp-lo

* * *

o ~ 2 0 u o y 2 0

oyp.

I t follows that, in the formula for @ 2 p - l ( & , . . . ,u, . . . ,y p ) , the only terms that occur come from cyclic permutations of this order. Since cyclic permutations of an odd number of elements are even, and since the trace is unaffected by a cyclic permutation, this implies that

@2p-1(Pp, . . . , u, . . . , y p ) = ( 2p - 1) tr Bp . . . a . 0

0

y p = 2p - 1.

Hence C$2p--1 # 0.

Q.E.D. Assign (VL(n)+),,, the even gradation of sec. 6.1. Theorem IV: (1) T h e elements @ B p - l (1 i p 5 n) form a basis for the primitive space of L(n). In particular, L(n) has rank n. (2) (VL(n)")&Q is the symmetric algebra over the graded subspace C spanned by the elements C,, . . . , C,?. Similarly, (VL(n)")&o is the symmetric algebra over the subspace T spanned by the elements T r l , , . . , Tr,, . (3) The PoincarC polynomial for H*(L(n))and the PoincarC series for (VL(n)")&O are given, respectively, by

fi ( 1 4-

and

t2p-')

p=1

fi (1 -

@)-I.

p= 1

Proof: (1) Since the @2p-l are nonzero (cf. Lemma VIII) and have different degrees, they are linearly independent. Since they are primitive, and since n

n

C deg @2p-1

C

=

p=1

(2p - 1) = n2 = dim L(n),

p=1

they form a basis of the primitive space. (2) I n view of (l), formulae (6.14) and (6.15) show that there are linear isomorphisms, homogeneous of degree 1, N

zc: PL(n)

C

and

tT:PL(n)4

T,

255

6. Cohomology of the classical Lie algebras

such that p L ( , , )0 zC = 1 and p L ( , < )0 zT = 1. Lemma VII, ( l ) , sec. 6.13, implies that these maps are transgressions. Now apply Theorem I, sec. 6.13. (3) This follows immediately from (1) and (2). Q.E.D. Example: The primitive space of L(2) is spanned by the elements G1 and Q 3 , where Q1(a)= tr (Y

and

%(a,

B, r) = tr(a 0 [B, rl),

a, B, Y E L(n).

Thus the PoincarC polynomial for H " ( L ( 2 ) ) and the PoincarC series for (VL(2)*),,, are given by (1

+ t)(l + t3) = 1 + t + + t3

t4

and (1

-

t y ( i - t4)-1

=

1

+ + + + 3ts + 3t10 + . . . . t2

2t4

zt6

I t follows that

6.20. The Lie algebra Lo(n). Let Lo(n) be the Lie algebra of linear transformations of X (dim X = n ) with trace zero. Denote the restrictions of Tr, , C,, and @2p--1 to Lo(n)by T r i , Ci, and @!&l. Then the direct decomposition L(n) = (c) @ Lo(n)together with Theorem IV, sec. 6.19, yields (with (VLO(n)*),=, graded as in sec. 6.1)

Theorem V: (1) The elements @gpPl (2 I p 5 n) form a basis of the primitive space of Lo(n).In particular, Lo(n)has rank n - 1. (2) (VLO(n)*),,, is the symmetric algebra over the graded subspace spanned by the elements C!j', . . . , C:. I t is also the symmetric algebra over the graded subspace spanned by T r t , . . . , Tr;. (3) The PoincarC polynomial for H*(Lo(n))and the PoincarC series for (VL0(n)*),,, , respectively, are given by

Corollary: The Lie algebra Lo(n) is simple.

256

VI. The Weil Algebra

Proof: Apply Proposition V, sec. 5.20.

Q.E.D. 6.21. The Lie algebra Sk(n). I n this and the next two sections we consider Euclidean spaces. T h e results obtained, however, (and the proofs) are valid for any inner product space over an arbitrary field of characteristic zero. (Although, in that setting, the Lie algebra depends on more than the dimension of the space!) Let ( X , ( , )) be an n-dimensional Euclidean space, and denote by Sk(n) the Lie algebra of skew linear transformations of X . According to Example 2, sec. 4.6, Sk(n) is reductive. Let j : Sk(n) -+ L(n) be the inclusion map, and write (cf. sec. 6.19)

1 4p 5 n.

C,so - j L O ( C p ) , TrpSO = jLo(Trp), @f$-l = $ = 0 ( @ 2 p - 1 ) ,

(We use the superscript SO because Sk(n) is the Lie algebra of SO(n) -cf. sec. 6.29.) Since $=o maps primitive elements to primitive elements (cf. sec. 5.18), the are primitive. Remark: Let K E (Vz Sk(n)*)o=obe the Killing form of Sk(n). An easy computation yields

K

=:

-(n - 2)CfO.

Lemma M: (1) I f p is odd, then Cg0 = 0, TrpSO = 0, and

(2)

@gl # 0,

3 5 2p

+ 1I n.

@fgl = 0.

Proof: (1) Since for odd p, C p ( v )= 0 and tr rpp = 0 (p E Sk(n)), it follows that CpSo = 0 and TrpSO = 0 in this case. Now Proposition 11, sec. 6.7, and formula (6.15), sec. 6.19, imply that @f$-l = 0. (2) Choose an orthonormal basis e,, e l , . . . , enPl of X . Define skew transformations ai and PA by ai(x) = (x, ei)eo - (x, e,)ei,

i = 1, . . . ,2p,

PA(.)

A = 2, . . . ,2p.

and = (x,

ede1 - (x, e l h ,

We show that @f2-l(al,. . . ,a z p ,P 2 , . . . , P2p) f 0. First observe that for distinct ( i , j ,K) and distinct (A, p, Y), ai

0 aj 0

ak

= 0,

PA 0

P, 0 By= 0,

and

is,

o cli o

By = 0.

2.57

6. Cohomology of the classical Lie algebras

Now let p be a transformation obtained by composing all the ui and all the in some order, and assume that tr p f 0. Then all the transformations obtained by cyclically permuting the given order, and then composing, have the same trace, and so are nonzero. Hence it follows from the relations above that p is obtained from a product of the form Y = ail

0

aia

0

Pa,

0

PA, . . aiZp-, uiB1)B A ~ ~ - ~ . Since tr y = tr v # 0, it follows that y f 0.

Paa 0 ai,

by a cyclic permutation. Next observe that

0

ui,

0

PA,

0

0

0

0

whence ui 0 P a 0 a k = 0 unless i = 1, k = 1 or i = A, k must have one of the following forms: Either Y

= ~1 = a1

0

aia

0

Pi,

0

Pi,

0

ai3 0

ai, 0 Pi,

0 *

0

=

1. Thus y

. . 0 aiaP-l0

0

Piap>

or y = pz = uiap0 ( L i z

0

pi, pis 0

0

ai,

0 * * * O %p-l

O

a1

O

Piap

But tr y1 = 1, and y1 is obtained from u l ,. . . , a Z p, P Z , . . . ,PZ, by an even permutation; while tr y z = -1, and y z is obtained from u l , . . . , az,, Pz , . . . , PZp by an odd permutation. Now write /-Ij = uzp+i-l(2 5 j 5 2p). Our arguments above show that for every permutation (T, tr a,(1)o

--- o

=0

or

8,.

Hence, since tr y1 f 0,

6.22. The skew Pfaffian. Let (X, ( , )) be a 2m-dimensional Euclidean space. An isomorphism u : Sk(2m) % A2X is given by <.(p), x

AY> =


v E Sk(2m)y

x, Y

E

x.

It is inverse to the isomorphism ,4 defined in sec. A.5. Now fix an orientation in X and let e be the unique unit vector in A2'IEX which represents the orientation. Then the skew Pfafian is the invariant skew symmetric (2m - 1)-linear function Sf E (AZm-lSk(2m)*),,, ,

VI. The Wed Algebra

258

On the other hand, consider the Pfaffian Pf E (VmSk(2m)*)o=0, of the oriented Euclidean space X defined in sec. A.7. I t follows directly from Proposition IV, sec. 6.8, that

( - l ) m - l ( m - I)! Sf. @ S k ( 2 w ~ ) (= ~ ~ ) 2*-l(2rn - l)!

(6.17)

In particular (cf. Theorem 11, sec. 6.14) Sf is primitive. Lemma X:

Sf is nonzero, and linearly independent of #f$-l.

Proof: Write X = (2,)@ Y where x, # 0 and Y is the orthogonal ~ of Y and define complement of x,. Let xl, . . ,x ~ be~ a - basis pli E Sk(2m), i = 1, . . , 2 m - 1, by

.

.

Pi = 8(%A Xi). Then a simple computation shows that

Hence Sf f 0. T o prove the second part of the lemma choose the x i (i = 0, . . . , 2m - 1) mutually orthogonal. Then vi(xj) = 0 if i # j ( j = 1 , . . . , 2m - 1). It follows that for distinct i , j , k, v i0 tpj 0 pk = 0. Thus

and so Sf is linearly independent of @f$-l.

Q.E.D. 6.23. The cohomology of Sk(n). Assign (VSk(n)*),,, dation of sec. 6.1.

the even gra-

259

6. Cohomology of the classical Lie algebras

+

Case I : n = 2m 1. Lemma IX, together with a word by word repetition of the proof of Theorem IV, sec. 6.19, yields

Theorem VI: (1) The elements @f[-l (1 5 p 5 m) are a basis of 1). I n particular Sk(2m 1) has the primitive space for Sk(2m rank m.

+

+

+

(2) (VSk(2m l)x)o=o is the symmetric algebra over the graded subspace spanned by the elements Cf: (1 < p 5 m). It is also the symmetric algebra over the subspace spanned by the elements Trfg m). (1 I p I 1)) and the PoincarC (3) The PoincarC polynomial for H"(Sk(2m series for (VSk(2m respectively, are given by

+

+

fl (1 +

n (1

m

m

t4p-')

and

p=1

Corollary: Sk(2m

- t4p)-'.

p=1

+ 1) is simple.

Proof: Apply Proposition V, sec. 5.20.

Q.E.D. Case 11: n = 2m. I n view of Lemmas I X and X, the same argument used to prove Theorem IV, sec. 6.19, establishes Theorem VII: (1) T h e elements @fspO-l (1 5 p < m) and Sf are a basis of the primitive space for Sk(2m). In particular, Sk(2m) has rank m.

(2) (VSk(2m)")o=ois the symmetric algebra over the graded subspace spanned by the elements Cfz (1 5 p < m) and Pf. It is also the symmetric algebra over the subspace spanned by Trf; (1 p < m) and Pf. (3) The PoincarC polynomial for H"(Sk(2m)) and the PoincarC series for (VSk(Zm)*),=, are respectively given by

<

(1

+

tZm-l)

5'(1 +

and

(1 - t Z m ) - l

p=1

Corollary: If m

n (1

m-1

t4p-l)

- t4p)-'.

p=1

> 2, then Sk(2m) is simple.

Proof: Apply Proposition V, sec. 5.20.

Q.E.D.

VI. The Wed Algebra

260

Examples: 1. 8/43): T h e PoincarC polynomial for H"(Sk(3)) is

1

+ t3, and (VSk(3)"),=0 is a polynomial algebra in a single indeterminate

of degree four. Thus, in view of sec. 5.20, a basis element of (VSk(3)");=, is given by the Killing form:

x, y E Sk(3).

K(x, y ) = t r ad x 0 ad y,

A straightforward computation shows that (cf. sec. 6.21)

K = -CsO. 2

+ +

2. Sk(4): T h e PoincarC polynomial for H"(Sk(4)) is 1 2t3 ts. Thus (cf. Proposition V, sec. 5.20), Sk(4) is the direct sum of two ideals I, and I,. Identify W4 with the space of quaternions (cf. sec. 6.30). Then according to [4; p. 2481 every skew transformation, q? determines unique quaternions p, q E ( e ) l such that

q?x = p x - xq. The ideals I, and I, consist respectively of the transformations of the form yx = p x

and

rpx

=

-xq.

Now the adjoint representation of Indefines an isomorphism of Lie algebras Id Sk(3). Moreover, since Sk(4) = I, 0Iz, (VSk(4)")e=o = (VIf)e=o 0(VIF)e=o whence (VSk(4)*):=, = {(VIF)Lo0I} @ (1 0(VI,x)~=o} = (Kl) 0(KA.

(Kdis the Killing form of I d ,) A straightforward computation shows that 4Pf = I(, - K 1

and

2Ci0 = - ( K l

+ K,).

6.24. The Lie algebra Sy(m). Let ( X , ( , )) be a symplectic space (cf. sec. 0.1). Then dim X is even, dim X = 2m. T h e Lie algebra Sy(m) of skew symplectic transformations is the Lie algebra of those linear transformations of X which are skew with respect to ( , ). Note that dim Sy(m) = m(2m 1). It follows from the example in e c . 4.8 that Sy(m) is reductive.

+

6. Cohomology of the classical Lie algebras

Let j : Sy(m)

--f

L(2m) be the inclusion, and set

As in sec. 6.21, it follows that the @bz","-l Lemma XI:

261

(1) (2)

Tr:y

CpSy,

are primitive elements.

and @&j-l are zero if p is odd.

f 0, 1 5 p I m.

@9&1

Proof: (1) This follows in exactly the same way as Lemma IX, (l), sec. 6.21. (2) Choose a basis a l , . . . , a,, b,, . . . , b,, of X so that

(bi ,b j ) = 0,

(ai,a j ) = 0,

and

( a i , b j ) = dij.

Define w , c y i , /Ii, y i , S i E Sy(m) by

4.)

=

+

(x, b,)a,

ai(x) = (x, b i ) a i - ,

(2,

a121

+ (x, a i - l ) b i ,

B d x ) = - (x, ai>ai ,

y d x ) = (x, O b i

9

i = 2, . . . ,p i = 1, . . . , p i = 1, . . . , p

and di(x) = (x, bi-l)ai

+ (x, a i ) b i P l ,

i = 2,

. . ., p ,

xE

X.

We show that

I n fact, denote these transformations (in this order) by E l , and, for each permutation c E S 4 P - l , write Fa = E m

O *

-

* O

. . . , E4p-1 ,

Eo(4p-1).

We show by induction on p that tr qu = E,

or 0.

(6.19)

VI. The Weil Algebra

262

If p = 1, tl = w , t2= P I , f 3 = y l , and (6.19) is clear. Assume p 2 2 and that (6.19) holds for p - 1. First observe that following a permutation G by a cyclic permutation changes neither tr yu nor e, (since 4p - 1 is odd). Now choose any G so that tr qu # 0. I n view of this remark we may assume that E,(4p--P) = Y P ' Because q, f 0, the relations

imply that

Consider the first case. (The second case is treated in the same way.) Among the remaining ti the only one which satisfies Eip, # 0 is a p . Hence FU(4P-4)

=a p ,

and so pu is of the form

On the other hand, write yt = [z(l) 0 the relations above for X show that

- - - ts(4p-5). Because y,, = yt 0

Finally, observe that rank yz 5 1 (because each tr pu# 0, it follows that y r ( u i )= 0,

if p -1

and

y,(bj) = 0 all j .

Thus, in view of our induction hypothesis, t r % = tr y r = E,

Pi has rank

= E,,

0

X,

1). Since

6 . Cohornology of the classical Lie algebras

263

which proves (6.19). Formula (6.18) follows since

where q i

=

Pi o T i di. 0

Q.E.D. Now assign (VSy(m)*)o=othe even gradation of sec. 6.1. In view of Lemma XI word by word repetition of the proof of Theorem IV, sec. 6.19, yields Theorem VIII: (1) T h e elements 1 5 p 5 m, are a basis of the primitive space for Sy(m). In particular Sy(m) has rank m.

(2) (VSy(m)*)o=, is the symmetric algebra over the subspace spanned by C,s,Y(m), 1 < p 5 m, and over the subspace spanned by Tr!$(m), 1
fi (1 + p=1

n (1 m

t4p-l)

and

- t@)-'.

P-1

Corollary: The Lie algebra Sy(m) (m 2 1) is simple.

Proof: Apply Proposition V, sec. 5.20.

Q.E.D.

$7. The compact classical Lie groups 6.25. Complexification of a real Lie algebra. T h e complexijication of a real Lie algebra E is the complex vector space Ee = C 0E, together with the Lie product

[~@x,p@oy]=1pO[x,y1,

A P E @,

X,oyEE.

Evidently, dimc EC = dim E. T h e symbols (Ec)", AE", V E e will denote the dual space, the exterior algebra, and the symmetric algebra over the complex space E@. Regard the elements of (Ec)* as complex linear functions in E @ ; by restricting them to E we obtain a linear isomorphism

L(E; C).

(E")"

On the other hand, there is a canonical isomorphism

C @ E",

L ( E ; C)

(cf. sec. 0.1). Combining these yields an isomorphism

C 0E*

(EC)#.

It identifies E" with the set of elements in (E")" taking real values in E. This isomorphism extends to the isomorphisms of graded algebras a : C @ AE" % A(Ec)"

/?: C @ VE" 5 V(Ee)",

and

given by a ( l @ xr

A

- -.

b(iz 0xf

v

- - v x;)

A

A

- -

A

= AX? v

* * *

v x;,

xp") = 1xf

*

xp"

and *

iz E @,

X? E

E".

Thus A(E@)"(respectively, V ( E @ ) +is) identified with the skew symmetric (respectively, symmetric) complex p-linear functions in E. 264

7. The compact classical Lie groups

Since 0(1 @ x) 0 a = a o d(x) and O(1 @ x) a and /3 restrict to isomorphisms r~e,o:

C 0(AE*),=o

pe=o:

c 0(VE*),=,

N

o

265

p = p c O(x)

(x E E ) ,

(A(E')*),,o

and

Further, we have a 0 ( L 06,)

=6

------*

(V(EQ)*),=o.

, ~0 a, and so a induces an isomorphism

I n particular, H * ( E ) and H*(Ec) have the same PoincarC polynomials. Now assume that E is reductive. Then ag,o restricts to an isomorphism of graded vector spaces up:

C @ PE

N

PEc

as follows immediately from the definitions. Clearly, the diagram

I

c @ (V+E*),=o-2% (V+(E@)*),=, 1aeE

@ PE

1

N

eEc

N

PEc

'YP

commutes . Finally, let Q be a graded subspace of (VE+),,, and use p to define an inclusion C @ Q + (V(E")*),,,. Then the induced homomorphism

is an isomorphism if and only if the homomorphism

is. (This follows from the fact that ,Be=ois an isomorphism.) 6.26. Linear groups. I n this section T = R or 6.Recall from sec. 2.5 and sec. 2.6 (volume 11) that GL(n; is the Lie group of linear automorphisms of an n-dimensional TTector space rnover

r)

r.

VI. The Weil Algebra

266

T h e Lie algebra of GL(n;I') is the Lie algebra L(n; F ) of linear transformations of rn.Since (obviously) GL(n;F ) is an open submanifold of L(n; I'),the tangent bundle of GL(n;F ) is given by TQL(,;r,=

GL(n;F ) x L(n; T).

With this identification, left translation in TQL(n;r, by an element GL(n;F ) is given by

Lr(u,a ) = (t 0 u, t 0 a ) ,

uE

GL(n;I'), a

E

t

in

L(n; I'),

(cf. Example 2, sec. 1.4, volume 11). A closed subgroup of GL(n;I') (which, by Theorem I, sec. 2.1, volume 11, is a Lie group) is called a linear group. Let G be a linear group with Lie algebra E c L(n; I'). Then the tangent space of G at t is given by

Hence if Q, cs APE*, the corresponding left invariant form G (cf. sec. 5.29) is given by

(.z'@)(t;

011,

.

* *

, a,)

=

CP(t-1 0 a1,

tE

. .. ,

t-1 0

txl(Q,) in

ap),

G, a j E T,(G).

(6.20)

In particular, if @ E (APE*)e_;O,then the corresponding p-form on G is biinvariant and hence closed. 6.27. The group V(n). Consider the compact, connected Lie group U ( n ) of isometries of an n-dimensional complex vector space C'&with Hermitian inner product ( , ). Its Lie algebra Sk(n; C)is the real Lie algebra consisting of the complex linear transformations a of @" which satisfy x, y E Cn. (ax, y ) (2, ay) = 0,

+

For simplicity denote Sk(n; C) by E. Next recall the elements C, , Tr,, @ j 2 p - 1 defined in sec. 6.19. Observe that the restrictions of these multilinear functions to E take only real (respectively, only purely imaginary) values if p is even (respectively, odd). Hence elements

7. The compact classical Lie groups

267

are defined by

and

T h e element @2up-l extends to a unique closed, biinvariant form on U(n). We denote this form by @g-las well; it is given explicitly by

@,u,-dt;a1 . . 9

9

azp-1)

(cf. sec. 6.19 and formula (6.20), sec. 6.26). T h e cohomology class represented by @.&l is denoted by [@p-,]. Let PCT(n) denote the primitive subspace of H ( U ( n ) ) (cf. sec. 4.12, volume 11, and sec. 5.32). Recall that E = Sk(n;

e).

(1) The elements @g-l(1 < p 5 n ) are a basis of = A(@f, . . . , @g-l). (2) (VE"),=, = V(Cy, . . . , C,V) = V(TrP, . . . , Tr:). (1 < p 5 n ) are a basis of (3) The cohomology classes PUW, and so H(U(n))= A([@?], * - .> [@E-ll). Theorem IX:

P E , and so (AE"),,,

(4) T h e PoincarC polynomial for (AE8)8=0,H"(E), and H ( U ( n ) )is

fi (1 +

t2p-1).

v=l

Proof: (1) Observe that an isomorphism of complex Lie algebras

C @ E Z L ( n ; C) is given by il @ q

H

ilp, il E

C,q.~E E. This isomorphism induces an

VI. The Weil Algebra

268

isomorphism (as described in sec. 6.25)

which restricts to an isomorphism

But by definition,

Now it follows from Theorem IV, (l), sec. 6.19, that the @&. , basis of P E .

are a

(2) This follows in exactly the same way as ( 1 ) from Theorem IV, (2), sec. 6.19 (as described in sec. 6.25). (3) According to sec. 5.32, the isomorphism a U ( n )(AE"),,, :

5H(U(n))

restricts to an isomorphism from PE to Pucn). Since ( Y U ( ~ ) ( @ ~ = -~) [@g-,],1 < p I:n, (3) follows from (1). (4) This follows at once from (1) and (3) (cf. sec. 5.32). Q.E.D. 6.28. The Lie group SU(n). Recall from Example 4, sec. 2.6, volume 11, that SU(n) is the closed subgroup of U(n)consisting of the isometries with determinant 1. Denote its Lie algebra by E. Let @'"El denote the restriction to E of the element @gWl of sec. 6.27, and let @El also denote the corresponding biinvariant form on SU(n). The latter is the restriction to SU(n) of the biinvariant form @gPl on

0).

Let C i u and TriU denote the restrictions of Cg and Trp" to E. Then the argument of the previous section, applied to Theorem V, sec. 6.20, yields Theorem X: (1) The element's @fE1(2 ( PE, and SO (AE"),,o = A(@fu, , @:El).

(2) (VE")e,,

= V(CiU,

p 5 n) are a basis of

.. . . . . ,CiO)= V(TrfU, . . . ,Triu).

7. The compact classical Lie groups

269

( 3 ) The cohomology classes [@&K1] ( 2 I p 5 n ) are a basis of PSlJ(?l,,and so H ( S U ( n ) )= A([@'], . . . ,

[@fzl]).

(4) The PoincarC polynomial for (AE"),=, , H"(E), and H ( S U ( n ) )is

n (1 + n

t2p-1).

p=2

6.29. The Lie group SO(n). Recall from Example 3, sec. 2.5, volume 11, that SO(n) is the compact connected Lie group of proper rotations of Euclidean n-space. Its Lie algebra is the Lie algebra Sk(n) described in sec. 6.21. Recall the primitive elements @f$-lE PSk(*) ( 3 5 2p 1 5 n ) introduced in sec. 6.21. Their extensions to biinvariant closed forms on SO(n) (also denoted by @f$-l) are given explicitly by (cf. sec. 6.26)

+

so

@dp-l(t;

a1,

. . . ,a4p--1)= C

E,

tr(z-'

o

0

+

-

o (t-l

0

UES4P-1

zE

SO(n), aj E T,(SO(n)).

(6.22)

Moreover, if n = 2m, the skew Pfaffian Sf defined in sec. 6.22 extends to a biinvariant form (also written Sf) on SO(n). We may use sec. 5.32 to translate Theorems VI and VII, sec. 6.23. This yields Theorem XI: T h e cohomology classes [@f$-l] (1 I pI m ) are a basis of Ps0(2m+l). Thus

H(SO(2m

+ 1 ) ) = A([@fO],

and the PoincarC polynomial of H(SO(2m

fi (1 +

*

-

3

[@fg-i]),

+ 1)) is

t4p-1).

p=1

Theorem XII: T h e cohomology classes [@fipPl] [Sf] are a basis of Psoczm). Thus

(1 5 p < m ) and

H ( S O ( 2 m ) )= A( [@fO], . . . , [@fLi], [Sf]), and the PoincarC polynomial of H ( S O ( 2 m ) )is

(1 + P - 1 )

fil(1 + p=1

t4p-1).

VI. The Wed Algebra

270

6.30. The Lie group Q(n). Let I! denote the algebra of quaternions. Recall from sec. 0.2, volume 11, that H is an oriented 4-dimensional Euclidean space with a multiplication defined as follows: Choose a unit vector e and give the Euclidean 3-space (e)l the induced orientation (i.e., a basis e, , e 2 , e3 of (e)" is positive if and only if the basis e, e l , e , , e3 of H is positive). Now set

pe=p=ep,

and Pq = - ( P , 4>e

P E H

+ P x 4,

P, 4 E (ell,

where x denotes the cross product in ( e ) l . Observe that every quaternion p is uniquely of the form p = l e p l (A E W,p , E ( e ) l ) ; the conjugate of p is defined by 9 = l e - p , . T h e identity element of H is e. Now choose a fixed unit vector i E (e)l. Then i2 = -e and so the subspace of H spanned by e and i is a subalgebra of H , canonically isomorphic to C.We identify this subalgebra with C and write

+

.H = C @ CJ-.

Note that 0 is stable under left multiplication by elements of C and hence it is a (1-dimensional) complex vector space. Next, let (W",{ , IBP.) be an n-dimensional Euclidean space and consider the quaternionic vector space X = H ORRn with scalar multiplication given by p(4 0x) = pq

0x,

p,qE

ff, x E

P.

Define a quaternionic inner product in X by setting (P

0x , 4 O r >= pq<x,YIw,

p , 4 E ff,

X,

y E Rn.

Since C c H, restriction of scalar multiplication to C yields a 2ndimensional complex vector space X,. Use the decomposition H = C 0 C'l to write (u,

0)

= (u, v>c

+ (u,

O)@,

u, ZJE

x.

Then ( , )e is a Hermitian inner product in X,. Finally, fix a unit v e c t o r j c C .. Then the complex bilinear function ( Y ) given by u, v E X C , (u, v ) = (u , j v h , makes X , into a complex symplectic space (cf. sec. 0.1).

271

7. The compact classical Lie groups

Now recall from Example 4, sec. 2.7, volume 11, that the quaternionic group Q(n) is the compact connected Lie subgroup of GL(X,) consisting

of those transformations t(pu) =p t ( u )

and

t

which satisfy ( t u , t v ) = (u, v),

u, v E

X, p

E

H.

Its Lie algebra Sk(n; H)is the real subalgebra of Lx, consisting of the transformations a such that a ( p u ) = p(au)

and

+ (u, a v ) = 0,

(au, v )

Denote this Lie algebra by E. Next observe that the inclusions Q(n) in fact inclusions

Q(n)+ U(2n)

--j

E

and

p

E

H,

GL(X,) and E

-+

u, v E

-

X.

Lx, are

Sk(2n; C)

(with respect to the Hermitian inner product ( , ),). Thus we may restrict the elements (1 < p 5 n ) in PSk(2n;C) to E ; the resulting primitive elements will be written @?&I

E PE,

1 5 9 5 n.

Moreover, the extension of @4&pp-l to a biinvariant (4p - 1)-form on Q(n) coincides with the restriction to Q(n) of the biinvariant form @gPl on U(2n). Finally, let C$ and Tr$ (1 < p 5 n ) denote the restrictions of C g and Tr& to E (cf. sec. 6.27). Theorem XIII: ( 1 ) The elements @fpp-l, 1 5 p 5 n, are a basis of PE; in particular, (AE"),=, = A(@!, . . . , @4&n-1). (2) (VE"),=o = V(C,Q, . . . , C$) = V(Trt, . . . , T&). (1 5 p 5 n ) are a basis of (3) T h e cohomology classes [@fP-J PQh), and so

H(Q(n))= ~([@,31,*

*

-

9

[@2%-11).

(4) The PoincarC polynomial of (AEX)B,O,H"(E), and H(Q(n)) is

lq (1 + n

t4p-1).

p=1

Proof: (1) First observe that if v E Lxc , then cp is in E if and only if y is skew with respect to both ( , )e and ( , ); i.e.

E

= Sk(2n; C )

n Sy(n; C).

272

VI. The Weil Algebra

It follows that we have a commutative diagram .Sy(n; C)

=

C@E

-

C @ Sk(2n; C)

L(2n; C )

N

in which the vertical arrows are the obvious inclusions, while the horizontal arrows are the isomorphisms of complex Lie algebras given by A@gJH1gJ. This leads (cf. sec. 6.25) to the commutative diagram

C@Pg-

4

el

-

PSy(n;c)

N

1

r

@ PSk(2n;C)

yp

7 PL(2n;C) -

Thus (cf. sec. 6.27 and sec. 6.24)

1

0%-1

=741

@a

0

= Y1%((-1)P@4p--l) =

&l((-l)p@$;-l).

Now, since c1 is an isomorphism, (1) follows from Theorem VIII, (l), sec. 6.24. (2) This follows from Theorem VIII, (2), via the isomorphism C @ (VE")e-o zz (VSy(n; C)")e=o,

(as described in sec. 6.25). (3) and (4): In view of sec. 5.32, these assertions are immediate consequences of (1) and (2).

Q.E.D. Example: d = Q ( 1 ) . In this case E is the Lie algebra of pure quaternions, and a simple calculation shows that the Killing form K satisfies K = 4Cf. Now in Example 2, sec. 6.23, we have I Ag E, 1 = 1, 2. Let Cf)E (VIf)8-o correspond to Cf,under these isomorphisms. Then

Pf

= Cf)-

C;')

and

Cf"= -2(C4"

+ Cf)).

Chapter VII

Operation of a Lie Algebra in a Graded Differential Algebra $1. Elementary properties of an operation 7.1. Definition. An operation of a Lie algebra in a graded dagerential algebra is a 5-tuple ( E , i, 8, R,S) where:

(1) E is a finite dimensional Lie algebra, and (R, 6) is a graded anticommutative differential algebra R = CP>,, Rp. 0 is a representation of E in the graded algebra R; that is, for (2) each x E E, O(x) is a derivation in R, homogeneous of degree zero, and e([X,YI) =

0(x) O 0 ) - fqy) O @),

X,Y

E

E.

(3) i is a linear map from E to the space of antiderivations of R, such that each i ( x ) is homogeneous of degree -1. (4) The following relations hold:

i ( x ) 2= 0, i([X,Yl)

=

(7.1) O

i ( Y )- i ( Y )O

and 8(x) = i ( x ) o 6

+6

o

(74

x,y

i(x),

E

E.

(7.3)

Suppose ( E , i, 8, R,S) is an operation of a Lie algebra E in a graded differential algebra (R,S). Applying 6 on the left and on the right hand sides of formula (7.3) we obtain

s qx)=

S,

x

E

E.

(7.4)

This relation shows that 8 is a representation of E in the graded dijferentiul algebra (R, 6). Moreover, condition (7.3) implies that 0 induces the zero representation in the cohomology algebra H ( R ) . If ( E , i, 8, R, S) is an operation of E, it follows from formula (7.1) 273

VII. A Lie Algebra in a Graded Differential Algebra

274

that i extends to a unique linear map

i: AE -+L ( R ; R), such that i ( a A b ) = i ( b ) o i ( a ) , a, b E AE, and i(1) = 1. I n particular, i(X1 F\

-

*

*

A X p ) = i(X,)

-

0 * *

0

i(Xl),

Xi

E

E,

so that i ( a ) is homogeneous of degree -p when a E APE. Assume that operations of E in graded differential algebras ( R , 6,) and (S, 6,) are given. Then a homomorphism of operations P:

(E, iR, O R , R, 6,)

is a homomorphism rp: ( R , 6,) such that rp

o

O,(x)

= O,(x)

o a,

--f

( E , is, 8.9, s,6,)

+ (S, 6,)

of graded differential algebras

y~ o iR(x) = ).(si

and

0

xE

rp,

E.

Clearly Q' o

i,(a)

= is(a) o

rp,

aE

AE.

Observe that we define only a homomorphism of operations of a fixed Lie algebra. Clearly the composite of two homomorphisms of operations is again a homomorphism of operations. 7.2. Important formulae. In this section we obtain some relations connecting the operators i ( a ) , 8(x), and 6 in R (for an operation ( E , i, 8, R , 6)). They generalize formulae developed in article 1, Chapter V. Recall the representation 8E of E in AE (sec. 5.1) and the differential operator a E (sec. 5.2). T h e relations to be established are

i ( a )-).(i

e(x) = i ( e - y x ) a ) ,

i ( x , A . . . ~ x ~ ) O d (-l)p-160i(x1A +

E

E , a E AE,

... AX,)

P

=

(7.5)

C (-l)y-li(xp) o - . - o i(xYfl)o O ( x y )o i ( ~ , , o- ~. - ). o i(xl),

x i E E,

Y=l

i(a)6

(7.6)

+ (-l)P-lSi(a)

= --i(aEa)

+ (-1)P-'6i(a)

= i(aEa)

+ Ce O(e,)i(i,(e*e)a),

aE

APE, (7.7)

and

i(a)6

+ C i(iE(e*@)a)O(ee),a E APE. 0

(Here e"e, ep denotes a pair of dual bases for E" and E.)

(7.8)

275

1. Elementary properties of an operation

Proof of (7.5) and (7.6): O(x)

0

--

i(xl A

=

f i(xp)

A

x p ) - i(xl

o

.

Use formula (7.2) to obtain A

- .

o i ( x v + l )0

A

xp)O(x)

[ e ( x > i ( x y ) - i ( X y ) ~ ( x )01i(xv-l) 0

.-

+

0

i(xl)

v=l

=i(P(x)(x, A

..

A

xp)).

Formula (7.6) is proved in exactly the same way, with the aid of (7.3). Without loss of generality we may assume that a = x1 A - . A x p , xi E E. Then, in view of formula (7.6), we have only to prove that Proof of (7.7):

-

+ C O(ee)o i(iE(e*e)a)

-i(a,a)

e P

=

C ( - 1 Y - 5 ( x p ) o . . . o qXv) o ...

o

i(xl).

v=l

It remains to be shown that

But this follows at once from formula (5.6) sec. 5.3. Proof of (7.8): Formula (7.5) and formula (5.5) of sec. 5.3 yield

C [O(e,)

0

i(i,(e"e)a)

- i(iE(e*e)a)O(ee)]=

e

Now (7.8) follows from (7.7).

C i(OE(e,)iB(e+e)a)

e = 2 4 8,a).

VII. A Lie Algebra in a Graded Differential Algebra

276

7.3. The invariant, horizontal, and basic subalgebras. Let (E, i, 8, R, 6) be an operation. Since 8 is a representation of E in the graded differential algebra (R, S), the invariant subspace

Re=O =

n ker O(x),

XEE

is a graded subalgebra, stable under 6. It is called the invariant subalgebra. The inclusion map Re=,--+ R induces a homomorphism

Since 8 induces the zero representation in H ( R ) , Theorem IV, sec. 4.10, gives Proposition I: Let (E, i, 8, R, 6) be an operation of E such that the representation 8 is semisimple. Then the homomorphism

is an isomorphism. Next consider the subspace

Rise =

n ker i ( x ) .

r€E

Since the operators i ( x ) , x E E, are homogeneous antiderivations, Ri,, is a graded subalgebra of R. Moreover, it follows immediately from formula (7.2) that Ri=,is stable under the operators e(%),x E E. Hence 8 restricts to a representation of E in Rime. Ri,o is called the horizontal subalgebra of R. It is simple to verify the formula

-

~ ( u ) ( uV ) = (;(a).)

*

a E AE,

V,

u E R, v E Ri-0.

(7.9)

Observe that the horizontal subalgebra is, in general, not stable under 6. Finally consider the subspace (RiPO)B=O (= Rinon R8=o).I t will be denoted by Ri=o,e=o. Since the operators 8(x), x E E, are homogeneous derivations in Ri=o,Ri,o,e=ois a graded subalgebra of R. I t is called the basic subalgebra. The basic subalgebra is stable under 6. In fact, let z E Ri,o,e-o. Then formulae (7.4) and (7.3) yield 8(x)dz

=

68(x)z = 0

whence 62 E Ri=o,e=o.

and

i(x)dz = 8(x)z - Gi(x)z = 0,

1. Elementary properties of an operation

277

T h e inclusion map eR: Ri=o,e=o + Re-0 induces a homomorphism eg: H(Ri=o,e=o)

-

--+

H(RO=o)-

Now let p: (E, iR, OR, R, dR) ( E , is, OS, S, 6,) be a homomorphism of operations. Then qj restricts to homomorphisms qje=o:

Re-0

-+

So=, ,

pi=o: Ri=o+ S i = o ,

and

as follows from the definitions. Moreover, pe=o and pi=o,e=o are homomorphisms of graded differential algebras. Thus we have the commutative diagram

.j

H(Ri=o,e=o)

Cto,e=o

___*

H(Si=o,e=o)

$2. Examples of operations 7.4. Examples. 1. Actions of Lie groups: Let MxG + M (or Gx M -+M )be a smooth action of a Lie group on a manifold M. This action induces an operation of the Lie algebra of G in the algebra of differential forms on M.This particular example is discussed in detail in article 6. 2. The operation on AE": Let E be a Lie algebra and recall the definition of the operators i E ( x ) ,8,(x), and 6, in AE" (cf. secs. 5.1, 5.2). Then formulae (5.1) and (5.3) imply that ( E , i,, 8,, AE", 6,) is an operation of E in AE". I n this case we have

( A E * ) ~ == , ( A E * ) ~ = ~=, r. ~,~ 3. Restriction: Let ( E , i, 8, R , 6) be an operation of a Lie algebra E. Let F be a subalgebra of E. Restricting i and 8 to F, we obtain an operation ( F , i F , O F , R , 6) of F in ( R , 6), which will be called the restriction of the original operation to F. T h e invariant, horizontal, and basic subalgebras for ( F , i F ,O F , R , 6 ) will be denoted by ROF=,,,RiF=O,and RiF=o,e,=o*

A homomorphism of operations v: ( E , i, 8, R , 6) + ( E , be considered as a homomorphism of operations of F.

r, 6,8,8) may

4. Subalgebras: Let F be a subalgebra of a Lie algebra E. Then the restriction of the operation ( E , i,, 8,, AE*, 6,) to F is denoted by ( F , i F , O F , AE", 6,). This operation will be studied extensively in Chapter X. 5. The tensor product operation: Let ( E , i R ,O R , R, 6,) and ( E , is, Os, S , 6,) be operations of E. Then an operation ( E , i, 8, R 0S , 6) is defined by

278

2. Examples of operations

279

(where mR denotes the degree involution in R). I t is called the tensor product operation. 6. The operation in the Weil algebra: Let W ( E )be the Weil algebra of a Lie algebra E. Then ( E , i, O w , W ( E ) ,),S is an operation of E (cf. formula (6.5), sec. 6.4). T h e horizontal and the basic subalgebras are given by

W(E),=,= VE" @ 1

and

W(E)i=o,e=o = (VE")o=O@ 1.

7. Let eT be a representation of E in a graded anticommutative algebra T and let 6 denote the induced representation in T @ AE"

e ( x ) = &(x)

0

+

0e,(x),

x E E.

+

Define a differential operator S in T @ AE" by S = S E So (cf. sec. 5.25). Then ( E , i E , 8, T @ AE", S ) is an operation, as follows from formula (5.17), sec. 5.25. I n this case we have (5" @ AE")<=, = T

01

and

( T 0AE"),,o,,=o

=

TeZ0@ 1 .

7.5. The associated semisimple operation. Let ( E , i , 8, R, S ) be an operation of a reductive Lie algebra. We shall construct a graded subalgebra R, of R with the following properties: (i) R, is stable under the operators i(x), e(x) (x E E ) , and 6. (ii) If e,(x) denotes the restriction of e(x) to R, (x E E ) , then €Js is a semisimple representation of E.

A subspace X c R will be called admissible if (i) dim X < 00, (ii) X is stable under e(x), x E E, and (iii) The restrictions e,(x) of O(x) to X (x E X ) define a semisimple representation of E in X . Given two subspaces X c R and Y c R we shall denote by X the subspace spanned by the products u ZI (u E X , v E Y ) .

-

-

Y

Lemma I: ( 1 ) If X and Y are admissible subspaces of R, then so are X Y and X Y. (2) If X is an admissible subspace of R, then so is ep(X), p = 0 , Rj. 1, . . . , where Q P : R + R* is the projection with kernel

+

-

VII. A Lie Algebra in a Graded Differential Algebra

280

Proof: (1) Addition and multiplication define surjective linear maps

X@Y-+X+Y

and

X@Y-+X.Y.

Moreover, these maps are E-linear with respect to the representations OXey and Oxey determined by Ox and O y (cf. sec. 4.2). But it follows from Theorem 111, sec. 4.4,that Oxoy and Oxey are semisimple representations, since Ox and O y are, and E is reductive. Hence X Y and X . Y are admissible. (2) Observe that e p restricts to a surjective E-linear map from X to

+

ep(x).

Q.E.D.

Lemma 11: Let X c R be an admissible subspace. Then X is admissible.

-+

+ 6(X)

Proof: Apply the argument of Lemma I to the linear map X @ X X 6 ( X ) given by

+

U @ V H U + 6 V ,

U,VEX. Q.E.D.

Lemma 111: Let X c R be an admissible subspace, and suppose /?:E @ X -+ R is the linear map given by p(x @ u ) = i(x)u,

x E E,

uE

X.

Then I m p is admissible. Proof:

Use formula (7.2), sec. 7.1, to show that

p 0 (ad x @ L + L @ O(x)) = O(x) o ,8,

x

E

E

and apply the argument of Lemma I. Q.E.D.

X

A vector z E R is called admissible if there exists an admissible subspace c R such that z E X.

Proposition 11: The admissible vectors form a graded subalgebra R, of R. This subalgebra is stable under the operators i ( x ) , O(x) (x E E), and 6.

281

2. Examples of operations

Proof: I t follows immediately from Lemma I that the admissible vectors form a graded subalgebra Rs of R. Clearly, R, is stable under the operators O(x) (X E E ) . Lemmas I1 and I11 show that R, is stable under the operators 6 and i ( x ) . Q.E.D.

I t follows from Proposition I1 that the representation 8 restricts to a representation Bs of E in R,. Proposition 111: The representation Os is semisimple. Proof: Let X c R, be any stable subspace. Choose a subspace Y c R, which is maximal with respect to the properties

X n Y =0

and

O,(x)(Y) c Y ,

XE

E.

We shall show that X @ Y = R,. Let z E R, and choose an admissible subspace, 2 c R,, such that z E 2. Then 2 n ( X @ Y ) is an E-stable subspace of R,. Since Oz is semisimple, there is an E-stable subspace U such that 2 = 2 n ( X 0Y ) 0 U .

This implies that ( X @ Y ) n U = 0, and hence the maximality of Y yields U = 0. I t follows that z E 2 c X @ Y , and so R, = X @ Y . Q.E.D. Definition: Let is(x), 6,(x) ( X E E) and 6, denote the restrictions of i ( x ) , 6(x), and 6 to R,. Then the operation (E, i s , 8,, R,, 6,) is called the semisimple operation associated with (E, i, 6,R, S ) .

Since every element z E I?,=,, generates a 1-dimensional admissible subspace, we have Re=o c Rs. This relation implies that

Re-0

=

(Rs)O=o

and

Ri-o,e=o = (Rs)i=o,e=o.

-

Because 8, is semisimple, the homomorphism H(ROzo) H ( R s ) induced by inclusion is an isomorphism (cf. Proposition I, sec. 7.3).

7.6. Tensor products. Let (E, i K ,O R , R, 6,) be an operation of a Lie algebra. Let 0, be a representation of E in a graded differential

282

VII. A Lie Algebra in

a

Graded Differential Algebra

space ( M , aM). Define operators e(x) (x E E ) and 6 in M

0R by

where wM denotes the degree involution in M. Proposition IV: Assume that E is reductive and that at least one of the representations 8, and 8, is quasi-semisimple (cf. sec. 4.3). Then the inclusion map

g: Me=o 0Re-o

-

( M 0R)e=o

induces an isomorphism

Proof: We show first that

where ( E , is, B,, R,, 6,) denotes the associated semisimple operation (cf. sec. 7.5). I n fact, clearly

On the other hand, let Y E ( M 0R)e=o.Then Proposition I, sec. 4.3, shows that there is a finite dimensional subspace 2 c R, stable under the operators O,(x>, and such that Y' E ( M 0Z)e=o, and the induced representation 82 of E in 2 is semisimple. Thus !P E ( M 0Rs)e=o, and so (7.10) is proved. Since also (Rs)o=o= Re=o,we may replace R by R,. Thus in view of Proposition 111, sec. 7.5, we may assume that the representation On is semisimple. On the other hand, the equations

imply that each B,(x)# = 0. Now the proposition follows from Theorem V, sec. 4.11 (with M = Y and R = X ) . Q.E.D.

2. Examples of operations

283

Corollary: If E is semisimple, ( E , in, O n , R, 8,) is any operation, and OM is a representation of E in a graded differential space ( M , dM), then the map

-

P :f w & = o ) 0 x H(Re=o) H ( ( R 0W O = O ) is an isomorphism. Proof: Observe (via Theorem I, sec. 4.4)that every representation of a semisimple Lie algebra is quasi-semisimple and apply the proposition.

Q.E.D.

§3. The structure homomorphism I n this article (E, iR, O R , R, 6,) denotes an operation; oR denotes the degree involution in R.

7.7. The structure operation. Consider the (skew) tensor product 0AE", and define operators ~ R B E ( ~ )@, R @ E ( x ) of graded algebras (x E E), and 8RmE in R 0AE" by iABE(x) = WR

Here

0~ E ( x ) ,

ORBE(%)

= eR(x)

01

a8 is the operator defined in sec. 5.25 60

=

+

0O E ( ~ ) ,

x E E,

(with R = M); i.e.,

C ~ R O R ( ~0 , ) ,4eQV). V

Lemma IV: The 5-tuple (E, i R g , E , eration.

e R @ E yR

0AE", B R B E )

is an op-

Proof: Clearly e R @ E is a representation. Relations (7.1), (7.2), and (7.3) follow from an easy computation (cf. sec. 5.1 and sec. 5.25). Thus = 0. we have only to verify that Evidently 6; = 0. Next, observe that the equations

Moreover, in sec. 5.25 it was shown that

Q.E.D. 284

3. The structure homomorphism

285

T h e operation ( E , i R B E ,O R B E , R 0AE", SROE) is called the structure operation for (E, i,, OR, R, 8,). On the other hand, form the tensor product operation for

(E, i R ,OR, R,S,)

( E , i E ' ,b ,AE", S E )

and

(cf. Example 5, sec. 7.4). I t is given by ( E , i, 0, R @ AE", d), where i ( x ) = iR(x) @ 1 f

WR

@ iE(x),

e ( x ) = eR(x) @ 1

L

@ e,J$(X),

and

S = S&C+

wR@SE.

We shall construct an isomorphism of operations

(E, i, 0 , R

0AE", S ) 25 ( E , i R @ E , O R B E , R 0AE", ~ R B E ) .

First, define a derivation a homogeneous of degree zero in R a

=

0AE",

by

C wRiR(e,) @ p ( e X u ) , V

where e",, e, is a pair of dual bases of E" and E. Since a p = 0 ( p > dim E), we can form the map

It is an automorphism of the graded algebra R @ AE". T h e following explicit form for /? will be useful: For a E APE, define j ( a ) :R -+ R by j(u)(z) = (- 1)p(q-l)iB(u)(z), z E Rq. Then an easy computation yields

j ( u A b ) =j ( u ) 0 j ( b ) . Lemma V: Then

(7.11)

(1) Let @,an be a pair of dual bases for AE" and AE.

B = Ca i ( 4 0p.(@"-

(2) Let rp be a linear transformation of a finite-dimensional vector space X , and let x , , x+A be a pair of dual bases for X and X". Then

C Vx' 0x"' a

=

C 0p " ~ " ' . XA

1

VII. A Lie Algebra in a Graded Differential Algebra

286

In particular, if X

AE, then

==

0P ( W

Cj(Y-"I> I

=

CA.1) 0P.(Q?"@'").

Proof: (1) First observe that the expression Cij(al)@ p(@) is independent of the choice of dual bases. Now fix a basis e l , . . . , en of E and extend it to the basis {ei, A A ei,}, i, < < ip of AE. T h e dual basis is A . A e 8 i D } , i1 < . . < ip, where ( P ,e j ) = 6j. Evidently

-

-

-

+

f2 =

c i < e " )0p(e"9. V

I t follows (via formula (7.11)) that a p =

-.

Cj(e,,A

A

e y p )0p(e"Y1

C..c"pozj(eYl A .

*

A

. . - A e*v,),

. , . , v P ) with

where the sum is over all p-tuples ( v l , i = 1, . . . ,p . Thus 1 (XP = P! 1

A

0p(e"Y1 A - . -

evp)

GI<.

1 5 pi 5 n,

A elup),

and so (1) follows. (2) This is obvious.

Q.E.D. Proposition V: With the notation above ,B is an isomorphism of operations

,B: (E, i, 8, R @ AE", 6 ) -5(E, iHOE, 8R@B,R

0AE", B R @ E ) .

Proof: We must check that fi converts the operators i ( x ) , 8(x), and 6 into i f i & ~ ) , e R @ E ( X ) , and B R R O E , respectively.

(1)

The operator i ( x ) : Observe that (wR

0iE(x)) =

-

f2

(wR

8i E ( x ) )

iR(ev) @ [iE(X)p(e*') Y

= iR(X)

+

/4(e"v)iE(x)1

01.

It follows by induction on p that (wR

@ iE(x)) 0 CP = U P

0

(wR

@ iE(x))

+ pap-'

0

( i B ( x )@ L ) ,

p 2 1.

3. The structure homomorphism

287

Hence

+

(2) The operator O(x): L

@ O,(x).

Observe that O(x) = OAeE(x)= BR(x) @ 1

Moreover

(by Lemma V, (2) with X = E and

'p =

ad x). Hence

( 3 ) The operator 6 : Recall from sec. 7.2 that iR(a)SR

+(-l)P-lSRiR(a)

= -iR(dEa)

+ C eR(ee)iR(iE(e*e)a), 0

I t follows that

Now apply Lemma V, (1) and (2), to obtain

On the other hand, Lemma V, (l), also yields

and

a E APE.

288

VII. A Lie Algebra in a Graded Differential Algebra

Adding these three relations, we find that

7.8. The structure homomorphism. Define a homomorphism

Proposition V, sec. 7.7, shows that the map

is a homomorphism of operations. I t is called the structure homomorphism for the operation (E, i R , O R , R, dR). If v: R -+ S is a homomorphism of operations, then the diagram

yRl

R @ AE"

+s

I

R

I

-

ys

S @ AE"

commutes, as follows from Lemma V, (l), sec. 7.7. Examples. 1. The operation (E, i s , Ox, AE", E be the addition map

8E):

Let p : E

-+

Ax,y) =X

+Y ,

x, Y E

Then p induces a homomorphism pA:AE" It follows frorh the definitions that

Since E" generates AE", this implies that

-+

E.

AE" @ AE".

0E

3. The structure homomorphism

2. The homomorphism phism YR~,,.,E*:R

YRBAE'

289

: Consider the structure homomor-

@ AE"

+R

@ AE" @ AE".

Since ~ R B E ( z )= O R

0~ E ( x ) ,

xE

E,

it follows that YRBAE*

=1

0YM* = 1 0pA

(cf. Example 1). Thus the commutative diagram above, applied with q~ = yR and S = R @ AE", yields the commutative diagram

R

YR

yR1 This diagram shows that y R makes R into a comodule over the coalgebra AE". 7.9. The cohomology structure homomorphism. I n this section we assume E to be reductive. Since the structure homomorphism is a homomorphism of operations, it induces a homomorphism

On the other hand, we have the inclusion g : R0-o 0(AE"),,, + ( R @ AE"),=,. Since E is reductive, Proposition IV, sec. 7.6 (applied with R = M and AE" = R ) , implies that g induces an isomorphism

g*: H(R,=,) @ (AE*)O=o -5 H ( ( R @ AE")e=o, ~ R B E ) of graded algebras. (Note that dRoE reduces to dR @ i - w R @ d E in ( R @ AE"),,, .) Thus we may compose (g#)-' with (yR)jLo to obtain a homomorphism

Definition: BR is called the cohomology structure homomorphism for the operation ( E , i R , O R , R, d R ) .

290

VII. A Lie Algebra in a Graded Differential Algebra

Suppose q ~ :R diagram

-+

S is a homomorphism of operations of E. Then the 4Ll

H(Rf3-0)

H(Se=o)

i.1

(7.12)

19.

H(Re=o)0(AE")e=o v g p H(Se-0) 0(AE")o-o commutes, as follows from the analogous property for y R and ys (cf. sec. 7.8). Proposition VI: Let q : H(R6,o) @ (AEX)o,O -+ H(Re,o) be the projection with kernel H(Re,o) @ (A+E")e,o. Then q 0 f R = L. I n particular, f R is injective. Proof: Consider the projection

p : R @ AE"-+R with kernel R @ A+E". Then p induces a homomorphism

pe",o: H ( ( R @ AE*),,o, Since (clearly)

p yn 0

~RBE)

-+

H(Ro-0).

= L, it follows that #

Pe=o

0

#

( ~ ~ ) o ==o 1.

Thus q0

=

# q 0 (g"1-l 0 (rn)e#=o= Po=,

0

*

(YR)e=o = 1,

and the proposition is proved. Q.E.D. Example: The operation ( E , i E ,0 E , AE", dE). T h e cohomology structure homomorphism for this operation coincides with the comultiplication in (AE*),=o as defined in sec. 5.17, fAE*

=

(AE"),=o

-+

T o see this, let q : AE" @ AE" jection with kernel

(AE"),,,

+

(AE*),=, @ (AE"),=o be the pro-

~ E @ E ( A E@" AE") = O(AE") @ AE" Then y E = q o pj=o = q

o

0(AE"),=o.

+ AE" @ 0(AE").

(yAEl)e-o(cf. Example 1, sec. 7.8).

291

3. The structure homomorphism

On the other hand, note that 7o

(s,

@ 1)

= 0,

q

0

S0 = 0,

and

q

0

(WE

0)6,

= 0.

Remark: This example provides a second proof that the comultiplication y E is an algebra homomorphism (cf. sec. 5.17).

$4. Fibre projection I n this article (E, iR,OR, R,8,) denotes an operation of a reductive Lie algebra. T h e algebra H(Re,o) is assumed to be connected. 7.10. Definition. Recall from sec. 7.9 the cohomology structure homomorphism f R : H(Re=o) H(Re,o) @ (AE")e=o. +

Since H(R,=,) is connected, we have the canonical projection

-+r;it induces a homomorphism

H(R0-0)

n ~ H(Re=o) : @ (AEX)e=o-+ (AE')e=o.

Hence a homomorphism eR: H(Re-0)

is given by Since YR

@R = nR

eR(2) =

0

fR

--+

(AE"h-0

. It is called theJibre projection for

B(. @ 1) = @ 1 = g(. @ I),

the operation.

x E Ri=o,e,o,

(cf. Lemma V, (l), sec. 7.7), it follows that @R o (e$)+ = 0. S is a homomorphism of operations (with H(S0-0) conIf 9:R nected), we have -+

@8

# %=O

= @R

(cf. sec. 7.8 and 7.9). 7.11. Projectable operations. The operation of E in R will be called projectable if there is a linear map q: R -+ r (not necessarily an algebra homomorphism) which satisfies the conditions (i) q ( x ) = 0, x E Rf, (ii) q(1) = 1, and (iii) q o O(x) = 0, x E E. Thus the operation is projectable if and only if 1 4 O(R0). 292

4. Fibre projection

293

In particular, the operation is projectable I one of the following three conditions hofds : (1) Ro = E=o@ O(Ro); (2) the representation of E in Ro is semisimple; (3) R is connected; i.e., Ro = .'l Now assume the operation (E, in, O R , R, 6,) projection q : R -+ r. Consider the linear map

q @ 1 : R @ AE"

+

is projectable, with

AE".

It satisfies OE(X)

0

(Q 01 )

=

( q 01 )

0

xE E

OR@E(X),

and BE

0

(q 01 ) = ( q 01 )

0

~R@E.

Hence it induces a linear map

(q 01)8#=o: H ( ( R @ AE")e=o

~RBE)

(AE")e=o.

-+

Proposition VII: T h e fibre projection of a projectable operation is given by @ R = ( q @ l)e#=O (YR)B#=O*

Proof: Observe that q restricts to a linear map that n R

Qe=o:

Re,o

+

r, and

= qe#=o@ 1 .

Hence @ R = (q,Eo0L ) 0 (g")-' On the other hand, clearly

0

(q 0c)e=o whence ( q 0L)S#=~ = (q,jz0 @ 1 ) @R = (Q 0L K = O O (YR)B#=O.

(YR)~#=~.

g = Pe=o @ 1, o

(g")-'.

This in turn implies that

Q.E.D.

Example: The operation ( E , i E ,O E , AE", dE). Since AE" is connected, this operation is projectable. We show that its fibre projection is the identity map @AE* = 1 :

(AE"),=o -+ (AE")e=o.

VII. A4Lie Algebra in a Graded Differential Algebra

294

I n fact, let q : AE* + r be the projection. Since (cf. Example 1, sec. 7.8) = p, we have

YAE*

(q @ c,

= c*

YAE'

Now Proposition 1711 implies that

@AE'

= 1.

7.12. The operators i,(a)*. Fix an element a E (APE)O=o and consider the operator iR(a)in R (cf. sec. 7.1). I n view of formula (7.5), the invariant subalgebra is stable under this operator. Moreover, formula (7.8) implies that

+ (-1)P-161&R(a)(~) = 0,

z E RB=O.

iR(a)Sfi(~)

Thus iR(u) induces an operator

homogeneous of degree -p. Proposition VIII:

(1) T h e cohomology structure homomorphism

f R satisfies f R

iR(a)" =

(Wj

@ iE(a))

a E (AE)O=o

f R

(cf. sec. 7.9). (2) T h e fibre projection satisfies

i,(a)*

@R

= iE(a) 0 @ R e

(3) If a is primitive (i.e., if a E P"(E)-cf. is an antiderivation. (4) ~R(u)# 0 eg = 0, u E (A+E)O,O. Proof: ( 1 ) Since tions, we have YR

iR(a)

YR:R

iRBE(a)

-

R

0AE"

Y R = (WR @

sec. 5.14), then i R ( a ) *

is a homomorphism of opera-

iE(a))

YR

,

-

E

(AE)o=o-

On the other hand, the inclusion g: ROZo 0(AE")B,o ( R 0AE")O=, commutes with wR 0iE(u),and so (1) follows. (2) This follows from (I), and the relation @ R = nR 0 9~ (cf. sec. 7.10). (3) This is an immediate consequence of (l), together with the fact

4. Fibre projection

295

that i E ( a )is an antiderivation (cf. Lemma VIII, sec. 5.22) and the fact that f R is injective (cf. Proposition VI, sec. 7.9). (4) This follows from the relation iR(a)0 eR = 0. Q.E.D. 7.13. The Samelson subspace. Identify APE with (AE"),,, under the canonical isomorphism x E of sec. 5.18. Then the fibre projection for the operation (E, iR, O R , R, 6,) is a homomorphism @ R : H(Re-0)

+

Definition: The graded subspace

p,

P R = PE

APE. c PE given by

n Im Q R

is called the Samelson subspace for the operation (E, i R , O R , R, 68). Theorem I: Let (E, i R ,O R , R, 6,) be an operation of a reductive Lie algebra, with H(Ro=o) connected. Then the subalgebra Im p R is the exterior algebra over the Samelson subspace

Im

eR

AP,.

:

Proof: Let i p ( a ) :APE + APE ( a E P % ( E ) )be the substitution operator. I n view of Lemma IX, sec. 5.22, and Proposition VIII, sec. 7.12, we have u E P"(E). @ R 0 iR(a)" = &(a) 0 Q R ,

Thus the subalgebra Im en is stable under the operators ip(a). Now it follows from Proposition I, sec. 0.4, that Im

@R =

A(Im

eR n P E ) = A P R . Q.E.D.

7.14. Basic factors. A graded subalgebra A c H(Re,o) will be called a basicfactor for the operation (E, iR, OR, R, 6,) if it satisfies the following conditions:

(1) Imeg c A. (2) There is an isomorphism of graded algebras

where A @ ApR denotes the skew symmetric tensor product.

296

VII. A Lie Algebra in a Graded Differential Algebra

(3) The diagram

W L o )

-@I3

QR

commutes. Two basic factors A , and A, will be called equiwalent if there is an isomorphism of graded algebras A , 5 A, which restricts to the identity in Im ex. I t follows immediately from the definition that such an isomorphism extends to an automorphism of H(Ro,o),which makes the diagram W O - 0 )

commute. Theorem 11 (reduction theorem): Let (E, i R , OR, R , 6,) be an operation of a reductive Lie algebra E with H(Rozo)connected. Then the operation admits a basic factor, and any two basic factors are equivalent. Proof: Existence: Identify ApR with Im eR via x E , so that @R becomes a surjection onto A P R . Let P c P*(E) be a graded subspace dual to PR (cf. sec. 5.21). Then it foIIows from Proposition VIII, sec. 7.12, that

( 1 ) @ R : H(RO,o)+ APR is a surjective algebra homomorphism, and satisfies u E P. @R 0 ~ R ( u ) # = &(a) 0 p R ,

4. Fibre projection

297

(2) The operators iR(u)# (u E P) are antiderivations and satisfy (iR(a)#)2= 0. (3) Im ej$ c ker iR(u)#.

naEP

Thus the hypotheses of Proposition X, sec. 7.16, in article 5 below are satisfied. It follows that the subalgebra

A

keriR(a)#

= aeP

is a basic factor.

Uniqueness: Suppose A , and A, are two basic factors. Then we have the commutative diagram

40

which yields the commutative diagram

Now the equivalence of A , and A , follows from Proposition XI, sec. 7.18, in article 5, below. Q.E.D. 7.15. The cohomology structure homomorphism. In this section we express the homomorphism f B (cf. sec. 7.9) in terms of the operators

VII. A Lie Algebra in a Graded Differential Algebra

298

iR(a)#.Recall from sec. 5.22 that the canonical isomorphisms x*:

AP,(E) -% (AE),,,

-

-

and

xE:

APE -=-+(AE")e,o

preserve scalar products. As in sec. 7.7 define a derivation d in the algebra H(Re=o)0(AE")o=o by d=

c o;iR(a")# @

p(a""),

V

where a*v, a, is a pair of dual bases of PE and P#(E).Then an automorphism fl of H(Roc0)@ (AE"),,, is defined by

Proposition IX: With the notation above, the cohomology structure homomorphism is given by

gR(t)

==

&t o 11,

t E H(Re-0).

Proof: A simple computation (as in the proof of Proposition V, sec. 7.7) shows that

(40& ( a ) ) b(t 01) = B(iR(.)*tO 11, O

a E P"(E), 5 E H(Re=o). On the other hand, by Proposition VIII, (l),

Now let 5 E HP(R6,O). We proceed by induction on p . For p = 0, the proposition is obvious. Assume now that it hoIds for 1;7 E H * ( R O , ~ ) , s < p . Then, for a E P#(E),

(40iE.(a))(\(5 01) - f R ( 5 ) )

=

&i(a)#t

01) - BR(;B(a)"C)

= 09

4. Fibre projection

299

is the projection with kernel H(R,=o) @ (A+E")e=o. Since (clearly) q(&[ @ 1)) = [ and (by Proposition VI) q ( f R [ )= 5, it follows that

B(5 @ 1) - f R ( [ )

=

5 @ 1 - 5 @ 1 = 0.

This closes the induction and completes the proof.

Q.E.D.

$5. Operation of a graded vector space on a graded algebra 7.16. Operation of a graded space. Let X = CkX k be a finitedimensional graded space satisfying X k = 0 for even k, and let M be a graded anticommutative algebra. An operation of X in M is a linear map i: X

-+ Lryr

such that

( 1 ) i ( x ) is an antiderivation, homogeneous of degree -k (x E X k ) , and (2) i(x)Z = 0, x E

x.

An operation of X in M determines, as in sec. 7.1, the operators i(a) (a E AX) in M given by

i(x, A

. . . A x p ) = i ( x p ) - .- o i ( x l ) , o

xi E

x.

Given an operation of X in M, set

Mi=, =

n ker i ( x ) .

.r€M

Since, for a homogeneous element x, i ( x ) is a homogeneous antiderivation in M, it follows that Mi,,is a graded subalgebra of M. Next, consider the dual graded space X" = &(Xk)"and let i x ( x ) (x E X ) denote the substitution operator in AX".

Proposition X: Suppose X operates in a connected positively graded anticommutative algebra M. Assume that there is a surjective homomorphism of graded algebras @: M + AX", satisfying @ 0

i ( x ) = i&)

0 @,

xE

x.

Then there is an isomorphism of graded algebras

f:Mi,,@ AX* 5 M , 300

301

5. Operation of a graded vector space on a graded algebra

which makes the diagram

M commute. For the proof we first establish Lemma VI: Assume in addition to the hypotheses of Proposition X that all the vectors of X are homogeneous of some degree K ( K odd). Then the conclusion of Proposition X is correct. Proof: Choose a linear map X : X" + M k such that e 0 X = 1. Since M is anticommutative and k is odd, X extends to a homomorphism X,: AX" + M . Define

f:Mi=, @AX* f(z @ @)

=z

*

+

M

X,(@).

Then the diagram of Proposition X certainly commutes. It remains to show that f is an isomorphism. First observe that, since M o = r a n d X = X k , i ( x ) o X maps X * into r. Hence i ( x ) X ( x * ) = e i ( x ) X ( x * ) = i x ( x ) ( x * ) = (x*, x),

X*E

X",

X E

x.

(7.13) Since i ( x ) is an antiderivation, it follows that

f 0 (w, @ i&))

= i ( x ) 0 f,

xE

x,

where w, denotes the degree involution. This yelds

f 0 ( W S 0i A T ( U ) ) = i ( a ) 0 f,

aE

APX.

(7.14)

(1) f is injective: Choose a nonzero element SZ E Mi,o @ A X * .

VII. A Lie Algebra in a Graded Differential Algebra

302

Then, for some p 2 0 and some a ( W L

0i A Y ( U ) ) Q

E

ApX,

=z @

z # 0.

1,

Hence, in view of (7.14),

i(u)f(.n)=f(z @ 1) = z f 0. This shows that f ( Q ) # 0 and so f is injective.

(2) f is surjective: Define an operator Y : M

Y ( z )= C x(erY) i(et,)z,

z

+M

E

by setting

M,

Y

where e*u, e, is a pair of dual bases for X" and

i(x)Y - Yi(x)= i ( x ) ,

X.Use (7.13) to obtain

xE

x.

Conclude that

i(a)Y- Yi(a)= p Next, define subspaces FP

F p

= (Z E

c

i(a),

u E ApX.

(7.15)

M by

M I ~ ( u )= z 0, u E A'X}.

Then

M a=O .

-

F1 c F2 c

Moreover, i ( x ) : F p + l

+F p ,

t

xE

Fn+l= M

(n = dim X).

X.It follows that

Y:F P + '

-+I m f

. Fp.

On the other hand, relation (7.15) implies that 1

- Y - 1: Fp+l-+

P

FP.

These equations yield Fp+lcIrnf.Fp+Fp=Imf.Fp, whence Fn+lc Im f

. F'.

prl,

But Fn+l= M and F1= Mi=,, c I m f. Thus

f is surjective. Q.E.D.

303

5. Operation of a graded vector space on a graded algebra

7.17. Proof of Proposition X: We proceed by induction on dim X .

Let

K be the least integer such that X k f 0. Write Y = Xk

and

Z

CA

=

Xp.

IC

Then

X = Z 0 Y and X" = Z" 0 Y", whence AX

Let p : AX"

=A

2 0 AY

-+AY"

and

AX"

= AZ"

0AY".

be the corresponding projection. Then

Finally, define a graded subalgebra N c M by

N

= {Z E

M I i ( r )=~0, y E Y } .

Then, applying Lemma VI, sec. 7.16, to the action of Y on M , we obtain an isomorphism of graded algebras

g: N @ AY" 5 M , which makes the diagram

(7.16)

M commute. Since e 0 i ( x ) = i x ( x ) 0 e, i.e.,

e

restricts to a homomorphism N

e: N Moreover, since (AX")k= Y",

--+

AZ".

-+

304

VII. A Lie Algebra in a Graded Differential Algebra

These equations together with (7.16) imply that the diagram

N @ AY"

Mi-,

/

PO1

AZ" @ AY"

-

0

N

\ = M

+

P

AX"

commutes. In particular, the hypotheses of the proposition are satisfied for the operation of Z on N. Thus, by induction hypothesis, there is an isomorp hism h: Mi,, @ AZ" N. N

(Note that Mi,, = N,,,,.) Now define an isomorphism N

f:Mi,, @ AX" LM , by f = g o (h @ 1 ) . It follows from the construction that f makes the diagram of Proposition X commute.

Q.E.D. 7.18. Proposition XI:

Let A,, A,, B, S be connected positively graded anticommutative algebras and suppose iA: B -+ AA( A = 1,2) are homomorphisms homogeneous of degree zero. Assume that

B

/ \

N -

(7.17)

is a commutative diagram of homomorphisms of graded algebras in which el, e2 are the obvious projections, and is an isomorphism. Then there A, of graded algebras such that y 0 i, = i2. is an isomorphism y: A,

5. Operation of a graded vector space on a graded algebra

305

Proof: Define a homomorphism of graded algebras

v,: A , @ S +

A, @ S

by setting

Then the diagram (7.17) still commutes if p is replaced by p, . Now we show that p1 is an isomorphism. I n fact, write

Then pz(a @ 1) = a @ 1, a E A,. Moreover, el 0 q, = el, and so

v,(l @ z ) - 1 @ z E A: @ S ) I t follows that for a

Now set

Fp =

E

A<)x

E

z

E

s.

S,

A{ @ S. Then the relation above shows that

Thus vz is a filtration preserving map and induces the identity in the associated graded algebras. Hence, Proposition VII, sec. 1.14, implies that q z is an isomorphism. Thus so is p, = p 0 p2. Finally observe that

Since p,(l @ z ) = 1 @ z , z morphism

E

S , it follows that pl restricts to an iso-

p,: A , @ S + z A , @ S+.

Thus it induces an isomorphism between the factor algebras. Now the

306

VII. A Lie Algebra in

a

Graded Differential Algebra

proposition follows from the commutative diagram

where j , (A = 1,2) denotes the isomorphism induced by the inclusion map A, -+ A l @ S.

Q.E.D.

§6. Transformation groups 7.19. Action of a Lie group. Let G be a Lie group with Lie algebra E. Suppose M x G + M is a right action of G on a manifold M . Recall

from sec. 3.9, volume 11, that this action associates with every vector h E E the fundamental vector field Zh on M given by

d Zh(f)(Z) =

f ( z exp th),-o ?

Moreover the map E -$Y(M)

ZEM, feY(M).

given by

h-zh, is a homomorphism of real Lie algebras (cf. Proposition IV, sec. 3.9, volume 11). Now consider the algebra A ( M ) of differential forms on M and define operators i ( h ) , B(h) i n A ( M ) by

where i(Z,J (respectively, O ( 2 , ) ) is the substitution operator (respectively, Lie derivative) in A ( M ) associated with the vector field 2,. I n sec. 3.13, volume 11, we established the formulae:

Moreover, i ( h ) (respectively, B(h)) is an antiderivation of degree - 1 (respectively, a derivation of degree zero) in A ( M ) . It follows that (E, i, 0, A ( M ) , 6) is an operation of E in the graded differential algebra ( A ( M ) ,6). I t is called the operation of the Lie algebra associated with the action of G on M . 307

308

VII. A Lie Algebra in a Graded Differential Algebra

Next, let N x G -+ N be an action of G on a second manifold N. Suppose q: M -+ N is a smooth equivariant map: V(Z

*

a ) = ( q ~ * )a,

z E M , u E G.

Then the induced homomorphism v*: A ( M ) +. A ( N ) is a homomorphism of operations (cf. sec. 3.14, volume 11). 7.20. The invariant subalgebra. Let T: M x G -+ M be a smooth action of a Lie group G on a manifold M , and let (E, iM,, ,e A ( M ) , dM) be the associated operation of the Lie algebra E of G. Recall from sec. 3.12, volume 11, that the invariant subalgebra of A ( M ) is given by

A,(M)

=

{@ E A ( M ) I T,*@= @, u E G } .

Proposition XII: A,(M) c A(M),=o. If G is connected, then A,(M) = A(M)e=o.

Proof: This is Proposition VI, sec. 3.13, volume 11.

Q.E.D. Proposition XIII: If G is compact, then the inclusion map A(M)o-o A ( M ) induces an isomorphism

--+

Proof: Let Go be the 1-component of G ; then A(M)e-o is the algebra of differential forms invariant under GO.Now apply Theorem I, sec. 4.3, volume 11, to the action of Go on M.

Q.E.D. 7.21. Example: Consider the multiplication map pa: G x G 3 G of a Lie group as a right action of G on itself. Then the fundamental vector field corresponding to a vector h E E is the left invariant vector field X,, generated by h. Let (E, i a , 0, ,A(G),6,) denote the corresponding operation of E. Now consider the algebra AL(G)of left invariant differential forms on G. According to sec. 4.5, volume 11, the algebra AL(G)is stable under the operators i ( X h )and O(X,,).Thus restricting these operators to AL(G), we obtain a second operation of E, called the left invariant operation

6. Transformation groups

309

and denoted by (E, i,, OL , AL(G),d L ) . Clearly the injection ,?G:

A,r,(G) -+ A ( G )

is a homomorphism of operations. On the other hand, as we observed in sec. 5.29, an isomorphism t,: A,(G) -3AE" is defined by tL(@) = @(e). Moreover, under this isomorphism iL(h), OL(h),and BL correspond respectively to i ~ ( h )B,E ( h ) , and 6,; thus tI,:(E, iL,

e,, A,(G), 8,) 2% (& i , ~O,E , AE", 6,)

is an isomorphism of operations. I n particular, the homomorphism is a homomorphism

= lG

0

ti1(defined in sec. 5.29)

of operations. Thus we obtain the commutative diagram

H"(E)

\\\. I G

(7.18)

H(G)

(cf. sec. 5.29). Proposition XIV: If G is a connected Lie group with reductive Lie algebra E, then the map

is an isomorphism. Proof: Since G is connected, and since O(h) = O(Xh) where XI, denotes the fundamental field associated with the right action of G on itself, we have A(G)e=o=

(A,(G) is the algebra of right invariant differential forms on G).

310

VII. A Lie Algebra in a Graded Differential Algebra

Consider the map

G -+ G given by

Y:

= x-l. The isomorphism

Y(X)

P:A ( G ) +- A ( G ) restricts to an isomorphism A,(G) Z - A,(G) of graded differential algebras. Clearly the diagram

AdG)0 AdG)

commutes, where i and j are inclusion maps and w is the degree involution (cf. Lemma V, sec. 4.9, volume 11). Since A(G)e=o= A,(G), we may pass to cohomology in the commutative diagram above to obtain the commutative diagram TI

AdG) -N AdG) N (AE")o=o -

i'l

I

I

l j"

H(A(G),,,)

I

I

ng

H"(E).

H(A,(G))

Tf,

Because E is reductive, nE is an isomorphism (cf. Theorem I, sec. 5.12); = i# o t~1. it follows that so is i#. But

Q.E.D. Remark: If G is compact and connected, then all the maps in diagram (7.18) are isomorphisms (cf. diagram (5.19), sec. 5.29).

-

7.22. Fibre projection. Let M x G M be a smooth action on a connected manifold M . Let A,: G + M be the smooth map given by &(a) = x

*

U,

,

u E G,

where z is a given point in M . According to sec. 4.2, volume 11, the homomorphism A,#:H(G) +- H ( M ) is independent of the choice of z.

311

6. Transformation groups

On the other hand, we have Lemma VII: T h e graded algebra H(A(M),=o) is connected. Proof: Observe that

HO(A(M)o,o)c ZO(A(M),S) = HO(M). Since M is connected, H o ( M )= R (cf. p. 177, volume I). It follows that HO(A(M),=,) = R.

Q.E.D. Now assume the Lie algebra E of G is reductive. Then, in view of the lemma, the fibre projection @A(M,Z f

wM)o=o)

-

W")o=o

9

is defined. Proposition XV: Let M x G - M be a smooth right action of a connected Lie group on a connected manifold M. Assume the Lie algebra E of G is reductive. Then the diagram

H(A(M)e=o)

eAiM~

___*

(AE")o=o

commutes. Remark: Suppose G is also compact. Then (cf. Theorem I, sec. 4.3, volume 11, and sec. 5.29) the vertical arrows of the diagram are isomorphisms. Proof: Since A,(&) = AL(a). b, a, b E G, it follows that A , is equivariant with respect to the actions of G on M and on itself (by right multiplication). Hence, according to sec. 7.19, A; is a homomorphism of operations

VII. A Lie Algebra in a Graded Differential Algebra

312

On the other hand, E ~ AE" : + A(G)is also a homomorphism of operations (cf. sec. 7.21). Thus it follows from sec. 7.10 that the fibre projections satisfy @A(C)

(%)B#=O

and

= @AE*

@A((?) (A?)e#=O

= @A(?&*

Since G is connected and E is reductive, Proposition XIV, sec. 7.21, is an isomorphism. Moreover, according to the example shows that of sec. 7.11, &,EL

= 1:

(AE"),,,

Thus the first equation above yields this in the second, we find

+

(AE"),,,. = [(&G)&,]-'.

Substituting

The proposition follows from this, and the commutative diagram

7.23. The main theorem. Let M x G -+ M be a smooth action of a compact connected Lie group G on a connected manifold M . Let 0

xE:

APE -5H(G)

be the canonical isomorphism (cf. sec. 5.29 and sec. 5.32), and let

A,#:H ( M ) + H ( G ) be the homomorphism defined above. Then we have as an immediate consequence of Theorem I, sec. 7.13, Theorem 11, sec. 7.14, and Proposition XV, sec. 7.22:

Theorem 111: Let M XG + M be a smooth action of a compact connected Lie group G on a connected manifold M . Then the isomorphism a. o x E identifies Im A,# with AP where P is a graded subspace of PE.

6. Transformation groups

313

Moreover, there is a graded subalgebra A c H ( M ) and an isomorphism of graded algebras

A@AP~H(M) which makes the diagram

commute.

Chapter VIII

Algebraic Connections and Principal Bundles In this chapter (R, 6,) denotes a positively graded anticommutative differential algebra. W , denotes the degree involution in R.

$1. Definition and examples 8.1. Definition: An algebraic connection for an operation (E, i R ,O R , R, 6,) is a linear map X : E* R1which satisfies the conditions ---f

iR(x)X(x*)= (x*, x),

x* E: E*,

X E

E,

and eR(x)

0

X =X

0

e,(x),

X

E

E.

(Here, as usual, 0, denotes the representation of E in AE*.) If p: (E, i R ,O R , R, 6,) 3 (E, i s , B s , S, 6,) is a homomorphism of operations, and if XR is an algebraic connection for the operation of E in R, then XS

= p 0 X,

is an algebraic connection for the operation of E in S . In fact, for x* E E" and x E E , we have

Examples: 1. Let (P,n,B, G) be a principal bundle. Then there is a principal action PXG -+ P . The corresponding operation of the Lie algebra of G on the algebra of differential forms on P admits an algebraic connection (cf. article 5 at the end of this chapter). 314

1. Definition and examples

315

2. The identity map E" E* is an algebraic connection for the operation (E, i g, dg, AE*, 6,) (cf. Example 2, sec. 7.4). --+

3. The inclusion map X : E* --+ W(E)given by X(x*) = 1 @ x* is the unique algebraic connection for the operation (El i, O w , W(E),6,) of E in the Weil algebra W(E) (cf. Example 6, sec. 7.4).

Subalgebras. Let F c E be a subalgebra of a Lie algebra E. Recall from Example 4, sec. 7.4, the definition of the operation (F, iF, OF, AE*, dE). Let X : F* + E* be an algebraic connection for this operation, with dual map %*:F t E. Then: 4.

(i) E = ker X* @ F. (ii) X* is the projection onto F induced by this decomposition. (iii) ker X* is stable under the operators adE(y), y E F. (adE denotes the adjoint representation of E-cf. sec. 4.2). Conversely, assume a direct decomposition E = H @ F, where H is a subspace stable under the operators adE(y), y E F. Let n: E F denote the corresponding projection. Then the dual map --+

*

n*:F"

-+ E"

is an algebraic connection. To establish these statements, let X be any algebraic connection. Then for y E F, y" E F* (y*, X"y> = (Xy*,y) = iF(Y)XY+= ( Y * , Y ) . Hence X*y = y, y E F, and so (i) and (ii) follow. Since X O 4 Y E F, we can dualize this relation to obtain

mJ)

0

B,(y)

=

I t follows that ker X* is stable under the operators adE(y), y E F. Conversely, assume that E = H @ F satisfies the conditions above, and let 7c: E --+ F be the projection. Then for y* E F" and y E F,

iF(y)n"y* = ( y " ,

v)=
On the other hand, since H is stable under each adE(y), y have y E F. 7c o adE(y) = ad y o n,

E

F, we

316

VIII. Algebraic Connections and Principal Bundles

Dualizing we obtain

Hence,

AE",

7c*

is an algebraic connection for the operation (F, iF, O p ,

6E).

5. Let E be a reductive Lie algebra and let (E, i, 0, R, 6,) be an operation. Recall from sec. 7.5 the definition of the associated semisimple operation ( E , is, Os , Rs , bS). We shall show that an algebraic connection for (E, i, 8 , R,6,) is also an algebraic connection for (E, is, 8,, Rs , ds), and conversely. In fact, let X : E* --+ R' be an algebraic connection. Since E is reductive, the representation OE is semisimple. Moreover, X is an E-linear isomorphism of E" onto Im X . I t follows that the restriction of 8 to Im X is semisimple and so Im X c R i . Thus X can be regarded as a linear map of E" into Rk and so it is an algebraic connection for (E, is, Os, Rs, ds). The converse is trivial.

8.2. Surjective fibre projection. Let (E, iR, O R , R, 6,) be an operation of a reductive Lie algebra and assume that H(Re-0) is connected. Proposition I: If the fibre projection @R (cf. sec. 7.10) is surjective, then the operation admits an algebraic connection. Proof: Choose nonzero elements p E Hn(Re-o) and @ E (AnE")e,o (n = dim E ) so that e,(p) = @. Let x E Zn(R,,o) be a cocycle representing 8, and let a E (AnE),=obe the unique element such that (@, a ) = 1. Define a linear map X: E* --+ R1 by X(X")

= (-1)n-'iR(iE(x*)a)z,

X"

E

E".

Since a and x are invariant, the formulae of sec. 7.2 show that

Moreover, since a A X = 0 (x E E ) we have i,(X)X(X*)

= i,(X

h iE(x*)a)x = ( X I ,

x)iR(a)z,

XE

E,

X*E

E".

Thus to show that X is an algebraic connection, we must show that i&)z = 1.

1. Definition and examples

317

But the formulae of sec. 7.2 show that iB(a)x is an invariant cocycle of degree zero. Since H(R0,,,) is connected it follows that, for some

a E r,

iB(a)z= 1.

It follows that

(cf. Proposition VIII, sec. 7.12). Hence iR(a)z= 1.

Q.E.D. 8.3. Transformation groups. Let G be a compact Lie group acting smoothly from the right on a manifold M. Consider the induced operation ( E , i, 0, A ( M ) , 6 ) of the Lie algebra E of G in the algebra of differential forms of M (cf. sec. 7.19). We shall show that this operation admits an algebraic connection if and only if the action of G is almost free (for the terminology cf. sec. 3.4, volume 11). I n fact, assume that X: E" --+ A 1 ( M ) is an algebraic connection. Fix h E E (h f 0) and let 2, denote the corresponding fundamental vector field on M . Choose h" E E" so that (h", h ) # 0. Then for z E M , we have

X(h")(z;Z,(z)) = (i(h)X(h"))(z) = (h", h ) f 0, whence

Z,(X) f 0,

h E E, z E M.

This implies (cf. sec. 3.11, volume 11) that the action of G is almost free. Conversely, assume that the action of G is almost free. According to sec. 3.11, volume 11, we can form the fundamental subbundle F M of t M . Since G is compact, it follows from Example 1, sec. 3.18, volume 11, that

, under the action of G. where 7 is a subbundle of t Mstable Now let e: tM + FM be the strong bundle projection induced by the decomposition above and let

318

VIII. Algebraic Connections and Principal Bundles

be the induced map of cross-sections. Since Fryrand 7 are G-stable, it follows that

e" ([Z,, Z ] ) = [Z* eJ1,

h E E,

1

z E F(W.

(8.1)

On the other hand, according to sec. 3.11, volume 11, a strong bundle isomorphism a:M x E FM, N

is given by a(x, h ) = Zh(z),

x E

M , h E E.

Moreover, a ( ~[h, , k]) z= [Z,, Z ~ ] ( Z ) , z E M , h, K

Now define a linear map X: EY

(Xh")(x; Z(Z))= (h", a;1

0

--+

E

E.

A 1 ( M )by setting

ez(Z(X))),

z E M, Z E F ( M ) .

Then

i(h)X(h")(z)= X(h")(X; Z,(X)) =

(h",h),

Z E

M , h E E,

h"E

E".

Finally, it follows easily from formulae (8.1) and (8.2) that

0(h) 0 X

=

(8.2)

X o 0,(h),

h E E.

Hence X is an algebraic connection for ( E , i, 0, A ( M ) , 8 ) .

$2. The decomposition of R 8.4. The decomposition of R as a tensor product. Let X be an algebraic connection for an operation (E, iR, O R , R, ax). Since Im X c R1 and since R is anticommutative, we have X ( X " ) ~ = 0,

X"

E

E".

Hence X extends to a homomorphism of graded algebras X,: AE"

R.

-+

It satisfies the relations

iR(a) o X, = X, o iE(a), a E AE, In fact since iR(xl A . is sufficient to show that

-

A

and

O,(x)

x p )= i R ( x P ) 0

iR(x) 0 X, = X, 0 iE(x),

0

...

X, = X, 0 O,(x), 0

E. (8.3 1

xE

i R ( x l ) (cf. sec. 7.1), it

x E E,

in order to establish the first relation. Now the operators on both sides are X,-antiderivations. They coincide by definition in E X; hence they are equal. On the other hand, O,(x) o X, and X, o OE(x) are both X,-derivations, and they coincide by definition in E". Hence these operators are equal. Theorem I: Let (E, i, O R , R, 6,) be an operation admitting an algebraic connection X . Then an isomorphism of graded algebras

f:Ri-o @ AE" % R is given by

f (X @ @)

=x

*

x

X,(@),

E

(Ri=,@ AE" is the skew tensor product.) 319

Riz0, @ E AE".

320

VJII. Algebraic Connections and Principal Bundles

This isomorphism satisfies

Proof: The commutation relations are immediate consequences of the definition and the analogous formulae for X, (cf. formula (8.3)). To show that f is an isomorphism, we proceed as in Lemma VI, sec. 7.16.

(1) f is injective: Choose a nonzero element SZ E Ri=o@ AE". Then for some u E APE ( p 2 0 ) ,

(WE 0iE(a))S2= x

1,

where x # 0. It follows that

Hence f ( Q ) # 0 and so f is injective.

( 2 ) f is surjective: Define an operator Y:R --+ R by

Y(x)= C x(e*.) - iR(ev)z, Y

where eQv, e, is a pair of dual bases for E" and E. Then

as follows from the relation iR(x)X(x*)= (x+, x). Now an induction argument gives

iR(u)Y -

YiR(u)= p i R ( ~ ) ,

uE

APE.

Next define subspaces FP c ' R by

P = {X E R I ~ R ( u ) z = 0,

uE

APE).

Then

Rino = F' c F 2 c

--

c

P+l =

R

( n = dim E).

(8.4)

2. The decomposition of R

Since, clearly, i R ( x ) :Fp+l + Fp, p

. . . , n, it

1 , 2,

=

321

follows that

On the other hand, formula (8.4) implies that

($Y -

I)

: Fp+1 -+ Fp.

The last two equations yield

FP+' c I m f . FP+ Fp c Imf

-

- Fp,

It follows that P+'c Im f F1. Since Fn+l f must be surjective.

p = 1, ... ,n.

=R

and F1 = Ri=d c I m f,

Q.E.D. Corollary I: The homomorphism X,: AE"

+

R is injective.

Corollary II: R is generated by the subalgebras Ri=o and I m X,. Corollary III: R is the direct sum of the subalgebra Ri=, and the ideal generated by X,(A+E"):

R

= Ri=,@ R

*

X,(A+E").

-

Corollary IV: R is the direct sum of the subalgebra Ro I m X, and the ideal generated by Ri',o:

R

=R

R t o @ Roe Im X,.

Finally, let p: ( E , iR, eR, R, 13,) + (E, is, O s , S, 6,) be a homomorphism of operations. Assume that XR is an algebraic connection for the first operation and let Xs = p o XR be the induced algebraic connection in the second operation (cf. sec. 8.1). Then the induced isomorphisms fR:

RiEo@ AEX 5 R

and

fs: Si=o0AE" 5 S

VIII. Algebraic Connections and Principal Bundles

322

make the diagram

Ri=o0AE"

w

j

fR -

R

1.

Si,o 0AE" ---E.--. S fs

commute, as follows from the definitions. 8.5. Covariant derivative. Let (E, in, O R , R, S,) be an operation with an algebraic connection X. Then the direct decomposition

(cf. Corollary I11 to Theorem I, sec. 8.4) induces a linear projection

zHis "called the horizontal projection associated with X .

-

Since Ri=o is a subalgebra of R while R X,(A+E") is an ideal, it follows that zHis a homomorphism of graded algebras. Evidently,

Now consider the operator V : R

+R

given by

It is called the covariant derivative in R associated with X. It follows from the definition that V is homogeneous of degree 1 and that it satisfies the relations

Since SR is an antiderivation, V is a nH-antiderivation

v(z

*

W ) = VZ

-

XHW

+ (-1)pn~z

*

vw,

z E Rp,

W

E

R.

In particular, the restriction of V to Ri=, is an antiderivation; it is denoted by G-0.

2. The decomposition of R

323

Proposition 11: T h e covariant derivative has the following properties :

VZ = BRz, z E Ri=o,e=o. V z = BRz - CyX(e*.) O,(ev)z, z E Rise. (V 0 XA)@ = 0, @ E Cj>Z HE". (3) (1) (2)

-

Proof: ( 1 ) follows that

If

X E

Ri=o,e=o,then d R z € Ri=o,O=o(cf. sec. 7.3). I t vz = nH6h.z

=

6,z.

(2) Let z E Ri=o. Then O,(e,)z E Ri=, and so X(e*.)

- OR(e,)z

=

eR(x)z- OR(x)z= 0, x E E.

It follows that

Thus

(3) Since V is a nH-antiderivation and since that for j 2 2

(v

0

X,)(X?

=

A

* * *

X

= 0, it follows

ZH

0

* * *

A ~CHX(X~)

AX:)

i

C (-1)v-13Z~X(X?)

A

* * *

Vx(Xp)

v-1

= 0.

Q.E.D.

8.6. Curvature. T h e curvature of an algebraic connection X is the linear map X : E" + RL0

given by X=VoX,

where

V is the covariant derivative associated with X.

VI 11. Algebraic Connections and Principal Bundles

324

Thus (cf. sec. 8.1 and sec. 8.5)

i&)

o

I

=0

e,(x)

and

'1

I

=

2 0 O,(x),

x E E.

Proposition 111: The curvature of an algebraic connection X satisfies the following identities:

(1) I = 6,o X - X, 0 S E (equation of Maurer and Cartan). ( 2 ) V I = 0 (Bianchi identity). ( 3 ) V 2 z = -Cv I ( e " y ) - OR(eV)z,z E Ri=,. (Here eXv, e, is a pair of dual bases for E" and E . )

Since S E : E" + A2E", Proposition 11, sec. 8.5, implies that VX,,SE = 0. (3) Proposition 11, sec. 8.5, yields 17%

C V ( X ( e * v ) . OR(ev)z) = - C ( V X ( e # V ) ) - OR(ev)z+ C nHX(eaV). V(O,(ev)z) =

(V 0 S,)(z)

-

V

V

V

Q.E.D. 8.7. The operation of E in Ri=-,@AE*. Let ( E , iR, O R , R, 6,) be an operation with an algebraic connection X. Denote the curvature and

2. T h e decomposition of R

325

the covariant derivative by X and V , respectively. We shall construct an operation ( E , i, 8, Ri=, @ AE", d ) in Ri=, @ AE". First we define i and 8 by and

i(x) = wR@ iE(x)

@1

8(x) = O,(x)

+

1

@ 8,(x),

xE

E.

To define d we introduce four antiderivations in Ri=,0AE", all homogeneous of degree 1. First we have the operator wR

04

@ dE =

p(e " Y)e E(e ~ ), Y

where PV, e, is a pair of dual bases for E" and E. Next, recall from sec. 5.25 that the representation OR of E in Ri=, induces an antiderivation a8 in Ri=o0AE". It is given by 68

=

c

w,8,(e,)

0p(e"9.

Y

T h e third operator, denoted by h,, is defined by

h,(z @ 1 ) = 0 and

I n particular,

h,(l @ x")

= X(X")

01 .

I n terms of a pair of dual bases for E" and E we can write

h, = C w ~ pXe"V) ( @ iE(eV). Y

Finally, the fourth operator dH is defined by SH =

Vi=, @ 1.

Now set d = m n @ d ~ + do+h,+ Recall from sec. 5.26 that in (Ri=,@ 2WR

@ SE

+ 68 = 0.

VIII. Algebraic Connections and Principal Bundles

326

Hence, the restriction of d to this subalgebra is given by

Theorem 11: Let ( E , i,, O R , R, 6,) be an operation admitting an algebraic connection X. Then (in the notation just defined):

( I ) (Ri=,@ AE", d ) is a graded differential algebra. @ AE", d ) is an operation. (2) ( E , i, 8, Ri=, (3) The isomorphism f: R i = , @ AE" 2R of Theorem I, sec. 8.4, is an isomorphism of operations. Proof: Since f is an isomorphism of algebras, the entire theorem will follow once it has been shown that

f

0

i ( x ) = iR(x) 0 f ,

fo

xE

8(x) = OR(x) 0 f,

E,

and

fod=S

R

0

f*

The first two relations are proved in Theorem I, sec. 8.4.It remains to establish the third. Since d and 6, are antiderivations, we need only show that ( f o d - dRof)(l @ ~ " ) = O = ( f o d -

d,of)(~@l),

x" E E",

zE

Ri=,.

But Proposition 111, sec. 8.6, yields

6,f(l

0x")

= 6nx(x") = =

x(x")

+ x,dJ$(x")

f(x ( x " ) @ 1 + 1 @ S J j ' ( X " ) ) .

Since 6,(1 @ x") = 0 = 6,(1 @ XI), it follows that 6, f ( 1 @ x")

=f

0

(h,

+

OR

@ 6,)(1 @ x") = ( f 0 d ) ( l

0x").

On the other hand, Proposition 11, sec. 8.5, shows that

6,f(z @ 1) = 17,

+ C x(e*u) Y

= (fo

BR(ev)z= f

0

(6,

+ 6,)(x 01 )

d ) ( x @ 1).

This completes the proof.

Q.E.D.

327

2. The decomposition of R

8.8. Homomorphisms. Let a,: ( E , iR, OR, R, 6,) + (E,is, 8,, S , 6,) be a homomorphism of operations. Assume that XR is an algebraic connection for the first operation and let Xs = a, 0 X, be the induced connection for the second operation. Recall from sec. 8.4 the commutative diagram RiZo@ AE" R

-

I

It follows from this diagram that a,,

Z H = 7dH

Yi-0

where nH denotes both horizontal projections. Hence the covariant derivatives VR,V, and the curvatures ZR, Zs are related by vi=o 0 VR = Vs

0

and

9

pi=,,

o

XR

=

2s.

These relations imply that (vi=O

0L ,

0sE) = @ sE) (vi=O@ L ) , (a,i=o 01 ) 6, = 6 , (a,i=o 011, (a,i=o 01 ) h, = h, (a,i=o 011, (wR

O

O

O

O

and (qi=O @ 1 )

SH = BH

(pi+ @ 1 ) .

(Here we have used do, h,, and 6, to denote the appropriate operators both in Ri=o@ AE" and Si=, 0AEX, cf. sec. 8.7.)

The Wed algebra: Consider the operation of a Lie algebra E in its Weil algebra W(E) (cf. Example 6, sec. 7.4). Let X be the algebraic connection given by x* H 1 @ x", x" E E". The horizontal subalgebra is VE" @ 1 and the horizontal projection is induced by the direct decomposition 8.9. Examples: 1.

W(E)= (VE" @ 1 ) @ (VE" @ A+E"). In particular, the curvature is given by %(x") = ZHdw(1 @ x") = nH(x" @ 1

i.e., X ( x " ) = x" @ 1,

+10

BE%?")

x* E

E".

= XIc @

1;

328

VI 11. Algebraic Connections and Principal Bundles

of V to VE" @ 1 is zero. Hence the operators

T h e restriction

h, and BH are given by h,

=h

and

BH=

0.

Thus the decomposition

used to define 6, in sec. 6.4 coincides with the decomposition given in sec. 8.7. Now Proposition 111, sec. 8.6, implies that

0=

= p( %(e+")) 0 O,(ev).

Since W(E),,, = VB" @ 1 and W(e"y) = eryv@ 1, we can rewrite this as

where ,uLs(e"v)denotes multiplication by ebv in the algebra VB". Let E be a Lie algebra and let F be a subalgebra. Assume that the operation (E,iF , OF ,AE", 6,) admits an algebraic connection X : F" --t E" (cf. Example 4, sec. 8.1). We shall compute the corresponding curvature 2. Let e l , . . . , embe a basis of F, and let em+,, . . . , en be a basis of ker X". Then e l , . . . , e n is a basis of E. If e*l, . . . , e t n denotes the dual basis of E", then e"', . . . , esm is a basis of I m X, while e"m+l, . . . ,e"" is a basis for FL. It will now be shown that the curvature is given by 2.

n

Y

=

4 C

,u(e*")0 O,(e,)

0

X.

v=ni+l

In fact, by Proposition 111, sec. 8.6, we have for y"

E

F",

OR, R,6,) be an operation which admits an algebraic connection X. Recall the definition 8.10. The structure homomorphism. Let (E, i,,

2. The decomposition of R

329

of the structure homomorphism +R

yli; R

@ AE"

(cf. sec. 7.8). On the other hand, we have the structure homomorphism YAE.:

AE"

-+

AE" @ AE",

for the operation ( E , iE, e E , AE", 8,) as well as the isomorphism f : Rime @ AE" 5 R of sec. 8.4. Proposition IV: T h e isomorphism f makes the diagram

commute. Proof: Since all the operators are algebra homomorphisms, we need only check the diagram for elements of the form z @ 1 ( z E Ri-0) and 1 @ X + (x" E E"). Recall from sec. 7.8 that y R ( z )= b(z @ l), z E R. This implies (in view of Lemma V, sec. 7.7) that YR(z) =

@ 1,

E Ri-0,

whence (YRf

)(x

01) = Y R ( Z ) = z 01 = (f@

1 ) O (1

0YAE*)(Z 0I), E Ri-0.

On the other hand, Lemma V, sec. 7.7, shows that YR(X(X*))

=~

( X ( X " ) @ 1) = X ( X " ) @ 1

+ 1 @ x",

X"

E

E".

It follows that (YRf)(l @ x")

= X(X")

@1

+ 1 @ x" 1 0 1 0%")

=

(f@L)(l

=

(f@ l) @ Y A E * ) ( l @ X"),

(cf. Example 1, sec. 7.8).

@X*@l+ (1

Q.E.D.

VI 11. Algebraic Connections and Principal Bundles

330

Corollary: If @ E AE", then

( y Ro X,)(@)

-

1 @ @ E R+ @ AE".

Proof: It follows from Example 1, sec. 7.8 that Y,+,p(@)- 1 @ @ E A+E" @ AE".

Hence, (L

@ y A p ) ( l @ @) - 1 @ 1 @ @ E Ri-0 @ A+E* @ AE".

Applying f @ 1 to this relation and using the proposition, we find that (yRf)(l @ @) - 1 @ d, E R+ @ AE";

i.e., (yaXA)(@)- 1 @ @ E R+ @ AE".

Q.E.D.

$3. Geometric definition of an operation 8.11. Assume that ( E , iR, O R , R, 6,) is an operation with an algebraic connection X R . I n articles 1 and 2 we constructed the following "geometric" objects:

(1) The horizontal subalgebra Ri=,and the restriction of the representation of E to Ri=,. (2) The covariant derivative (restricted to I?,=,,) and the curvature

q-,:Ri=,

+

R+,

and

XR:

E'

+

R:,,.

I n sec. 8.5 and sec. 8.6 it was shown that the following properties hold:

( 1 ) &=, is an antiderivation in Ri=,, homogeneous of degree 1. (2) IR 0 8,(x) = 8,(x) 0 XR, and Gi=, 0 e R ( x ) = OR(x) 0 q-,,x E E. (3) q - 0 %R = 0. (4) K2=, = - ~ Y , u ( I R ( e b Y ) ) ~ R where ( e y ) , e"", e, is a pair of dual bases for E" and E. T h e purpose of this article is to reverse this process; in particular we shall establish Theorem In: Let 8T be a representation of a Lie algebra E in an anticommutative positively graded algebra T. Assume that linear maps VT:

T+ T

and

X : E"+ T 2

are given, subject to the following conditions :

(1) 0, is an antiderivation, homogeneous of degree 1. (2)

X

0

8,(x) = 8,(x)

0

.X, and

0,o

&(x) = e,(x)

0

V,,

x E

E.

(3) v, 0 I = 0. (4) 0," = -Cy pr(Ie*wy)O,(e,) (where e*', e, is a pair of dual bases for E", E, and pl.(z) denotes left multiplication by z in T). Then there is a differential operator d in T @ AE'

(E, i, 8, T @ AE*l d ) such that (i)

( T 0AE")i=, = T @ 1. 331

and an operation

332

VI 11. Algebraic Connections and Principal Bundles

(ii) The restriction of 8 to T @ 1 is OF. (iii) The inclusion map X : E" + 1 @ E" is an algebraic connection for the operation. (iv) X is the curvature of the co,nnection X . (v) T h e restriction of the covariant derivative to T @ 1 is 0,. Moreover, the operators i ( x ) , 8(x) (x by conditions (i)-(v).

E

E ) , and dare uniquely determined

Remark: If ( E , iR, OR, R, 8,) is an operation with a connection XB and we apply Theorem I11 with

T = Rieo,

8, = O R ,

0, = Visa,

and

X = Xa,

then the resulting operation coincides with the operation defined in sec. 8.7, as will be obvious from the definition. Hence, in view of Theorem 11, sec. 8.7, the operation obtained from Theorem I11 in this case is isomorphic to the original operation. 8.12. The differential algebra (T@ AEW,d). I n this section we use the notation of Theorem 111. I t is assumed that the algebra T and the operators OT(x) (x E E ) , VT , and X satisfy the hypotheses of the theorem. We wish to construct an antiderivation of degree 1,

d: T @ A E " - T @ A E " , such that d2 = 0. T o do so we introduce four antiderivations in T @ AE", all homogeneous of degree 1 precisely as in sec. 8.7. First we have the operator wT

@ 8E

= OT

@4

p(e"Y)eE(ev), Y

where esv, e, is a pair of dual bases for E" and E, and wl1 denotes the degree involution in T . Next recall from sec. 5.25 that the representation 6 T of E in T induces the antiderivation 8, in T 0AE" given by 88

=

wTeT(ev) V

0p(e"v)*

Thirdly, define an antiderivation h, of degree 1 by

h,(X @ 1) = 0,

x

E

T,

3. Geometric definition of an operation

333

and

...

X F E E"

Tq,

X E

( i = 1, . .

I n particular,

h,(l @ x")

= iE(x")

x* E

@ 1,

E".

If e"', e, is a pair of dual bases for E", E we can express h, in the form h,

=

C wTpT(Ze"")

@ iE(e,).

b

Finally, since 0, is an antiderivation of degree 1 in T , the operator 8H = 0 , @1 is an antiderivation of degree 1 in T @ AE". Now we define d by

d= Proposition V:

wT

@ 8E

+ So + h, + 8 H .

(T@ AE", d ) is a graded differential algebra.

Proof: Clearly d is an antiderivation homogeneous of degree 1. Thus we have only to show that d2 = 0. According to sec. 5.25, (wT

@ 8 E $-

86)'

= 0.

Moreover, it is immediate from the definitions that zE

hi(z @ 1) = 0 = hi(1 @ x"),

T,

X" E

E".

Since hi is a derivation, it follows that

hi Next we show that the equation 6H

0

h,

= 0.

VT

+ h,

0

0

(8.9) X

=0

8, = 0.

implies that

(8.10)

VIII. Algebraic Connections and Principal Bundles

334

I n fact, dH 0 h, (dHh,

+ h,

0

dH is a derivation. Moreover,

+ h,6,)(.

@ 1) = h,(V+ @ 1) = 0,

x

E

T,

and

Hence (8.10) is correct. Now we shall use the relations

V; = - 1p T (Xe*V)O,(e,)

and

I 0 e,(x)

= e,(x)

0

I,

x

E

E,

Y

to prove that

I n fact, denote the left-hand side by cr. Then cr is a derivation, as is -6;. Moreover, for X E Tg, U(X

@ 1) = h,S,(z @ 1) = h, = Cp,(ae*~)e,(e,>z @ V

On the other hand, for x*

But if x , y E E, then

E

E",

1 = -&(z @ 1).

3. Geometric definition of an operation

335

This yields

and hence

o(1 @ x")

=0 =

-&(1

0x").

Formula (8.11) follows. Finally, we use the equations

to show that

Denote the left-hand side by we have

t. Then T

is a derivation. For z E Tq

On the other hand, clearly

t ( l @ x")

= 0,

X" E

E",

and so (8.12) follows. Now adding relations (8.8)-(8.12) yields d2 = 0, and so the proof of the proposition is complete.

Q.E.D. 8.13. Proof of Theorem 111: We first construct the operation ( E , i, 8, T @ AE", d ) . Recall the definition of the graded differential algebra ( T @ AE", d) in sec. 8.12.

Now define operators i ( x ) and 8(x), x E E , by

Then each i ( x ) is an antiderivation of degree - 1, while 8 is a representation of E in the graded algebra T @ AE". Moreover, i ( ~ ) 2= L

0~ E ( x ) '= 0,

xE

E

336

VIII. Algebraic Connections and Principal Bundles

i(x)0

(OT @

+

6,)

06,)

0

i(x)= 1 0 O&),

i ( x ) o d8

o

i ( x ) = O,(x)

i(x) 0

0

i ( ~=)0,

o

i ( x ) = 0,

(WT

+ 68 h, + h,

and

i ( x ) o BH

+ BH

0L , x E E.

These relations imply that

Hence, ( E , i, 0, T 0AE", d ) is an operation. Next it will be shown that this operation satisfies conditions (i)-(v) of Theorem 111. The first three are immediate consequences of the definitions. T o verify (iv), observe that the horizontal projection nH corresponds to the decomposition

T@AE"= (T@l)@(T@A+E"). Hence we have

V+")

= (ZH

0

d)(l

= T(x") @

0x")

1,

x*

= E

(nfi0 h,)(l @ x")

E+;

i.e., 3 is the curvature for X. T o prove (v) let x E T . Then

V ( x @ 1) = n,d(z @ 1 )

= VTZ @ 1,

whence q=,,= V,. It remains to be shown that the operators i ( x ) , O(x) ( X E E ) , and d are uniquely determined by conditions (i)-(v). Let (E, i, 8, T @ AE*, d )

337

3. Geometric definition of an operation

be a second operation which satisfies these conditions. Then

z E T , x E E,

i ^ ( ~ )@ ( z 1) = 0, and

i^(x)(l @ x")

= i ^ ( x > X ( ~ "= ) (x",

x),

X" E

E", x E E.

It follows that [(x) = i ( x ) , x E E. T h e same argument shows that 6(x) = O(x), x E E. Finally, we prove that d = d. Let z E T. Then Proposition 11, sec. 8.5, yields

d(z 01 ) = V T z

1

+ 1 X(e*v)

. e(e,)(z

1)

V

= d(z

On the other hand, if x"

d(l @ x")

=

@ 1).

E

E", Proposition 111, sec. 8.6, yields

($0

X)(X")

= qx")

+ X,6,(.")

= d(l @ x").

Since d and d are antiderivations, it follows that d = d. This completes the proof.

Q.E.D. 8.14. The algebra VE* @ R. Let (E, i,, O R , R,6,) be an operation. Recall that E" is the graded space which coincides with E" as a vector space and all of whose elements are homogeneous of degree 2 (cf. sec. 6.1). Recall further that each -(ad x)" (x E E ) extends to a derivation Os(x) in the graded algebra VE", and that the map x H O,(x) defines a representation of E in VE". Now consider the graded anticommutative algebra T = VE" @ R . We can define a representation 6~ of E in this algebra by setting &(X) = &(X) @ L 1 @ 6 ~ ( x ) ,X E E. Next let hR be the antiderivation in VE" @ R given by

+

where PV, e, is a pair of dual bases for E", E, and ps(Y) denotes multiplication by Y in VB". Then

hR e,(x)

=

e+)

hR,

xE

E.

(8.13)

VIII. Algebraic Connections and Principal Bundles

338

+

A computation shows that Cu(8E(x)e*V 0e, PU@ [x, e,]) = 0, and so formula (8.13) follows. Finally, define an antiderivation D Rof degree 1 in VIE* @ R by setting

D, = L @ 8,

- hn.

Then

D, e,(x)

=

e,(X)

0

x E E,

DR,

(8.14)

and

Di = - Cp(e*'

(8.15)

@ 1) o 8,(ev).

In fact, the first relation follows at once from formula (8.13). T o prove the second, observe that hk = 0 = S i . Hence 0: = -hR =-

0

(1

@ S,)

-

((06,)

hR

c ,Us(e"') 0

e,(e")

V

=-

0

2 Y

01)&(%) +

c Ps(e*v)eS(~r)0

1.

V

But according to formula (8.7) in Example 1, sec. 8.9, X up5(e+y)8s(eu)= 0. Thus formula (8.15) follows. Now consider the graded algebra VE" 0R @ AE*. Proposition VI: There exists a unique differential operator d in VE* @ R @ AE+ and a unique operation ( E , i, 8, V B W@ R 0AE", d ) satisfying the following conditions :

(i) ( V P @ R @ AE*)i,o = VE" 0R @ 1. (ii) The restriction of 8 to VE" @ R @ 1 is 8,.

3. Geometric definition of an operation

339

(iii) The inclusion map X : x" H 1 @ 1 ox", x" E E", is an algebraic connection for the operation. (iv) The corresponding curvature I is the inclusion map given by X " H X " @ 1 @ 1. (v) The restriction of the covariant derivative to VE" @ R is given by DR. Proof: We shall apply Theorem 111, sec. 8.11 with

T

=P VI

0R,

eT = eTT, vT = D R

and

I ( x " ) = X" @ 1,

X"

E

E".

T o verify that the hypotheses of Theorem I11 are satisfied, observe that (1) is obvious, (2) follows from formula (8.14), (3) is obvious, and (4)is relation (8.15).

Q.E.D.

$4. The Weil homomorphism 8.15. Definition: Let ( E , iR, O R , R, 6,) be an operation admitting an algebraic connection X with curvature 2: E* + Ri=,. Regard 2 as a linear map 2: E" RiXo -+

homogeneous of degree zero. Since Rise is a graded anticommutative algebra, it follows that

X(X")

I ( y * ) = I(y*) * X ( X * ) ,

X*,Y" E E".

Hence 3 extends to a homomorphism of graded algebras 2,: VE* + Ri=o.

Recall that

I

0

e,(x)

= e,(x)

w,

x E E,

and that the extension of O,(x) from E" to a derivation in VE* is denoted by O,(x). Evidently, wv e,(x) = e,(x) 3,. In particular, X, restricts to a homomorphism (Iv)~-o:

(VB*),=o -+ Ri-0,e-o-

Next, recall that the Bianchi identity (cf. Proposition 111, sec. 8.6) states that 6 - 0 0 I = 0, where Fxodenotes the restriction of 1;7 to Ri=,. Since derivation, it follows that q - 0 0 I, = 0.

KX0is an

anti-

Finally, recall from sec. 8.5 that the restriction of K'i=o to Ri-o,e-o coincides with b R . Hence the above equation restricts to 8R

(&)+o

340

= 0.

4. T h e Weil homomorphism

341

Thus composing (Iy)e=owith the projection Z(Ri,8,6_,) + H(Ri,o,e=o), we obtain a homomorphism X # : (VE"),=,

-

H(Ri,,,&,)

of graded algebras. It will be shown in sec. 8.20 that I# is independent of the mnnection. The homomorphism I# is called the W e d homomorphism of the operais called the tion. T h e image of I*, which is a subalgebra of H(Ri=O,B=o), characteristic subalgebra. I n the next section we give another interpretation of I,, (Iv),=,, and X # . 8.16. The classifying homomorphism. Again, let ( E , i R , O R , R,d,) denote an operation with algebraic connection X . Then X determines the homomorphism of graded algebras Xfr: W ( E ) + R

given by Xl,(!P

@ @)

=

Z,(!P)

*

Y EVE",

X,(@),

@E

AE".

XW is called the classifring homomorphism of the operation (E, iR, O R , R , 8,) corresponding to the connection X.

*PropositionVII: With the notation above Xlv is a homomorphism of operations : xW:

( E , i, %, W ( E ) ,6,)

-

( E , i R , OR , R, 8,).

Proof: T h e relations i B ( x ) X W ( l @ x") = iR(x)X(xx)= (x", x) = XWi(x)(l @ x"),

and

iR(x)Xw(x" 01) = i R ( x ) Z ( x x )= 0 = XIVi(x)(x"@ l), x E E, x* E E", show that iR(x)Xfvcoincides with XHri(x)in (E" 01) @ (1 @ E"). But this space generates the algebra W ( E ) , and X,i(x) and iR(x)& are XWantiderivations. Hence, iR(x)Xw = Xwi(x).

342

VIII. Algebraic Connections and Principal Bundles

Similarly the relations

It remains to prove that 6, 8.6, yields

aRxIr(1@ x")

0

XIr

=

XFr

0

X(x")

=

&X(x")

=

XIVSII.(l0x").

=

dlV. Proposition 111, sec.

+ X,(S,x")

On the other hand, Proposition 11, sec. 8.5, and Proposition 111, sec. 8.6, imply that

Since BN XIv and Xlr6,, are XIr-antiderivations, these relations yield

Corollary: XFr induces homomorphisms ( Xw)i=o, ( X l r ) g 0 , 8 = 0 . They are given by

(Xw)i=o,e=o,

and

Proposition VIII: Let (E, iR , O R , R, 6,) be an operation with a connection X and let eR: Ri=o,e=o + Re=, denote the inclusion map. Then

eg

0

( X * ) + = 0,

and so the ideal generated by Im(X*)+ is contained in the kernel of eg.

4. The Weil homomorphism

343

Proof: Since the classifying homomorphism XFv is a homomorphism of operations, the corollary above yields the commutative diagram

Thus the proposition follows from the relation ff+(W(E),,o) = 0 (cf. Proposition I, sec. 6.6).

Q.E.D. 8.17. The differential algebra W E * @ R)e=o. In this section we establish a theorem which will enable us to show that the Weil homomorphism is independent of connection. Let (E, i R ,O R , R, 6,) be an operation. Recall that in sec. 8.14 we introduced the graded algebra T = VE" @ R, the representation O T , and the antiderivation D R = 1 @ SR - h R . Moreover, it was shown that

and

Di

=-

C ,u(e*"

@ 1) 0 O,(e,).

Y

These relations imply that D R restricts to an antiderivation D, in (VE* @ R)e=O, and that in this algebra Df = 0. Hence ((VE* @ R)e=o, D R ) is a graded differential algebra. Now consider the injection

Since hR =

i.e.,

En

xu,uS(e*,)

0iR(e,) it follows that

hR

0

ER =

is a homomorphism of graded differential algebras.

0. Hence,

V I I I. Algebraic Connections and Principal Bundles

344

Theorem IV: With the notation above, assume that the operation (E, i R ,O R , R, 6 , ) admits a connection. Then the map

-

H(Ri=o,e=o) H((VE' @ R)o=o,D R ) , is an isomorphism. Proof: Let X denote the connection and consider the corresponding operation (E, i , 8, RiXo@ AE", d ) (cf. sec. 8.7). According to Theorem R is 11, sec. 8.7, an isomorphism of operations f:Ri=, @ AE" given by N

. f(x @ @)

==

x

*

x

(X,,@),

E

Ri=o, @ E AE".

It follows that an isomorphism of graded algebras

is given by

g(.

Y @ @ )= Y @ x * ( X , @ ) ,

x

E

Ri=o, Y EVE",

@ E AE".

Identify these algebras via this isomorphism. Then (cf. Theorem 11, sec. 8.7, and sec. 8.14)

Now filter the algebra (Ri=, @ VB" Fp =

C

AE"),=o by the ideals

(RCo@ VE" @ AE"),=,.

Ir 2 P

T h e formulae above show that DR (= 6 ,

-

h R ) is filtration preserving.

4. The Weil homomorphism

345

On the other hand, filter Ri=o,s=o by the ideals

Then E~ is a homomorphism of graded filtered differential algebras. Hence it induces a homomorphism (ER)~:

(Bi, di)

+

(Ei,di),

i 2 0,

of spectral sequences (cf. sec. 1.6). We shall show that ( E ~ is) ~an isomorphism; in view of Theorem I, sec. 1.14, this will imply that ef: is an isomorphism. Consider the operator, OR

@ 1 86 B f

@ 0p(e"')

~ R ( e v ) ~ R1

-

hR >

Y

in R 0VE" 0AE". It commutes with O,(x), x E E, and hence restricts to an operator 6, in ( R @ VE" 0AE*),=,. Moreover, the formulae above imply that

6,: (Ri",o@ VE" @ AE"),,o

-+

(Rf'o @ VE" @ AE"),=o,

p = o , 1 , ..., while DR - S,: Fp -+ Fpf'. Thus it follows from sec. 1.7 that 6; = 0, and that there is a canonical isomorphism

H((R,=,@ VB" @ AE*),=o, 6,) 22 El. Similarly there is a canonical isomorphism N

Ri=o,e=o =+ El. Moreover, according to sec. 1.7, these isomorphisms identify the homomorphism Ri=o,e=o

+

H ( ( R , = ,0VE"

0AE")o=o

(ER)'

with

6,)

induced by E ~ (It . will be denoted by E ; . ) We have thus only to show that EE is an isomorphism. But this is an immediate consequence of the two lemmas in the next section. 8.18.

Observe that (Rixo@ VE" @ AE")B=O= (Ri=o@ W(E))e=o.

346

VIII. Algebraic Connections and Principal Bundles

Lemma I: The operator

Proof: I n

BT is given by

W(E)),=,we have

(cf. formula (5.4), sec. 5.3). Thus the lemma follows at once from the definitions of dT and d,,. (cf. sec. 6.4).

Q.E.D. Lemma 11: The homomorphism err (end of sec. 8.17) is an isomorphism.

Proof: Since

it is sufficient to show that

+

is a linear In view of Lemma I, sec. 6.6, the map d = &k automorphism of W+(E).Evidently, 1 @ d restricts to an automorphism of (Ri=o@ W+(E)),,,. It follows from the definition y = (1 of A that y commutes with wR @ drr and that

Since y is an automorphism, these relations imply (8.16).

Q.E.D. 8.19. The inverse isomorphism. In this section we shall give an explicit expression for the isomorphism

-

H ( (VE+ @ R)e=o DR)

N -

H(Ri=o,e=o)

4. The Weil homomorphism

347

inverse to E$ (cf. Theorem IV, sec. 8.17). Define a homomorphism of graded algebras a,: VE" @ R + Ri=o by

0Z) = X y ( ! P )

a,(Y

- ~H(z),

!P E VE",

'

zE

R,

where x H denotes the horizontal projection onto Ri=o (cf. sec. 8.5). Lemma 111: T h e homomorphism a, satisfies

e,(x)

ax

= e,(X)

E

xE

0

and a,

0

DR

=

V

001,.

Proof: T h e first equation is immediate from the relations zHo

eR(x)= e,(x)

o

X

and

zH

0

O,(x)

=

x

e R ( x )0 X ,

E

E

(cf. sec. 8.5 and 8.6). T o prove the second, observe first that, in view of the Bianchi identity in Proposition 111, sec. 8.6, a , o D R ( ~ " @ l ) = O = (Voa,)(x"@l), Next, if z

E

X"E

E".

Ri=o,then (a, 0 D,)(1 @ Z ) = ",(I @ ~ R z )= T c H ( ~ R z ) =

Finally, if x"

G

V ( Z ) = (V oa,)(l BZ).

E" then the definition of X in sec. 8.6 yields

(a, 0 D,)(1

0X X " )

= a,(l

@ B R X ( X " ) - X"

=

(nf&q(x")

-

=

(VX - #)X"

=

01 )

X(x")

0.

Since ax(l 0 X X " )

= ZHX(X")

= 0,

we obtain OL, 0 DR( 1 @

XX")

=

0

=

( V 0 a,)( 1 @ XX").

Now observe that VE" @ R is generated by E" @ 1 , 1 @ Ri=, and 1 @ X(E"). Since a, 0 D, and c7 0 a, are a,-derivations, the relations

348

VIII. Algebraic Connections and Principal Bundles

above imply that ax 0 DR = V o a x .

Q.E.D. I n view of the lemma, ax restricts to a homomorphism of graded differential algebras (ax)e=o:

(WE* 0R)e=o3 DR)

+

(Ri=o,e-o9 8 ~ ) .

Proposition IX: Let ( E , iR , O R , R, 6,) be an operation admitting an algebraic connection. Then the induced homomorphism

-

H((VE* @ R)e=o) H(R,=o,e=o) is the isomorphism inverse to the algebraic connection.

6%.

is independent of

In particular,

Proof: Clearly, (ax)e=o 0 cR = 1, whence IV, sec. 8.17, E# is an isomorphism. Thus isomorphism.

0

62 = 1. By Theorem

(a,)e#=omust

be the inverse Q.E.D.

8.20. Independence of connection. Theorem V: The Weil homomorphism Zf of an operation (E, i R ,O R , R, 6,) which admits a connection XR is independent of the connection.

denote the inclusion map :

In view of the definition of a,, we have the relation (ax)e=o

Moreover, clearly graded algebras

DR o

0

6~= ( Z d V , e = o .

lR= 0 and so lBinduces a homomorphism of

tj$:(VE*),=o

-+

H((VE* 0R)e=o,DR).

4. The Weil homomorphism

349

It follows that (ax)B#=o 0

51 = 28.

Since the maps (GI~)B#=~ (= ( E $ ) - ~ ) and algebraic connection, so is %I.

[I

are independent of the Q.E.D.

Corollary I: Let q R :VIE* @ R V+E* @ R. Then VR

--f

D R ==

R be the projection with kernel

BR

qR

9

and the diagram

commutes. Proof: We have noted above that the left hand triangle commutes. Clearly l j l 0~ E~ = e R , and so the right hand triangle commutes too. Q.E.D. Corollary n: Let v: (E, i R ,O R , R,6,) + (E, is, &, as) be a homomorphism of operations which admit algebraic connections. Then the diagram H(Ri=O,&O)

s,

Proof: Let XR be an algebraic connection for the first operation. Then Xs = q o X R is an algebraic connection for the second operation

VIII. Algebraic Connections and Principal Bundles

350

(cf. sec. 8.1). With this choice of connections we have

(cf. sec. 8.8). I t follows that (2,)" = piso 0 (2&, whence # 2," = Qi=o,f+o

0

2;.

Since 2: is independent of the connection, the proof is complete. Q.E.D. 8.21. Cohomology sequence of a regular operation. Definition: An operation ( E , i,, O H , R, 6,) will be called regular if:

(i) E is a reductive Lie algebra. (ii) H(Re=o) is connected. (iii) T h e operation admits an algebraic connection. Let ( E , in, 6,, R, 6,) be a regular operation. Then there are homomorphisms :

ex: H(Re=o)-+

(AE")o=o

eg: H(Ri=o,o=o) -+ H(R8-0)

(fibre projection), (induced by the inclusion map),

and 2:;

(VE")e=o

+

H(Ri=o,o=o) (Weil homomorphism).

The sequence

will be called the cohomology sequence of the operation. According to Proposition VIII, sec. 8.16, and sec. 7.10,

Next, let y : (E, iR, O R , R, 6,) -+ (E, is, O,, S , 6,) be a homomorphism of regular operations. Then it follows from sec. 7.10 and Corollary

4. The Weil homomorphism

351

I1 to Theorem V, sec. 8.20, that the diagram

-

H(Ri=O,O=O)

ew

H(Ro=o)

commutes. This diagram shows that a homomorphism of regular operations induces a homomorphism of the corresponding cohomology sequences.

$5. Principal bundles This article will make consistent reference to Chapter VI, volume 11.

G denotes a connected Lie group with Lie algebra E. = (P, n, B, G ) be a principal 8.22. Principal connections. Let ,9' bundle, with principal action, T : P X G -+ P, (cf. sec. 5.1, volume 11). Let 2, E p ( P ) be the fundamental vector field generated by h (h E E) (cf. sec. 7.19), and denote i(2,) and O(2,)simply by i ( h ) and 8(h). Then (E, i, 8, A ( P ) ,6) is an operation of E in A ( P ) . It is called the operation of E associated with the principal bundle ._P. T h e remark at the end of sec. 6.3, volume 11, shows that the homomorphism n*: A ( B ) -+ A ( P ) can be regarded as an isomorphism

n": A ( B )3 A(P)i=o,e-o.

Now let V be a principal connection i n . 9 ' (cf. sec. 6.8, volume 11). Thus V is a G-equivariant strong bundle map in zp, which projects the tangent bundle onto the vertical subbundle. T h e bundle map H = L - V is the projection of tp onto the corresponding horizontal subbundle. I n sec. 6.10, volume 11, it was shown that principal connections are in one-to-one correspondence with the E-valued 1-forms w on P which satisfy i(h)w = h and 8(h)w = -ad h(co), h E E. The E-valued 1-forms on P satisfying these conditions will be called connection forms on P. T h e correspondence between connections and connection forms is given, explicitly, by Z w W = w

VC),

5 E T,(P),

27

E

p.

is called the connection form for V. On the other hand, let w be a connection form, and define a linear map o+:E"

+A 1 ( P )

by

w"(h")(z; 5') = (h",

O(Z;

t)), 352

h" E E",

5 E T,(P), z E P.

5. Principal bundles

The relations above for

w

353

imply that

i(h)(w*(h"))= (h", w (2 ,))

=

(h", h )

and

5')

[8(h)(w"(h"))l(z;

(h", (O(h)w)(z;5')) = ( b ( h ) ( h " ) ,w ( z , t)> = [wq(8E(h)h")1(X; C), h~ E , h" E E", [ E T,(P), Z E P. =

Hence W" is an algebraic connection for the operation of E in A(P). Conversely, if X is an algebraic connection for the operation, then a connection form w for the principal bundle is defined by

(h",

W(Z;

C))

=

X(h")(z;C),

h"

E

E",

[ E T,(P), x E P.'

Hence we have a bijection between algebraic connections for the operation of E in A ( P ) and principal connections in .6. In particular, since a principal bundle always admits a principal connection (cf. sec. 6.8, volume 11), the operation ( E , i , 8, A ( P ) ,S) always admits an algebraic connection. Finally, let V be a principal connection in .P with corresponding algebraic connection, a". Then the isomorphism

f:A(P)i=o@ AE" 35 A ( P ) , induced by w" (cf. sec. 8.4) is given by

f ( Y 0@)k; 51 9

* * *

, Cp+J

8.23. Horizontal projections. Fix a principal connection V in ,p with connection form w and corresponding algebraic connection w *. Let H = L - V be the projection on the horizontal subbundle. I n sec. 6.1 1, volume 11, we defined a corresponding operator

H": A ( P ) --* A(P)i=o by

(H"Q)(z; C i ,

. . . , C p ) = Q ( z ; HC1, . . Hcp), sz E AP(P), C i E T*(P), *

9

H" is called the horizontal projection associated with V .

z

E

P.

354

VIII. Algebraic Connections and Principal Bundles

On the other hand, in sec. 8.5 we defined the horizontal projection associated with the algebraic connection w" : nI{:A ( P ) + A(P),=o.

We show now that a z= ~ H".

(8.17)

I n fact, both operators reduce to the identity in A(P),=,. Moreover, by definition, nH0 w* = 0, while

H"(w"(h")) = (h", H " w ) = 0,

h*

E

E".

Hence H" 0 w" = 0 as well. Since (cf. Corollary I1 to Theorem I, sec. 8.4) the algebra A ( P ) is generated by A(P)i=oand I m w*, formula (8.17) follows. 8.24. Covariant derivative and curvature. Fix V , w, w + as in sec. 8.22. In sec. 6.12, volume 11, we defined the covariant exterior derivative Vp: A ( P ) 4 A ( P ) corresponding to V by

a,

= H"

0

6.

On the other hand, in sec. 8.5 we defined the covariant derivative corresponding to w" by = ng 0 6.

V

v

Since, in view of formula (8.17), nH = H", it follows that

v = 0,.

(8.18)

Next, recall from sec. 6.14, volume 11, that the curvature 52 E A 2 ( P ;E ) for V is defined by

52 = vp3,

where c7, is regarded as an operator in A ( P ;E ) . On the other hand, the "algebraic curvature" I of w * is defined by

I=

Vow",

Since V = 09,it follows that

(h", Q ) = C"((h", a))

=

(V o ~ " ) ( h " ) , h"

E

E".

Hence 52 and Z are related by

X(h")

=

(h", Q),

h"

E

E"

(8.19)

5. Principal bundles

355

8.25. The operator h,. Again fix V , w , a*. Recall from sec. 8.7 the definition of the antiderivation h, in A ( Q = ,@ AE". Under the isomorphism f induced by w" this operator corresponds to an antiderivation (again denoted by hn) in A(P). I t is given, explicitly, by ( h z @ ) ( z ;5 0 , =-

* * * 9

5,)

C ( - ~ ) ~ + j @ ( zzo(z:ct,cj) ; ,

i
50

9

*

@ E AP(P),

ti

* *

- tj

* * *

, 5,))

T i E T,(P), z

E

P.

I n fact, let h*v, h, be a pair of dual bases in E" and E. Then

h,

=

c

p( iEhXv)i(hv).

V

Moreover, for z

E

P and

Cl, Ca E T,(P),

These relations yield

(4@)(z;

50

9

* * * ?

5,)

8.26. The Weil homomorphism. In sec. 6.16 through sec. 6.19, volume 11, we defined the Weil homomorphism

h p : (VE"),

+

H(B)

for a principal bundle 9. (The actual definition is in sec. 6.19.) On the other hand, we have the Weil homomorphism X # : (VE*)O==O+ H(A(P)i,o,,=o)

for the operation of E in A(P) (cf. sec. 8.15).

VIII. Algebraic Connections and Principal Bundles

356

Since G is connected, (VE"), = (VE"),,,. = (VE"),=,. tions) T h e purpose of this section is to prove

Hence, (forgetting grada-

Theorem VI: T h e diagram

=I Proof: Let o be a connection form for 9with ,algebraic connection w". Recall that in sec. 6.17, volume 11, we defined an algebra homomorphism y : (VE*) -+ A(P)i=o,

which restricted to a homomorphism

Then we set h9 = y $ , where y B : (VE")), + A ( B ) was defined by 72

#

'YB=YI*

Now recall that AB(P)= A(P)i,o,e=o.Hence to prove the theorem it is sufficient to show that 71 = ( a 4 = o . (8.20) This in turn will follow from the relation y = I,.

(8.21)

Since y and 2, are homomorphisms, it is enough to prove that

y ( h # ) = I(h"),

h" E E".

But in view of the definition of y (sec. 6.17, volume 11) we have

Y(h")(z;51, Cz) = (h", Q ( x ;51, C,)),

h"

E

E", 5a E T,(P), x E P.

Now formula (8.19), sec. 8.24, implies that y(h*) = I(h").

Q.E.D.

5 . Principal bundles

357

8.27. The cohomology sequence. Assume that P is connected, and let G, denote the fibre over x E B. For each x E B , the inclusion map j,: G, -+ P induces a homomorphism

j $ : H(G,)

+

H(P).

Now use a local coordinate representation for 9 ' to identify G, with G and H(G,) with H ( G ) . Then the resulting homomorphisms

j $ : H ( G )+ H ( P ) all coincide (since P and G are connected). We denote this common homomorphism by

ep: H ( G ) + H ( P ) . It is called the Jibre projection for the principal bundle 9. Observe that if x E P and if A,: G -+ P. denotes the inclusion map given by a H z . a, a E G, then A: = e p . Now suppose E is reductive. Then we can apply Proposition XV, sec. 7.22, to obtain the commutative diagram

H(A(P)e=o)

H(P)

e4P1

QP

(AE")o=o

H(G).

This diagram relates the fibre projection for the operation to the fibre projection for the bundle. Combining the diagram above with Theorem VI yields the commutative diagram

358

VIII. Algebraic Connections and Principal Bundles

The lower sequence is called the cohomology sequence for the principal bundle 9. Thus the diagram is a homomorphism from the cohomology sequence of the operation to the cohomology sequence of the bundle. Moreover, if G is compact, then all the vertical maps are isomorphisms, so that the diagram is an isomorphism between cohomology sequences (cf. Theorem I, sec. 4.3, volume 11, and sec. 5.29).

Chapter IX

Cohomology of Operations and Principal Bundles $1. The filtration of an operation 9.1. Definition: Let ( E , i B ,O R , R , 8,) be an operation. Define subspaces Fp(RQ)c Rq ( p < q ) by

Fp(Rq). q=P

Evidently the spaces FP(R) define a filtration of R , so that R becomes a graded filtered space. This filtration is called the Jiltration of R induced by the operation (E, i R , O R , R , 42). Proposition I: T h e spaces Fp(R) are stable under the operators iR(a) (u E AE), O,(x) (x E E ) , and B R , Moreover,

P ( R ) * P ( R ) c Fp+q(R),

p , q 2 0,

(9.1)

and so the above filtration makes R into a graded filtered differential algebra. Proof:

It follows immediately from the definition that i R ( x ) :FP(Rq) + Fp(Rq-l),

x E E,

and so the P ( R ) are stable under the operators iR(u) (u E AE). Formula (7.5), sec. 7.2, implies that the P ( R ) are stable under the operators O,(x). On the other hand, it follows from formula (7.7), sec. 7.2, that FP(R) is stable under d R . 359

360

IX. Cohomology of Operations and Principal Bundles

To prove formula (9.1), let z E Fp(Rg) and w E Fr(R8)).Then i~(X1 A

- w)

Xq+8-p-r+l)(z

*

(Xi

E

E)

is a sum of terms of the form

-

(iR(a)z) ( i R ( b ) ~ ) ,

It follows that z

ME, b E A",

+ 1 or 1 2 s - r + 1. Hence either iR(u)z= 0

where either k 2 q - p or iH(b)w= 0, and so i,(X,

uE

A

*

*

A Xq+8-p-r-1)(X

*

= 0.

W)

- w E FP+7(Rq+a). Q.E.D.

Corollary: The subspaces Fp(R) are ideals in R . Proposition II: Let (E, i R ,O H , R,S,) be an operation. Then the basic subalgebra B R of R with respect to the induced filtration coincides with the basic subalgebra of the operation:

BR = Ri=o,e=o. Proof: Recall from sec. 1.13 that an element z E Rp is contained in BR if and only if it satisfies

z E Fp(Rp)

and

6,z E FP+*(Rp+l).

It is immediate from the definition that

F p ( R p )= REo. Hence, z E Bg if and only if z E R k o and d R z E

RE:. But for z E Ri=, ,

X E

OR(X)X = i~(X)dn(Z),

E,

and so the proposition follows.

Q.E.D. Suppose now that

v: (E, i R

9

OR 3

R , 6,)

-

(E, iS

3

OS 9

s, 6,)

is a homomorphism of operations. Then it follows at once from the

361

1. T h e filtration of an operation

definition that

p7

preserves the induced filtrations:

v: Fp(R*) + Fp(Sq). Thus 9 is a homomorphism of graded filtered differential algebras. T h e induced homomorphism F~ of the basic subalgebras is given by q R = p7z=o,B=o, as follows from Proposition 11, above. Finally, suppose that the operation ( E , iR , O R , R, S,) admits an algebraic connection. Consider the operation (E, i, 8, Rise @ AE", d ) defined in sec. 8.7. It follows immediately from the definitions that

FP(RiSO @ AE") =

C RCo @ AE". lL

>P

Hence, since f:Ri=o@ AE" 5 R is an isomorphism of operations, it restricts to isomorphisms

This shows that F p ( R ) is the ideal generated by

Let ( E , iR , O H , R, S,) 9.2. The filtration of Define a filtration in Resoby setting

be an operation.

F'(RB=o) = F p (R ) n &=O. It follows from Proposition I, sec. 9.1, that this filtration makes R&O into a graded filtered differential algebra. T h e corresponding spectral sequence will be denoted by

(Ei(R,=o), a,),

i 2 0,

and called the spectral sequence of the operation. Proposition 11, sec. 9.1, implies that Ri=o,o=o is the basic subalgebra of Re=,,with respect to this filtration. If F: R S is a homomorphism of operations, then RO,O So=, is a homomorphism of graded filtered differential algebras. Now assume that the operation ( E , in, OR, R , 6,) admits a connection, and consider the associated operation ( E , i, 8, Ri=, @ AE", d ) (cf. sec. 8.7). Then the corresponding filtration of (Ri=,@ AE*),=, is given by

-

-

FP((Ri=o@ AE"),=o) =

2P

/ I

0AE")t3=13.

362

IX. Cohomology of Operations and Principal Bundles

Hence the isomorphism f restricts to isomorphisms

Example: T h e filtration of W(E),,, defined in sec. 6.12 is the filtration induced by the operation (E, i, O w , W(E),dW).

s2. The fundamental theorem

In this article E denotes a reductive Lie algebra with primitive space P E . We shall identify APE with (AE*),=, under the isomorphism x E of Theorem 111, sec. 5.18. Further, t : PE

---f

(V+E")@=o

denotes a fixed transgression in W(E),=, (cf. sec. 6.13), and (Y:

PE + W(E)o=o

denotes a fixed linear map, homogeneous of degree zero, such that 8,,a(@)

= z(@)

a ( @ )- 1 @ @ E (V+E* @ AE"),,,,

and

@1

(cf. sec. 6.13). 9.3. The Chevalley homomorphism. Let (E, i,, O R , R, be an operation with a fixed algebraic connection X,. Define a linear map, homogeneous of degree 1,

Recall from sec. 8.15 that 8,

0

8R

( XR)v,e=o TR =

=

0. This implies that

0.

It follows that (Ri=o,o=o, 8,; z R )is a ( P E ,8)-algebra (cf. sec. 3.1). Definition: The ( P E ,8)-algebra (Ri=o,o=o, 8,; z R ) is called the ( P E ,8)-algebra associated with the operation via the connection X R and the

transgression z. Since (Ri=o,o=o, 8,; complex

tR)

is a ( P E ,8)-algebra we can form the Koszul

(Ri=O,O=O

@ APE 363

OR)

IX. Cohomology of Operations and Principal Bundles

364

(cf. sec. 3.2).' V is given explicitly by VR(Z

@ 1) = 6RZ @ 1

and

Recall from sec. 3.4 that a filtration of this graded differential algebra is defined by

FP(Ri=o,e,o0A P E )

=

1 Ry=o,e-o@ APE.

P ZP

The corresponding spectral sequence will be denoted by {Ei ,di}i>o. Next (cf. sec. 8.16) let X,: W ( E )-+R be the classifying homomorphism for the algebraic connection X,. Consider the linear map

6 R = (Xw)e=o 0 : P E

+

Re=o.

Then 6, is homogeneous of degree zero. Hence, since R is anticommutative and P$ = 0 for even K, 6, extends to a homomorphism of graded algebras 6 R : APE -+ Re=o. Finally, extend 6R to a homomorphism of graded algebras

8, Ri-0,e-o @

+

Re=o

by setting ~ R ( z@

@) = z

. 6,(@),

z E Ri=o,e=o, @ E APE.

8, is called the Chevalley homomorphism associated with the operation (E, i, ,OR, R, 6,) via the algebraic connection XR and the linear map a. Since

2. The fundamental theorem

365

it follows that 8Ro V' = d R 0 8,. Hence 8,
0APE

9

OR)

-

(&=o

t

8~).

Theorem I (Fundamental theorem): Let ( E , i R ,O R , R, 8,) be an operation of a reductive Lie algebra, which admits an algebraic connection. Then the Chevalley homomorphism GR has the following properties:

(1) The induced homomorphism # 8B: H(Ri=o,o=o0APE 0')

+

H(Ro-0)

is an isomorphism of graded algebras. ( 2 ) 8, is filtration preserving. T h e induced homomorphisms ( 8 ~ ) Ei i:

+

Ei(Ro-0)

of the corresponding spectral sequences (cf. sec. 1.6) are isomorphisms for i >_ 1. (3) GR(z @ 1) = X, x E Ri=o,o=o,and 8~(0 1 @) - ( X R ) ~ ( @ ) E F'(Re=o),

@ E APE.

Proof: ( 3 ) The first statement is obvious. T o prove the second, it is sufficient to consider the case @ E PE. Recall that XI,.is a homomorphism of operations, whence

(1) and ( 2 ) We show first that GR is filtration preserving. By defini@ APE) is the ideal genertion, R&+O c FP(Ro=,).Since FP(Ri,o,e=o ated by CpSp R:=o,e=o @ 1, and since FP(Ro,o)is an ideal, this implies that GR preserves filtrations. Now, in view of Theorem I, sec. 1.14, it only remains to show that the map ( 8 ~ ) i4 : 3 &(Re=o)

is an isomorphism. This is done in the next section.

IX. Cohomology of Operations and Principal Bundles

366

9.4. Proposition 111: With the notations and hypotheses of Theorem I, (6R)l:

E l --+

El(RfJ=O),

is an isomorphism. Proof: Since f:Ri=o@ AE" --% R is an isomorphism of operations (cf. Theorem 11, sec. 8.7), we may assume that

R

= Ri=, @ AE*

XH(x*) = 1 @ x*,

and

x* E

E".

In this case we have FP(R,,,) = C,,,(R&, @ AE*),=,. Define bigradations in the algebras Ri,o,e=o@ APE and Re=oby setting

(Ri=o,e=o@ APE)','

= RPt=o,e=o

0(APE)',

and

R{A%= (Rip_,@ AqE*)e=o. Then these bigraded algebras are the bigraded algebras associated with the filtrations. Hence they coincide with E, and Eo(Re=o) (as bigraded algebras). Next, we show that the differential operators do in Eo and 2, in Eo(Re,o) are given by

do=O

do= -wR@SE.

and

(9.2)

In fact, it is immediate from the definitions that

I t follows that do -- 0. On the other hand, recall from formula (8.6), sec. 8.7, that the restriction of SR to (Ri=,@ AE*),,, is'given by S R = -wR

@

f h,

+ 88.

By definition the operators ha and dH are homogeneous of bidegrees (2, -1) and (1,O) respectively, while w R @ 6 , is homogeneous of bidegree (0, 1). It follows that Jo = -wR @ d ~ . Finally, we show that

(Wo:

Eo --+ Eo(Ro=o)

2. The fundamental theorem

is simply the inclusion map

j : Ri=o,e=o0APE

-

367

(Ri=,0AE"),=,.

(9.31

I n fact, j is homogeneous of bidegree zero. Thus we need only show that

(8, - j ) : Rf==,,e=o @ APE

+ Fp+'(R,=o),

p 2 0.

(9.4)

But j(z

0@) = x (%,),,(a),

xE

&=o,o=O,

@E

APE,

and so property (3) of Theorem I, sec. 9.3, yields (for x

E

R$O,o,O,

@ E APE)

(6, - j ) ( z

0@) = z -

(8R(@) -

(x,),,@)

R,P,o,e=o - F'(R,=o)

c FP'-l(R,=O).

-

Thus (9.4) is established. Since ( 8 R ) i(:E i , di) (Ei(Re=o),d j ) is a homomorphism of spectral sequences, we have the commutative diagram

H(E0 do) 9

(a&

H(Eo(Ro=o),do)

(cf. sec. 1.6). Thus to prove the proposition we need only show that (8,)o# is an isomorphism. In view of formulae (9.2) and (9.3) above it has to be shown that the inclusion map

-

j : Ri=o,,+o 0(AE*)e=o

(Ri=,0AE"),=,

But since E is reductive, this follows immediately from Theorem V, sec. 4.11 (applied with ( Y ,6,) = (Rice, 0) and (XIdx) = (AE", 6 ~ ) ) . T h e proof of the proposition (and hence the proof of the fundamental theorem) is now complete.

Q.E.D.

368

IX. Cohomology of Operations and Principal Bundles

9.5. Corollaries of Theorem I. Corollary I: The graded differen8,) and (Ri=o,eso @ APE, G'), are c-equivalent (cf. tial algebras (Re=O, sec. 0.10). Corollary II: The c-equivalence class (and hence the cohomology algebra) of the differential algebra (Re=,,6,) depends only on

(1) the differential algebra (Ri=o,e=o, d,), (2) the Lie algebra E, (3) the Weil homomorphism Zg. Proof: Assume that (Ri,o,8=0, S,), E, and 2; are given. Choose a transgression t in W(E),,, and let

be a linear map, homogeneous of degree 1, and such that f* = t. Then (Ri=o,e=o, dR; 2) is a (PE, 8)-algebra. Denote the corresponding Koszul complex by (Ri=o,e=o @ APE, Now we have

v).

Hence, in view of Proposition IX, sec. 3.27, the fundamental theorem yields

(Re-c, 8,)

7 (Ri-o,e-o 0APE ck) 7 (Ri-o,e-o 0APE, 8). Q.E.D.

Corollary 111: The first three terms of the spectral sequence for an operation are given by

Proof: Apply the fundamental theorem, and the observations of sec. 3.4. Q.E.D.

2. The fundamental theorem

369

Corollary IV: Assume that the graded differential algebra (Ri=o,o=o, 6,) is c-split. Let (H(Ri=,,o=o) @ A P E , V i ) denote the Koszul complex of the PE-algebra (H(Ri=,,,=,);T$), associated with (Ri=o,e=o, 6,; tn) (cf. sec. 3.3). Then

0APE V g ) 7 (Re=@aE)*

(H(Ri=O,O=O)

7

f

I n particular, the cohomology algebras are isomorphic. Thus in this case the algebra H(R,=,) depends only on E, H(Ri=O,B=O), and @. Proof: Apply the example in sec. 3.29.

Q.E.D. Corollary V:

H(R,,,)

is connected if and only if H(Ri,o,o,o)is.

Corollary VI: H(R,,,) has finite type if and only if H(Ri,o,o,o)has finite type. In their case the PoincarC series are related by

Proof: Apply Proposition V, sec. 3.18.

Q.E.D. Corollary VII: Suppose H(Ri=o,o=o) has finite dimension. Then H(R,=o) has finite dimension, and

dim H(R,=,) 5 2'

a

(Y = dim PE). dim H(Ri=,,,=,)

Moreover, in this case the Euler-PoincarC characteristic of H(R,=,) is zero. Proof: Apply the corollary to Proposition V, sec. 3.18.

Q.E.D.

3 70

IX. Cohomology of Operations and Principal Bundles

9.6. Homomorphisms. Let v: (E, i,, O R , R, 6,) -+ ( E , is, O,, S, 6,) be a homomorphism of operations of a reductive Lie algebra E. Suppose that X, is an algebraic connection for the first operation, and recall that then X, = q o XR is an algebraic connection for the second operation. Moreover, with this choice of connections,

(cf. sec. 8.8). Hence,

v

O

( X d W = (XS)W.

These relations show that (in the notation of sec. 9.3)

is a homomorphism of ( P E ,d)-algebras (cf. sec. 3.1). On the other hand, for @ E P E ,

Thus the diagram

commutes. Hence so does the diagram

$3. Applications of the fundamental theorem In this article the notation established at the start of article 2 remains in force. In particular PE is the primitive space of a reductive Lie algebra E ; t :PE -+ (VE"),,, is a transgression; and a : PE + W(E),=,satisfies

G@(@)

= T(@)

@1

and a ( @ )- 1 0@ E ( V + P 0AE"),=,. 9.7. The cohomology sequence. Recall from sec. 8.21 that an operation ( E , iR, OR, R, 6,) is called regular, if E is reductive, H(R,=,) is connected and the operation admits a connection. For such operations, the cohomology sequence was defined to be the sequence

(VE"),=o

rf:

H(Ri=,,e=o)

+

e;f

+

H(R,g,o)

QR +

(AE")o=o.

On the other hand, the choice of an algebraic connection X R in a regular operation determines the (PE, 8)-algebra (Ri=o,e=o,8 R ; tR), with tR = (XR)V,8=0 o t (cf. sec. 9.3). The corresponding cohomology sequence, as defined in sec. 3.14, reads

Finally, in Theorem 111, sec. 5.18, Theorem I, sec. 6.13, and Theorem I, sec. 9.3, we established isomorphisms of graded algebras

(Note that the isomorphism 83 depends on the choice of an algebraic connection XR .) 371

372

IX. Cohomology of Operations and Principal Bundles

Theorem II: Let (E, i R ,O E , R, S,) be a regular operation and let XK be an algebraic connection. Then the diagram I*

(Q):

."IZ

V ~ E

H(Ri=o,e-o)

H(Ri=o,e-o0A P E )

I

G%

(VE")e=o

G

H(Ri-o,e-o)

exR

P* +

1

8;

APE N

H(Re=o)

eR

1

XE

(AE"1e-o

commutes. Proof: The commutativity of the left-hand square follows from the relation t R= ( XR)V,@=O0 t. The commutativity of the centre square is a consequence of property (3) in the fundamental theorem (sec. 9.3). That the right-hand square commutes is proved in Proposition IV, below. Remark: Theorem I1 also permits us to apply the Samelson and the reduction theorems (sec. 3.13, sec. 3.15) to operations. The resulting theorems, however, would coincide with Theorem I, sec. 7.13, and Theorem 11, sec. 7.14. Thus we do not restate them here. Proposition IV: Let (E, i R ,OR, R , 6,) be a regular operation with algebraic connection X R . Then (in the notation above) XE

0

e* = @ R

0

8%.

Lemma I: Assume that an element a E H(Re,o) is represented by a cocycle 52 of the form

52 = (XR)*(@) Then

@R(a) =

+ Z,

z E F'(R0,o).

@ E (AE")e-o,

0.

Proof: Recall from sec. 7.8 that the structure homomorphism is a homomorphism of operations YR:

(E, iR

OR

R, 6,)

(E, ~ R Q E, ORBE R

-+

9

0AE", 8 ~ 8 ~ ) .

According to the corollary of Proposition IV, sec. 8.10, Y E ( ( XR)n@) -

1 @ @ E (R+ 0AEX)f3-O*

3. Applications of the fundamental theorem

373

Moreover, y R preserves filtrations since it is a homomorphism of operations. Evidently,

F'((R @ AE"),,,) and

SO ~ R E z

fi E

(R+@ AE")e=o,

(R+@ AE"),=o. It follows that yRQ

where

=

1@ @

+ 0,

(R+@ AE"),=o. We also have = S ~ ~ ( Y R Q=) Y R ( ~ R Q ) =

dRa&

0-

Finally recall from sec. 7.10 that the fibre projection @R = Z R

(g")-'

(YK)$O

@R

is defined by

9

where

g#: H(R,,,) @ (AE"),=, 5 H ( ( R @ AE*)o=o) is the isomorphism induced by the inclusion map g: Re-0 @ (AE"),=, (R @ AE*),=,, and

-+

zR:

H(Re=o)@ (AE"),,,

+

(AE")o=o

is the projection. Since g# is an isomorphism we can write

a,

Let @ , and (SRoEsZ,)o denote the components of the elements Q,, Q 2 , and BROE(Q2) in (Ro @ AE*)o,o. Then

a E 1 @ (AE"),=, Moreover, in ( R @ AE"),=, , (SR&2,)0

=

-(wR

+

and SRoE =

@ d,)Qg

SR @ c E

(SROEQz)O

= 0.

- W R @ BE.

Thus

(RO @ O(AE")),=,.

These equations imply that L$ = 0; i.e., Q1 E Z+(RB,o) @ (AE")e=o.

374

IX. Cohomology of Operations and Principal Bundles

It follows that 12, represents a class a1 in H+(Ro=o)0(AE"),,,. But, clearly (YR)B#=o(a)= ##(a, 1 0@). Thus

+

Q.E.D.

0A P E ) and let Y be a Proof of Proposition IV: Let B E H(Ri=o,o=o representing cocycle. Then, by Lemma I, sec. 3.13, Y = Y,

+ 1 0,O*p,

where !PI E R&o,e=o 0APE (= F'(Ri,o,e=o0APE)). Since OR is filtration preserving, it follows that

8RY

-

8,( 1 0@"p)E F'(Re=o)*

Now identify APE with (AE"),=, via fundamental theorem asserts that

xE.

Then statement (3) in the

&?(0 1 e"B) - ( X R ) h " B ) E F1(Re=o)l whence

6nY

-

(X,),(e#B)

Thus, since 6,(y/) represents

@(B),

en@(B)

=

E

vL=o).

Lemma I above yields

e"(0

Q.E.D.

Corollary: Assume that (Ri=o,e=o, S,) is c-split. Then

(RO=O6 R ) 7 (H(Ri-o,o=o) @ 3

9

vR#),

and the induced isomorphism of cohomology algebras makes the diagram H(H(Ri-o,o=o)@ APE)

commute.

APE

3. Applications of the fundamental theorem

375

Proof: Apply the example of sec. 3.29.

Q.E.D.

9.8. Homomorphisms. Recall that a homomorphism y : A + B of graded vector spaces is n-regular if y : A p -+ B p is an isomorphism for p 5 n and injective for p = n 1.

+

Theorem 111: Let y : ( E , in, O R , R,6,) + (E, is, 6,, S , 6,) be a homomorphism of operations of a reductive Lie algebra E. Assume that the first operation admits a connection. Then

-

ygo,o=o: H(Ri,o,o=o>

is n-regular if and only if &a:

H(R,=,)

H(S,=o,,=o)

H(S,=o)

is n-regular.

Proof: It follows from sec. 9.6 that Q)B#=~ is n-regular if and only if (pi=o,o=o

@:')i

H(Ri=o,e=o@ A P E )

+

H(Si=o,e=o@ APE)

is n-regular. Now the theorem follows from Theorem I, sec. 3.10. Q.E.D. Corollary:

Q)B#=~ is an isomorphism if and only if

is.

Applying Theorem I11 to the classifying homomorphism (cf. sec. 8.16) yields Theorem IV: Let (E, in, O R , R, 6,) be an operation of a reductive Lie algebra admitting a connection. Then the Weil homomorphism

is n-regular if and only if

H0(R,=,)= r

and

Hp(R,=,)

= 0,

1 < p 5 12.

(9.7)

Proof: Recall that Ho(W(E),=,)= r and H+(W(E),=,)= 0 (cf. Proposition I, sec. 6.6). It follows that formula (9.7) holds if and only if ( Xw)tT0: H(W(E),=,)+ H(R,=,) is n-regular. Hence the theorem follows from Theorem 111. Q.E.D.

376

IX. Cohomology of Operations and Principal Bundles

Corollary I: X$ is an isomorphism if and only if Ho(Re,o) = r and H+(Re=o)= 0. Corollary 11: Suppose Ho(Re=o)= T and HP(R0-0) = 0, 1 < - p 5 n. Then the Betti numbers bp = dim HP(Ri,O,e=O)(0 5 p 5 n) are the (1 - tQ+')-', where '& Pfis the coefficients of t in the series nj=l Poincark polynomial for PE. Proof: This follows from the isomorphism (VE*)e=o

VPE.

Q.E.D. 9.9. N.c.z. operations. Let (E, iR, O R , R, 6,) be an operation of a reductive Lie algebra, and assume that H(R8,o) is connected. Then we say (AE*)e=ois non cohomoZogous to zero in Reso (n.c.2.) if the projection eR:H(Re-0) + (AE+)e=Ois surjective. For the sake of brevity we shall often simply say that the operation is n.c.2. According to Proposition I, sec. 8.2, an n.c.2. operation admits an algebraic connection. Thus every n.c.z. operation is regular. Theorem V: Let (E, iR, O R , R, 6,) be a regular operation. Then the following conditions are equivalent :

(1) eR is surjective. (2) There is an isomorphism of graded algebras

H(Ri,o,e=o) 0(AE*),+.o ES H(Re=o) which makes the diagram

H(&o,e-o)

0(AE*)e-o

H(Re-0) commute. (3) There is a linear isomorphism of graded vector spaces H(R+,,,O=~) 0(AE*)e=o-5 H(RO=,)which makes the diagram (9.8) commute.

3. Applications of the fundamental theorem

(4)

377

e t is injective.

( 5 ) (2%)'= 0. (6) There is a homomorphism

of graded differential algebras such that f # is an isomorphism making the diagram (9.8) commute. (7) The spectral sequence for Re+ collapses at the &-term. Proof: In view of the fundamental theorem (cf. sec. 9.3) and Theorem 11, sec. 9.7, this result is simply a translation of Theorem VII, sec. 3.17. Q.E.D. Theorem VI:

Let (E, iR, O R , R, S,) be a regular operation. Then

(1) H ( R i ~ o , , = ,is) of finite type if and only if H(R,=,) is. In this case their PoincarC series are related by r

-fH(~o_,)

5 -f~(Ri=~,e=,,) fl (1 i=l *

+

tgt)

xi

where t g l is the PoincarC polynomial for P E . Equality holds if and only if the operation is n.c.2. (2) Suppose that H(Ri=o,o=,,)is finite dimensional. Then so is H(R,=,). In this case dim H(Rt?,o)5 2'

. dim H(Ri,,,,=,).

Equality holds if and only if the operation is n.c.2. Proof: In view of the fundamental theorem and Theorem I1 the theorem is a translation of Proposition V, sec. 3.18, and its corollary. Q.E.D.

$4. The distinguished transgression Let E be a reductive Lie algebra. Recall from sec. 6.10 the definition of the distinguished transgression z E :PE -+ (VE*)o,o. It is immediate from the definition of tEthat there is a linear map a : PE + W(E),=,, homogeneous of degree zero, such that 61ra(@) =

tE@

01

and

a ( @ )- 1 @ @ E W+(E)+,,,=,, @E PE.

(Recall that W+(E)il=o,o=o consists of the invariant elements D in W+(E) which satisfy i(a)G = 0 for a E (A+E)o=,.) In this article a denotes a fixed linear map, satisfying the properties listed above. In particular, the pair t E ,a satisfies the properties listed at the beginning of article 2. 9.10. The operator i,(u)#. Let (E, i,, O R , R, 6,) be a regular operation and let P*(E)c (A+E),=, be the primitive subspace defined in sec. 5.14. Recall from sec. 7.12 that the operators iR(a) (a E P*(E)) restrict to operators in Re=,and induce operators iR(a)* in H(R0=,). On the other hand, since P*(E)= (PE)*,the elements a E P*(E)induce ordinary substitution operators i p ( a ) in the exterior algebra APE. According to Lemma IX, sec. 5.22, the isomorphism x E : APE --+ (AE*),=, identifies iF(a) with i E ( a ) ; we use the latter notation. Now extend i&(a)( a E P*(E))to an antiderivation, i(u),in the algebra Ri,o,o=o 0APE by setting N

;(a) = O R @ iE(a).

Then (cf. sec. 3.2)

i(a)V'

+ V'i(a) = 0,

aE

P*(E),

and so we obtain antiderivations, ;(a)*, in H(Ri=,,,=, @ APE). The purpose of this article is to prove Theorem VII: Let ( E , i,, 8, , R, 6,) be a regular operation. Let XR be an algebraic connection for the operation and let

8 R : (Ri=o,O=O@

9

378

0,)

-

(RO=O,

sR)

4. The distinguished transgression

379

be the corresponding Chevalley homomorphism constructed via Then

sg 0").(i

= &(a)#

0

sg,

aE

tE and

a.

P*(E).

Lemma II: Assume that the Weil homomorphism 2ig of the operation (E, iR ,OR, R, 8,) is trivial: (%$)+= 0. Then the assertion of Theorem

VII holds. Proof: Evidently, t$ 2%0 tE = 0. Thus we can apply the results of sec. 3.17 to the ( P E ,8)-algebra, (Ri=o,e,o, 8,; tR). Let ,8 and y be elements of H(Ri,o,o,o @ APE) which admit representing cocycles of the form =I

z @ 1 ( z E Ri-o,8-o)

and

w

01 + 1 @ @

(w E Ri=o,e=o, @ E PE)

respectively. (Since t f := 0, the corollary of Theorem VII, sec. 3.17, implies that H(Ri,o,o,o@ APE)is generated by cohomology classes with this property.) On the other hand, according to Proposition VIII, (3), sec. 7.12, if a E P*(E)then iR(a)#is an antiderivation in H(ROao).Since i(a)# is an antiderivation in H(Ri-o,e-o@ APE), it is sufficient to verify that = W(a)#(B),

aE

P*(E),

(9.9)

i R ( a ) " s l ( y ) = %i(a)*(r),

aE

P#(E).

(9.10)

i,(a)"sj$(D)

and

Since /? is represented by z @ 1, @(B) is represented by z E Ri=o,s=o. But i(a)(z01) = 0 = i,(a)z, a E P*(E), and (9.9) follows. T o prove (9.10), observe that since @ E P E ,

i(.)(w

01 + 1 @ @)

= (@, a ) ,

aE

P*(E).

Hence @$i(a)#(y)= (@, a ) . On the other hand, %(W

01

+ 10

@) = w

+

(WW,O-O(~(@N.

But a was chosen so that

(iE(a)0 a)(@)= iE(a)(l 0@)

= (@,

a).

380

IX. Cohomology of Operations and Principal Bundles

It follows that

and (9.10) is proved.

Q.E.D. 9.11. Proof of Theorem VII: Recall the definition of the structure 8R@E,R @ AE", 8 R B E ) in sec. 7.7. The horizontal operation ( E , iR@E, and basic subalgebras for this operation are given, respectively, by

( R @ AE")i,o

=R

@1

and

( R @ AE")i,o,,=o

= Re=o @

1.

Moreover, the map 2: x* + 1 @ x' (x" E E') is an aIgebraic connection for this operation. Since ~ R @ E ~ ( X *= )

1@

EX'

x* E

== ,f,,(d~x"),

E",

it follows that the curvature for the connection 2 is zero (cf. Proposition 111, sec. 8.6). Hence the Weil homomorphism of this operation is trivial. On the other hand, the structure homomorphism y R : R -+ R @ AE" is a homomorphism of operations. The corresponding base homomorphism is given by (YR)i=O,tJ-O

= eR : R i - O , O = O

RO=O*

Moreover, since yR is a homomorphism of operations, the map yR 0 XR is an algebraic connection for the structure operation. Let 8 ~ 8Re,o ~ :@ APE ( R @ AE"),,,

f

=

-+

be the corresponding Chevalley homomorphism. Then (cf. sec. 9.6) the diagram Ri-0,e-o

@ APE

Re-o

eR@a __*

(YRb-0

Re-0 @ APE

( R 0AE*)e-o

4. The distinguished transgression

381

Since the structure operation has trivial Weil homomorphism, Lemma I1 yields 8Z@E

i(a)" = iR@E(a)*

ag@E

,

a E P*(E).

This, together with the equation above, yields (YR)@=o # 0

(8: i(a)"

- iR(a)"

0

8;)

= 0.

Since (cf. Proposition VI, sec.7.9) (yR)z=o is injective, the theorem is proved. Q.E.D. Next recall that the inclusion g: R,=, @ (AE*)e=o--+ ( R @ AE*),=, induces an isomorphism in cohomology (cf. sec. 7.9) and set

PR = (g")-'

(YR);==O: H(&=o)

--+

H(&=,) @ (AE*)o=,.

Corollary: Let A,: APE -+APE @ APE be the homomorphism defined by A,(@) = @ @ 1 1 @ @, @ E P E . Then the map

+

L

@ A,: Ri=o,e=o@ APE

-+

Ri=o,+o @ APE @ APE

Proof: T h e first equation follows from a simple computation. The commutative diagram is a consequence of Proposition IX, sec. 7.15, and the theorem. Q.E.D.

$5. The classification theorem In article 2 we associated with every regular operation (El iR , OR , R, 8,) a (PE , 6)-algebra (Ri=o,e=o , B R ; tR)whose associated cohomology algebra was isomorphic to the cohomology of Re=,. In this article we reverse this process to show that every ( P E ,8)-algebra can be obtained in this way. In this way we obtain a “cohomology classification theorem” for operations. In this article E again denotes a reductive Lie algebra with primitive space P B . (VE*),=, will be identified with V P E via a fixed transgression z in W(E),,, (cf. Theorem I, sec. 6.13). 9.12. The induced operation. Theorem VIII: With the notation above let ( B , B B ; tB)be a c-connected ( P B ,8)-algebra. Then there is a regular operation (E, iR , OR, R, 8,) together with an algebraic connection XR with the following properties:

(1) There is an isomorphism of ( P E ,8)-algebras

A: (BY 8,

Y

;T B )

(Ri-o,e=o

9

8,; (P&,e=o

0 7).

(2) Let (El is, B s , S, 6,) be an operation admitting a connection X , and let

w: (B, 8 B ; T B )

-

(si=O,O=O,

(XS)v,O=O

).

be a homomorphism of ( P E ,8)-algebras. Then there exists a homomorphism 9:( E , i R i

OR,

R, 6R)

-

( E , is,O S ,

s, 8S)

of operations such that

(9.12) T h e proof of this theorem occupies the next three sections. I n sec. 9.13 below we construct the operation, the connection, and the homo382

5. T h e classification theorem

383

morphism 1. In sec. 9.14 it will be shown that A is an isomorphism. Finally, property (2) is established in sec. 9.15. 9.13. Construction of the operation. Consider first the operation where

( E , i, 8, B @ W(E),

i ( x ) = wB 0i E ( x ) ,

e(x) = 1 0 e,(X),

and dH@lV = dB

@

+

x

E

E,

0b-

(wB denotes the degree involution in B . ) Observe that

f : x"

H

1 @ 1 @ x",

x" E

E",

is an algebraic connection for this operation. Now denote by X the subspace of B @ VE" whose elements have the form @ E PE. z = T B ( @ ) @ 1 - 1 0T ( @ ) , Denote by I the graded ideal in B @ W ( E ) generated by X @ 1. Then

is a graded algebra. Let n: B @ W ( E )-+ R denote the projection. Next observe that X @ 1 c Z ( B ) @ (VEX),,,, @ 1. It follows that for z E X @ 1, x E E, i ( x ) z = e(x)z:= dB,,(z)

= 0.

Since these operators are either derivations or antiderivations, the ideal I must be stable under them. Hence they induce operators iR(x),

eR(x),

and

dR

in the factor algebra R. Since n is a surjective algebra homomorphism and ( E , i, 8, B @ W ( E ) ,8 B B W ) is an operation, it follows that ( E , iR, O R , R, 6,) is an operation. Moreover, n is a homomorphism of operations. In particular, since the linear map f is an algebraic connection, it follows that XR = n 0 R is an algebraic connection for ( E , iR, O R , R , d ~ ) . Next observe that the representation Ofi is semisimple. In fact, since E is reductive, the representation OW is semisimple and hence so

3 84

IX. Cohomofogy of Operations and Principal Bundles

is the representation 8. Since n is surjective, it follows that OB is semisimple, Finally, note that B @ 1 @ 1 c ( B @ W(E))i,o.e=o.Hence a homomorphism of graded differential algebras

is defined by 1(b) = n(b @ 1 @ l ) , b E B. To show that 1 is a homomorphism of ( P E ,8)-algebras, we must prove that (9.13) 1 0 zg = (.xR)v,e=o 0 t. But, since

?G

is connection preserving,

Moreover, it follows easily from Example 1, sec. 8.9, that (iv)e-o:

(VE*)tJ=o

-

( B 0WE))O==O

is the inclusion map !P t+ 1 @ Y @ 1. Since tg(@) @ 1 - 1 @ t(@) E X , @ E PE, this yields

whence (9.13). 9.14. Proposition V: The homomorphism 1: B morphism.

-

-+

Ri=o,e-o is an iso-

Lemma III: Let J denote the ideal in B @ (VE*),=, generated by X . Then the inclusion 5 : B B @ (VE*)8=o,induces an isomorphism,

5. T h e classification theorem

385

Since r ( P E )generates (VE*),=, (cf. Theorem I, sec. 6.13) this implies that e(B @ (VE*),=o) = E,(B) . e(1 0(VE'%)o=o) c Im 61. Thus t1 is surjective. On the other hand, define q : B

q(b 0y)

b

*

1

[(tg), 0

0(VE*),=,

-

B by

( T ~ ) - ' ( Y ' ) ] , b E B, Y'

E

(VE*),=,

Then q ( X ) = 0 and so q ( J ) = 0. Thus q induces a homomorphism

Since (clearly) ql

0

= 1,

El is injective. Q.E.D.

Proof of Proposition V: We show first that

ni=o,e=o: B @ (VE")o=o

-j

Ri=o,e=o

(9.14)

is surjective and that ker ni=o,e=o

= J.

(9.15)

In fact, the algebraic connection X R determines an isomorphism @ AE" 5 R (cf. Theorem I, sec. 8.4). Moreover, since n is connection preserving, it follows that the diagram

commutes. Since n is surjective. this diagram implies that ni,o is surjective. Next recall that 0 and OR are semisimple representations. Hence

is surjective, these relations imply that so is ni=o,,=o. Since ni=o

3 86

IX. Cohomology of Operations and Principal Bundles

Moreover, it follows from the commutative diagram (9.16) that ker ziE0= ( B @ VB*) . X . Since the representations are semisimple, it follows that

Formulae (9.14) and (9.15) show that 7ci=o,B=o induces an isomorphism ~

Clearly 1 = z1o

1

[B : 0(VE*)e=o]/J5 ~ i - o , e = o .

E l , and so Lemma 111 implies that 1is an isomorphism. Q.E.D.

Consider the operation (E, i, 8, B 0W(E),BBOl1.), defined in sec. 9.13 and let ( Xs)lv denote the classifying homomorphism for X s . Define a homomorphism of operations y : B 0W ( E )+ S by 9.15. Proof of Theorem VIII, (2):

Then, for @ E P E ,

This implies that y ( X 01) = 0. Hence y factors over commutative diagram of algebra homomorphisms

R In particular, p is a homomorphism of operations. Moreover, for b E B,

TC

to yield a

5. The classification theorem

387

Thus property (2) is established and the proof of Theorem VIII is complete.

Q.E.D. 9.16. Classification theorem. We shall say that an operation (E, i,,

O R , R, 8,) is cohomologically related to an operation (E, is,Os, S, 6,) if there is a homomorphism of operations p: R S , which induces an isomorphism pe#-o: H(Re=o)5 H(Se=o). -+

In this case we write

Two operations will be called cohomologically equivalent (c-equivalent) if they are equivalent under the equivalence generated by the above relation. Now let (E, i R ,O R , R, 6,) be a regular operation and let X R be an algebraic connection. Then a ( P E ,8)-algebra (Ri=o,e=o, d n ; 7), is determined, where tR = ( %R)v,e=o 0 t and t is the transgression in W(E),,o fixed at the start of this article. T h e associated PE-algebra is given by (H(Ri=o,s=o);2%0 t). By Theorem V, sec. 8.20, this P,-algebra is independent of the algebraic connection. It follows that the c-equivalence class of the (PE, 6)-algebra (Ri=o,e=o, 8,; t,) is independent of the algebraic connection (cf. the corollary to Proposition XI, sec. 3.29). Hence, to every regular operation corresponds a well-defined c-equivalence class of (PE, S)-algebras. Lemma IV: Suppose (E, i,, O R , R,8,) and (E, is,O,, S, S,) are c-equivalent regular operations. Then the corresponding (PE, 8)-algebras are c-equivalent. Proof: It is sufficient to consider the case that there is a homomorphism of operations 9 : R S, such that q,fo is an isomorphism. It follows that (cf. Theorem 111, sec. 9.8) -+

&o,e=o:

is an isomorphism.

H(&=o,e=o)

--+

H(S,=o,e=o)

IX. Cohomology of Operations and Principal Bundles

388

Now we can apply Proposition XI, sec. 3.29, to obtain (Ri-o,e=o,

6,;

TR)

7 (Si-o,e-o,

6s; t s ) .

Q.E.D. In view of Lemma IV, there is a set map:

-

c-equivalence classes of c-connected classes of {c-equivalence } { (PE, 6)-algebras regular operations

a:

Theorem IX (Classification): With the notation above, a is a bijection. Proof: Theorem VIII, sec. 9.12, shows that a is surjective. T o show that a is injective, let (E, iR , O R , R, 6,) and (E, is, O,, S, 6,) be regular operations with algebraic connections XR and X,. Assume that the cor, responding (PE, 6)-algebras, (Ri=o,e-o, SR; tR) and (Si=o,e-o, 6,; t ~ )are c-equivalent. We must show that the two operations are c-equivalent. By hypothesis there are ( P E ,6)-algebras ( B j ,S j ; r j ) j = 1, . . . , q such that (1) ( 4 , S i ; and

ti.)=

(Ri-o,e-o, 6 ~tR), ; (Bq, 6,;

tq)

= (Si-o,e-o, 6s; t s )

(2) For each j (1 I j 5 q - 1) either

(Bj,6 j ; rj)

6j+l; tj+l) or

(~j+l,

( ~ j + l P

6j+l; rj+l) A

(~j, 6,; r j ) .

Now according to Theorem VIII, sec. 9.12, there are regular operations (E, ij , Oj , T j , dj), and algebraic connections Xj such that

( B j , d j , rj) EZ ((Tj)i=o,e=o, dj; (Xj)",e=o

0

7).

Since it is sufficient to prove that

(E, i ~ , e ~ , ~ , 6 ~ ) y ( ~ T , i1i, d, el ) iy ,. - .7 ( E , i q , e q , T q , d q )

7 (E,is , 0s

9

s, as),

we need only consider the case

5. The classification theorem

389

Thus we may assume that there is a homomorphism, y : ( R i = o , 8 =6,) o, (Si=o+o, S,), such that y 0 t, = tS, and such that y # is an isomorphism. Let ( E , /, 0, 8, 8) be the operation constructed in sec. 9.13 from the ( P E ,6)-algebra (Ri=o,e=o, SA; tR).Since tR = ( % H ) v , o = O 0 t , property (2) in Theorem VIII, sec. 9.12 (applied with y = 1 ) yields a homomorphism of operations y : ( E , z", 8, R, 6) (E,iR, O K , R,SR), --f

--+

such that q ~ ~ = is~ an , ~isomorphism. = ~ Hence, yi#=o,e=o is an isomorphism. Now Theorem 111, sec. 9.8, implies that I&, is an isomorphism; i.e., these two operations are c-equivalent. Finally, an analogous argument yields a homomorphism of operations @:

( E , /, 0, 8, 8)

-

(E, is, Bs, S , 6 s )

(induced from y ) such that @Lo is an isomorphism. Thus these two operations are also c-equivalent. It follows that

(E, if?, OR R, )6, 9

7 ( E , is, Bs, s, w Q.E.D.

§6. Principal bundles 9.17. The structure of H(P). Let 9= ( P , n,B, G ) be a smooth principal bundle whose structure group G is compact and connected. Denote the Lie algebra of G by E. Since G is connected, we have (VE")I =

(VE"),,,.

Let V be a principal connection in ,P.Recall from sec. 6.19, volume 11, and sec. 8.26 that the curvature form of the connection can be used to construct a homomorphism y e : (VE"), + Z ( B ) , such that =hp, where h p denotes the Weil homomorphism for.9'. Let z: PE + (VE*)o=o be a fixed transgression in W(E)o=o.Consider the ( P B ,B)-algebra ( A ( B ) ,6; tg) given by te

=Ye

z,

O

and let ( A ( B )@ APE, 0) denote the corresponding Koszul complex. Theorem X: Let ,9' = (P, n,B , G ) be a smooth principal bundle whose structure group G is compact and connected. Let ( A ( B )0APE, 0 ) be the Koszul complex constructed above via a principal connection. Then there is a homomorphism of graded differential algebras

-

6 : ( A ( B )0APE,

r)

( A ( P ) ,6)

such that

(1) 6 # :H ( A ( B )@ APE)+ H ( P ) is an isomorphism. (2) If B is connected, the diagram

VPE

(r&

H(B)

+-^II

WE"),

h 9

H(B)

I*

H ( A ( B )0APE, 0 ) N

n*

1

-

B*

H(P)

e*

N

ep

I

APE

H(G)

commutes. Remark:

See sec. 8.27 for the lower sequence in (9.17). 390

a~

(9.17)

391

6. Principal bundles

Proof: Let X be the algebraic connection for the operation (E, i, 8, A(P), S) obtained by dualizing the connection form to (cf. sec. 8.22). Then according to formula (8.20), sec. 8.26, yr = ( X,)o=o, whence

It follows that n*: A ( B ) z A(P)i=o,e-o may be considered as an isomorphism of ( P E ,6)-algebras. Thus we obtain an isomorphism of Koszul complexes n*@I

( A ( B )@ APE, r)_N_ (A(P)i=o,e=o 0APE

G(P))-

On the other hand, the Chevalley homomorphism

(cf. sec. 9.3) induces an isomorphism of cohomology, as follows from the fundamental theorem (cf. sec. 9.3). Finally, since G is compact and connected, the inclusion 1: A(P)e=o

--+

A(P)

also induces a cohomology isomorphism (cf. Theorem I, sec. 4.3, volume 11). Thus a homomorphism of graded differential algebras

is given by 6 = 1o @ A ( p ) o (ne@ L), and evidently 6* is an isomorphism. Finally, the commutativity of diagram (9.17) follows at once from diagram (8.22), sec. 8.27, together with Theorem 11, sec. 9.7. Q.E.D. Corollary I:

The

graded

differential

( A ( B )@ APE, 17) are c-equivalent. Corollary 11: If (A(B),6) is c-split, then

algebras ( A ( P ) , 6) and

392

IX. Cohomology of Operations and Principal Bundles

If B is connected, the induced isomorphism of cohomology makes the diagram H ( H ( B ) @ A P B , 17') APE

"C

Proof:

Apply the corollary to Theorem 11, sec. 9.7. Q.E.D.

Corollary 111: If H ( B ) has finite dimension (in particular if B is compact), then H ( P ) has finite dimension and the Euler characteristic of P is zero. Proof: Apply the corollary to Proposition V, sec. 3.18.

Q.E.D. Theorem XI: Let 9= (P,x, B, G) be a smooth principal bundle with compact connected structure group G. Then the following conditions are equivalent:

(1) T h e Weil homomorphism h p : (VE"), H ( B ) is m-regular. (2) H o ( P )= R and HP(P) = 0, 1 I p 5 m. ---f

Proof: I n view of the commutative diagram (8.22), sec. 8.27 (which identifies h p with W * ) as well as the isomorphism H(A(P),=,) H ( P ) , the theorem follows from Theorem IV, sec. 9.8, applied to the operation (E, i, O,A(P),6). Q.E.D. 9.18. Fibres noncohomologous to zero. Let .9' = (P,x, B, G) be a principal bundle with connected base. G will be called noncohomologous to zero in P (n.c.2.) if the fibre projection ep:

is surjective (cf. sec. 8.27).

H(P)

-

H(G)

6. Principal bundles

393

Theorem XII: Let 9= (P,z, B , G) be a principal bundle with connected base and compact connected structure group G. Then the following conditions are equivalent :

(1)

ep

is surjective. (2) There is an isomorphism of graded algebras

H ( B ) @ H ( G ) -3H ( P ) which makes the diagram

(9.18)

H(P) commute. (3) There is a linear isomorphism of graded vector spaces H ( B ) @ H(G) % H ( P ) which makes the diagram (9.18) commute. (4) z* is injective. ( 5 ) T h e Weil homomorphism is trivial: h& = 0. (6) There is a c-equivalence (A (Bx G), S ) 7 ( A ( P ) ,6) such that the induced isomorphism of cohomology makes the diagram (9.18) commute. Moreover, if H ( B ) has finite dimension, then these conditions are equivalent to dim H ( P ) = dim H ( B ) dim H ( G ) . Proof: In view of Theorem X, sec. 9.17, the theorem is a direct translation of Theorem VII, sec. 3.17, and the corollary to Proposition V, sec. 3.18. Q.E.D.

9.19. Homomorphisms. Theorem XIII: Let p: (P, n, B, G) + be a homomorphism of principal bundles with compact

( p ,5, s, G)

394

IX. Cohomology of Operations and Principal Bundles

connected structure group G, and let pB:B -+fl denote the induced map between base manifolds. Then the homomorphisms 6 and 8 of Theorem X, sec. 9.17, can be chosen so that the diagram

A ( B )@ APE

A ( B )@ APE

commutes. In particular,

6* 0 (p$ @ L ) #

= p# 0

8..

Proof Since p is a homomorphism of principal bundles, it is equivariant with respect to the principal actions of G. It follows that the fundamental vector fields & on P and Z h on P are p-related (cf. sec. 3.9, volume 11). This implies that yP: A ( P )c A(P) is a homomorphism of operations (cf. sec. 3.14, volume 11). Moreover, since f 0 p = 9 j B 0 n, we obtain the commutative diagram

The theorem follows now from sec. 9.6.

Q.E.D. Corollary: The homomorphism p#:H ( P )+- H ( P ) is m-regular if and only if the homomorphism p: H ( B ) c H ( f l ) is rn-regular. Proof: Apply Theorem I, sec. 3.10.

Q.E.D. 9.20. The fibre integral. Let .P= ( P , n,B, G) be an oriented principal bundle with compact connected fibre. I n Proposition IV, sec. 6.5,

6.. Principal bundles

395

volume 11, we obtained the commutative diagram

I

(9.19)

where ladenotes the fibre integral. Here E E AnE (n = dim E ) is determined by (d(e), E ) = 1, where d is the unique invariant n-form on G satisfying Je d = 1. Further, w is the involution given by m(@) =

(-l)pn@,

@ E Ap(P).

Next, recall from sec. 3.2 that i ( ~is )also an operator in A ( B ) @ APE. Moreover, since E is of degree n, we may regard i ( e ) as a linear map, homogeneous of degree --n, from A ( B ) @ APE to A ( B ) : i(&)(Y@ @)

( - l ) ~ ' ( @ ,E ) Y ,

@ E APE,

Y EAp(B).

I n particular, i(~)(yI @ @) = 0 if deg @ < n. It follows from sec. 3.2 that i( E ) induces an operator,

i(~)*: H ( A ( B )@ APE)

+

H(B).

We show now that this operator is related to the fibre integral by the commutative diagram

where 6# is the isomorphism of Theorem X , sec. 9.17. In fact, recall that 6 = A o 6Ail(p, o (7c* @ L ) , where 6-4.1cr, is the Chevalley homomorphism determined by an algebraic connection X. Thus,

396

IX. Cohomology of Operations and Principal Bundles

in view of the commutative diagram (9.19), it is sufficient to show that the diagram

A(P)i=o,e=o@ APE

*4P)

A(P)e=o

commutes. Use the algebraic connection to write

Then Theorem I, (3), sec. 9.3, implies that for @ E (APE)'J

8&P)(1 @ @)

-

1

0@ E F'(AQ(P)&O).

Now

Finally, let z E AP(P)i,o,e=o and @ E APE. Then

aAAP)(z) =z

so formula (7.9), sec. 7.3, yields

i(E)8Afp)(z @ @) = (-1)p"z =

This completes the proof.

(-l)pn(@,

- i(~)@~@ ( ~@) )(l E)Z

= i(E)(z@ @).

and

$7. Examples 9.21. Principal S0(2rn)-bundles. Let E denote the Lie algebra of the Lie group SO(2m) (cf. Example 3, sec. 2.5, volume 11). Then (cf. Theorem VII, sec. 6.23),

(VE*)fJ=o = V(u4, us ,

*

9

U4m-4

, v2,11),

where

(Note that the subscripts of the ui and v2, denote the degrees.) It follows (cf. Theorem 11, sec. 6.14) that the elements x , j - l = PE(244j)y j = 1, . . . , m - 1, and y2m--1 = eE(v2m) are a basis of PE; whence

(AE")o=o = APE = A(x, In particular, a transgression T(X+~)

=u

3

x7 9

..

* 9

x4,,-5

t: PE + (VE*)o=ois

, ~ and

7

~2m-1).

defined by

t(y2m-1) = vZm.

Now let 9= ( P , n,B, SO(2m)) be a principal bundle, and form the associated oriented Riemannian bundle E = ( P x W2",z EB, , In sec. 9.4 and sec. 9.13 of volume I1 we defined the Pontrjagin classes and the Pfaffian class of E by

pi([) = kE(~4j)E H"(B),

j

=

1, . .

. , m,

and pf( 6) = kE(vzm)E HZ"(B).

(kEis the characteristic homomorphism for 6.) Further, according to Proposition IX, sec. 9.12, volume 11, pf(5)' = (- 1)"pm( E ) . On the other hand, the Gauss-Bonnet-Chern theorem (sec. 10.1, volume 11) asserts that pf(6) coincides with the EuIer class X, of the associated sphere bundle. Q 8 , . . . , Q4"-,, and Yzmbe any closed forms on B Now let representing p,( 0, . . . ,pm-l( 6) and pf( E ) . Let ( A ( B )@ APB, F') be 397

IX. Cohomology of Operations and Principal Bundles

398

the Koszul complex of the ( P E ,d)-algebra ( A ( B ) ,6; u) given by

u(x4j+) = Q4j,

j' =

1,

. . . ,m

-

and

1,

U(Y~,-~ =) Y,,

Proposition VI: There is a homomorphism of graded differential algebras '8: ( A ( B )8APE 17) ( A ( P ) ,6) 9

-

such that 6 # is an isomorphism, and the diagram

commutes. Proof: Consider the ( P E ,6)-algebra ( A ( B ) ,6; tg)of sec. 9.17. Since (by Theorem VII, sec. 8.24, volume 11) k, = h p , we have

~'(x4j-l) z=

pj(

6)

= hp(u4j) = h p

0

z(x~-1)= t$(~qi-l), j = 1, . . . , m - 1.

Similarly a#(y,,-,) = ~ j $ ( y , ~ - and ~ ) , so G # = 3 . Now combine Theorem X, sec. 9.17, with Proposition X, sec. 3.27, to achieve the proof. Q.E.D.

+

9.22. Principal SO(2m lkbundles. Let E denote the Lie algebra of SO(2m 1) (cf. Example 3, sec. 2.5, volume 11, and sec. 6.21). Then (cf. Theorem VI, sec. 6.23)

+

(VB")O-O

= v(u4 9

3

* *

'

?

u4m),

where uU = ((-l)j/(2n)2j)C~~. ~ ( j = 1, . . . , It follows (cf. sec. 6.14) that the elements x ~ =- e,(uG) m) are a basis of PB;whence

(AE")O,o

= APE = A(x,

X,

---

xqna-l).

7. Examples

Thus a transgression

t:PE + (VE"),,,

Z(X+~)

399

is given by

j = 1, . . . , m.

= uPj,

+

Suppose now that 9= (P, n,B, SO(2m 1)) is a principal bundle with associated vector bundle 5 = ( P x ~ ~Rzm+', ~ ~ ~5 B,, ~ + ~ Let Q4, . . . , @4m be closed forms on B representing the Pontrjagin classes p j ( 5 ) . Let ( A ( B )@ APE, V ) be the Koszul complex of the ( P g , 8)-algebra ( A ( B ) ,8; o), where = @4j. Then the argument of the preceding section also establishes Proposition W: There is a homomorphism

6 : ( A ( B )@ APE, V,

-

( A ( P ) ,8,

of graded differential algebras, such that 6* is an isomorphism, and the diagram

H ( A ( B )@ A P E )

+

commutes. 9.23. Principal U(rn)bundles. Let E be the Lie algebra of the unitary group U(m).Then (cf. Theorem IX, sec. 6.27) (VE")e=o

= V(uz

9

a4 3 %3

3

.

*

9

uzm),

where = (- 1/2n)jC?. It follows that the elements x2j-l = @ E ( u , j ) ( j = 1, . . . , m) are a basis of PE; whence

(AE")e,o = APE

= A(x1, x,,

xg,

In particular, a transgression z: PE + (VE"),,, = uZj,

j

=

1,

. . . ,X ~ ~ - I ) . is given by

. . . , m.

)

400

IX. Cohornology of Operations and Principal Bundles

Now consider a principal bundle 9= (P,n,B , U ( m ) )with associated nt , B , I n sec. 9.18, complex vector bundle 5 = ( P x u(,n)em, volume 11, we defined the Chern classes of 5 by

ern).

cj(

t )= &(Cj),

j = 1,

. . . , m,

where & is the modified characteristic homomorphism for 5. I n view of sec. 9.16, volume 11, c j ( t ) = m6(uZj),

j = 1, . . . , m,

where m, is the characteristic homomorphism for t, regarded as a bundle with Hermitian inner product. Let Qz, Q 4 , ... , QZm be closed forms on B representing cl(t), . . . , cm([), and consider the ( P E ,8)-algebra (A(B),6; a), where a is given by

(cf. Theorem VII, sec. 8.24, volume 11). Thus, exactly as in sec. 9.21, we have Proposition VIII: Let ( A ( B )@ APE, V ) be the Koszul complex for ( A ( B ) ,6; a). Then there is a homomorphism of graded differential

algebras

(W) 0APE, V )

6:

+

( A ( P ) ,8)

such that 6 # is an isomorphism, and the diagram

H ( A ( B )@ APE)

commutes.

APE

P

401

7. Examples

9.24. Manifolds with vanishing Pontrjagin classes. Let B be an oriented connected Riemannian n-manifold, and 1 e t . P = ( P ,n,B, SO(n)) be the frame bundle associated with the tangent bundle tBof B (cf. Example 3 , sec. 8.20, volume 11). We shall compute the cohomology algebra of P in the case that all the Pontrjagin classes of the tangent bundle vanish (examples are given below). First notice that if B is compact and even dimensional, then the Euler class XTB of t Bis given by xrB

=

XB ' O B ,

where wg is the orientation class of B, and XB is the Euler-PoincarC characteristic (cf. Theorem I, sec. 10.1, volume I). If B is not compact, then H n ( B ) = 0 (cf. Proposition IX, sec. 5.15, volume I). Hence X,, = 0 in this case. Now consider the following three cases. Case I: n = 0 for all

=

2m

+ 1.

Apply Proposition VII, sec. 9.22. Since pk(rB) @4j = 0, j = 1, . . . , m. Thus V = 6 @ i

K, we may choose

and

H(P)

H ( A ( B )@ APE, 6 @ i)

= H(B)@

APE

H ( B ) @ H(SO(n)).

I n particular, SO(n) is n.c.2. in P . Case 11: n = 2m, and either B is not compact or X,, = 0 and Proposition VI, sec. 9.21, gives

H(P)

XB =

0.

I n this case

H ( B ) @ H(SO(n)).

In particular, SO(n) is n.c.2. in P. Case 111: n = 2m, B is compact, and XB f 0. Apply Proposition VI, sec. 9.21. Since the Pontrjagin classes vanish we may choose @4j = 0 ( j = 1, . . . , m - 1 ) . For Yn,we may choose any n-form satisfying J B Yn= X B . Thus the Koszul complex of Proposition VI has the form

( ( A ( B )0AYm-1)

0A(%,

* * * 9

%n,--5),

I7

011,

where

V ( @ @ 1 f y @yzm-l)

6@ @ 1

+ 6y @Yzrn-l + yn

y @ 1, @, Y E A ( B ) . A

402

IX. Cohomology of Operations and Principal Bundles

Now apply sec. 3.6 to obtain a linear isomorphism

H ( W ) 0AYw-1)

= H ( H ( B )0Ay2,-,

, V#),

-

+

where V # ( a @ 1 ,9 @y2m-1) = wB ,9 @ 1. It follows (since top degree in H ( B ) ) that

LOB

has

ker V # = ( H ( B )@ 1) 0 ( H + ( B )@yzm-l) and I m .V* = Hn(B) @ 1. Thus we obtain linear isomorphisms of graded spaces

and

H(P)

(7' Hj(B) 0 5 Hj(B) j=O

j=1

@ J J ~ = -@ ~ )A(x,, x,,

. . ,x4m--s).

In particular, the PoincarC polynomials for H ( P ) and H ( B ) are related by m-1

Examples: 1. Suppose B admits a linear connection with decomposable curvature. Then the Pontrjagin classes of B vanish (cf. sec. 9.8, volume 11.) 2. If for some r, tg 0E~ = tntr ( E * is the trivial bundle of rank q over B ) , then the Pontrjagin classes of B vanish (cf. sec. 9.4, volume 11).

3. If B can be immersed into Rn+l(n = dim B ) , then zB 0E = cn+l and so the Pontrjagin classes are zero. 4. Let B = G/S, where S is a torus in a compact connected Lie group G. Then the Pontrjagin classes of B vanish. T h e Euler class vanishes if and only if the torus S is not maximal (cf. sec. 5.12, volume 11).

9.25. The frame bundle of CP" Recall from Example 3, sec. 5.14, volume 11, that @Pa is the manifold of complex lines in Cn+l, and is diffeomorphic to the homogeneous space U(n l ) / ( U ( l ) x U(n)).Moreover, if u E H 2 ( C P n )is the Euler class of the canonical complex line

+

7. Examples

403

bundle 6, then the elements 1, a, 2,. . . , an are a basis of H(CPn) (cf. sec. 6.24, volume 11). In particular, an is an orientation class for CPn. Next observe that (A(@Pn),6) is c-split. Indeed, if @ E A 2 (CP n )is a closed form representing a, then a homomorphism 1: (H(@P"),0) -+ (A(CP"),6) is defined by 1(aP) = @p,

< n.

1< p

Clearly 1#= L and so 1 is a c-splitting. A second graded differential algebra, c-equivalent to (A(CPn),S), is given as follows: Let ( a ) and (zzntl) be one-dimensional graded vector spaces generated by vectors a and z ~ , of ~ + degrees ~ 2 and 2n 1, respectively. Consider the Koszul complex (V(a) @ A(zgnfl),V ) defined by

+

V(a @ 1)

=

0

V(1 @ z,,+~)

and

=

ant1@ 1.

Then a homomorphism of graded differential algebras

P: (V(a>0&2n+1),

V)

-

(H(CP"),0)

is defined by y(a @ 1) = a

and

y(l

0z

~ ~=+ 0.~ )

Evidently, p # is an isomorphism. T o determine the Pontrjagin classes of @Pn observe that

5"@.**@E"~qp8@&

(&=

CPXC)

n+l factora

as follows easily from sec. 5.16, volume 11. Proposition XIII, sec. 9.20, volume 11, implies that the total Chern class of 6 is simply 1 a. Now it follows from Example 3, sec. 9.17, and Example 3, sec. 9.20, both of volume 11, that

+

Finally, the Euler class of @Pa is given by

x = (n + l)or%.

IX. Cohomology of Operations and Principal Bundles

404

Next, consider the orthonormal tangent frame bundle 9 ' = (P, x,

@Pn,SO(2n)) over CPn, and assume n 2 2.

Proposition IX: (1) The graded differential algebra (A(P),S) is c-split. (2) There is an isomorphism of graded algebras

H(P)

A 0A ( ~ z n + 1 >x7 ,~

--

1 1 9

9

x4n-6

9

~2n-1),

+

where zZnfl, x i ,yznP1 are homogeneous vectors of degrees 2n 1, i, and 2n - 1, and A is the truncated polynomial algebra V ( a ) / ( a 2 ) .(Thus A H(CP') = H ( S 2 ) . )

(3) The PoincarC polynomial of H ( P ) is

and dim H ( P ) = Zn+l. Proof: Denote Sk(2n) by E and H(@Pn) by S. Then a PE-algebra ( S ; a) is defined by

4 x 4 =p k ( @ P ) , 1 5 k I n - 1,

a(y2n-.l)= X.

and

It is the associated PE-algebra of the (PE, 6)-algebra defined in sec. 9.21 (with B = CP). Thus, since ( A ( C P ) ,6) is c-split, the example in sec. 3.29 shows that the Koszul complex ( S @ APE, E) is c-equivalent to the Koszul complex ( A ( C P )@ APE, V ) of sec. 9.21. Combining this with Proposition VI, sec. 9.21, we obtain (A(P), 6, Next observe that, since n

7 (S @ APE,

(9.20)

Vu).

2 2,

a(x4p-l)E S+ . a(x3), p 2 2,

a(y2n-1)E S+ . a(x3).

and

Since a(xs) # 0, Proposition IV, sec. 2.13, shows that the Samelson space for ( S ; a) is spanned by x,, xll, . .. ,x ~ ~yznA1. - ~ , Hence the reduction theorem of see. 2.15 yields the relation

(S @ APE

9

V ) 7 (S 8 A(x,), Fb) @ (A(x7

9

* * *

9

Xpn-5

9

Yzn-l),

0). (9.21)

7. Examples

Now consider the graded space P (V(a); T) defined by

+,)

=

-(n

+ l)d,

405

= ( z ~ , + x3), ~,

T(Zzntl) =

and the P-algebra

a"+'.

Its Koszul complex (V(a) @ A ( z ~ ~ x3), + ~ Or) , is also the Koszul complex of the ((x3), S)-algebra, (V(a) 0A ( z ~ ~ + ~a);, $), where f is given by q x 3 ) = -(n l)a2 @ 1. But the homomorphism rp: V ( a ) @ A ( Z ~ , + ~-+) S defined above satisfies y ( Zx,) = u(x,). Hence

+

is a homomorphism of graded differential algebras. Moreover, because p# is an isomorphism so is y # (cf. Theorem I, sec. 3.10). It follows that

Finally the reduction theorem of sec. 2.15, applied to the Koszul .complex on the right-hand side of (9.22) yields

Moreover a c-equivalence from (V(a) @ A(xQ), Vr)to ( A ,0) is given by Thus formulae (9.20), (9.21), (9.22), and (9.23) give

a t - t a , x,++O.

The proposition follows.

Q.E.D. 9.26. The frame bundle of

CP" x CP". In this section we compute

+

H(P), where 9= (P,n, CPnx CP",SO(2n 2m)) is the bundle of (positive) orthonormal tangent frames of CPnx CP". Let a E H2(CPn)and /?E H2(CPm)be the canonical generators (as in sec. 9.25). Then the Kunneth theorem (sec. 5.20, volume I ) yields

H(CP"X C P )= H ( C P ) @H(CP") = ( l , a , . . . ,CP)@(l, p, . . . ,#I"). In particular, (A(CPnx CP"),6) is c-split.

IX. Cohornology of Operations and Principal Bundles

406

Next, note that Proposition 11, (Z), sec. 9.4, volume 11, together with sec. 9.25 shows that the total Pontrjagin class of C P n x CP* is given by

p ( C P x CPm) = (1 - a2)fi+l(1 - pp", whence

(Observe that the ith term in this sum is zero unless k - m - 1 5 i 5 n 1.) As in sec. 9.25, the Euler class is given by

+

x

=

(n

+ l ) ( m + 1)ayP.

We assume throughout that n 2 2 and m 2 2. Denote Sk(2n 2m) by E (cf. sec. 9.21) and denote H(CP1Lx C P ) simply by S. Define a P,-algebra ( S ;o ) by

+

o(X4k-1) =p k ( c P n X

cp"), k

=

1,

. . . ,n + m - 1

Then, exactly as in sec. 9.25, we have

Next observe (as follows from the formulae above for p k and X ) that

4 x 4 = -(n

+ 1)a2- (m + 1)p2

and

Thus since m 2 2 and n 2 2, o(y2n+2m-l) and o ( ~ 4 k - 1 ) (k 2 3 ) are in the ideal S+ . o(PE). (Note that if n = m = 2, then a4 = p4 = 0.) Thus we can apply the simplification theorem of sec. 2.16 as follows: Set

7. Examples

407

Now let (a, b ) be a two-dimensional space with basis a, b, both homogeneous of degree 2. Let P be the four-dimensional graded space ( x 3 , x 7 , . z ~ , + ~w, ~ , ~ +(subscripts ~) still denote degrees). Consider the P-algebra (V(a, b ) ; t), where

+ l)a2 - (m + l)b2,

t ( x 3 ) = -(n

and

= an+1,

+2n+1)

T(X7) t(W2m+l)

= b4, = b"+l.

Define a homomorphism of graded differential algebras Y: (V(a, b ) 04

x 3

, x7

7

x 2 n + 1 9 WZrn+l),

ot)

-

( S 0A(x3 , x7), 0,)

by

It follows exactly as in sec. 9.25 that y # is an isomorphism, whence ( S 04 x 3 2

x7), V y )

7 (V(a, b ) 04 x 3 , x7

9

Z 2 n + l , W2rn+l),

K). (9.26)

Now we distinguish three cases: Case I:

n

2 3 and m 2 3 . Set

Then the Samelson subspace for (V(a,b ) ;T ) is given by P = (Zpnfl, G,, Thus the reduction theorem yields

(V(a, b ) 04

7(V(a,

G) b ) 04%x7)1 K) 0( 4 5 2 2 n + l > 5 2 r n + l ) ,

x 3

9

x 7 , X 2 n + l , W2m+l), Y

0).

(9.27)

408

IX. Cohomology of Operations and Principal Bundles

Moreover, it follows from (9.26) and (9.27) that H(V(a, b) 0A(x,, x,)) has finite dimension. Now the corollary to Theorem VIII, sec. 2.19 (1) = 5)) implies that H+(V(a,b ) 04 x 3 9 x7)) = 0.

(9.28)

Finally, let f c "(a, b ) be the ideal generated by (n+l)a2 and b4. Define a homomorphism

In view of (9.28) it is easy to see that

Proposition X:

H(P)

+ (m+l)b2

v# is an isomorphism. Thus

If n 2 3 and m 2 3, then ( A( P) ,6) is c-split, and

V(a, b ) / f 0A(xll

9

* * 9

x4n+4m-6

9

Y2n+Zm-l,

%n+I

9

62m+1)

(as graded algebras). The PoincarC polynomial for H ( P ) is

(1

+ t2)yi +

t4)(1

+ tzn+zm-i )(1 + P+l)(l +

n+m-1

P+l)

(1 + t4i-1),

j=3

and dim H ( P ) = Pfnai3. Proof: T h e first part of the proposition follows from the c-equivalences (9.24), (9.25), (9.26), (9.27), and (9.29). The second part follows from sec. 2.20 (formula (2.16)) applied to the Koszul complex M a , b ) 04 x 3 xo), K). Q.E.D. 9

Case II: n = 2 for (V(a, b ) ; T) is ideal generated by as given in Case I

and m 2 3. I n this case the Samelson subspace given by fi = (x,, Let J c V(a, b ) be the 3a2 ( m l)b2 and a3. Then the same argument establishes

+ +

Proposition XI: If n = 2 and m 2 3, then

(A(P), 6) is c-split and

7. Examples

409

The PoincarC polynomial of H(P) is

(1

+ tz)(i + + t2

t 4 y

+

t2"+3

)(1

+

P + l )

31(1 +

t4j-1),

j=2

and dim H(P) = 3

. 2m+3.

Case 111: n = 2 and m = 2. define zl:Pl + V ( a , b) by tl(x3)

=

-3(d

Let P, = (x3, x7, xll, y 7 , z5,w5) and

+ b2),

T~(x,) = 6 4

and

Then it follows from (9.24), (9.25), and (9.26) that (A(P), 4 7W(a, b ) 0A P l ,

KJ.

On the other hand, the Samelson subspace P, for (V(a, b); zl) is given by PI = 7.( 9 Y7 9 x11). Hence dim P , < dim P,

-

dim(a, b).

Thus Theorem XI, sec. 3.30, shows that (V(a, b) 0AP,, Kz,) is not c-split. It follows that ( A ( P ) ,S) is not c-split. To compute H(P) it is easiest to use (cf. equation (9.25)) the Koszul complex (S @ A(x3, x7), Ky). I n this case y ( x 7 ) = -384

= 0,

and so the reduction theorem gives

Thus formulae (9.24) and (9.25) imply that

(as graded algebras). A simple calculation shows that the cocycles

410

IX. Cohomology of Operations and Principal Bundles

and

represent a basis for H(S @ A(x3)). We have thus established Proposition XII: If n = m = 2, then (A(P), 6) is not c-split. The PoincarC polynomial of H ( P ) is

(1

+ 2t2 + 2t4 + + 2tfi + tii)(i +

and dimH(P) = 80.

2t7

t7)yi

+ P),

Chapter X

Subalgebras $1. Operation of a subalgebra 10.1. Lie algebra pairs. A Lie algebra pair ( E , F ) consists of two Lie algebras E, F and an injective homomorphism j : F + E. We shall identify F with its image j ( F ) and consider F as a subalgebra of E. Recall from sec. 4.4that a subalgebra F of a Lie algebra E is called reductive in E if the adjoint representation of F in E is semisimple. I n this case the induced representations of F in AE", VE", and W ( E )are semisimple (cf. Theorem 111, sec. 4.4). A Lie algebra pair ( E , F ) will be called a reductivepair if E is reductive and F is reductive in E. Let (E, F ) be a Lie algebra pair. Then the inclusion map j : F + E induces a homomorphism of graded algebras

9:AF" +-

AE".

The kernel of 9 is the ideal IFi generated by the orthogonal complement

FJ. of F in E*. Since j is a homomorphism of Lie algebras, j . restricts to a homo-

-

morphism

j L , : (AF")8=o (AE"),=, (cf. sec. 5.6). Moreover, 9 o BE = BF 09; i.e., j . is a homomorphism of differential algebras. Hence it induces a homomorphism of cohomology algebras j # : H"(F) +- H"(E), and the diagram A 0

(AF"),=, t--(AE")o=O

I

I

H"(F) jtr H " ( E ) commutes. 411

412

X. Subalgebras

If E(respectively, F ) is reductive, then the appropriate vertical arrow in the diagram above is an isomorphism (cf. Theorem I, sec. 5.12). If both E and F are reductive, then j$=o= Aj,, where P E ,PF are the primitive subspaces of (AE")~,O,(AF"),=o (cf. sec. 5.19) andj,: PE-+ Pp is the restriction of is-,. Finally, j induces a homomorphism

j v : VP"

+ VE"

which restricts to a homomorphism

10.2. The operation associated with a pair. Let (E, F) be a Lie algebra pair. Then an operation (F, iF, O F , AE", 6,) of F in the graded differential algebra (AE", 6,) is defined (cf. Example 4, sec. 7.4). The invariant, horizontal, and basic subalgebras for this operation will be denoted (AE"),,,, , (AE")iF=o, and ( ~ \ E X ) + . 0 , 6 F = 0 , respectively. We shall abuse notation, and denote the cohomology algebra

H((AE")~F-o,~p=o 9 'E) by H(E/F). (Observe that E/F is not equipped with a differential operator!) Consider the inclusions

they are all homomorphisms of graded differential algebras. Moreover, k = i 0 e, and so the diagram

1 H((AEX)6F=~

9

6E)

7 H"(E)

commutes. Note that if F is reductive in E, then i* is an isomorphism, as-follows from Proposition I, sec. 7.3.

413

1. Operation of a subalgebra

10.3. Fibre projection. Let ( E , F ) be a Lie algebra pair with F reductive. Then (cf. sec. 7.8 and sec. 7.10) the structure homomorphism YEIF:

AE"

---+

AE"

0AF",

and the fibre projection

p : H((AE")O,=o)

+

(AF")o=o

(for the operation ( F , i F ,O F , AE", 6,)) are defined. On the other hand, let a,: AE" -+AE* @ AF" be the homomorphism extending the linear map a : E" + E" @ F" defined by a(%") = x" @ j"x" = x"

01 + 1 0j Q x X ,

Let

j#F=o:H((AE")O,=,)

-

x" E E".

(AF")o=o

be the homomorphism induced by j . Proposition I: With the hypotheses and notation above,

Proof: (1) It is immediate from the definition that a, and ys,p agree in E"; since they are both homomorphisms, they must coincide. AFX be the projection. Then q 0 a, = j.. (2) Let q : AE" @ AF' Now Proposition VII, sec. 7.11, yields --f

= q8#F=0

'#

( y E I F ) e # = O == J O F = O

Q.E.D. Corollary I:

The diagram

(AE")O=O

i*

commutes.

414

X. Subalgebras

Corollary II: If (E,F ) is a reductive pair, then the Samelson subspace of the operation ( F , iF,O F , AE*, 6,) is the image of the mapj,: Px + PF. Proof: In view of Corollary I and sec. 5.19, we have the commutative diagram

-I

H((AJwoF=U)

I

'I

e'

(AF")o=o

It follows that PF n Im p

= PF n I m

Aj,

= I m j,.

Q.E.D. 10.4. The Samelson subspace for a pair. Let ( E , F ) be a Lie algebra pair with E reductive and let P E be the primitive subspace of (AE"),,,. We shall identify H " ( E ) with the exterior algebra APE under the canonical isomorphism ~f (cf. sec. 5.18). Then Im k # is a graded subalgebra of APE. Definition: The graded subspace

P=

p

of PE given by

I m k # n PE

is called the Samelson subspacefor the pair (E, F ) . A Samelson complement for ( E , F ) is a graded subspace p c PB such that

Observe that the Samelson subspace for the pair (E, F ) is a subspace of PE while (if F is reductive) the Samelson subspace for the operation of F in AEX is a subspace of P f l ! Theorem I (Samelson): Let ( E , F ) be a Lie algebra pair with E reductive. Then the subalgebra Im k # c H"(E) is the exterior algebra over the Samelson subspace of the pair ( E , F ) :

I m k* = Ap.

1. Operation of a subalgebra

415

Proof: Fix a E (APE),=,. Formula (5.8), sec. 5.4, implies that

+

iE(a)dE

(-l)p-ldEiE(u)

=

0.

Hence, iE(a) induces an operator i E ( a ) #in H*(E). Moreover, clearly iE(a) commutes or anticommutes with the operators e E ( x ) and iE(x), x E E. I t follows that (AE")iF=o,eF=o is stable under iE(a).Hence iE(a) induces an operator i E I F ( a ) #in H(E/F). By definition, k 0 iE(u) = iE(a)0 k, a E (AE),=,, and so

On the other hand, it follows from Lemma IX, sec. 5.22, that i E ( u ) " = ip(a),a E P,(E). Hence

and so Im K # is a subalgebra of A P E , stable under the operators i p ( a ) , a E P"(E). Since P# ( E ) is dual to P E , it follows from Proposition I, sec. 0.4 that Im k#

= A(Im

k# n P E ) = AP. Q.E.D.

10.5. Algebraic connections. Let ( E , F) be a Lie algebra pair and consider the associated operation (F,iF , O F , AEQ, dE). Recall from sec. 8.1 that an algebraic connection for this operation is a linear map X: F" + E" which satisfies the conditions

( 1 ) iF(y) 0 X = X 0 i(y), and (2) &(Y) O x = x O q y ) , Y E F. Let X be an algebraic connection. Then the dual map

is a projection onto F, and the kernel of X" is stable under the operators adEy, y E F (cf. Example 4, sec. 8.1). Moreover, according to that example, the correspondence X Hker X" is a bijection between algebraic connections for the operation, and F-stable complements for F in E. I n particular, if F is reductive in E, then the operation always admits a connection.

X. Subalgebras

416

Proposition II: The curvature I of an algebraic connection X is given by

(IJ'", x1 A Xz> = ( y " , [X"xi X"xz] 9

- %"[xi

y" E

9

xz] >,

F",xl,

x2 E E.

In particular, I = 0 if and only if X" is a Lie algebra homomorphism. Proof: In fact,

Subtracting these relations yields the proposition.

Q.E.D. Corollary I: Let x1 E ker X" and x2 E E. Then

Corollary II: The dual map I": A2E + F is given by

Corollary III: The operation of F in AE" admits a connection with zero curvature if and only if F is complemented by an ideal in E.

Again, assume X is an algebraic connection. Set ker X* = S. Then the scalar product between E" and E restricts to a scalar product between F'- and S. Since (AE")i,=o = AFl, it follows that the scalar product between AE" and AE restricts to a scalar product between (AE")i,,o and AS. The induced (algebra) isomorphism AS" (AE*)i,,o is F-linear and so restricts to an isomorphism between the invariant subalgebras. Thus the algebraic connection determines isomorphisms N

(AE")+, -% AS"

and

N

(AE+)ip=O,~p-O (AS")eFpos

Under the first isomorphism the curvature X of X corresponds to the linear map Is: F" + A'S"

417

1. Operation of a subalgebra

given by

as follows from Corollary I to Proposition 11. Finally let e l , . . . , em be a basis for F and let em+l, . . . , en be a basis for S . Let e"', . . . , e*fl be the dual basis of E". Then, according to Example 2, sec. 8.9, the curvature is given by

x

=

4

2

p(e*e)

0

e,(e,)

0

e=m+l

X.

10.6. The cohomology sequence. Let F be a reductive subalgebra of a Lie algebra E and assume that the operation (F, i F ,O F , AE", 6,) admits an algebraic connection. Then the Weil homomorphism

X*:

(VP),_, + H ( E / F )

is defined (cf. sec. 8.15) and is independent of the choice of connection (cf. Theorem V, sec. 8.20). The sequence of homomorphisms .v

(VE"),j,O

38-0 --+

(VF"),,,

x' --+

'k

H(E/F)

--+

'.9

H " ( E ) --+ H"(F)

is called the cohomology sequence for the pair ( E , F). If F is reductive in E, then

H"(E) = H((AE"),,=o),

K#

= e*,

and

j #= p

(cf. sec. 10.2 and sec. 10.3). Thus in this case the last part of the cohomology sequence of the pair coincides with the cohomology sequence for the operation of F in (AE", 6,) (cf. sec. 8.21). Proposition 111: T h e cohomology sequence of a reductive pair ( E , F ) has the following properties :

P

The image of (j:=,)+ generates the kernel of X # . The image of (2.). is contained in the kernel of A*. T h e image of A# is an exterior algebra over the Samelson subspace the image of (k*)+ is contained in the kernel of j*. (4) T h e image of j * is an exterior algebra over a graded subspace

(1) (2) (3) and

of PF.

X. Subalgebras

418

Proof: (1) and (2) are proved in Corollary I11 to Theorem 111, sec. 10.8. (3) is proved in Theorem I, sec. 10.4 (the last part of (3) is obvious). (4) follows from Corollary I to Theorem 111, sec. 5.19.

Q.E.D. Remark: I n Theorem VII, sec. 10.16, we shall give necessary and sufficient conditions for the image of (3")' to generate the kernel of k#. Theorem X, sec. 10.19, contains necessary and sufficient conditions for the image of (A*)+ to generate the kernel of j * .

10.7. The bundle diagram. Let (E, F ) be a Lie algebra pair with F reductive in E. Let X be an algebraic connection for the operation of F in (AE", S,), with curvature homomorphism (Xv)e=o:

(Vp*)e=o

-+

(cf. sec. 8.15). Fix a transgression v: PF + (VF"),,o linear map

-

YEIF: ' F

(AE")iF=o,eF=o (cf. sec. 6.13) and consider the

(AE")iF=O,~F=O

given by VEIF = (X,)e=o 0 v. Then the triple ((AE")iF=O,oF=O, 8,; v E / F ) is the ( P F ,S)-algebra associated with the operation via the connection X (cf. sec. 9.3). Its Koszul complex will be denoted by ((AE")jF=,-,eF=o 0A p F , V E / F ) ; thus V,/F(x @ @o

A

*

.

*

A

QP)

= 8 ~ Z @ @ o / \ * * A' @ p P

+(-I)*

(-l)jvS/F(@j) A Z

@ @o

j=O

zE

A '

* '

dj

* * *

A @p,

( A 4 E " ) ~ F = 0 , ~ F@=j 0E, P F .

T h e cohomology sequence of this ( P F ,8)-algebra (cf. sec. 3.14) will be written

VPp

- (YEIF):

H(E/F)

1% F

H((AE")iF=O,eF=O 0APF)

ektP ___+

APF.

Next, recall that in sec. 9.3 we defined the Chevalley homomorphism

8:((AE*)iF=o,eF=o 0APF 7

-

G / F ) ((AE")e,=o, S E ) .

1. Operation of a subalgebra

419

Composing 6 with the inclusion i : (AE")oF,o morphism

AE" yields a homo-

-

%IF:

((AE")i,=o,e,=o

0APF,~ E , F )

+

(AE", 6,)

of graded differential algebras. Theorem 11: Let ( E , F ) be a Lie algebra pair with F reductive in E . Let 6 E I F be the homomorphism above. Then

(1) z Y ; , ~ : H((AE*)i,=o,e,=o@ A P F ) -% H* (E ) is an isomorphism of graded algebras. (2) The bundle diagram

VPF

U(E,F)!:

H(E/'F) *-

&F

H((AE")iF=o,e,=o@ APp)

e t F

A P p

commutes. Proof: ( 1 ) Theorem I, sec. 9.3, shows that 6* is an isomorphism. Since F is reductive in E, i# is an isomorphism. Hence so is (2) Apply Theorem 11, sec. 9.7, and Proposition I, sec. 10.3.

Q.E.D.

Remark: T h e bundle diagram identifies the last part of the cohomology sequence for ( E , F ) with the cohomology sequence of the associated (P*7,6)-algebra.

10.8. "he base diagram. Let (E, F ) be a reductive Lie algebra pair with inclusion j : F + E. Fix a transgression t: P E -+ (VES)e=o (cf. sec. 6.13). Define a linear map u: PE

+

(vF*)e=o ,

homogeneous of degree 1, by u = j L o PE-algebra (cf. sec. 2.4).

o t.

Then ((VFS),,o; u ) is a

Definition: The pair ((VF*)o=o;a) is called the PE-algebra associated with the pair (23, F) wia t.

The Koszul complex for this PE-algebra is given by ((VF"),,, @ APE, K), where

vo(y@ @o A

* * *

A

QP)

P

(-1)jP

=

A

V U(@j)

@0

* *

0 A

j-0

!€' E (VF*),=o,

@j

* *

@j E

A

aP,

PE.

I t is called the Koszul complex for the pair (E, F ) determined by t. The cohomology sequence for ((VF*),=o; u ) (cf. sec. 2.14) will be written uv

V~'E

--+

(VF*)e-o

If -+

H( (VF*)e=o

0M E )

ef +

APE.

Now we state the main theorem of this article, whose proof will be given in the next three sections. Theorem III: Let (E, F) be a reductive pair and let t be a transgression in W(E),=o.Then there is a homomorphism of graded differential algebras

9:((VF*)o=oO APE -K)

-

((AE*)i,=o,e,=o, 6,)

with the following properties :

(1) e, induces an isomorphism,

v*: H((VF*),,o

APE) 5 H ( E / F )

of graded algebras. 420

2. The cohomology of (AEX)+o,eF=a

(2) The base diagram OV

VPE

(VF"),,,

I#

H((VF")o=,0APE)

42 1

e*

APE

commutes. Remark: Theorem 111 shows that the base diagram identifies the first part of the cohomology sequence of ( E , F ) with the cohomology sequence of the associated PE-algebra (cf. sec. 10.6). Corollary I: The Samelson subspace P for the pair (E, F ) (cf. sec. 10.4) coincides with the Samelson subspace for the PE-algebra ((VP*),=o;a) (cf. sec. 2.13). Corollary 11: An element CD E PE is in

P

if and only if

Proof: Since a = jV,=,o t, Proposition IV, sec. 2.13, shows that the / equation above characterizes the Samelson subspace for ((VF"),c+,; a). Now apply Corollary I. Q.E.D. Corollary 111: (1) The image of (jV,,o)+generates the kernel of Z#. (2) The image of (2")' is contained in the kernel of k#. Proof: Apply Proposition V, sec. 2.14, to ((VF"),=o;0).

Q.E.D. T h e homomorphism p of Theorem I11 will be constructed as follows. In sec. 10.9 we shall apply the results of sec. 8.17, sec. 8.18 and sec. 8.19 to the operation of F in AEX,This will yield a graded differential algebra ((VP" @ AE*)oF=o,D ) and a homomorphism, a x :((VF*

@ AE")eF=o,0) ((AE*)iF=o,eF=o, SE) +

of graded differential algebras, such that a: is an isomorphism.

X. Subalgebras

422

Next, in sec. 10.10, we shall construct a homomorphism of graded differential algebras

6 : ((VF*),,o @ APE, -K) such that

t?*

((VF" @ AE")BF=O,D)

--j

is an isomorphism. Finally, p will be defined by p

= az 0

6.

10.9. The Weil algebra of a pair. T h e graded algebra

W(E,F ) = VF" @ AE" will be called the Weil algebra of the pair ( E , F ) . Thus in particular, W(E,E ) = W ( E ) and W(E,0) = AE". A representation (again denoted by 0,) of F in the algebra W(E,F ) is given by

eF(y)

=e

(y) @1

+

@ eF(y),

Y

E

The corresponding invariant subalgebra will be denoted by W(E,F)o,=o. Now define an operator hF in W(E,F ) by P

hp(Y/@X:

A

AX,")

A

(-l>'j"(X:)V

=

Y @ O Z A

- * .

Xt

* * .

AX:

i =O

and Y E VF",

h F ( y @ 1) = 0,

Xr

E

E".

Then hF is an antiderivation, homogeneous of degree 1. It satisfies the relation hF

BF(y)

=eF(y)

hF,

y

E

F,

and so it restricts to an operator in W(E,F)oF=o.In terms of a pair of dual bases for E" and E we can write hF = Cvp( j * P ) 0i(ev). Denote the operator i @ BE in W(E,F ) simply by B E , and define an operator D in W(E,F ) by

D

=

BE

- hp.

Then according to sec. 8.17 (applied to the operation ( F , i F , O F , AE", 6,)) the pair (W(E,F)sF=o,D)is a graded differential algebra. Next, consider the inclusion map, E:

-

(AE*)iF=o,OF=o W E , F)BF=O

2. The cohomology of (AEX)LF=o,eF=O

423

given by E ( @ ) = 1 @ @. According to Theorem IV, sec. 8.17, F * is an isomorphism. Finally, let X be an algebraic connection for the operation ( F , iF, O F , AE", S E ) . Then a homomorphism of graded differential algebras a,: ( W(E>F ) O ~ = O

9

D,

-

((AE*)i~=O,B~=O

> 'E)

is given by a,( !P @ @)

=

(X,!P)

A

!P E VF",

(nH@),

@ E AE",

(cf. sec. 8.19). By Proposition IX, sec. 8.19, a," is the isomorphism inverse to &#. Moreover (cf. Corollary I to Theorem V, sec. S.ZO), the diagram

(VmkO

5'

l)*

H ( W ( E , F)Bp=O)

H((AE")Op=0)

commutes. (Here 6 : (VP),,, W ( E , QFZO is the inclusion and 1;1: W(E,F)BF=O + (AE"),,=o is the projection.) --f

10.10. The Chevalley homomorphism. Recall from sec. 10.8 that t:PE -+ (VE"),,, denotes a transgression in W(E),=,.Hence there is a h e a r map a : P E W(E),,, , homogeneous of degree zero, such that

-

a ( @ ) - 1 @ @ E (V+E" @ AE"),,,

and aW(a(@)) = T(@)

01,

Extend 01 to a homomorphism a,: @ L restricts to a homomorphism

j . @ 1 : W(E),+o

APE +

-+

@E

PB.

(10.2)

(10.3)

W(E),,, and observe that

W(E,F)oF=O.

Now define a homomorphism of graded algebras,

X. Subalgebras

424

We show that 6 is a homomorphism of differential algebras:

6 0 (-E)

=D

0

(10.4)

6.

In fact, both sides give zero when applied to Y @ 1 ( Y E (VF8)o,0). On the other hand, if @ E P E , then

6 0 (-Vu)(l @ @) while

D 0 6(1

=

0@) = ( D

- j L 0 ( T @ ) @ 1, 0

( j .0&))(a@).

Since (cf. sec. 6.4) dW reduces to h - 8 E in W(E),=,,it follows that

D 0 ( j .@ 1 )

=

- ( j . @ 1) 0 8w.

Hence (cf. formula (10.3))

D 6(1 @ @) 0

=

- ( j . @ L)(T@ @ 1 ) = 6 0 (-Vu)(1 @ @).

This shows that D o 6 agrees with 6 o (-E) in 1 0PE; since these operators also agree in (VF"),,, @ 1, and since they are 6-antiderivations, they must coincide. This proves formula (10.4). The homomorphism 6 is called the Chevalley homomorphism for the pair (E, F ) (determined by t and a). Proposition IV: The Chevalley homomorphism 6 has the following properties :

(1) 6* is an isomorphism. (2) The diagram e*

commutes. ( t #and q# are defined in sec. 10.9.) Proof: ( 1 ) Filter the algebras (VFIW)BFO @ APE and W(E,F)oF=o respectively by the ideals

Zp =

C (VF');,,

jZP

@APE

and

Pp =

,Z ((VF")'@ AE"),F=o.

3ZP

2. The cohomology of (AEX)iF,o,e,=a

425

Then 6 is filtration preserving and so induces a homomorphism

6,:( E i , di)+ (Ei, $1 of spectral sequences. In view of sec. 1.7, we have (Eo do) = ((VF*)e=o @ A P E , 0)

and

(&,

do) = ( W ( K F)e=o, 6 ~ ) -

Moreover, formula (10.2) implies that 6, is simply the inclusion map

6,(Y @ @) = Y @ @,

Y E (VF"),,,

,

@ E APE.

(Identify APfl with (AE*),=, via the isomorphism x E of Theorem 111, sec. 5.18.) Now Lemma I, below, implies that 68 is an isomorphism. It follows that 6, is an isomorphism, and hence so is 8#(cf. Theorem I, sec. 1.14). (2) Evidently 6 0 1 = [ and so 6 # 0 I# = [#. Moreover, if A: APE is the inclusion, then formula (10.2) shows that A 0 e = q 0 6. Hence A# 0 e# = q# 0 6#.

+ (AE*),,=,

Q.E.D. Lemma I: The inclusion

induces an isomorphism of cohomology. Proof: Observe that 8, is the composite of two inclusions

&: (VF*),=o @ APE

+

@ (AE*),,=o,

(VF"),,o

and

1,: (VF"),=o @ (AE")O,,o

+ -

( V P 0AE"),,=o.

Since the representations of F in VF" and AE" are semisimple (because F is reductive in E ) Proposition I, sec. 7.3, and Theorem V, sec. 4.11, imply, respectively, that Al# and A$ are isomorphisms. Hence so is 88. Q.E.D. 10.11. Proof of Theorem 111: Composing the Chevalley homomorphism 6 : (VF*),=o @ APE W(E,F)OF=O +

(cf. sec. 10.10) with the homomorphism

426

X. Subalgebras

(cf. sec. 10.9), we obtain a homomorphism of graded differential algebras

v: ((vp*),=O @ “ E ,

-%)

-

((AE*)iF=0,8F=0,

‘El’

This homomorphism has the properties stated in Theorem 111. I n fact, since a t and 6* are isomorphisms, so is p*. Moreover, combining diagram (10.1) of sec. 10.9 with part (2) of Proposition IV, we see that the right-hand two squares in the diagram of Theorem I11 commute. The commutativity of the left-hand square follows immediately from the definition of 6.

Q.E.D.

§3. The structure of the algebra H(E/F) I n this article we shall study the algebra structure of H ( E / F ) , where ( E ,F ) is a reductive pair, with Samelson subspace P and a Samelson complement P (cf. sec. 10.4). I n view of Theorem I11 (sec. 10.8) and the reduction theorem (sec. 2.15) there is a sequence of homomorphisms

AP

-

H((VF"),=, @ A P ) @ AP -% H((VF"),=, @ APE) -3H ( E / F )

which imbeds AP into H ( E / F ) as a subalgebra. O n the other hand, we have the characteristic subalgebra I m 2# c H ( E / F ) . We shall construct a graded algebra A such that I m %*c A c H ( E / F ) and an isomorphism,

A @ AP % H(E /F ) . 10.12. Characteristic factors. A graded subalgebra A c H ( E / F )will be called a characteristic factor if it satisfies the following conditions :

(1) A is the direct sum of I m X # and a graded ideal I in A. (2) There is an isomorphism of graded algebras g: A 0AP'H ( E / F ) ( A @ AP denotes the anticommutative tensor product) which makes the diagram A 0AP AP

-

Theorem IV: Let (E, F) be a reductive Lie algebra pair. Then H ( E / F )contains a characteristic factor. Moreover, if A and B are charac427

428

X. Subalgebras

teristic factors, then there is an isomorphism A % B of graded algebras, which reduces to the identity in I m Z# and extends to an automorphism of the graded algebra H(E/F). Proof: Existence: Fix a transgression t in W(E),,, and consider the linear map 0 : P E --+ (VF"),,,

given by 0 =j;,, o t (cf. sec. 10.8). Let p be a Samelson complement for the pair (E, F).Corollary I to Theorem 111, sec. 10.8, shows that P is also a Samelson complement for the PE-algebra, ((VF"),=,; 0 ) (cf. sec. 2.13). Set (VF*)6=o@ Ap = N . Then, by Corollary I1 to the reduction theorem (sec. 2.15), there is a commutative diagram of algebra homomorphisms

H ( N ) @ AP

c

AP

in which f* is an isomorphism. Combining this with the commutative diagram of Theorem 111, sec. 10.8, yields the commutative diagram

H ( N ) @ AP

-

AP

where g = v # 0 f #. Define a subalgebra A of H ( E / F ) by setting A = g(H(N) @ 1). Theng and A satisfy condition (2). Moreover, Im Z # c A . To construct the ideal I observe that

H ( N ) = Im a*

0H + ( N )

3. The structure of the algebra H ( E / F )

429

and that H + ( N ) is a graded ideal in H ( N ) (cf. sec. 2.5). Set

I = g ( H + ( N )01). Then I is a graded ideal in A and

A = g(Im l# @ 1 ) @ g ( H + ( N )@ 1) = Im X # @I. Thus A is a characteristic factor.

Uniqueness: Let A and B be two characteristic factors. Then the diagram A@AP

commutes. Hence so does the diagram,

B @AP N

Now Proposition XI, sec. 7.18, yields an isomorphism A A B which reduces to the identity in I m X # . It follows from property (2) for characteristic factors that this isomorphism extends to an automorphism of

H(E/F).

Q.E.D.

430

X. Subalgebras

Corollary: If ( E , F ) is a reductive pair, then there is a homomorphism of graded algebras Q : H ( E / F ) I m I# which reduces to the identity in Im X#. --+

Proof: Identify H ( E / F )with A @ AP viag, and let 7c1: A @ AP + A be the projection. Let n2:A I m I* be the projection with kernel I. Then set = n2 0 nl. --+

Q.E.D.

§4. Cartan pairs Let (E, F ) be a reductive pair. I n view of Theorem IV, sec. 10.12, the following conditions are equivalent:

(1) There is an isomorphism of graded algebras, Im %# @ AP 5 H(E/F) which makes the diagram

(10.5)

WEIF) commute. (2) Im X # is (the unique) characteristic factor. (3) dim H ( E / F ) = dim f m X # dim AP. In this article we shall derive necessary and sufficient conditions for the pair ( E , F ) to satisfy the above properties. 10.13. Deficiency number. Let ( E , F ) be a reductive pair. Then the integer def(E, F) = dim P E - dim P F - dim P

is called the deficiency number for (E, F ) . (Here PEand PFare the primitive subspaces and P is the Samelson subspace for the pair (E, F ) . ) If def(E, F ) = 0, or equivalently, if dim PE = dim PF

+ dim P,

then (E, F ) is called a Cartan pair. Theorem V:

Let ( E , F) be a reductive pair. Then def(E, F ) 2 0; i.e.,

dim PE 2 dim PF 431

+ dim P.

X. Subalgebras

432

Moreover, equality holds if and only if there is an isomorphism of graded algebras Im I * @ AP 5 H ( E / F ) which makes the diagram (10.5) commute. Corollary I: If (E, F) is a reductive pair, then

dim P, 5 dim PE. Corollary 11: Assume (E, F) is a Cartan pair. Then the EulerPoincarC characteristic of H ( E / F ) is given by

0

xmE'F)

=

{ dim H ( E / F )

if if

I* is not surjective, I # is surjective.

Corollary III: If (E, F) is a Cartan pair, then the cohomology algebra H ( E / F )is the tensor product of an exterior algebra (over a space with odd

gradation) and the factor algebra of a symmetric algebra (over an evenly graded space). Proof: Simply observe that Im 2# is a factor algebra of the symmetric algebra (VFs)e=o(cf. Theorem 1, sec. 6.13).

Q.E.D.

Corollary IV: Let (E, F) be a Cartan pair. Let I. and I, denote the ideals in H + ( E / F )generated respectively by the elements of even and odd degrees. Then there is a linear isomorphism of graded vector spaces

P 5 H+(E/F)/Io and an isomorphism of graded algebras Im I # -3H ( E / F ) / I , . Remark: In article 5 , Chapter XI, it will be shown that a reductive pair is not necessarily a Cartan pair. 10.14. Proof of Theorem V: We wish to apply Theorem VII, sec.

2.17, to the PE-algebra ((VFb)e=o; a) defined in sec. 10.8. Observe first that, since F is reductive, Theorem I, sec. 6.13, implies that (VFs)o,o V B F . Hence ((VF"),=O; a) is a symmetric PE-algebra. Moreover, Theorem 111, sec. 10.8, yields the relation dim H((VF"),=, @ APE) = dim H ( E / F ) < 00. Thus the hypotheses of Theorem VII are satisfied.

4. Cartan pairs

433

Since the Samelson subspace for the pair ( E , F ) coincides with the Samelson subspace for ((VP),=,; 0)(cf. Corollary I, sec. 10.8), it follows from Theorem VII, sec. 2.17, that dim P E 2 dim PF

+ dim P.

T o prove the second part of the theorem, observe that the conditions

(1)

dim PE = dim PF

+ dim P

and

(P a

(2) H+((VF"),,, @ AP) = 0

Samelson complement)

are equivalent (cf. Theorem VII, sec. 2.17). Thus we must show that (2) holds if and only if Im X # is a characteristic factor for H ( E / F ) . Write (VF*),=, @ AP = N . I n sec. 10.12 we constructed an isomorphism, g : H ( N ) @ AP -5H ( E / F )

such that g ( H ( N ) @ 1) was a characteristic factor and g(H,(N) @ 1) = Im X # . Hence, if (2) holds, Im X * is a characteristic factor. Conversely, assume that Im X # is a characteristic factor. Then dim H,(N)

=

dim Im X *

=

dim H ( E / F ) dim AP

=

dim H ( N ) ,

and so H+(N)= 0.

Q.E.D. 10.15. Poincark polynomials. Theorem VI:

pair with Samelson space

P.

Let ( E , F ) be a Cartan

Let

be the PoincarC polynomials for P E , P F , and P. Then the PoincarC polynomials for I m X * and for H ( E / F )are given by

n (1

=

- tgt+l)

n (1 S

fH(B,F,

=

i=l

- tZi+')-',

i=l

i=l

and

n (1 R

8

fImp

n (1 R

- tgt+1)

i=l

n (1 + Pi). T

- tlt+1)-1

i=s+l

434

X. Subalgebras

Proof: Recall again from Theorem I, sec. 6.13, that (VF*),,, = VPP and observe that the PoincarC polynomial of Pp is given by

fPF

=

tl"1.

Thus, in view of the isomorphism H ( E / F ) H((VF"),=, @ APE), (cf. Theorem 111, sec. 10.8) the theorem follows at once from the results of sec. 2.20. Q.E.D. Corollary: The dimension and the Euler-PoincarC characteristic of Im $ # are given by

10.16. I n this section we translate Theorem VIII, sec, 2.19, to obtain conditions for a reductive pair to be a Cartan pair. Let (E, F ) be a reductive pair with Samelson subspace P and choose a Samelson complement p . Let ((VF*),=,; 0)be the associated PE-algebra.

Theorem VII: With the notation above the following conditions are equivalent : (1) ( E , F ) is a Cartan pair. (2) The kernel of k* coincides with the ideal generated by Im($#)+. (3) The map F :(VF"),,, + H((VF"),,, @ AP) is surjective. (4) There is an isomorphism of graded algebras,

g: I m

$ # @ AP -3 H ( E / F )

which makes the diagram (10.5) commute. (5) There is an isomorphism of graded p-spaces

h : I m Z# 0VP 5 (VF*)o,, such that the diagram

commutes.

435

4. Cartan pairs

(6) H,((VF"),+, @ A P ) = 0. (7) H+((VF")o,o 0A P ) = 0. (8) The characteristic factors of H ( E / F )are evenly graded. (9) The characteristic factors of H(E/F) have nonzero Euler characteristic.

(10) The restriction 3,: V P + (VF")e,o of a,to V P is injective. Proof: This is a straightforward translation of Theorem VIII, sec. 2.19, via Theorem 111,sec. 10.8, and the proof of Theorem V, sec. 10.13. Q.E.D.

10.17. Split and c-split pairs. A reductive Lie algebra pair (E, F ) will be called split (respectively, c-split), if the graded differential algebra ((AE*)iF=o,sF=o, S,) is split (respectively c-split) (cf. sec. 0.10). Theorem VIII: a Cartan pair.

A reductive pair ( E , F ) is c-split if and only if it is

Proof: I n view of Theorem 111, sec. 10.8, this is an immediate consequence of Theorem XI, sec. 3.30. Q.E.D.

Now suppose ( E , F ) is a Cartan pair, and consider the PF-algebra ( H ( E / F ); YZ/F), where VgIF

=

'z

0

Y,

and Y: PF + (VP),,, is a transgression (cf. sec. 10.7). Denote its Koszul complex by ( H ( E / F )@ APF, V$,,F). Since ((AEQ)iF=O,oF=O, S,) is c-split, we can apply the example of sec. 3.29 to the results of sec. 10.7 to obtain a commutative diagram

H ( H ( E / F )@ ApF , VZ,F)

&(h'IF)

ApF

-

Y

in which the vertical arrows are isomorphisms of graded algebras.

§S. Subalgebras noncohomologous to zero 10.18. Definition: Let (E, F ) be a Lie algebra pair such that F is reductive in E. Then F will be called noncohomologous to xero in E (or simply n.c.z. in E ) if the homomorphism,

p:H*(E) + H"(F) is surjective. Theorem M: Let (E, F ) be a Lie algebra pair with F reductive in E. Then the following conditions are equivalent :

-

n.c.z. in E. homomorphism A#: H ( E / F ) H * ( E ) is injective. characteristic homomorphism Z# for the operation ( F , iF, is trivial: (a#)+= 0. (4) There is an isomorphism of graded cohomology algebras H W ( E ) H ( E / f ' )@ H*(F), which makes the diagram

(1) F is (2) The (3) The OF, AE*, 6,)

H*(E) commutative. ( 5 ) dim H"(E) = dim H ( E / F ) dim H"(F). Proof: The theorem follows immediately from Theorem 11, sec. 10.7, Theorem VII, sec. 3.17, and the Corollary to Proposition V, sec. 3.18. Q.E.D. 436

5. Subalgebras noncohomologous to zero

437

10.19. Theorem X: Let ( E , F ) be a reductive pair, with Samelson P c PE. Then the following conditions are equivalent:

subspace

( 1 ) F is n.c.2. in E. (2) $=o is surjective. (3) ji=ois surjective. (4) There are transgressions fying 'V

Je-0

( 5 ) kerjp

=

0

Y

and

t =V

t

in W(F),,o and W(E),=, satis-

Je=o.

0 *'

P.

(6) The kernel of j # coincides with the ideal generated by Im(K#)+. (7) There is an isomorphism of graded spaces

(8) There is an isomorphism of graded algebras H(E/F)

AP.

(9) The algebra H ( E / F ) is generated by 1 and elements of odd degree. Proof: The equivalence of conditions (1)-(4) is established in Proposition VIII, sec. 6.15. To complete the proof we show that

(1) * ( 5 ) => (6) => (7) =2- (8) * (9)

-

(1).

(1) =2- ( 5 ) : Since F is n.c.2. in E, it follows from Theorem I X that (2.). = 0 and that K* is injective. Hence the kernel of k# coincides (trivially) with the ideal generated by Im( X # ) - k . Thus Theorem VII, (2), sec. 10.16, implies that (E, F ) is a Cartan pair: dim PE = dim P

+ dim P F .

(10.7)

Now consider the commutative diagram

I; jl Im k#-

H " ( E ) j*H " ( F )

(10.8)

438

X. Subalgebras

(cf. sec. 10.4 and Corollary I, sec. 5.19). Since, by Proposition 111, (3), sec. 10.6, j * 0 (k*)+ = 0, it follows from this diagram that

P

c kerj,.

(10.9)

On the other hand (since by hypothesisj* is surjective), the diagram shows that so is j p . This implies that

PE z ker j p @ P F .

(10.10)

Relations (10.7), (10.9), and (10.10) show that kerj, = P. (5) e- ( 6 ) : I t is elementary algebra that ker Aj, coincides with the ideal generated by kerjp. Hence, if (5) holds, ker Ajp is generated by P. Now the commutative diagram (10.8) shows that kerj* is generated by (Im K*)+. (6) ( 7 ) : Assume that (6) holds. Then it follows at once from the commutative diagram (10.8) that kerj, = P. Let be a Samelson complement. T h e n j p restricts to an injection f,: p P F . But, by Theorem V, sec. 10.13,

-

-

0 5 dim PE - d i m p - dim PF = d i m p - dim Pp. Hence, since fp is injective, it must be an isomorphism of graded vector spaces, and ( 7 ) follows.

(7) e- (8): Assume that ( 7 ) holds. Then (E, F ) is a Cartan pair, and so Theorem V, sec. 10.13, yields an isomorphism I m X # @ AP

H(E/F).

Now let p be a Samelson complement for ( E , F). It follows from the hypothesis that p and Pp are isomorphic as graded vector spaces. Thus the formula in Theorem VI, sec. 10.15, for the PoincarC polynomial of Im % # reads fIrnZ+ = 1.

-

It follows that ( X # ) + = 0 and so AP H ( E / F ) . (8) (9): This is obvious. (9) e- (1): The corollary to Theorem IV, sec. 10.12, yields a homomorphism ~ 1 H : ( E / F )-+ Im %#, which restricts to the identity in I m %#.

5. Subalgebras noncohomologous to zero

439

Since I m I # is evenly graded, it follows that

C

ITP(E/F) c kerv.

p odd

Now assume that (9) holds. Then the relation above implies that H + ( E / F )c ker v, whence Im( I # ) + = 0. Thus ( I # ) + = 0, and so Theorem I X shows that F is n.c.2. in E. Q.E.D. Corollary I: If ( E , F ) is a reductive pair, and F is n.c.2. in E, then ( E , F ) is a Cartan pair. Corollary 11: Let ( E , F ) be a reductive pair and let P be a Samelson Then F is n.c.2. complement. Consider the restriction 3 : P -+ (VP),,,. in E if and only if the homomorphism 6":V p + (VF*),=, is an isomorphism. Proof: If F is n.c.2. in E, then ( E , F ) is a Cartan pair and, in view of Theorem IX, (3), sec. 10.18, ( I # ) += 0. Hence Theorem VII, sec. 10.16, shows that d, is an isomorphism. Conversely, if d, is an isomorphism, then (T, is surjective. But (T, - *V 0 t, and s o j i = , must be surjective. It follows that F is n.c.2. in E. Q.E.D.

10.20. Semidirect sums. Let E be a Lie algebra which as a vector space is the direct sum of a subalgebra F and an ideal R:E = F @ R. Then E is called the semidirect sum of F and R. R is stable under the operators ad x, x E E. We define the adjoint representation of F in R by

(adR y). = [ y , Z],

y

E

F,

E

R*

This representation induces a representation OF of F in AR* (cf. sec. 4.2). A simple computation shows that Op(y)B,= 6R8p(y) ( 6 , is the differential operator corresponding to the Lie algebra R).Thus OF induces a representation 0: of F in H*(R). Proposition V: Let E be a Lie algebra which is the semidirect sum of a subalgebra F and an ideal R with F reductive in E. Then F is n.c.2. in E and there is an isomorphism of graded algebras

H*(E)

H*(R),F=~ @ H*(F).

440

X. Subalgebras

Proof: Consider the operation (F, i F ,O F , AE", dB). Corollary I11 to Proposition 11, sec. 10.5, implies that this operation admits a connection with zero curvature. Hence ( X # ) + = 0. Now Theorem IX, sec. 10.18, implies that F is n.c.z. in E, and that there is an isomorphism of graded algebras, H * ( E ) H ( E / F )0 x H"(F).

=

A straightforward computation shows that ( (AE")+o,e,=o, 6,)

((AR")e,=o,

d d

Thus, since F is reductive in E,

(cf. Theorem IV, sec. 4.10). T h e proposition follows.

Q.E.D. 10.21. Examples. 1. Levi decomposition: Let E be any Lie algebra. Then E is the semidirect sum of a semisimple subalgebra F and its radical R (Levi decomposition) (cf. [ 6 ;Theorem 15, p. 961). Thus Proposition V gives the cohomology of E in terms of

(1) the cohomology of the "Levi factor" F, (2) the cohomology of the radical R, (3) the representation 0%. The Lie algebra Lv @ V : Let V be a vector space and define a Lie algebra structure in Lv @ V by setting 2.

[(a,a), (B, b)l = ( [ a ,B1,

- B(a)),

a, B E L v , a, b E

v.

Then Lv @ V is the semidirect sum of the reductive subalgebra Lv and the abelian ideal V. Moreover, Lv is reductive in Lv @ V. T h e induced representation of Lv in AV" is given by

In particular, O ( i ) @ = -p@, @ E ApVx. Hence (ApV")~=o = 0 , p 2 1. Thus by Proposition V

H"(LV @ V ) ZZ H"(L,).

5. Subalgebras noncohomologous to zero

441

3. Let ( E , F ) be a reductive pair with F n.c.2. in E, and let p: E + E , be a homomorphism of reductive Lie algebras such that I m y is reductive in E l . Define a subalgebra Fl c E @ El by

Then ( E @ E l , F l ) is a reductive pair (cf. Proposition 111, sec. 4.7). Moreover, Fl is n.c.z. in E @ E l . I n fact, there is a commutative diagram of Lie algebra homomorphisms,

FT

E.

Since the map j * : H * ( E ) + H * ( F ) is surjective, the diagram implies that j? is also surjective.

$6. Equal rank pairs 10.22. Definition: A reductive Lie algebra pair (E, F ) will be called an equal rank pair if dim PE = dim P F .

I t follows from Theorem V, sec. 10.13, that an equal rank pair is a Cartan pair, with zero Samelson subspace. I n this article we establish the following Theorem XI: Let (E, F ) be a reductive Lie algebra pair. Then the following conditions are equivalent :

(1) (E, F ) is an equal rank pair. (2) The characteristic homomorphism %#: (VF+),,o + H ( E / F ) is surjective. (3) H ( E / F ) (VF")~,o/I, where I is the ideal generated by j;l=o(V+E")&o* (4) There is an isomorphism of graded spaces

g: H ( E / F )0(VE"),=o

= (VF"),=O, Y

which satisfies g(1) = 1, (%# og)(a 01) = a, and g ( a 0Y ) = g(a 01) . j;=o(Y), a E H ( E / F ) , Y E (VE")&O. ( 5 ) H ( E / F ) is evenly graded. (6) H ( E / F ) has nonzero Euler-PoincarC characteristic. (7) j L 0 :(VE"),=, (VF"),,, is injective.

-

Proof:

VPE

Recall from Theorem 111, sec. 10.8, the commutative diagram 0" __+

(VF"),=o

'2

H((VF"),=o 0APE)

442

e4

APE

443

6. Equal rank pairs

In view of this diagram, Theorem X I is a simple translation of the Corollary to Theorem VIII, sec. 2.19.

Q.E.D. Remark: I n article 5, Chapter XI, we shall construct reductive pairs with zero Samelson space which are not Cartan pairs. 10.23. Cartan subalgebras. Recall (from sec. 4.5) the definition of a Cartan subalgebra, and note that a Cartan subalgebra of a reductive Lie algebra E is abelian, and reductive in E. Theorem XII: Let F be a Cartan subalgebra of a reductive Lie algebra E. Then ( E , F ) is an equal rank pair. In particular, dim PB = dim F. Proof: Assume first that the coefficient field I' is algebraically closed. If E is abelian, then F = E and the theorem is trivial, Assume that E is not abelian, and let d denote the set of roots of E for the Cartan subalgebra F. Since E is not abelian and r is algebraically closed, we have (cf. sec. 4.5)

E=FO

c Ea,

A#0,

aEA

where E, is the 1-dimensional root space corresponding to a. In view of [ 6 ; p. 1201 there are r roots a l , . . . ,a, such that every root a E d can be uniquely written in the form

We shall call a root even (respectively, odd) if k , is even (respectively, odd). Let A (respectively, B ) be the collection of even (respectively, odd) roots. Since a , E B we have B f 0. Every odd root is of the form a

= pal

+ C kiai,

ki E Z, p odd.

i22

T h e odd number p will be called the degree of a. Now set

T=F@

C E, asA

and

S=

C E,.

B EB

X. Subalgebras

444

Then E = .T @ S (vector space direct sum). Since

[ E a 2 Ej31 c Ea+j33

% B E

4

it follows that T is a subalgebra, and that

[T, S ] c S,

[S,S ] c T.

In particular, the subalgebra AS c AE is stable under the operators

OE(Y),

Y E

T.

The following two lemmas will be established in the next section : Lemma 11: (E, T ) is a reductive pair. Lemma 111: Let h E F be a vector such that

a,(h) = 1, where a,,

i = 2, . . . , Y ,

ai(h) = 0,

. . , ,a, are the roots used above to define A ker O E ( h ) n ApS = 0,

and B. Then

p odd.

Now consider the operation (T, ir , eT, AE*, ~ 3 ~ )The . decomposition E = T @ S yields an isomorphism, (AE")+o,e,=o

(AS")e,=o

(cf. sec. 10.5). Since F c T , Lemma I11 implies that (AE*)iT=O,eTsO is evenly graded. Thus by Theorem XI, sec. 10.22, dim PT = dim PE. On the other hand, T is a proper subalgebra of E. Moreover, F c T and so F is a Cartan subalgebra of T. It follows by induction (on dim E ) that dim Pp = dim PT , whence dim PE = dim PB7= dim F. Now let r be arbitrary and let Q denote the algebraic closure of r. Then SZ @ F is a Cartan subalgebra of the reductive Lie algebra SZ @ E (over SZ). Moreover, clearly

SZ @ (AE")e,o

= A(Q @ E)faO

and

SZ @ P E = Po,,

445

6. Equal rank pairs

(and similarly for F). It follows that dim F = dim PF = dim, PQOI. = dim, PQOE= dim PE.

Q.E.D. Corollary: Let F be a reductive subalgebra of a reductive Lie algebra E. Then (E, F) is an equal rank pair if and only if F contains a Cartan subalgebra of E. Proof: Assume that (E, F ) is an equal rank pair, and let H be a Cartan subalgebra of F. Then, since H is reductive in F and F is reductive in E, H must be reductive in E (cf. the corollary to Proposition 111, sec. 4.7). Now Lemma 11, sec. 4.5,shows that H is contained in a Cartan subalgebra HE of E. But, in view of .Theorem XII,

dim HE = dim PE = dim PF = dim H. Thus H = H E ; i.e., H is a Cartan subalgebra of E. Conversely, assume that F contains a Cartan subalgebra H of E. Then ZF c H , and so ZF is reductive in E (cf. sec. 4.5). It follows (cf. sec. 4.4)that F is reductive in E. Moreover, by Theorem XII, dim PF = dim H

= dim

PE,

and so (E, F ) is an equal rank pair.

Q.E.D.

-

10.24. Proof of Lemma 11: Observe that T is the fixed point subalgebra of the involution w :E E given by W(X)

= x,

X E

T,

and

w ( x ) = -x,

XE

S.

Now apply Proposition V, sec. 4.8.

Q.E.D. Proof of Lemma 111: Choose in each EB (p E B ) a vector xp f 0. Then the vectors xp form a basis for S . Since the xp are eigenvectors for the transformation ad,(h): S + S , where h is the vector defined in the lemma, it follows that the products, xB1A . A xBr are eigenvectors for the transformation OE(h):A"S+ MS. Moreover, they form a basis of AkS. +

446

X. Subalgebras

Now for $ E B we have

whence A

In particular, since the products xp, A of OE h ) in AkS are all of the form

. . A xpk span AkS, the eigenvalues

E

3, =

C deg/51i.

i=l

Since deg pi is an odd integer (piE B ) , this implies that I f 0 for odd k. Hence ker OE(h) n AkS = 0, k odd.

Q.E.D. 10.25. Poincark polynomials. Let ( E , F ) be an equal rank pair. Let

the PoincarC polynomials of H"(E), H " ( F ) be given by

Then, in view of Theorem XI, sec. 10.22, and Theorem VI, sec. 10.15, the PoincarC polynomial of H ( E / F ) is given by

Thus

In particular, if F is a Cartan subalgebra of E ,

and

$7. Symmetric pairs 10.26. Definition: Let E be a reductive Lie algebra and le: o:E -+ E be an involutive Lie algebra isomorphism. Then a subalgebra F of E is given by

F

=

{Y

E

E I 4 Y ) = Y 1.

The pair ( E , F ) will be called a symmetric pair. According to Proposition V, sec. 4.8,F is reductive in E and so a symmetric pair is reductive. Now let ( E , F ) be a symmetric pair, and write

E=F@S, where S = {x E E I wx = -x}. Then S is stable under ad y , y E F. Hence the projection E + F determined by this decomposition dualizes to an algebraic connection

X: F"

-+E"

for the operation ( F , i F , B a , AE", S E ) (cf. sec. 10.5). X will be called the symmetric connection for the pair (E, F ) . Proposition VI: If ( E , F ) is a symmetric pair, then the restriction of BE to (AE+)ip=O,~F=O is zero. In particular, a symmetric pair is split (cf. sec. 10.17). Proof: Recall from sec. 10.5 that the symmetric connection X determines isomorphisms N

AS" S (AEa)i,=o

and

(AS")e,,o

N

(AE")i,,o,e,,o.

(10.11)

Next, observe that the involution CD extends to an isomorphism wA in AE". Since w restricts to the identity in F, it follows that wA is an automorphism of the operation of F in AE". In particular, w~ restricts to isomorphisms w;=o and w&o,e=o of the algebras (AE")iF=o and (AE+)i,=o,e,=o

-

447

448

X. Subalgebras

On the other hand, w restricts to an isomorphism ws of S Denote the induced isomorphism of AS" by m i ; then

~ b ( @ )= (-l)P@,

(ws =

-I).

@ E A'S".

It follows immediately from the definitions that w$ corresponds. to wb,, under the isomorphism (10.11). T h h implies, in particular, that

Next recall that, since w is a homomorphism of Lie algebras wA 0 aE 6, 0 wA. Restricting this relation to (AE+)iF=O,op=O, and using formula (10.12), we obtain (for @ E (APE")i,,o,eF=,) =

whence

BE@

= 0.

Q.E.D. Corollary: A symmetric pair is a Cartan pair. Proof: Apply Theorem VIII, sec. 10.17.

Q.E.D. Next, let (E, F)be a symmetric pair, with involution w . Then, since w is a Lie algebra homomorphism, wA restricts to a linear involution,

Define subspaces Pi c PB and Pi c PE by

Then PE = Pi @Pi. Proposition W: Let (E, F ) be a symmetric pair with involution w . Let zE: PE --+ (VE"),=o be the distinguished transgression in W(E),=, (cf. sec. 6.10). Then

P = Pi = kerj;;,,, where

P

0

tE,

denotes the Samelson subspace for the pair ( E , F).

449

7. Symmetric pairs

Proof:

It follows at once from Corollary I1 to Theorem 111, sec.

10.8, that ker($=, Let @ E

P. Then

(since @

Since

P

0

c Im

+ 6,yy

tE) c

P.

k#) for some Y E AE",

(AE")i,=o,e,=o.

E

Pk= 0 (k even) it follows from formula

+ d,Tj!wAF=

UA@

(10.12) that (10.13)

-(@ f 6EF).

Projecting both sides of this equation into (AE*),=, we find cop@=

-@;

i.e., @ E Pjj. This shows that

P

c

P;.

Finally we prove that

In fact, let @ E Pi. Since w is a Lie algebra automorphism it follows from Proposition VII, sec. 6.11, that rBo w p = o;soo t E ,whence W;,O(tg@)

=

-XE@.

Moreover, since w reduces to the identity in F it follows that j" 0 ov-y (where j : F -+ E is the inclusion). Hence 'V Je=o(z@)

= jv,=l)w;=o(z~@) =

-jL,(t@).

This shows that j;=,(tE@) = 0, and so

Pz

c

kerj;=,

0

tE.

It has now been proved that kerj;,,

0

tE

c

P

c

Pg c kerj;=,

o

zE,

and the proposition follows.

Q.E.D.

$8. Relative Poincar6 duality Consider a Lie algebra pair ( E , 8') 10.27. The isomorphism D,,,. where E is unimodular and F is reductive in E (cf. sec. 5.10). Fix a basis vector e in A"E ( n = dim E ) and recall from sec. 5.11 that the PoincarC isomorphism D : AE" 2 AE is defined by @ E AE".

D ( @ ) = i(@)e, I t satisfies

(D i ( y ) ) @ 0

=

(-l)~-'(p(y)

0

I))@,

@ E APE",

y E:

F.

Hence it restricts to an isomorphism N

D : (AEx)iF=o 5 (AE)/LF=O. (Here (AE),,=, denotes the ideal in AE consisting of the elements a for which y A a = 0, y E F.) Next, fix a basis vector eF in AnzF ( m = dim F ) . Then multiplication from the right by e F yields a short exact sequence

0 --+I,

- 1

PR(eF)

AE

(AE)pF=O -+ 0,

where I F denotes the ideal in AE generated by F and 1is the inclusion map. T h e sequence above determines an isomorphism N

y : AE/IF

(AE)PF=O.

Composing y-l with D we obtain an isomorphism

-

-

DE/F:

(AEX)iF=o ---t AEIIF.

It maps (ApE+)iF=O isomorphically to (AE/IF)"-m-P. In particular, it follows that dim(AE/IF)ri-m= 1 and the element eEIF

=

DE,F(l)

is a basis vector of (AE/IF)'i-m. 450

451

8. Relative Poincark duality

Definition: The isomorphism isomorphism for the pair ( E , F ) .

DEiF

is called the relative Poimare'

Now observe that, with respect to the scalar products between AE" and AE, (AE*)iF=O= (IF)'. Thus there is an induced scalar product between (AE*)i,=o and AEIIF.

In particular, (AE*)iF=ois a PoincarC algebra, and DEIF is the corresponding PoincarC isomorphism (cf. sec. 0.6). Proof: I t follows from the definition that, if a E AE is an element satisfying a A eF = i(@)e, then

Choose b E AE so that b A eF = e. Then i ( @ ) e = i ( @ ) ( b A e F ) = ( i ( @ ) b )A eF

Hence (Y, DE/F@) = (Y, i ( @ ) b ) =

{@A

Y, b )

10.28. T h e representation of F in AE (obtained by restricting O E to F ) will be denoted by O F . It induces a representation in AEII, , also denoted by OF. Since F is reductive, it is unimodular; thus eF E (ArnF),+,. I t follows that DE/F

OF(Y)

= OF(y) D E / F t

and so DEiFrestricts to an isomorphism

In particular eE/p E (AE/IF)tZ0.

Y

F,

X. Subalgebras

452

Moreover, since F is reductive in E, the representations OF and OF are semisimple. Thus the duality between (AE*)iF=oand AE/IF restricts to a duality between the invariant subalgebras. Lemma IV yields

This relation shows that (AE")iF=0,8F=0 is a Poincark algebra with Poincark isomorphism (DE/F)eF=O. Finally, the equations

eE(r), Y F, show that (IF)eF=Ois stable under the operator aE. Since F a&'p(r)f p(rIaE

=

is reductive

in E ,

(AE/IF)eF=o= (AE)oF=o/(IF)oF-o Thus aE induces a differential operator a E f F in (AE/IF)p-o. Clearly, the restriction of Q E to (AE*)iF,O,eF=O is the negative dual of a,,. Thus a scalar product is induced between H ( E / F ) and H((AE/Ip)p=O,~ E / F ) . Next observe that, since F is reductive, a F e p = 0 and so = 0. Now (the second) formula (5.8) of sec. 5.4 implies that the isomorphism %=O:

(AE/IF)OF-O

(AE)pF-O,OF.=~

satisfies yo=o0 aE,p = aE0 po=o. Since D 5.11), it follows that

9

0

6,:

0

w =aEoD

(cf. sec.

Thus (DE/F)&O induces an isomorphism

Evidently,
8. Relative Poincari. duality

453

I t follows that

10.29. Reductive pairs. Proposition VIII: Let ( E , F ) be a reductive Lie algebra pair and let A be a characteristic factor for H ( E / F ) (cf. sec. 10.12). Then A is a PoincarC duality algebra. Proof: According to sec. 10.28, H ( E / F )is a PoincarC duality algebra. Now the relation H ( E / F ) A @ AP

together with Example 2, sec. 0.6, implies that A is a PoincarC duality algebra. Q.E.D. Corollary: If (E, F ) is a Cartan pair,,then Im Z* is a PoincarC duality algebra. Proposition IX: Let ( E , F ) be a reductive pair and assume that Hfi-"(E/F) c Im X * (n = dim E, m = dim F ) . Then ( E , F ) is an equal

rank pair. Proof: In view of Theorem XI, sec. 10.22, it is sufficient to show that X * is surjective. According to the corollary to Theorem IV, sec. 10.12, there is a homomorphism 'p:

H ( E / F )+ Im Z*,

which restricts to the identity in Im X * . I n particular ker q~ n H"-"(E/F)

= 0.

But, since H ( E / F ) is a PoincarC duality algebra of degree n - m, every nonzero ideal in H ( E / F ) contains Hfi-"(E/F). Thus the equation above implies that ker cp = 0. Hence q~ is an isomorphism and so Z* is surjective. Q.E.D.

$9. SympIectic metrics 10.30. Definition: Let ( E , F ) be a Lie algebra pair. Then the space (A*E*)IF,Omay be identified with the space of skew symmetric bilinear functions in EIF. An element @ E (A2E")iF,ois called a symplectic metric in E / F if @ is nondegenerate; i.e., if the relation

@(a1,a) = 0 for fixed a1E EIF and all 5 E E / F implies that Zl= 0. Equivalently, @ is a symplectic metric in EIF if F = { y E E 1 i ( y ) @= O } . Elementary linear algebra shows that a symplectic metric exists in EIF if and only if dim E / F is even. Now assume that this condition is satisfied: dim EIF = 2k. Then an element @ E (A2E*)iF,ois a symplectic A @, K factors). metric if and only if Gkf 0 (Qk= @ A A symplectic metric @ for E / F is called closed if a

If @ is a closed symplectic metric, then

and so @ E (AzE*)i,=o,e,=o. Proposition-'X: Let ( E , F ) be a Lie algebra pair with E semisimple. Then EIF admits a closed symplectic metric if and only if for some h E E, F = ker(ad h). Proof: Assume that EIF admits a closed symplectic metric @. Since E is semisimple, H 2 ( E )= 0 (cf. sec. 5.20). Hence, for some h" E E", @ = 6Eh".

It follows that

455

9. Symplectic metrics

Now let a : E -ZE" be the isomorphism determined by the Killing form (cf. Theorem I, sec. 4.4)and set h = a-l(h*). Then a((ad x ) h ) = OE(x)h* = iE(x)@,

x E E.

Since @ is nondegenerate, this relation implies that F = ker(ad h). Conversely assume that F = ker(ad h), some h E E. Set hY = a(h). Reversing the argument above shows that dEh* is a closed symplectic metric in EIF. Q.E.D. 10.31. Reductive pairs. Theorem XIII: Let ( E , F ) be a Lie algebra pair with E semisimple. Then the following conditions ar.e equivalent:

F

(1) For some Cartan subalgebra H of E , and for some h E H , = ker(ad h).

(2) ( E , F ) is a reductive pair and E / F admits a closed symplectic metric @. (3) ( E , F ) is a reductive pair, dim E / F = 2k, and, for some a! E H 2 ( E / F ) ,ak# 0. Moreover, if these conditions hold, then ( E , F ) is an equal rank pair. Proof: First we show that (3) implies that ( E , F ) is an equal rank pair. Then we show that

Assume that (3) holds. Since E is semisimple, Pg = 0, q < 3, This implies that pq = 0 and pq = 0, q < 3, where f) is the Samelson subspace and p is a Samelson complement for ( E , F ) . Now according to Theorem IV, sec. 10.12,

H(E/F)

A @ AP,

where

A

=

Im X # @ I

and

I

H,((VF*)o=o@ AP).

This shows that H 2 ( E / F )c I m 2#. I n particular, a! E I m Z#. Since a k f 0 it follows that H Z k ( E / F ) c Im Z#.Now Proposition IX, sec. 10.29, shows that ( E , F ) is an equal rank pair.

456

X. Subalgebras

It remains to establish the equivalence of conditions (l), (2), and (3). (1)- (2): Since h is in a Cartan subalgebra of E, Proposition 11, sec. 4.5 implies that F is reductive in E. Moreover, by Proposition X, sec. 10.30, E/F admits a closed symplectic metric. (2) * (3): Let a E H 2 ( E / F )be the class represented by the closed symplectic metric @. Then dim E - dim F = 2k. Thus, in view of formula (10.14), sec. 10.28,

and so it follows that ak = Qk f 0. be a cocycle representing a. Then (3) * (1): Let Q E (A\2E+)iF~o,ep=o @k f 0 and so @ is a closed symplectic metric. Hence, by Proposition X, sec. 10.30, for some h E E,

F

= ker(ad

h).

Now let H be a Cartan subalgebra of F. I n view of the relation above, h E 2, and so h E H. But, by the first part of the proof, (3) implies that (E, F ) is an equal rank pair. Thus, in view of the corollary to Theorem XII, sec. 10.23, H is a Cartan subalgebra of E. Q.E.D.

Chapter X I

Homogeneous Spaces $1. The cohomology of a homogeneous space In this chapter, the results of Chapter X will be applied to homogeneous spaces. G will always denote a connected Lie group with Lie algebra E. K is a closed connected Lie subgroup with Lie algebra F. Recall from sec. 2.9, volume 11, that the left cosets aK ( a E G ) form a manifold .G/K. It is the base of the principal bundle 9= (G,TC, GIK, K ) , where n denotes the projection a I-+ aK and the principal action of K on G is Ky right multiplication (cf. sec. 5.1, volume 11). Finally, note that if G is compact, then so is K . I n this case both Lie algebras E and F are reductive. Moreover, Proposition XVII, sec. 1.17, volume 11, implies that the adjoint representation of K in E is semisimple. Hence so is the adjoint representation of F in E. Thus, if G is compact, then ( E , F ) is a reductive pair. 11.1. The operation of F in A(G). According to sec. 8.22 the prin-

cipal bundle 9determines an associated operation of F in A ( G ) ;it will be denoted by ( F , i K ,O K , A ( G ) ,&). Since the principal action is right multiplication, the fundamental vector fields are simply the left invariant vector fields xh (h E F ) . Thus

It follows that the operation ( F , iK, O K , A ( G ) ,6,) coincides with the restriction to F of the operation ( E , ic, O G , A ( G ) ,6,) defined in sec. 7.21. I n sec. 7.21 we considered the left invariant operation ( E , iL, O h , A,(G), 6,) of E in the left invariant differential forms on G. This, too, restricts to an operation ( F , iF , O F , A,(G), 6,) of F. Further, the inclusion

457

458

XI. Homogeneous Spaces

and the isomorphism tL:A,(G)

-5AE"

(cf. sec. 7.21) are homomorphisms of operations of E ; hence they may be regarded as homomorphisms of operations of F. It follows that the composite EG = ZG 0 ti1 is also a homomorphism of operations of F : '%:

( F , iF >

OF

9

AE",

sE)

-

( F ,i K

Thus cG restricts to a homomorphism ((AE")iF=O,OF=osE)

(EG)iF-O,BF=O:

9

-

eK

7

A(G),&)*

(A(G)i,=0,f3,=0,

dG)*

On the other hand, since K is connected, it follows from the results of sec. 6.3, volume I1 (applied to the principal bundle 9) that 7cx restricts to an isomorphism

( A ( G / K ) B, G i K ) 3 (A(G)i,-o,e,=o Composing morphism

(

E

~

)

~

~

9

8G)-

with ~ , the ~ ~ inverse = ~ isomorphism yields a homo-

=

~ G I K ((AE")iF=O,ep=o, : 6,)

+

( A ( G / K ) &/,i) ,

of graded differential algebras. I n view of sec. 6.29, volume 11, E ( ; / ~ can be regarded as an isomorphism

(AE*)iF-o,eF=o5 A,(G/K), where A,(G/K) denotes the algebra of differential forms on G / K invariant under the left action of G. Proposition I: -4ssume G is compact. Then

is an isomorphism:

E & ~

eglK : H (E / F ) 3 H( G / K) .

Proof: In view of the remarks above, it is sufficient to show that the natural homomorphism

HdGIK)

-

WG/K)

is an isomorphism. But this follows from Theorem I, sec, 4.3, volume 11, since G is compact and connected.

Q.E.D.

'

1. The cohomology of a homogeneous space

459

Corollary: If G is compact, then

11.2. The Samelson subspace. Assume that G is compact. Recall from sec. 4.12, volume 11, (or sec. 5.32) the definition of the primitive space Po c H+(G).According to sec. 4.12, volume 11, H ( G ) EZ Ape. Definition: The Samelson subspace Po for the pair (G, K ) is the graded space PG = I m n # n pG.

A complementary graded space pG in Pc is called a Samelson complement. Theorem I: Assume that G is compact. Then the image of the homomorphism z#: H ( G / K )+ H ( G ) is the exterior algebra over the Samelson space : I m n # = APo. Proof: Identify (AE#),=,, with H # ( E ) via nE (cf. sec. 5.12). Then ~jjcoincides with aG (cf. sec. 5.29). Thus sec. 5.32 shows that E$ restricts

to an isomorphism N -

:&;

P E S Pc.

Now observe that (by definition) the diagram

H ( E / F )k"H*(E)

&+ H(G/K)

=I E E

H(G)

commutes, and apply Theorem I, sec. 10.4.

Q.E.D. 11.3. Invariant connections. Suppose X is an algebraic connection for the operation ( F , iF, O F , AE", 8,). Then E~ 0 X is an algebraic connection for the operation ( F , iK , O K , A ( G ) ,8,). Let V be the corresponding principal connection in 9(cf. sec. 8.22).

XI. Homogeneous Spaces

460

It is easy to verify that Y is G-invariant. Moreover, Proposition XVIII, sec. 6.30, volume 11, together with Example 4, sec. 8.1 implies that the correspondence X H V is a bijection from algebraic conpections in (F, i p sBF, AE', 6,) to G-invariant principal connections in 9. Now assume that the operation of F in (AE", 6,) admits an algebraic connection. Then the Weil homomorphism 2*: (VH;"),,,

is defined. Since

-+

H(E/F)

is a homomorphism of operations, it follows that (Ea)i#=o,eF=o

0

Z # = $#,

where $* denotes the Weil homomorphism of the operation of F in ( 4 G h 47). Thus Theorem VI, sec. 8.26, applies to yield a commutative diagram I*

I=

(Vp*),_o

H(E/F)

1

(VF*),

&K

H(G/K),

where h p is the Weil homomorphism for the principal bundle 9.

11.4. The cohomology sequence. Let j,: K -+ G and j : F -+ E denote the inclusions (so that j = jh). Then the sequence of homomorphisms (VE"),

.?+(VF*), 3H(G/K)

n*

H(G)

3H(K)

is called the cohomology sequence for the pair (G, K ) . Now assume that the operation of F in E admits an algebraic connection and consider the diagram jJ-0

(VE"),,,

I

(VF")B=O

X*

I

WEIF)

4,K

- k#

H*(E)

j*

H"(F)

1. T h e cohomology of a homogeneous space

46 1

The left-hand square commutes by definition. The commutativity of the second square was shown just above, while the third square commutes, again by definition. That the fourth square commutes follows from sec. 4.7, volume 11. Thus the diagram (11.1) is commutative. It may be regarded as a homomorphism from the cohomology sequence of the pair ( E , F ) (cf. sec. 10.6) to the cohomology sequence of the pair (G, K ) . Moreover, if G is compact, then all the vertical arrows are isomorphisms, as follows from Proposition I, sec. 11.1, and sec. 5.29. Thus, in this case, the diagram is an isomorphism of cohomology sequences. Proposition 11: Assume that G is compact. Then the cohomology sequence of the pair (G, K ) has the following properties:

(1) The image ( j i ) +generates the kernel of h 3 . (2) The image of h& is contained in the kernel of n#. (3) The image of n # is an exterior algebra over the Samelson subspace p G ,and the image of (n*)+is contained in the kernel of j g . (4) The image of j g is an exterior algebra over a graded subspace of the primitive space P K . Proof:

In view of the isomorphisms

(cf. sec. 5.32) the proposition follows (via diagram (11.1)) from Proposition 111, sec. 10.6. Q.E.D.

$2. The structure of H ( G / K ) 11.5. The structure of H(G/K). In this section G is assumed to be compact, so that ( E , F ) is a reductive pair. As in sec. 10.8, fix a transgression z: PE+ (VE"),=o and define 0:PE + (VF")e=o

by (T = ji=oo t. Then the PE-algebra ((VF"),,o; (T)is called the PE-algebra associated with the pair (G, K ) . (It coincides with the PE-algebra associated with ( E , F ) . In particular, its Koszul complex ((VF"),,, 0APE, K) is the one defined in sec. 10.8.) Theorem 11: Suppose G is a compact connected Lie group with compact connected subgroup K. Then there is a homomorphism of graded differential algebras TG/l;:

((VF")O=O

8

--Gb)

( A ( G / K ) dG/K) ,

with the following properties :

(1) ~ 8 is ,an ~isomorphism. (2) The diagram

VPE

OV

(VF")e=o

I"

A

H((VF"),=o 0APE)

1-

N

1

&,K

e*

APE

1

EQ

commutes. Proof: Let q ~ :(VF"),,, @ APE -+ (AE")i,=o,e,=o be the homomorphism in Theorem 111, sec. 10.8, and set plClK = EG/K 0 q ~ . Then Q)G/K has the desired properties, as follows from Theorem 111, sec. 10.8, Proposition I, sec. 11.1, and diagram ( l l . l ) , sec. 11.4.

Q.E.D. 462

2. The structure of H ( G / K )

463

Theorem 111: Suppose G is a compact connected Lie group with compact connected subgroup K. Then there is a graded subalgebra A c N ( G / K )with the following properties:

(1) A as a vector space is the direct sum of the subalgebra I m h p and a graded ideal I in A,

A = ImhYO1. (2)

There is an isomorphism of graded algebras

g: A

0APG -5 H(G/K),

which makes the diagram

A @ APG

-

APG

H(GIK) 7 H(G) commute. (PG is the Samelson subspace-cf. sec. 11.2.) Moreover, if B is a second subalgebra of H ( G / K )with these properties, then there is an automorphism of H ( G / K )which restricts to an isomorphism A 4B, and induces the identity in I m h p . N

Proof: I n view of diagram ( l l . l ) , sec. 11.4, the theorem follows directly from Theorem IV, sec. 10.12.

Q.E.D.

Corollary: There is a homomorphism of graded algebras M: H(G1K) Im h p which reduces to the identity in Im h p .

-+

Theorem IV: Let G be a compact connected Lie group with compact connected subgroup K. Let pG be the Samelson subspace for the pair (G, K ) (cf. sec. 11.2). Then

dim PG

2 dim PK + dim PG.

Moreover, the following conditions are equivalent : (1)

(E, F) is a Cartan pair.

XI. Homogeneous Spaces

464

(2) dim Pa

= dim

PK

+ dim

PG.

(3) There is an isomorphism of graded algebras Im h p 0APG-% H(G/K), which makes the diagram

f w / K )IT*H ( G ) commute. (4) dim H ( G / K )= dim I m h p dim APG. (5) The kernel of n* is generated by the image of h&. is c-split. (6) T h e graded differential algebra ( A ( G / K ) S,),

-

Proof: I n view of the commutative diagram ( l l . l ) , the first statement and the equivalence of conditions (1)-(5) follows directly from the remarks at the beginning of article 4,Chapter X, together with Theorem V, sec. 10.13, and Theorem VII, sec. 10.16. Finally, Theorem VIII, sec. 10.17, asserts that (1) holds if and only if the differential algebra ((AE*)iF=O,eF-O, S,) is c-split. But, in view of the corollary to Proposition I, sec. 11.1,

and so (1) and (6) are equivalent.

Q.E.D. Corollary I: Let G / K be a homogeneous space with G and K compact and connected, Then the validity of the conditions in Theorem IV depends only on the smooth manifold structure of G/K (and so is independent of any Lie structure). Proof: Observe that this is correct for condition (6).

Q.E.D.

2. The structure of H(G/K)

465

Corollary 11: Assume that K , c G, and K , c G, are compact and connected Lie groups and let 9:G J K , + G,/K, be a smooth map such that p* is an isomorphism. Then the conditions in Theorem IV are satisfied by the pair (G,, K,) if and only if they hold for the pair (G, , &). I n this case there is a linear isomorphism of (graded) Samelson spaces PI5 P, , and an isomorphism of graded algebras Im hpl Im h p 2 . Proof: Since 9":A(G,/Kl, 6,) c A(G,/K, , 6,) is a c-equivalence, the first statement follows. T h e second assertion is a consequence of Corollary IV to Theorem V, sec. 10.13. Q.E.D. Example: Symmetric spaces: Let w be an involution of a compact Lie group G, and let K denote the 1-component of the subgroup left pointwise fixed by w . Then the corollary to Proposition VI, sec. 10.26, shows that the pair (G, K ) satisfies the conditions of Theorem IV. Theorem V: Let K be a compact connected subgroup of a compact connected Lie group G. Assume that the pair (G,K ) satisfies the conditions in Theorem IV. Let

,xtgt, T

fp, =

r

Y

fi.,

=

C tzt,

and

i=s+l

i=l

a=1

tgi

fpG =

be the PoincarC polynomials for Pc , PK , and P(;. Then the PoincarC polynomials for Im h p and for H ( G / K ) are given, respectively, by =

fIm

and

n (1

8

8

i=l

i=l

JJ (1 - tOi+l) JJ (1 - tli+1)-1,

i=l

r

S

8

fH(C,K) =

- tgtf')

JJ (1 i=l

- tZi+1)-1

JJ (1 + PI).

i=s+l

Proof: In view of the commutative diagram (ll.l), this follows from Theorem VI, sec. 10.15. Q.E.D.

11.6. Subgroups noncohomologous to zero. A subgroup K of G will be called noncohomologous to zero (n.c.2.) in G if the homomorphism j g : H(G) + H ( K ) is surjective.

466

XI. Homogeneous Spaces

Theorem VI: Let K be a compact connected subgroup of a compact connected Lie group G. Then the following conditions are equivalent:

(1) K is n.c.z. in G. (2) F is n.c.2. in E. (3) T h e homomorphism ? c # : H ( G / K )+ H ( G ) is injective. (4) T h e Weil homomorphism lzp is trivial; (i.e., h& = 0). ( 5 ) There is an isomorphism of graded cohomology algebras H(G) H ( G / K )@ H ( K ) , which makes the diagram

H ( G / K )0H ( K )

WG) commute. ( 6 ) dim H ( G ) = dim H ( G / K )dim H ( K ) . (7) j;l is surjective. (8) T h e kernel of j g coincides with the ideal generated by Im(n#)+. (9) There is an isomorphism of graded spaces PGz pG 0P K . (10) There is an isomorphism of graded algebras H ( G / K ) APG. (11) The algebra H ( G / K ) is generated by 1 and elements of odd degree. Proof: In view of the diagram (1 1.1) the theorem follows from Theorem IX, sec. 10.18, and Theorem X, sec. 10.19.

Q.E.D. 11.7. Subgroups of the same rank. Recall from sec. 4.12, volume 11, that the rank of a compact connected Lie group G is the dimension of PG. In view of sec. 5.32 we have

where E is the Lie algebra of G. Now let T be a maximal torus in G. Then according to Theorem IV, sec. 4.12, volume 11, dim T = rank G. On the other hand, it is easy to

2. The structure of H ( G / K )

467

check that the Lie algebra F of T is a Cartan subalgebra of E. Thus Theorem XII, sec. 10.23, (which asserts that dim F = rank E ) provides an algebraic proof that dim T = rank G. Theorem VII: Let K be a compact connected subgroup of a compact connected Lie group G. Then the following conditions are equivalent:

(1) G and K have the same rank. (2) T h e Weil homomorphism h p : (VF'), + H ( G / K ) is surjective. (3) H ( G / K ) (VFX),/J, where J is the ideal generated by

j,'((V+E")r). (4) H ( G / K ) is evenly graded, (5) T h e Euler-PoincarC characteristic of H ( G / K ) is nonzero. (6) j i is injective. (7) W"-m(G/K)c Im h p (n = dim G, m = dim K ) .

If these conditions hold, the Samelson space P, = 0 is zero and thus (n*)+= 0. Moreover if the PoincarC polynomials for Pc and PK are given by t g t and t l i , then the PoincarC polynomial for H ( G / K )is

x:=l

Jj (1 - Pf+l)/ i=l

(1 - t l i + ' ) 1=1

and

Proof: I n view of diagram (11.1) the equivalence of conditions (1)-(6) follows from Theorem XI, sec. 10.22. Proposition IX, sec. 10.29, shows that (2) is equivalent to (7). For the last part, apply Theorem V, sec. 11.5.

Q.E.D. Corollary I: Suppose some Pontrjagin number of the homogeneous space GIK is nonzero. Then G and K have the same rank. Proof: According to Proposition 111, sec. 5.11, volume 11, the tangent bundle of G / K has total space G x E/F. Thus formula (8.4), sec. 8.25, volume 11, shows that the characteristic classes of t & k ; are in I m h.p. Hence, by hypothesis, Im h9 3 H"-m(G/K), and so condition (7) of the theorem is satisfied. Q.E.D.

XI. Homogeneous Spaces

468

Corollary 11: Let T be a maximal torus of a compact connected Lie group G. Then H ( G / T ) is evenly graded. Its PoincarC polynomial is given by

xi-l

tg* is the PoincarC polynomial for P E . Moreover, if where the order of the Weyl group (cf. sec. 11.8) then

I W, I is

Proof: That I W, I = X,,, is shown in Proposition XIII, sec. 4.21, volume 11. The rest of the corollary follows from the theorem.

Q.E.D.

93. The Weyl group 11.8. The Weyl group. Let G be a compact connected Lie group with maximal torus T. Denote the corresponding Lie algebras by E and F. Let NT be the normalizer of T (in G). Then the factor group

W, = NT/T is a finite group (cf. sec. 2.16, volume 11). I t is, up to an isomorphism, independent of the choice of T and is called the Weyl group of G . A smooth right action @ of W, on G / T is defined by d E Wc,

@,-(zx) = @(zx, G ) z= z ( x ~ ) ,

E

G,

where a E NT is any representative of 5. Hence a representation @* of W G in H ( G / T ) is defined by @ # ( G ) = @$,

dE

w,.

The corresponding invariant subspace is denoted by H ( G/T)TfTG=l. On the other hand, the left regular representation of W, is defined as follows: Let V be the real vector space whose elements are the formal sums ~ c v . E I Y GAvGv with k E R (so that the elements of WG are a basis of V). Set

Proposition 111: T h e representation @* is equivalent to the left regular representation of W G . Proof:

It follows from [3; (2.6), p. 121 that it is sufficient to show that

tr @=:

0

if

c+if,

and

tr@# = I

W G

I.

Fix Z E W, and consider the Lefschetz number of the map @,- (cf. sec. 10.7, volume I). It is given by =

C (-l)p P

469

tr @,:

XI. Homogeneous Spaces

470

where @: denotes the restriction of @: to HP(G/T). Since H ( G / T ) is evenly graded (cf. Corollary I1 of Theorem VII), it follows that

L(Qb)= C tr @;

= tr @:

P

Now, if ii f 5, then has no fixed points and thus L(d&)= 0 (cf. the Corollary of Theorem 111, sec. 10.8, volume I). This shows that tr @:

=

ii # 6.

if

0

On the other hand, Proposition XIII, sec. 4.21, volume 11, shows that XG,T = I WQI and so tr

@f

= XG,T =

I WG I. Q.E.D.

Corollary: H(G/T)lv,,l = Ho(G/T). Proof: Clearly, Ho(G/T) is contained in the invariant subspace. But the invariant subspace of the regular representation has dimension 1. Q.E.D. 11.9. The image of j i = o . Observe that since T is normal in N T , the operators Ad a (a E N T )in E restrict to operators in the Lie algebra F of T. Thus a representation Y of WGin F is given by Y(Z)(y) = (Ad a)y,

u E NTT, y E F.

Denote the induced representation in V F X by Yv: YV(6) =

(Y(ii-1))v.

Denote the invariant subalgebra by (VF*),,=, . Now consider the inclusion j : F --+ E and the induced homomorphism *V VF". (Since F is abelian, VF" = (VF*)e=o.) je=o:

-

Theorem VIII: The homomorphism ji=o is an isomorphism of (VE"),,, onto (VF")w,=l:

Proof: As we remarked in the beginning of sec. 11.7, (E, F) is an equal rank pair. Thus by Theorem XI, (7), sec. 10.22, J:,=~ is injective.

3. The Weyl group

47 1

It remains to be shown that

It follows from the definition that an element Q

E

VPF" is in

(VPF+)WG=l if and only if

This implies that Im&, c (VF*)W,=~. T o prove the converse, consider the principal bundle 9= (G, n, G/T, 2'). Since (G, T)is an equal rank pair, Theorem VII, sec. 11.7, implies that the Weil homomorphism h p is surjective. Thus if X c VF" is any graded subspace satisfying VF" = X @ ker h p , then h p restricts to an isomorphism

X % H(G/T). According to Lemma I, below,

/zp

satisfies

Thus ker h p is WG-stable. Since W, is a finite group, the graded subspace X can be chosen to be stable under the operators ylv(ii), ii E WG (cf. [3; (l.l), p. 31). Let y : H ( G / T )+ VF" be the linear injection given by

y(hgaQ) = 12,

12 E x.

Then y is homogeneous of degree zero, and satisfies the following properties :

(i) y(1)

=

1,

(ii) h p 0 y

= 1,

(iii) y

0

@:

= Y"(ii)0 y ,

ii E

WG.

Now define a linear map

g: H ( G / T ) @ (VE*),=,

+

VF"

by g(a @ Q) = y ( a ) vj;=,(Q). According to sec. 2.9, g is an isomorphism of graded spaces. Moreover, it follows from (iii), above, that

g 0 (@$ @ 1 ) = y l V ( i i )

og.

472

XI. Homogeneous Spaces

Hence g restricts to an isomorphism

But the corollary to Proposition 111, sec. 11.8, states that H(G/T),,=, this isomorphism is exactly ji=o, Q.E.D.

= Ho(G/T);hence

11.10. Lemma I:

The Weil homomorphism h p satisfies

Proof: Fix a E N T . Define a commutative diagram of smooth maps,

GYBG

I. -in N

G/T

G/T,

?a

by setting yu(x) = a-lxa

and

pu(nx) = n(a-lxa) = u-l - n ( x a ) , x E G.

Since left translation of G / T by u-l is homotopic to the identity, q?$ = 0:.

Next, let @ = ( p , 32, G/T, T ) be the pullback of 9to G / T under 4pa. Then there is an isomorphism of principal bundles a : P + G. Define a smooth fibre preserving map &: G -+ P by GU = yu o a-l. This yields the commutative diagram

G/T-G/T-

?a

GI T.

Since yu(xy)= yu(x)yu(y)and a(xy) = a(x)y (x E G, y E T ) , it follows that

A(xr> = $U(X)YU(Y),

x E G, Y E T.

3. The Weyl group

473

Now denote the restriction of lya to T by @,and apply Theorem 11, sec. 6.19, volume 11, to a and Theorem 111, sec. 6.25, volume 11, to to obtain the relation h p o (@’>’= h& = pf; o h p .

Clearly (@’)”= Y”(5). Thus, since fp:

=

@ ,:

the lemma follows.

Q.E.D.

s4. Examples of homogeneous spaces Recall that in article 7, Chapter VI we computed the cohomology of certain compact Lie groups. The results are contained in the following table. (E denotes the Lie algebra of G.) G

SO(2n

+ 1)

SO(2n)

E

Sk(2n

+ 1)

Sk(2n)

basis of PE

so @ 4 p -1

SO

Sf, D4p-19 l < p < n

I

l l p l n

(VE*h=o rank E

V(C?,

. . . , czso ,)

so V(Pf, c?, . . . , C2@)

n

n

In this article we consider homogeneous spaces GIK, where G is one of the groups above, and K is a product of groups, each isomorphic to one of those above. As usual E and F denote the Lie algebras of G and K , and j : F 4E is the inclusion. In each case we shall determine

(1) the Samelson subspace for G / K , and (2) the homomorphism j;=o:(VE*),,, (VF*)O=O. +

Since the ranks of G and K can be read off from the table above, it will be possible for the reader to verify at once that each pair is a Cartan pair. Thus Theorem IV, sec. 11.5, and Theorem V, sec. 11.5, allow us to determine the cohomology of G / K . This information is contained in the tables at the end of this chapter. 11.11. G = U(n). Let V be a complex n-dimensional vector space with Hermitian inner product ( , ). Let V = Vl @ @ V , @ W be a fixed orthogonal decomposition of V into complex subspaces and let +

474

-

4. Examples of homogeneous spaces

dim Vi

= ki, (i =

47s

1, . . . ,q). Define an inclusion map

j,: U(k,) x

* *

x U(k,)

---+

U(n),

by

j u ( o l , . . . , o,)

=

u1 @

. - . 0oq 0c W ,

ui E

U(k,).

Its derivative,

is given by

Examples: 1. U ( n ) / U ( k ) : In this case q directly from the definitions that

jP(@&?j)=

=

1 and k, = k. It follows

@g;,1 5 p 5 k.

This shows that j , is surjective; hence so is j * . Thus (cf. Theorem VI, (2), sec. 11.6) U ( k ) is n.c.2. in U(n). Now Theorem X, ( 5 ) , sec. 10.19, implies that kerjp = f'. A simple degree argument shows that ker j p is spanned by the @%?I with k 1 I p 5 n ; thus

+

f' = (@2Uk+ll

>

@L).

Finally, observe that Proposition I , sec. A.2, yields

2. U ( n ) / ( U ( kx) U(n - k)): I n this case V = V , @ V,, dim V , = k and dim V , = n - k. Let ov be the isometry defined by ov = i in V , and av = - 1 in V,. Then 0% = i and so an involution o in U ( n ) is given by a ( . ) = ovtovl. The fixed point subgroup of o is U ( k )x U(n- k). Thus ( U ( n ) ,U ( k ) x U ( n - k)) is a symmetric pair. Moreover it is clearly an equal rank pair (cf. sec. 11.7 and sec. 10.22), and hence a Cartan pair with P = 0. Finally, note that

(VP),j=o = (V Sk(k; C)*),=, @ (V Sk(n - k ; C)")e=o.

XI. Homogeneous Spaces

476

Hence Proposition I, sec. A.2, gives j;&?pU(n))

=

c

C p )@ C y - k ) ,

qir--p

where C,U = 1 and C,Ocz) = 0 if s > 1.

xi

- -

3. U(n)/U(R,)x * x U(K,): Set K = Ki and consider the inclusion U ( k , ) X * . X U(k,) + U(n), defined above. There is a commutative diagram of smooth maps

-

Now let P and PI denote the Samelson subspaces for the pairs ( U ( n ) , U(k,) x ... x U(k,)) and (U(n), U(K)).The diagram shows that PIc P. On the other hand, by Theorem IV, sec. 11.5, dim P I rank U(n) - rank( U(k,) x

-

x U(k,))

= n - k = dimp,.

Hence, (cf. Example 1, above)

P = P, = (@El,. . . , @2",-,). Finally, observe that

this follows from the obvious generalization of Proposition I, sec. A.2, to direct decompositions into several subspaces. (Note that C,U = 1 and C PUV ( V = 0 if p , > A,.) 4. U(n)/SO(n): Write V = C @ X,where Euclidean space and


ox, p or)=

WX,

Y ),

a, p E

Then an inclusion SO(n) -+ U(n) is given by

X is an n-dimensional

c,

X,

G cf L

YE

0u.

x.

4. Examples of homogeneous spaces

477

Next, consider the (real) linear involution w v of V given by wv(A @ x) =

I@

x,

AE

c, x E x.

It determines the involution w of U ( n ) defined by w(.)

= wvfJwi1,

fJ E

U(n).

The 1-component of the fixed point subgroup for this involution is precisely SO(n). Thus ( U ( n ) ,S O (n )) is a symmetric pair and hence a Cartan pair (cf. sec. 10.26). Moreover, it follows at once from the definitions that

(m')&)(@g-l)

= (-l)p@g-l,

p

=

1,

. . . , n.

Thus Proposition VII, sec. 10.26, shows that P is spanned by the elements @&I (P odd). Finally, observe that

as follows directly from the definitions. U(2m)/Q(m): Consider V as the underlying complex space X c of an m-dimensional quaternionic space X as described in sec. 6.30. In particular, Q(m) c U(2m). Since, by definition, 5.

@&-l

=$=o(@g-l),

1I P 5 m,

it follows that j, is surjective, and so Q(m) is n.c.z. in U(2m). Hence f) = ker j,. A straightforward degree argument shows that ker j, is ( p odd). Thus f) = (@?, @f, @f, . . .). the space spanned by @&l Finally, combining Lemma XI, ( l ) , sec. 6.24, with the definitions in sec. 6.30, we find that

11.12. G = SO(n). Let ( X , ( , )) be an n-dimensional Euclidean space. An orthogonal decomposition of X leads (exactly as in sec. 11.11) to an inclusion

XI. Homogeneous Spaces

478

+

+

Examples: 1. SO(2m 1 ) / S 0 ( 2 k 1): Precisely as in Example 1, sec. 11.11, it follows that SO(2k + 1) is n.c.2. in SO(2m + 1) and that P is spanned by the elements @f"!l ( p = k 1, . . . ,m). Moreover,

+

SO(2m

2.

+ 1 ) / S 0 ( 2 k ) : I n view of the commutative diagram SO(2m + 1)

-

SO(2m + 1 ) / S 0 ( 2 k )

+ 1)

SO(2m + 1 ) / S 0 ( 2 k

+

+

the Samelson subspace f) for the pair (SO(2m l), SO(2k 1)) is contained in the Samelson subspace P for (SO(2m l), SO(2k)). Since

+

dim

5 rank SO(2m + 1) - rank SO(2k) = m - k

= dim

P,

(cf. Theorem IV, sec. l l . S ) , it follows that P = PI. I n particular (cf. Example 1 , above), f) is spanned by the elements @4spo_1 (k 1I p I m). The same formula for as in Example 1 continues to hold here. However, C$?2k) is not a generating element of (VF*),_,, and, in fact, Proposition VI, sec. A.6, yields

+

C g y ) = Pf v Pf. Thus

+

+

3. SO(2m 1 ) / ( S 0 ( 2 k )@ SO(2m - 2k 1 ) ) : Consider an orthogonal decomposition X = Y @ 2, where dim Y = 2k. Define involutions wx of X and co of SO(2m 1) by w x = 1 in Y , wx = - L in 2 and

+

o(o)

=

wxowzl,

0E

SO(2m

+ 1). +

Then the fixed point subgroup of w has SO(2k) x SO(2m - 2k 1) as 1-component, and so the pair is symmetric. Since it is also an equal rank pair, P = 0.

4. Examples of homogeneous spaces

479

Recall from Example 2, above, that C f f ( 2 k= ) Pf v Pf, and set C,"" = 1 and Cf?'" = 0, 2s > 1. Thus Proposition I, sec. A.2, yields

I

+

c

SO(2B) c2*

0c S O ( 2 m - Z k f l ) ,

P
27

4. SO(2m 1 ) / S 0 ( 2 k 1 )x . x S 0 ( 2 k q ) x S 0 ( 2 1 + l ) , C ki 2 1, 1 2 0: Set k = k i . Exactly as in Example 2 above it follows that the Samelson subspace for this pair coincides with the Samelson subspace for ( S O ( 2 m l ) , SO(2k 21 1 ) ) . Hence it is spanned by the elements CD~;)~, k 1 1 5 p 5 m. T h e formula for jg=gis the obvious generalization of the formula in Example 3, above.

xi

+ + +

5.

+ +

SO(2m)/SO(Zk), k < m : First observe that, as in Example 2,

above,

C:t(2k', SO(2m)

) = Pf v Pf, 10,

j:=o(C2p

p < k, p = k, p > k.

Moreover, it follows from Proposition VIII, sec. A.6, that ji=,(Pf)

=

0.

Next recall (cf. sec. 1 1 . 5 ) that the P,-algebra for this pair is given by ((VF"),,,; a), where c = jV,=, 0 t and t:PE 4 (VE*)o=ois a transgression. I n view of formula (6.15), sec. 6.19, and formula (6.17), sec. 6.22, we may choose t so that z(Sf) = I Pf

and

SO(2m) t(@4p-1 )-

I

,

CSO(2m) zp

1 I p < m,

where I and I , are nonzero scalars. It follows that a(Sf)

Thus these vectors are in dim P

SO(2m) a(@4p-l )

and

=0

P.

-

0, k

+ 1 I p <m.

On the other hand,

2 rank SO(2m) - rank S O ( 2 k ) = m - k

and so the elements basis of P.

@$?:m)

(p=k

+ 1, . . . , m - 1 )

and Sf form a

XI. Homogeneous Spaces

480

+

6. S0(2m)/S0(2k 1): In this case Proposition I, sec. A.2, and Proposition VIII, sec. A.6, yield

and jLo(Pf)

= 0.

Thus $Lo is surjective and so (cf. Theorem VI, (7), sec. 11.6) SO(2k 1 ) is n.c.2. in SO(2m). Exactly as in Example 5 it follows that t)is spanned by @fs2:m) (k 1 I p < m) and Sf.

+

+

7. S0(2m)/(S0(2k) x SO(2m - 2k)): As in Example 3, above, this is a symmetric equal rank pair and so P = 0. Proposition I, sec. A.2, and Proposition VIII, sec. A.6, yield

Note that in the first formula Cfo(2z) = 1, C$o(2z) = Pf v Pf, and 0, r > 1.

Cp(21)=

+

8. S0(2m)/(S0(2k 1 ) x SO(2m - 2k - 1)): As in Example 3, this is a symmetric pair (and hence a Cartan pair-cf. sec. 10.26). The involution ox of X reverses orientations, and hence

(w');=,(Pf) = -Pf (cf. Proposition VII, (l), sec. A.6). It follows (because Sf = i2eE(Pf)-cf. (o')$,,(Sf)

sec. 6.22) that

=

-Sf.

Hence, by Proposition VII, sec. 10.26, Sf E But dim P 5 rank SO(2m) - rank(SO(2k and so Sf is a basis of

P.

P.

+ 1) x SO(2m - 2k - 1)) = 1,

481

4. Examples of homogeneous spaces

Finally, as in Example 7,

and jL,(Pf)

= 0,

9. SO(2m)/U(m): Let X denote the underlying 2m-dimensional Euclidean space of a complex m-dimensional Hermitian space V. Then (S0(2m), U(m)) is an equal rank pair, and hence = 0. Moreover, &Lois given by

(11.2) and jXo(Pf) = cg,

(11.3)

where, as usual, C,U = 1 and Cf = 0, q > m. T o see this, observe first that the Hermitian inner product in V determines the R-linear isomorphism a: V V*, given by N

(.x,y>

=

X,Y

( y , x>,

E

V.

Define an isomorphism of complex spaces

e : e g x z vg v*, by setting

e(n g Now let

j(v) by

= (nx,

A+)),

;z E

c,

E

x.

v E Sk(m; C)(= Sk,). T h e n j ( v ) E Sk(2m) (= Skx). Denote L 0y is a complex linear transformation of C 0X ,

y. Then

and, evidently,

e

0

(L

= (p'

0-p*)

It follows that (cf. sec. A.2) C2,(W> = C2dL 0w) = C2,(P, 0 -P"> =

c

fl+r=2v

(-1)'C,(P>Cr(p'>

e.

XI. Homogeneous Spaces

482

whence

This establishes (11.2). Formula (11.3) follows from Example 3, sec. A.7. 10. S0(4k)/Q(k) : Regard X as the underlying 4k-dimensional Euclidean space of a k-dimensional quaternionic space V (cf. sec. 6.30). The corresponding inclusion Q ( k ) + SO(4k) is the composite of the inclusions

Q(k)

-

U(2k)

U(2k)

and

---f

SO(4k)

of Example 5, sec. 11.11, and Example 9, sec. 11.12. It follows that in this case

and

jL,(Pf)

=

c3,

where C,$?= 1 and Cz&,= 0, q > k. This implies that

Hence (cf. Theorem 11, sec. 6.14)

-

where A p is a nonzero scalar. This implies that there is a transgression T : PF (VF"),,,, such that t ( P F )is spanned by the vectors jV,=,(Cf;) (1 I P F k). But, in view of Theorem I, sec. 6.13, t ( P F ) generates (VF"),=,. This shows that j:=, is surjective and so Q ( k ) is n.c.2. in SO(4k) (cf. Theorem VI, sec. 11.6). I n particular, = kerj,. A simple degree argument shows that

4. Examples of homogeneous spaces

483

Moreover, kCSO 2k )

jP@E(ZPf- (- 1 )

=

@,(2jLO(Pf) - (- 1)kjL-oc:;)

= 0,

as follows from the formulae above for ji=o. Now use formula (6.15), sec. 6.19, and formula (6.17), sec. 6.22, to obtain (--1)"Zk-Z(2k - l)!@:J = 0. j,(Sf

+

Since dim P

= k, these

relations show that

(k + 1 C p 5 2k

-

1)

and

Sf

P is spanned by the elements

+ (--l)k22k-2(2k- l)!@f'-l.

Examples: 1. Q ( n ) / Q ( k ) : Exactly as in the unitary case (Example 1, sec. 11.11) it follows that Q ( k ) is n.c.2. in Q ( n ) , that is spanned by the elements (12 1 s p 5 ti) and that 11.13. G = Q(n).

+

2. Q ( n ) / ( Q ( k )x Q(n - A ) ) : As in Example 2, sec. 11.11, this is a symmetric, equal rank pair. Thus P = 0. Moreover, it follows from that example and the definition of C g that

3. Q(n)/U(n): Let ( Y , ( , ), be an n-dimensional Hermitian space Y where and consider the n-dimensional quaternionic space Z = H H is regarded as a complex vector space via multiplication by C on the rkht (cf. sec. 6.30) and the quaternionic inner product is given by

(P 0x, q 0Y ) = p(x, y h q ,

-

P,q E H,x, Y E y.

Thus we have the inclusion U(n) Q(n) defined by c H L 00. Evidently, (Q(n),U ( n ) ) is an equal rank pair and so P = 0. T o determine j:=o recall the inclusion Q(n) + U(2n) in Example 5, sec. 1 1 . 1 1. Let i: Sk(n; H)-+Sk(2n; C) denote the corresponding inclusion of Lie algebras and consider the composite inclusion 1 = i o j : Sk(n; C)--+ Sk(n; H)+ Sk(2n; C).

XI. Homogeneous Spaces

484

Recall from sec. 6.30 that H = @ @ C'L and that CL is stable under multiplication from the left and from the right by C. Let 2, denote the 2n-dimensional complex space underlying 2. Then

Now fix a unit vector j E

Use a to identify

@I

@L,

0,Y with

and define a @-linear isomorphism

Y+; then we have

z,= Y O Y* Moreover, with this identification, I is given by

It follows, as in Example 9, sec. 11.12, that

4. Q(n)/SO(n): Let X be an n-dimensional Euclidean space and set 2 = H @ X. Then the inclusion SO(n) -+ Q(n), given by u H L 0u is the composite

Wn)

-

u(n)

+

Q(n)

4. Examples of homogeneous spaces

485

of the inclusions in Example 4, sec. 11.11, and Example 3, above. Thus

(Here C,"" = 1, C2","= 0 if q > n ; and if n = Zm,C:: may be replaced by Pf v Pf.) Now an easy induction argument shows that Imj:=n is the subalgebra 1 5 2p 5 n. of ( V P ) , = , generated by the elements Cf,: In view of this, the formula above shows that jLo(Cz&,) E (ImjLo)+ . (Imj:=,)+,

Zp > n,

+

whence eE(Cz&,) E P. It follows that @4&,_, E P, n 1 5 Zp 5 2n. Now the standard argument on dimensions shows that these elements span P.

55. Non-Cartan pairs 11.14. The pair (SU(6), SU(3) xSU(3)). Recall from Example 2, sec. 11.11, that U ( 3 )x U ( 3 )is a subgroup of U ( 6 ) .T h e inclusion map restricts to an inclusion S U ( 3 ) xS U ( 3 )+ SU(6). Thus we can consider the homogeneous space

SU( 6)/(S U ( 3 ) x SU( 3 ) ) . Denote the Lie algebras of SU(6) and S U ( 3 ) by E and F. Then, by Theorem X, sec. 6.28,

Moreover, the homomorphism jgZ0:(VE"),,, is given by

+

(VF*),=, @ (VF"),,, (11.4)

(as follows from Example 2, sec. 11.11). Note that on the right-hand side C f u = 1 and Cf" = 0 if q + 0 , 2 , 3 . Now set

@E(C,"")

2 4 P 4 6,

= X2p-1,

and let Q be a graded vector space with homogeneous basis y 4 ,y e ,z, , 6' (subscripts denote degrees). Define a symmetric PE-algebra (VQ; cr) by =

cr(Xzp-l)

c

y2q

q+r=p

"

'zr

,

2I p I 6.

Then formula (11.4) shows that (VQ; cr) is the associated Pa-algebra for the pair ( S U ( 6 ) ,S U ( 3 ) x S U ( 3 ) ) (cf. sec. 11.5). Since cr(X3)

= Y4 cr(XCl)

f

'4

cr(X6)

7

= y4zS

f

= Ye

ySz4 >

486

+

z6

9

cr(X1l)

cr(X7)

= Y6'6

9

= Y4'4

5. Non-Cartan pairs

487

it follows that the essential subspace PIof PE (cf. sec. 2.22) is spanned by x7, x, , and xll. Moreover, the subspace Q1 c Q (cf. sec. 2.23) may be chosen to be the subspace spanned by y 4 and y s . Then the associated essential P,-algebra (VQ1; ul) is given by u1(x7)

=

-$ ,

GI(x9)

=

-2y4y6,

(T1(xll)

==

-$*

It is immediate that

is zero (cf. Proposition IV, sec. 2.13). and so the Samelson subspace Now Theorem X, sec. 2.23, shows that the Samelson subspace for (VQ; G ) is zero. This implies, in turn, via Theorem 11, sec. 11.5, that the Samelson subspace for ( S U ( 6 ) ,S U ( 3 )x S U ( 3 ) )is zero. Since the difference in ranks is 5 - 4 (= I), it follows that this pair is not a Cartan pair. I n particular, the differential algebra ( A ( S U ( 6 ) / ( S U ( 3 )SU(3 x j)), 8) is not c-split. Now we compute the cohomology algebra H ( S U ( 6 ) / ( S U ( 3x) S U ( 3 ) ) ) . Combining Theorem 11, sec. 11.5, with Theorem X, sec. 2.23, yields an isomorphism H(VQ, @ APl) -3 H ( S U ( 6 ) / ( S U ( 3 )x S U ( 3 ) ) ) . Thus we have to determine the algebra H(VQl @ APl). Let a0,a4,and as denote the cohomology classes in H,(VQ, @ APl) represented by the cocycles 1 @ 1 , y4 @ 1, and y s @ 1. Then, evidently these elements form a basis of H,(VQl @ AP1). Next observe that Hp(VQ1 @ AP1) = H p ( S U ( 6 ) / ( S U ( 3 )Sx U ( 3 ) ) )= 0,

p > 19.

Thus a simple degree argument shows that

H2(\/Ql @ AP1) = 0

and

H,(VQl @ AP,) = 0.

Finally, because dim Pl > dim Q1, the corollary to Theorem VIII, sec. 2.19, implies that H(VQl @ APl) has zero Euler-PoincarC characteristic. Since Ho(VQl @ APl) is evenly graded, while H,(VQl @ APl) is oddly graded, it follows that dim Hl(VQl @ AP,) = dim H,(VQl @ AP1).

XI. Homogeneous Spaces

488

Direct computation shows that the cocycles

represent nonzero cohomology classes a I 3 , a 1 5 ,and a19. Thus these classes form a basis of Hl(VQl @ APl). Hence they also form a basis of H+(VQi 0@i). This shows that the PoincarC polynomial for

is

Moreover, the algebra structure is given by (y.o!'.

''

=

i + j # 19, i + j = 19,

{:lg,

(for a < ,ajE H+(VQ, 0APl)). 11.15. The pair (Q(n),SU(n)). Recall from Example 3, sec. 11.13, that U(n)is a subgroup of Q(n) of the same rank n. Thus we can consider the homogeneous space

Q(n)/Sf-J(n). As usual, denote the Lie algebras by E and F and the inclusion by j . According to Theorem X, sec. 6.28, and Theorem XIII, sec. 6.30, (VE+)e=o = V(@,

. . . , C$)

and (VF")e=o = V(C,"", Cf",

. . . , C,"").

Moreover (cf. Example 3, sec. 11.13)~:=~is given by

j;=o(c2Q) =

1

(-1)'C,""

v

c;'",

1 5 p 5 n,

(11.5)

q+r=2p

where we set C,"" = 1 and Cf" = 0 if q = 1 or q > n. Let x4p--l (1 5 p 5 n) be the basis of PE given by xgp-1 = @ E ( c $ ) . Let Q be a graded space with homogeneous basis y 4 ,y6, . . . ,y z n (subscripts denote degrees) and define a PE-algebra (VQ; u ) by 4x4P-1)

=

1

p+7=2p

(-l)ty*qy2r,

p

=

1, . . . ,n.

(11.6)

489

5. Non-Cartan pairs

Then formula (11.5) shows that (VQ ; 0) is the associated P,-algebra of the pair (Q(n), S U ( n ) ) (cf. sec. 11.5). Direct computation shows that if n 5 4,then the Samelson subspace of this PE-algebra has dimension 1. Since rank Q(n) - rank S U ( n ) = 1, it follows that (Q(n), S U ( n ) ) is a Cartan pair if n 5 4. We shall show, however, that for n 2 5 the Samelson subspace is zero, and so (Q(n),SU(n)) is not a Cartan pair in this case. I n particular (A(Q(n)/SU(n)),8 ) is not c-split if n 2 5. We consider first the case n = 5, and then proceed by induction on n. Case I:

n = 5 . Then Q = ( y 4 , y 6 , y 8ylo) , and a is given by

4%) = 2Y4

4 x 7 ) = 2yYs

9

=

'('15)

+ Y: ,

+ +Y!

-2Yf3Y10

= 2y4y, - y;

+11)

a(x19) = -$O*

9

Thus the essential subspace Pl of Ps (cf. sec. 2.22) is spanned by xll, and x19. Moreover, the essential Pl-algebra (VQl; ol) (cf. sec. 2.23) may be chosen so that Q1 = (y6,y10) and

315,

=

gl(xll)

-$ ,

01(x15)

=

-2y6y10

al(x19)

=

-y;O*

As in sec. 11.14, if follows from Proposition IV, sec. 2.13, that the Samelson subspace pl is zero. Now Theorem X, sec. 2.23, shows that

P

= 0.

Case 11: n

pn

zz

2 6. Write PE= P,, Q = Q,, and a = a,. Then Pa-1

0( ~ 4 , - 1 )

and

Qn = Qn-1

0( Y Z ~ ) .

Observe that

as follows from (11.6). This implies that cn(pn-i)

*

V+Qn

c

V+Qn-i

0V(~z,r),

whence On(x4n-1)

@

This shows that x ~ @ ~p,; - i.e., ~

an(pfl-1)

p,

c

*

V+Qn*

Pflp1.

XI. Homogeneous Spaces

490

Next, let v: P, P,-l and y : Q, + Q,-l be the projections determined by the direct decompositions above. Then -+

VQn 0APn

WV

@ 9.5

VQn-10

APn-1

is a commutative diagram of graded differential algebras, as follows from formula (11.6). It follows that

m(pnin)= e - 1 . Now assume by induction that pn-l= 0 (n 2 6). Since pn-l c PnP1 and the restriction of 9 to is injective the relation p(pn)c pn-l implies that pn = 0. This closes the induction. 11.16. The frame bundle over CP2x CP2. Let GIK be any homogeneous space with G, K compact and connected. Denote the corresponding Lie algebras by E and F. Then the adjoint representation of K in E restricts to a representation in F1 (the orthogonal complement of F with respect to the Killing form). Thus we have a homomorphism

Adl: K

-+

SO(FL).

Now consider the inclusion y:K

-+

G x SO(F-L)

given by Y ( Y ) = ( Y , AWY)), Y E K. The corresponding homogeneous space is given by

I t may be identified with the total space of the associated principal bundle of the vector bundle

defined in sec. 5.10, volume 11.

5. Non-Cartan pairs

491

But according to Proposition 111, sec. 5.11, volume 11, E is the tangent bundle of G / K and so (Gx S O ( F l ) ) / Kis the total space of the tangent orthonormal frame bundle for G/K. I n particular, consider the case

G = U ( 3 ) x U(3),

K

=

U(1) x U ( 2 ) x U ( 1 ) x U(2).

Then GIK = C P 2 xCP2. According to Proposition XII, sec. 9.26, the algebra of differential forms on the manifold of frames over C P 2 xCP2 is not c-split. Now Theorem IV, sec. 11.5, implies that the pair

( U ( 3 ) x U ( 3 ) x SO@), U(1) x U ( 2 ) x U ( 1 ) x U ( 2 ) ) is not a Cartan pair.

TABLE I

G

K

=

U(n), E

=

u

u

Sk(n; C),PE = (01, 0 8 ,. . . ,

U(k),k < n

U ( k ) x U(n - k) (0 < k < n )

Basis of P

fIrn

x*

n

fHWK)

l-I

(1

+

tsp-1)

9-k+1

dim Im X x

dim H(G/K)

k!

1

2n-k

if k < n

0

XHWK)

0

k!

All

A,!

if k = n

yes if q = 1

n.c.2.

equal rank

* * *

no

no

if q

>1

no

if k


yes if k

symmetric pair

492

=

n

TABLE I (continued)

Q(m) (n = 2m)

1

+ tn

1

m

(1

+ t") n (1 + P-1

m+l tap-3)

n (1 +

t4p-3)

P-1

2

1

2m+1

2m+1

0

0

no

no

Yes

493

h

-+ x o

h

-+ N

*

h

a

g

0

E VI n,

VI

+

.-I

a

,.

8

0% E 4,

VI VI

+

d

* r(

s8i u

‘a,

9

u

+ d

-

+ r( Y

3

R

SEt

xx e

*;

8

t w

*;

3 h

I

*R u

+

Y CI

R

3

Ee:+ h

9u v

-+

n . I

I

*Ru + 3 v

I

R

EE;

N

-

494

u

0

I 9

E

N I

Y

-.,

E

+L* -

h

0

n

II

E E

v

4

J J N

E

O

v S

n

E

+ 9

N

I

N

9

0

8

0

8

cb

l

a a

h

E E

h

v II J S

sa E

n

V

S z!

0

I

3

E VI a, VI 3

3

$B6 3

E G m 3

I

E n,

VI VI 3

*+ 3

Qk 3

%

E G

v)

I

2

t

Ec

7;

Y

E

c

I v 3

2u

+

r(

Y

i

~

3 h

+

% w 3

h

I

3

E

u

+ v r(

u

I R

3 h

-+ v 3

3 h

Y

I

+

c h

+

1u

. v I

3

R

495

N

3

E

*+ N

0

8

C

0

3

V

n h l

Y

R

A

$ u

+

c

496

0

s s K

rl

n

8

cuE

E

W

h

w

E

II

E m

g

Y

E:

VI a, VI e

+

.4: ,+

8

cub

o

E

VI a, VI &

* + 4,

8

cub

ca, ru

”

h

I

,+

t

W

+

-

h

I

,+

R

4;

R

u

+

n ,+

t + v

4-.

,+

c

R u

n

R

n

+

. I

”

R

4 4

497

*

N

c

+

R

N

C

A? I

c

N

i:

N

4

E 6

%t

O

O

K

S

E

S

E

a

l

l

l

Chapter XI1

Operation of a Lie Algebra Pair $1. Basic properties 12.1. Definition: Let ( E , F ) be a reductive Lie algebra pair (cf. sec. 10.1) and assume that ( E , i, 8, R, S,) is an operation of E. This operation restricts to an operation ( F , i, 8, R , 6,) of F (cf. Example 3, sec. 7.4). T h e corresponding invariant subalgebras of R will be denoted respectively by Re,=o and ReFX0. We shall say that ( E , F, i, 8, R, 8,) is an operation of the pair (E, F ) in the graded dzflerential algebra ( R , 8,) if the inclusion map Re,=, + ReF=,induces an isomorphism H(Re,-o)

--%H ( R O F - o ) *

Given such an operation, we adopt the following notation conventions, to remain in force for the entire chapter:

(i) w R denotes the degree involution of R : wRz = ( - 1 ) p z , z E RP. (ii) T h e horizontal and invariant subalgebras for the underlying operations of E and F are denoted, respectively, by

(iii) The basic subalgebras for the operations are written

(iv) T h e obvious inclusions are denoted by

e E :B E + ReE=,,,

e F :B ,

+

R,,o

and

e: BE -+ BF.

and H(ReF=o) are identified (v) T h e cohomology algebras H(RoE=o) via the isomorphism induced by the inclusion map, and are denoted 498

1. Basic properties

499

simply by H(Reeo). I n particular, we have the commutative diagram

(vi) If the algebra H(R,=o) is connected, then the fibre projections associated with the operations of E and F are denoted respectively by eB:

-

H(ROxo)

(AE"),+o

and

-

&: H(ReSo)

(AF*),=,

(cf. sec. 7.10). (vii) T h e algebras APE, (AE*),=o, and H*(E) (respectively, APF, (AFY;)o,O,and H " ( F ) ) are identified via the isomorphisms X E and x i (respectively, xF and xf) of sec. 5.18 and sec. 5.19. (viii) The inclusion map of F into E is written j : F + E. It induces as described in sec. 10.1. homomorphisms j", j$=,,jv,and T h e basic subalgebra for the operation (F, iF, O F , AE", 6,) is (ix) denoted by (AEIF)(F,O,eF,O and its cohomology algebra is written H ( E / F ) . The inclusion map (AE*)iF=o,eF=o + AE" is denoted by k.

It is the purpose of this chapter to express H(BF) in terms of H ( B E ) and other invariants. 12.2. The associated semisimple operation. Consider an operation

(E, F, i, 8, R, 6,) of a reductive pair (E, F ) . T h e operation of E in (R, 6,) determines the associated semisimple operation ( E , i, 8, R, , 6,) as constructed in sec. 7.5. I n particular, 8 is a semisimple representation of E in R,. Since F is reductive in E, 8 restricts to a semisimple representation of F in R s , as follows from the definition of Rs and Proposition 111, sec.

4.7. Thus the inclusions

induce isomorphisms of cohomology (cf. Proposition I, sec. 7.3). Hence

500

XII. Operation of a Lie Algebra Pair

the same holds for the inclusion (Rs)e,-o -+ (Rs)e,-o, and so ( E , F, i, 6, Rs, 6,) is an operation of the pair (E, F). Finally consider the inclusion map Rs + R as a homomorphism of operations of F. Proposition I: Assume that the operation of F in Rs is regular. Then the inclusion induces an isomorphism

between the cohomology sequences of the two operations. Proof: In view of the corollary to Theorem 111, sec. 9.8, it is sufficient to show that the inclusion (Rs)e,-o -+ReF,, induces an isomorphism of cohomology. But this follows from the observation that ReE,, = (Rs)e,-o and the resulting commutative diagram

12.3. The structure operation. Let (E, F, i, 8, R , 6,) be an operation of a reductive pair (E, F). Recall from sec. 7.7 the definition of the structure operation

(E, iR@E

9

6R@E 9

8 AE",

associated with the operation of E in (R,6,).

dR@B)

1. Basic properties

so1

Proposition 11: The inclusion map

( ( R@ AE")o,=o

9

ROE)

-

( ( R0AE")o,=o

P

sRc3E)

induces an isomorphism of cohomology algebras; i.e., the structure operation is an operation of the pair ( E , F ) . Proof: Define an operator 6 in R @ AE" by setting

In sec. 7.7 we constructed an isomorphism of graded differential algebras @: ( R 0AE", 6)

( R @ AE",

BRBE)

satisfying @ 0 eROE(x)= ORBE(x)0 @,x E E (cf. Proposition V, sec. 7.7). Thus it is sufficient to show that the inclusion map induces an isomorphism

H ( ( R 0AE"),,,,

6) 5 H ( ( R 0AE")op=o,6).

which is induced by the obvious inclusions. In view of Proposition IV, sec. 7.6, the vertical arrows are isomorphisms. Our hypothesis on R implies that the upper horizontal arrow is an isomorphism. Hence so is the lower horizontal arrow. Q.E.D. 12.4. Fibre projection. Let ( E , F, i, 8, R, S,) be an operation of a reductive pair and assume that H(R,,,) is connected. In this section we shall define a homomorphism

to be called the fibre projection for the operation of the pair ( E , F ) .

502

XII. Operation of a Lie Algebra Pair

First, consider the inclusion map g:

0(AE")iF=o,eF=o

[R 0(AE")iF=o]eF=o.

-+

Since

SROE = 6~ 01 (cf. sec. 12.3 and sec. 7.7), and

60

+ +

WR

0SE

og = 0, it follows that

SROEog=go(81(@1+

WROBE).

Hence g induces a homomorphism

Lemma I: The homomorphism g # is an isomorphism. Proof: Filter the differential algebras by the ideals

Fp=

C R0E-o O (AjE")iR=o,~F-o

j2P

and

pp = C [R O (A'E")i,-o]eF-o. j2P

Then g is filtration preserving, and so it induces homomorphisms gi:

(Ei, 4 ) -* (EL , 4

between the corresponding spectral sequences. Now consider the commutative diagram

H { [ RO (AE")i,=oIe,=o,

SR

O11 7 4

where A1 and 1, denote the obvious inclusions (cf. formula 1.8, sec. 1.7). Our hypothesis on R asserts that 1T is an isomorphism. Proposition IV,

1. Basic properties

503

sec. 7.6 (applied with A4 = (AE"),,=, and ,S = 0) shows that A$ is an isomorphism. Hence g, is an isomorphism and so by Theorem I, sec. 1.14, g# is an isomorphism.

Q.E.D. Next, recall the structure homomorphism y R : R + R @ AE" defined in sec. 7.8. Since y R is a homomorphism of operations, it restricts to a homomorphism of graded differential algebras

For the sake of brevity, this map will be denoted simply by y R I p . On the other hand, since H(R0,,,) is connected, we have the canonical projection zR:

H(&=o)

0H ( E / F )+ H(E/F).

Definition: The jibre projection for the operation ( E , F, i, 8, R, SR) is the homomorphism

H(E/F)

P R : H(BF)

given by P R = XR

(g")-'

y$P*

12.5. Projectable operations. An operation ( E , F, i, 0, R,S R ) of a reductive pair ( E , F ) is called projectable if the underlying operation of E is projectable (cf. sec. 7.11). Let (E, F, i, 8, R, SR) be a projectable operation with projection q: R + Then the linear map

r.

q @ 1 : R @ (AE"),,,

+

(AE*)i,=o

induces a map

(cf. the relations above Proposition VII in sec. 7.11). Proposition 111: T h e fibre projection of a projectable operation is given by p R =

(q @ '!)e#,=O

Y&F*

504

XII. Operation of a Lie Algebra Pair

Proof: Consider the restriction qeF-o:ReF,, -+ T.Then ZdR

0L = (q 0L)&O

= q&=o

og#.

The proposition follows.

Q.E.D. Example: Consider the operation (F,i ~ , AE*, d ~ ) where , (E,F ) is a reductive pair. Since 8B7,

(AE*)e-o EZ H((AE*)+o, )6,

EZ H*(E),

it follows that this is an operation of the pair ( E , F ) in (AE", dE). In this case BE = 'I and BF = (AE*)+O,eF=O. Moreover, AEX is connected and so the operation is projectable. Now we show that the fibre projection PMr:

H ( E / F )-+ H ( E / F )

is the identity map. In fact, according to Example 1, sec. 7.8, the structure homomorphism y for AE* satisfies y(@) - (1 0@) E

(A+E*) @ AE',

@ E AE*.

Hence,

(40L)(Y@) = @, where q : AE* + r denotes the projection. Restricting this equation to (AE*)iF-O,BF-o yields

(48' ) e F - o

yAE*/F

= '*

Now pass to cohomology and apply Proposition 111, above. 12.6. Algebraic connections. Suppose ( E , F, i, 8, R, S,) is an operation of a reductive pair, such that the underlying operation of E admits an algebraic connection XE: E* -+ R'. Let X : F* + E* be an algebraic connection for the operation (F, iF, O F , AE*, 6,) (cf. sec. 10.5). Then the linear map Xp=

XEo

X :F"+R'

is an algebraic connection for the operation (F, i, 8, R, dR).

505

1. Basic properties

Definition: T h e triple (X, XE , X,) will be called an algebraic connection for the operation of the pair ( E , F ) .

I n view of Theorem I, sec. 8.4, the algebraic connections XF determine isomorphisms of graded algebras

f

=L

5 AE",

@ X, : (AE")+,o @ AF*

f E =L

@ ( X E ) , : Ri,,o @ AE"

f p =L

@ (XF), : Ri,,, @ AF"

X , X E , and

5R,

and

-

R.

Moreover, fE restricts to an isomorphism

and, clearly, the diagram

RiE..O @ (AE")iF=o@ AF" fE@t

I

$of -

RiE=o@ AE"

=

N

N

RiF,, @ AF"

I

f,

* R

fF

commutes. Thus an isomorphism fE,p

is given by f E , p

=fE

: RiE=O@ (AE")iF=o@ AF" -3R 0 (1

Of) =fB7

0

(fE @ L).

Lemma 11: Let ( X , X,, X,) be an algebraic connection for the operation of ( E , F ) . Then the curvatures of X , X E , and X F are related by

xF(y*) =

zE(

Xy")

+(

X E ) ~ z.Y"), (

y"

F**

XII. Operation of a Lie Algebra Pair

506

If we identify RiEZo@ (AE*)+, 0AF" with R via the isomorphism formula in Lemma I1 reads

fE,F, the

ZF(Y") = z ~ ( X y *@ ) 1 01

+ 1 @ Zy" @ 1,

y*

E

F".

(12.1)

An operation ( E , F , i, 8, R, 6,) is called regular if the pair (E, F ) is reductive, H(Ro=o)is connected, and the operation admits an algebraic connection. Thus an operation of a reductive pair ( E , F ) is regular if and only if the underlying operation of E is regular (cf. sec. 8.21). Proposition IV: Let ( E , F, i, 8, R, 6,) be a projectable regular operation, with algebraic connection (X, Xg , X F ) . Assume @ E (AE")i,=o,e,=o and Q E BF are homogeneous elements of the same degree, such that

(1) and

6E@ = 0

and

6R(Q

+

(XE)A@)

(2) fg!~(Q) E RtE=O@ (AE")i,=o

= 0,

01.

Let E E H(B,) be the class represented by Q represents p R ( & ) .

+ (XE),,@.

Proof: It follows respectively from the definition of corollary to Proposition IV, sec. 8.10, that

fE,F

Then @

and the

YnmQ E (R+0(AE*)i,=o)o,=o and Y R / F ( ( 'E)A@)

Hence, if q : R

--+

-

0@

(R+0(AE*)iF=O)@F=O'

r is the projection

Now apply Proposition 111, sec. 12.5.

Q.E.D. 12.7. The cohomology diagram. Let ( E , F , i, 8, R, 6,) be a regular operation. Then the Weil homomorphisms

-

zg: (VE*)@=o H(B,),

z:

(VP),,,

and

Z*: (VF")o=o -+ H ( E / F )

-

H(B,)

507

1. Basic properties

are defined. These, together with the homomorphisms introduced in sec. 12.1 and sec. 12.4 yield the diagram (VE")O=O

xH

H(BE)

It is called the cohomology diagram of the regular operation (E, F, i, 8 , R, b). The cohomology diagram combines e*

(1) the sequence H(BE) -+ H(B,) ..% H ( E / F ) , (2) the cohomology sequence for the operation of E in R, (3) the cohomology sequence for the operation of F in R, (4) the cohomology sequence for the pair ( E , F ) . In sec. 12.21 it will be shown that the cohomology diagram commutes. 12.8. Homomorphisms. Let ( E , F, i, 8,R, 6,) and ( E , F, i, 8, R, 6 ~ )

denote operations of the pair ( E , F ) . A homomorphism, v: R + R, of operations of E is automatically a homomorphism o f operations of F, and will be called a homomorphism of operations of the pair ( E , F ) . Such a homomorphism restricts to homomorphisms v B E : BE

+

8,

and

pBF

BB*+ B F

Moreover, if (E, F ) is reductive and H(R,=,) is connected, then the diagram

H(BF)

'"

H(&F)

commutes, as follows from the definitions.

508

XII. Operation of a Lie Algebra Pair

Finally, let ( X , XE , XF) be an algebraic connection for the first operation and set iE= pl 0 XB and = pl 0 X F . Then ( X , i E , i F ) is an algebraic connection for the second operation. In particular, a homomorphism between regular operations induces a homomorphism between the corresponding cohomology diagrams.

§2. The cohomology of BF 12.9. Introduction. In this article (E, F ) is a reductive Lie algebra pair, z: PE +. (VP),,, is a transgression (cf. sec. 6.13), and 'V

0 =/,3=o

'G

:P

E

* (VF*),9=0.

Then ((VF*),=,; a) is a PE-algebra, and the cohomology algebra H((VF*),=o @ A P E , --F) is isomorphic to H ( E / F ) (cf. Theorem 111, sec. 10.8). Let (E, F, i, 8, R, S,) be a regular operation with algebraic connection ( X , X E , X p ) (cf. sec. 12.6). Consider the map tR:P E + B E given by zR

=

(xE)v,O=O

t.

Then ( B E ,6,; zR) is a ( P E ,6)-algebra and the cohomology of the corresponding Koszul complex ( B E @ APE, VB) is isomorphic to H(Re=o) (cf. Theorem I, sec. 9.3). Now consider the tensor difference ( B E0(VF"),,,, BR; zR 0a), (cf. sec. 3.7). Its Koszul complex is given by

@ (VF*)i3=0 8

(BE

where

l7 = SR

0

1

@1

9

17),

+ K R - 17,

with O,,(Z

0Y / @ GoA - - - A QP) = (-1)s

c P

t,(@i)

i-0

- X @ p @00

A

A . * -

@i

GP,

A

* * -

and

ro(Z0p @ GoA

c P

= (-1)g

* * *

A

DP)

X @ a(@$)V p @ Q0A

A * * *

@i

* * .

A

@,

i=O

z E B%,

Y E (VF*)e=.O,

@i

E

PE.

12.10. The main theorem. In this section we state the main theorem of this chapter. It contains Theorem I, sec. 9.3, and Theorem 111, sec. 10.8, as special cases. 509

XII. Operation of a Lie Algebra Pair

510

Theorem I: Let ( E , F, i, 0, R, 6,) be a regular operation, with algebraic connection ( X , X E , X,) and let t be a transgression in W(E),=o. Then there are homomorphisms of graded differential algebras y : (BE @ 6R:

(VF9)8=0

(BE 0APE

9

3

G)

and Q,:

((VJ#),=o

-

8APE v )

63 APE, -%)

(BF

9

(Re,=o d ~ ) , 9

((AE8)iF=o,eF=o, d~),

with the following properties : (1) The induced homomorphisms y # , 8; , and p$ are isomorphisms of graded algebras. (2) The isomorphisms y # , t9;, p$, and the isomorphism t,:VPE

'c1

(VB")e=o

determine an isomorphism between the cohomology diagram

of the tensor difference, and the subdiagram

of the cohomology diagram of the operation.

2. The cohomology of BF

511

Remark: T h e rest of this article is devoted to the construction of 6, and y and the proof that 6; and y" are isomorphisms. T h e rest of (l), and (2), will be established in article 3. I n article 4, we shall derive corollaries and less immediate consequences of the theorem. In particular, the reader may skip directly to article 4,without going through the proof of Theorem I. T h e proof of the first part of the theorem is organized as follows. We construct a (noncommutative) diagram

BE

0(VF#),,o

@ APE

PP

BE @ APE

of graded differential algebras, which yields a commutative diagram of cohomology algebras. In particular, the lower half of the cohomology diagram is the commutative diagram in Corollary I , sec. 8.20. T h e difficulty is in the construction of 6 and 8;, this is done in sec. 12.13 and sec. 12.14. The homomorphism y is defined by y = aF 0 6, and the article concludes with the proof that y e is an isomorphism (sec. 12.16). 12.11. The algebra (( VF* 0R)oF=O, DF). We apply the results of secs. 8.17-8.20 to the operation of F in R. T h e antiderivation denoted there by D, will here be denoted by Dp:

DF = I, @ 6r( -

cP . s ( F ) 0

UP),

e

where fie, f, is a pair of dual bases for F" and F. According to sec. 8.17, ((VF" @ R)e,=o,D F ) is a graded differential algebra. Let c F : BF -+ ( V P @ R)eF=Obe the inclusion. I n sec. 8.19 we constructed a homomorphism (here denoted by aP) aF:

((vp"0R)BF=O D P ) 3

-

(BP

9

of graded differential algebras such that aF o cF = L. Moreover, by Theo-

XII. Operation of a Lie Algebra Pair

512

rem IV, sec. 8.17, and Proposition IX, sec. 8.19, a$ and isomorphisms. Finally, let

E$

are inverse

be the obvious inclusion and projection. Then Corollary I of sec. 8.20 yields the commutative diagram

(VJ'")e-o

rg

H((VF" @ R)e,=o)

G

H(Re,-o) (12.3)

12.12. The algebra VB" @ W(E). Consider the operation (E, i, O w , W(E),6,) (cf. Example 6 of sec. 7.4 and Chapter VI). Recall the de-

composition

dw = BE

-

+ 8, + h

and the projection nE:W ( E ) AE". Finally recall the canonical map (cf. sec. 6.7) @ E : (V+E")e=o ---* (A+E")e-o. Extend nE to a homomorphism nE:

VB" @ W ( E )-+ AE"

by setting nE(V+BW @ W ( E ) )= 0. Similarly define homomorphisms nj: VB" @ W ( E )

---f

and

W(E),

j = 1,2,

2. The cohomology of BF

513

T h e homomorphisms nE , ncl, and nz restrict to homomorphisms zE:

(VE" @ W(E)),=o+ (AE*),=,

and nj:(VE" @ W(E)),=,-+ W(E),=,,

j

=

1,2.

Now apply the results of secs. 8.17-8.20 to the operation (El i, O,,, W ( E ) ,dE). Consider the operator

in VE* @ W ( E )(P,e, a pair of dual bases for E" and E ) . According to sec. 8.17, Dw restricts to a differential operator in (VE" @ W(E)),=,. Moreover, combining Theorem IV, sec. 8.17, with Corollary I to Theorem V, sec. 8.20 yields the commutative diagram H((VE"

(Here E and

E

0w q ) o = o )

are the inclusion maps given by E ( Y )= 1 @ Y @ 1,

and

t(Y)= Y @ 1 @ 1

Recall from Theorem I, sec. 12.10 that

t

Y E (VE"),,,.)

denotes a transgression in

W E ) , = ,. Lemma 111: There is a linear map, homogeneous of degree zero, s:

PE

+

(VE" @ W(E)),=,

with the following properties: For @ E P E ,

(1) Dw(s@)= 1 @ T@ @ 1 - t@ @ 1 @ 1. (2) dwnj(s@)= T@ 01, j = 1, 2. (3) n&@) = @.

XII. Operation of a Lie Algebra Pair

514

Proof: The diagram above shows that for @ E P E , 1 @ t @ @l t @ @ 1@

-

1 I m~D l v .

Hence there is a linear map, homogeneous of degree zero, s: P E

(VE" @ w(E))o=o,

such that D,,(s@) = 1 @ t@ @ 1 - t@ @ 1 @ 1,

@E PE.

This establishes (1). A simple calculation shows that in (VE" @ W(E))o=o

n,Dlt-= Swn,

n,DIV= -&a2.

and

Substituting these relations in (1) we obtain (2). Finally, recall from Lemma V I I , sec. 6.13, that @ E 0 t = 1. In view of the definition of @ E it foIlows from (2) that (for @ E p E ) @

@Ex(@)

= n E ( n j ( s @ ) ) = nE($@),

j

=

1, 2.

Q.E.D. 12.13. The homomorphism 8. In this section we define a homomorphism of graded differential algebras

8:( B E0(VF"),,,

@ APE,

r)

+

((VF" @ R)oF=O,Dp)

First, consider the homomorphism

j" @ ( X E ) I ~ VE" : @ W ( E )+ VF" @ R, where ( X E ) I I . is the classifying homomorphism of the algebraic connection X E (cf. sec. 8.16). This homomorphism is F-linear (with respect to the obvious representations of F) and satisfies DF

(i'@ (XE)IIr) = (i'@ ( X E ) W )

DWa

Hence it restricts to a homomorphism

I : ((VE'

0W(E)),,=o, D,,.)

+

((VF"

0R)oF=O,DF)

of graded differential algebras. Lemma 111, (1) shows that

(DF o I ) ( s @ )= 1 @ tB@

-

00 @ 1,

@ E PE.

(12.4)

2. The cohornology of I?,

515

Now extend s to a homomorphism s,: APE

-+

(VEY @ W(E)),,=O,

and define 6 by S(Z 0Y @ @) = (Y @ 2) . I(s,,@),

xE

Y E (VF"),=o, @ E APE.

BE,

Then it follows at once from formula (12.4) that

60v

= D,

0

6,

and so 6 is a homomorphism of graded differential algebras. I n sec. 12.15 it will be shown that 6# is an isomorphism. 12.14. The homomorphism 4,. Let sl: PE + W(E),=obe the linear map given by s1 = 7c1 0 s. I n view of Lemma 111, sec. 12.12, we have drV(sl@) = T@ @ 1 and s ~ @ - 1 @ @ E ( V + P @ AEL),=o,

@ E PE.

Now define a homomorphism 6Rz

BE @ APE

+

Re,=,

by setting GR(Z

0@) = x

@

E BE,

' [(XE)IV(Si)~(@)],

Then GR is the Chevalley homomorphism associated with the algebraic connection XE and the linear map s1 (cf. sec. 9.3). InJarticular, 1 9 ~is a homomorphism of graded differential algebras. Moreover, by Theorem I, sec. 9.3, 6if is an isomorphism. 12.15. Proposition V:

8': H ( B E @ (VJ"),,o

The homomorphism

@ APE, V ) 4H ( ( V F " @ R)e,=o, D F )

induced by 6 is an isomorphism (cf. sec. 12.13). Proof: Filter the algebras BE @ (VF"),=o respectively, by the ideals

FP =

C B E @ (VFY)i=o@ APE

j2 P

and

0APE and ( V P @ R)o,=o, PP =

,x

[(VF')' @ R]ep=O.

3ZP

516

XII. Operation of a Lie Algebra Pair

Then 6 is filtration preserving and so we have an induced homomorphism of spectral sequences

6,:(Ei,di) -+ (Si, di),

i L 0.

In view of the comparison theorem (sec. 1.14) it is sufficient to show that

88: W E 0 , do)

-

f w o9

do)

is an isomorphism. First observe that

(Eo , do) = (BE 0( V F * ) e = o

O A P E , dB 01 06 + ER)

(cf. sec. 12.10), and that

commutes, where 1 is given by

and E ~ e2, are the obvious inclusions. Since 1 is an isomorphism, so is

According to sec. 12.14,

&?:

1.92 is an

isomorphism. By hypothesis

ff(K+=o)

-+

H(Re,-o)

is an isomorphism. Finally, Proposition IV, sec. 7.6, (applied with

2. The cohomology of B,

M

=

VF', 6,

(6

=

517

0, and the Lie algebra F ) shows that

@ E Z ) * : (VP"),=, 0H(R,,=o)

-

H((VF'

0R)B,=O, 1 08,)

is an isomorphism. I t follows that 68 (and hence 8#)is an isomorphism.

Q.E.D. Lemma IV: The diagram (12.5) commutes. Proof: Let

be the homomorphism given by

Then diagram (12.5) commutes (obviously) if 8, is replaced by 8. Hence it is sufficient to show that 6o = ,8; equivalently, we must prove that

6 - 6 : B E @ (VF')i=o @ APE

+

C

( ( V P ) ' @ R)Bp=O.

j>P+l

Since 8 and 6 agree in BE @ (VF*)e=o@ 1, and since both are homomorphisms, it is sufficient to check that for @ E P E

( 8 - 6)(1 @ 1 @ @)

E

(V+F' @ R)B,=o.

(12.6)

But

(8 - @)(I @ 1 @ @)

= =

1 @ ( x E : ) W ( s l @ ) - (i"@ ( x E ) I Y ) ( s @ ' > 1 @ n l S @ - s@).

(Y @ ( x E ) , ) (

In view of the definition of 1 @ n1Q - 52 E V+E" @ W ( E ) ,

SZ E VE' @ W(E).

Hence, in particular,

( p@ ( x E ) , ) (

1@

- $0)

[p@ (xE)fi'](V+E' @ w(E))O~=O c

( V + P @ R)BF=O.

This establishes formula (12.6).

Q.E.D.

XII. Operation of a Lie Algebra Pair

518

12.16. The homomorphism the homomorphisms CZF:(

9.

Recall from sec. 12.11 and sec. 12.13

V P @ R)e,=o -+ Bp

and @; BE

0(VF*),=o 0APE

+

(VF" @ R)e,=o.

Their composite will be denoted by y : (BE@ (VF")o=o @ APE

9

v)

+

( Bp 8,).

I t follows from sec. 12.11 and Proposition V, sec. 12.15, that y # is an isomorphism. Next, consider the diagram (12.2) of sec. 12.10. It is immediate from the definitions and formula (12.6), sec. 12.15, that the upper half commutes. Thus in view of the commutative diagram (12.3) of sec. 12.11 the diagram of cohomology algebras induced by (12.2) is commutative. In particular we obtain the commutative diagram

H(BE @ (VF")e=o @ APE)

2H(BE @ APB) sz 8%

.

(12.7)

$3. Isomorphism of the cohomology diagrams 12.17. T h e purpose of this article is to prove the rest of Theorem I, sec. 12.10. We carry over all the notation developed in article 2. Recall from sec. 12.14 and sec. 12.16 the isomorphisms

8:: H(BE @ APE) -5H(Re,o) and N_

y # : H(BE @ (VF*),,o @ APE)

H(B,).

T o finish the proof of Theorem I, we have to construct a homomorphism of graded differential algebras Q)F:

((VF*)e=o@ A P E , -K)

--t

((AE*)iF=o,eF-o, 6~1,

with the following properties: is an isomorphism. (2) The isomorphisms @, p f , y*, and t, define an isomorphism from the cohomology diagram of the tensor difference to the cohomology diagram of the operation (E, F,i, 8, R,6,). I n other words, the following diagrams commute: (1)

Q)$

( 7 ~ ) :

vL?E

7vlz

(VE*),=o

7v

I

__+

G

H(BE)

,I=

H(BE)

6

H(BE @ APE)

i

r 8;

4

1-

APE (12.8)

(AE*)e=o H(Re=o)7

9

eE'

=

519

520

XII. Operation of a Lie Algebra Pair

N

8;

(12.10)

.

(12.11)

First observe that, in view of the definition of t!lR(sec. 12.14), it follows from Theorem 11, sec. 9.7, that (12.8) commutes. That (12.10) commutes is established in sec. 12.16. Moreover, clearly y o mg = e, and so diagram (12.11) commutes. Finally, in sec. 12.18 we shall construct p F , prove that 93 is an isomorphism, and show that (12.9) commutes. I n sec. 12.19 we show that diagram (12.12) commutes. 12.18. The homomorphism cpF. Define s2: PE -+ W(E),=oby

(cf. sec. 12.12). Lemma I11 of sec. 12.12 implies that for @ E PE: 81ysz(@) = r@ 01

and

s,Q, -

1 0@ E ( V + P @ AE*),,,.

521

3. Isomorphism of the cohomology diagrams

The corresponding Chevalley homomorphism for the pair ( E , F ) (as defined in sec. 10.10) is the homomorphism

8,: (VF"),,, @ APE

+

(VF" @ AE")ep=o,

given by 8F(Y @ @) = (Y @ 1)

. ( j .@ L ) ( ( s ~ ) ~ @ ) , Y E (VF"),=o,

@ E APE.

Recall that (X,X E , X,) is a fixed algebraic connection for the operation of ( E , F ) in R. In sec. 10.9 we constructed a homomorphism a x :(VF"

@ AE*)ep,o

+

(AE*)iF=o,oF=o,

induced by the algebraic connection X. Now set V F = fiB7

0

ax : ((VF"),=o

@ A P E , -K)

-+

((AE")ip=o,oF=o,8 E ) .

It follows from Theorem 111, sec. 10.8 (as proved in sec. 10.11) that cpF is a homomorphism of graded differential algebras, and that q$ is an isomorphism making (12.9) commute. 12.19. Proposition

VI: The diagram (12.12) commutes; i.e., =V$OPif.

PROY*

Proof: Use the algebraic connection (X, X E , X,)

AE"

=

R

(AE"),,=, @ AF",

to write

= RiEZmo @ (AE")i,,,

@ AF"

and BF

= (RiE=O

@ (AE")iF

=O)Bp=O

8

as described in sec. 12.6. Define a homomorphism

$: B E @ (VF"),,o

@ APE

+

(Ri,=o @ (AE")i,=o)e,=o

by setting +(w @ Y @ @) = w @pp(Y @ @).

Now let t E H ( B E @ (VF"),,, @ A P E ) . According to sec. 3.19, can be represented by a cocycle of the form

x = 1@ 0

+ 0,

C

XII. Operation of a Lie Algebra Pair

522

where

0 E (VP),,,

@ APE is a cocycle representing

0 E B i @ (VF"),,,

pfit, and

@ APE.

Next, write y(z) = y(Q)

+ (y

-

00)+ 1 0V P Q .

+)(I

Then (by the definition of y )

Moreover, Lemma V, sec. 11.20, (below) shows that (Y'

-

$)(l @ 1'

LR$E=0

@ (AE")iF=O]oF=O'

Thus, Ijp the operation is projectable, Proposition IV, sec. 12.6 (applied with CD = v F ( 0 ) ) shows that plF(0) represents p R ( y # C ) .Hence

FP(P8C) = p R ( Y ' # C ) , and the proposition is proved in this case. Finally, suppose ( E , F , i, 8, R, 6,) is any regular operation. Let ( E , F , i, 8, R s , 6,) denote the associated semisimple operation (cf. sec. 12.2). Note that

(Rs)O,=, = ROE=,

(Rs)jE=O,OE=O = BE.

and

Since XE(EQ)c R ; , X E may be regarded as an algebraic connection j E for the operation of E in Rs. Set

11,

=fE

0

x.

Next, observe that the inclusion map A: Rs

+R

is a homomorphism

of operations. Thus it restricts to a homomorphism

Since the operation of ( E , F ) in Rs is regular, the construction of articles 2 and 3 may be carried out with Rs replacing R and with (X, f E , replacing (X, X E , XF), but with the same linear map s: PE -+ (VE" @ W(E)),=o. This yields a homomorphism

xF)

@: BE @ (VP*)e=o

0APE

+

(Rs);,=o,e,=o.

3. Isomorphism of the cohomology diagrams

523

denote the fibre projection for the operation of (E, F ) in Rs. Then (cf. sec. 12.8) PR, = P R

~i#,=O,f3p=O.

Moreover, since the representation of E in R, is semisimple, the operation of ( E , F ) in R, is projectable. Thus, by the first part of the proof, P R s O P =

91$

O P B .

It follows that

This completes the proof of Proposition VI and hence the proof of Theorem I.

Q.E.D. 12.20. Lemma

V: Let I be the ideal in BF given by I

=

C ( R t = o@ (A\Eb)ip,O)~F,O.

j 2 Z

Then Im(y

-

3) c I.

Proof: It is sufficient to establish the relations:

(v - $)(z @ 1 @ 1) E I , (y' +)(1 @ Y 01) E I ,

zE

Y E (VP),,,.

(12. 4 )

(v-$)(1@1@@)EI,

@EPE.

(12. 5 )

-

BE.

(12. 3)

It follows from the definitions that y and $ agree in BE @ 1 01, whence formula (12.13). Moreover if Y E(VF"),,,, then

(v-$)(1 @ y @ 1 ) = a F ( y @ l ) -

1 @)a,(p@1)

(12.16)

524

XII. Operation of a Lie Algebra Pair

(cf. sec. 12.11 and sec. 12.18). Further, a p and morphisms aF: VF" --+

a x :VF"

and

Ri,,o @ (AE")i,=,

ax extend

3

to the homo-

(AE")iF-o

given by ap(y") = TF(y+)

a x ( y * ) = T(y"),

and

y*

E

F".

Now formula (12.1), sec. 12.6, shows that

In view of this, relation (12.16) implies formula (12.14). T o prove (12.15) observe that s@

=

c !Pi i

where

xi!Pi@ Qi = s2@ and

1 @ @i

+

521,

SZ, E VE* @ V+BX@ AE". Since

( XE)W: V+B"

-+

C RiB-0, j 22

it follows that 6(1 @ 1 0@) - XPYi @ 1 @ @i E VF" @ i

Next, write Qi = $i

4i E

+

&i,

(AE")iF=o@ 1

1 R!Ez,-,@ AE".

j 22

(12.17)

where and

&i

E

(AEX)iF=o @ AfF".

Then

Thus applying formula (12.17) we find

~ ( @1 1 @ @)

-

Ci [(2p),(j"!P{)]

*

(1 @ di) E I .

Now it follows from formula (12,1), sec. 12.6, that

(12.18)

525

3. Isomorphism of the cohomology diagrams

In view of formula (12.18), this implies that

y(lOIO@)-lOC(~vjY(Yi)).~i€z. i

But

$(I

01 0@) = 1 0[a,(? 0l)(h@)l = 1 @ax(CjYYi0Qi) a

=

1

2gjYi)

*

di,

i

and so (12.15) is established.

Q.E.D.

$4. Applications of the fundamental theorem 12.21. Immediate consequences. Corollary I: The cohomology diagram of a regular operation of a reductive pair commutes. Proof: According to Proposition VI, sec. 3.20, the cohomology diagram of a tensor difference commutes. Thus, in view of Theorem I, (2) we have only to show that the diagram

H(RO=O) QR

-

H " ( E ) 7H " ( F )

commutes. R @ AF" be the structure homomorphism for the operaLet y g : R tion ( F , i, €',R, 8 R ) . Then, it follows from the definitions of y R (sec. 7.8) and /I (sec. 7.7) that the diagram

R

R @ AE"

3R @ AF"

commutes. This yields the commutative diagram

526

527

4. Applications of the fundamental theorem

Corollary 11: There is a spectral sequence converging to H(B,) whose &-term is given by

ESP#l IV = HP ( B E ) @ H 9 ( E / F ) . Proof:

Apply the results of sec. 3.9, recalling from sec. 10.8 that

H ( E / F )GZ H((VF*)O=o@ APE). Q.E.D. Corollary 111: There is a spectral sequence converging to H(B,) whose E,-term is given by

El'' Proof:

(VF*)S=o@ Hq(Re=o).

Apply the results of sec. 3.9, recalling from sec. 9.3 that

H(R8-0) EZ H ( B E @ APE). Q.E.D. Corollary IV: Assume ( B E ,8,) alences

( H ( B E )O

is c-split. Then there are c-equiv-

, E:#,) 7(Re,=o , 8,)

and

(H(BE) @ (VF")O=O@ APE

9

v$

- 0,) 7 ( B F , 8 R ) *

They can be chosen so that the induced cohomology isomorphisms, together with the identity map of H(BE), define an isomorphism from the cohomology diagram of the operation to the diagram

Proof: This follows from the fundamental theorem, together with Proposition XI, sec. 3.29, applied to a c-splitting. (In applying Proposi-

528

XII. Operation of a Lie Algebra Pair

tion XI, copy the example of sec. 3.29, replacing BE @ APE by BE @ (VF*),,o @ APB, as in sec. 3.28.) Q.E.D. 12.22. N.c.z. operations of a pair. Let (E, F, i, 0, R, 8,) be a regular operation. Then (AE*)iF=o,eF=o is called noncohomologous to zero (n.c.z.) in BF if the fibre projection PR: H(BF)

H(E/F)

is surjective. Theorem 11: Let (E, F, i, 0, R, 6,) be a regular operation. Then (AE*)i,=o,e,=o is n.c.2. in BF if and only if there is a linear isomorphism of graded vector spaces

f:H(BE) @ H ( E / F )5 H(BF)

-

such that f(a @ /?) = e#(a) f(1 @ 8) and the diagram

commutes.

WBF)

Proof: Apply Theorem VIII, sec. 3.21.

Q.E.D.

Theorem 111: Let (E, F, i, 0, R, 6,) be a regular operation. Then:

(1) If H ( B E )has finite type, then so does W(BF)and the corresponding PoincarC series satisfy

I~H(BJH(EIF).

~ H ( B F )

Equality holds if and only if (AE*)i,-o,e,=o is n.c.2. in B,. (2) If H ( B E )has finite dimension then so does H(B,) and dim H(BF) 5 dim H ( B E )dim H ( E / F ) . Equality holds if and only if (AE*)iF-O,,F--O is n.c.2. in B p .

4. Applications of the fundamental theorem

529

(3) If H(B,) has finite dimension, then

XH(B,)

=

XH(B~)XH(B/F).

I n particular, the Euler-PoincarC characteristic of H ( B F )is zero unless (E, F ) is an equal rank pair. Proof: Apply Corollaries IV, V, and VI to Theorem VIII, sec. 3.21, together with Theorem XI, sec. 10.22. Q.E.D.

12.23. Algebra isomorphisms. Proposition VII: Let ( E , F, i, 8 , R, S R ) be a regular operation, and assume that (@)+ = 0. Then there are c-equivalences

( B E @ (AE*)ip=O,Bp=O,

8R

@4

+

WR

@

7(BP?

and

( B E @ APE? S R @ L ,

7 ( R f 3 ~ = 0 ,S R ) *

Moreover, these can be chosen so the induced isomorphisms of cohomology make the diagrams

(12.19)

and

530

XII. Operation of a Lie Algebra Pair

Proof Since (Zg)+ = 0, the ( P E , 8)-algebras ( B E , 8,; 0) and ( B E ,8,; (ZE)"0 T ) are equivalent (cf. sec. 3.27). Hence (cf. the corollary to Proposition XI, sec. 3.29) they are c-equivalent. Now it follows from the remarks at the end of sec. 3.28 that there are

c-equivalences

such that the induced isomorphisms of cohomology define an isomorphism between the cohomology diagrams. The proposition follows now from Theorem I, sec. 12.10. Q.E.D. Theorem N : Let ( E , F, i, 8, R, 6,) be a regular operation. Then the following conditions are equivalent :

(1) There is a homomorphism A: (VF*),,, graded algebras, which makes the diagram

-+

H ( B E )0I m Z# of

commute. (2) There is a c-equivalence

such that the induced isomorphism of cohomology makes the diagram (12.19) commute.

4. Applications of the fundamental theorem

531

(3) There is an isomorphism of graded algebras H(BE)

8H ( E I F )

H(BF),

which makes the diagram (12.19) commute. Proof Apply Theorem IX, sec. 3.23, together with Theorem I, (Z), sec. 12.10. Q.E.D.

If ( X $ ) +

Corollary I:

= 0,

then the conditions of the theorem hold.

Corollary 11: If (BE, 6,) and ((AE*)lF=o,oF=o, 6,) are c-split, and if the conditions of the theorem hold, then ( B F ,6,) is c-split. Theorem V: Let (E, F , i, 8, R, 6,) be a regular operation. Assume F is abelian and E is semisimple. Then the following conditions are

equivalent : ( X $ ) + = 0. (2) There is an isomorphism H ( B E ) @ H ( E / F )-% H(B,) of graded algebras such that the diagram (12.19) commutes.

(1)

Proof Since F is abelian, (VF*),=, = VF" and so (VF*),=, is generated by elements of degree 2. Since E is semisimple, Pk = 0 for k < 3. Hence (VE")$=, = 0 for k < 4, and so

jL,(VE"),=, c (V+F'),,,

. (V+F"),=,.

This shows that ((VF*),=,; (T) is an essential symmetric PE-algebra. Now apply the corollary to Proposition VIII, sec. 3.25. Q.E.D. Theorem VI: Let ( E , F , i, 0, R, S,) be a regular operation which satisfies the conditions of Theorem IV. Let z be a transgression in W(E),=,and define a subspace P , of PE by

P,

=

{@ E Px

I je"=,(Z@)

c (V+F*),=,

*

(V+F"),=,}.

Then P , is contained in the Samelson subspace for the operation ( E , i, 8, R,

w.

532

XII. Operation of a Lie Algebra Pair

Proof: Apply Theorem X, sec. 3.26.

Q.E.D. 12.24. Special Cartan pairs. Let (E, F)be a Cartan pair with Samelson subspace P c PE. Let t be a transgression in W(E),,o and consider the linear map *V

(T

=je=o

0

t

-

: PE

(VF"),=o,

(cf. sec. 6.13 and sec. 10.8). The pair (E, F) will be called a special Cartan pair if u ( P ) c (Imji,o)+

Observe that if and so

t'

- (Imji=o)+.

is a second transgression, then

(t -

(12.21)

eE

0

(t- t') = 0

r')(PE) c (V+E")e=o * (V+B")e=o

(cf. Lemma VII, sec. 6.13, and Theorem 11, sec. 6.14). This shows that the condition (12.21) is independent of the choice of t. Clearly, an equal rank pair is a special Cartan pair. Moreover, Theorem X, (4) and ( 5 ) , sec. 10.19, show that if F is n.c.2. in E, then (E, F) is a special Cartan pair. Finally, Proposition VII, sec. 10.26, implies that a symmetric pair is a special Cartan pair. Lemma VI: Let (E, F) be a special Cartan pair. Then there is a transgression t in W(E),,, such that (jL0

0

.)(P)

= 0.

Proof: Let tl be any transgression in W(E),j=o. Then it follows from relation (12.21) that there is a linear map a: P

-+

(V+E*),,o

- (V+P),,

,

homogeneous of degree 1, such that

(]Looa)(@) Write PE

=

P @P

= ( j L o0 tl)(@),

and define

@ E P.

t:PE + (VE")e-o

by

4. Applications of the fundamental theorem

Then Theorem 11, sec. 6.14, yields ee 0 sec. 6.13, z is a transgression. Clearly,

(J-LO

0

t = 1;

533

thus, by Lemma VII,

z ) ( P ) = 0.

Q.E.D.

A transgression satisfying the condition of Lemma VI will be called adapted to the special Cartan pair ( E , F ) . Next, consider the Koszul complex, ((VF"),=, @ APE, -Vo), where a =jV,=, 0 z and z is adapted. Let P be a Samelson complement and let 5 denote the restriction of a to P. Then, since a(P) = 0,

((VF*),=o@ APE, -Vm) = ((VF'),,,

@ AP,

--E) @ (AP, 0).

Moreover, because (El F ) is a Cartan pair, we have

(VF*)o=O= VPF

and

dim P

=

dim P F .

Thus Theorem VII, sec. 2.17 yields

H((Vp*),,o @ A P ) == Ho((VF")~=o @ AP) =

(VF*)o,o/(VF"),,,

o PE g

I m I*,

where

I*: (VF*),=o + H((VF*),=, @ APE) is the .homomorphism induced by the inclusion map. I t follows that

H((VF*),,, @ APE)= Im I# @ AP. Finally, consider the isomorphism

(cf. sec. 12.18). It determines (via the equation above) the isomorphism

yF: AP @ Im I # -ZH ( E / F ) given by

534

XII. Operation of a Lie Algebra Pair

Evidently, the diagram

AP @ I m 1*

commutes (cf. diagram (12.9) in sec. 12.17). 12.25. Operation of a special Cartan pair. Let ( E , F, i, 8, R, 6,) be a regular operation of a special Cartan pair. Assume that the (PE, 6)algebra (BE, 6,; t,) and the P,-algebra ((VF*),=,,; 0) are defined via an adapted transgression t (cf. sec. 12.9). Let p be the Samelson subspace for ( E , F) and let P be a Samelson complement. Observe that VB restricts to a differential operator V’ in BE 0AP. Moreover, the inclusion

i: (BE 0AP, V B ) + (BE 0APE, V’) is a homomorphism of graded differential algebras. On the other hand, since a(P) = 0, the inclusion

iF: (BE @ Ap, VB) + (BE@ (VF*),=,, @ APE’,V ) is also a homomorphism of graded differential algebras. Hence so is i F : (BE

0@, V B )

-

(BF

9

where y is the homomorphism in Theorem I, sec. 12.10. Now let a : Im 1# -+ (VF*),=, be a linear map, homogeneous of degree zero, such that 1# o a = 1. Thenp, 0 1$ 0 a = 1 as follows from the cohomology diagram in sec. 12.7. Theorem VII: Let ( E , F, i, 8, R, 6,) be a regular operation of a special Cartan pair. Then a linear isomorphism of graded spaces

f:H(BE0AP) 0I m 2# -3H ( B F )

4. Applications of the fundamental theorem

535

Moreover, this isomorphism makes the diagrams

H(BF) and

€€(BE 0AP)

,Ig

Im I#

PR

-

H(BE @ AP)

i*

=

W E )

I

€€(BE @ APE) eH

(12.23)

' H(Ro=o)

e;

commute (cf. sec. 12.24 for y F and sec. 12.14 for 82). Proof: T h e commutativity of the diagrams (12.22) and (12.23) is an immediate consequence of the definition off and Theorem I, sec. 12.10. It remains to be shown that f is a linear isomorphism. Identify I m X # with Im Z* via y$ (cf. sec. 12.24 and sec. 12.17). Then a becomes a linear map a : Im I*

--+

(VF*),,,

satisfying I # o w = L. Now define a linear map

g: H(BE @ A P ) 0I m I*

-+

H(BE @ (V8'*),=0

@ APE)

by

g( C

0q ) = i;( C ) . (mf

0

a)(q),

C E H(BE 0AP), q E 1x11 I # ,

(cf. sec. 12.10 for mg). I n view of Theorem I, sec. 12.10, we have

f = y # o g and so it is sufficient to show that g is a linear isomorphism.

536

XII. Operation of a Lie Algebra Pair

Consider the (P, d)-algebra (BE@ AP, t$;TR) and the P-algebra ((VP*),=o; 5 ) given by fR(@) = z R ( @ @ )

1

and

5 ( @ )= o(@),

@E

P.

The Koszul complex of their tensor difference

((BE @ A P ) @ (VF*)e,o

0AP, p)

coincides with the Koszul complex ( B E @ (VP),c,=o @ AP @ AP, V ) under the obvious identification (since a(P) = 0). With this identification, iFbecomes the base inclusion for the tensor difference. Since (VF*),zj,o

=

VPF

and

dim PF = dim P

it follows (as in sec. 12.24) that

H((VP),cj=o@ A P ) = Im T'"

=

I m I#.

Now Corollary I1 to Theorem VIII, sec. 3.21, shows that g is an isomorphism. Q.E.D. Again let (E, F, i, 8, R, 6,) be a regular operation of a special Cartan pair. Let I be the ideal in H ( B E @ AP) @ (VF")e-o generated by the elements of the form f$@@1-1@5@,

@€P,

and let z:

H ( B , @ AP) @ (VF")+o

+

(H(BE @ A P ) @ (VP)+o)/I

be the corresponding projection. Consider the homomorphism y : H(B.q

@ A P ) 0(VF*),,o

+

H(BF)

given by

r(C

y ) = (W

0

iF)*(C)

*

%(yU),

C E H(BE

AP),

E

(VF*)e=o.

Proposition VIII: With the hypotheses above, y factors over z to yield an isomorphism of graded algebras (H(BE

8AP) 0 (VF*)e-o)/I 2H(BF).

4. Applications of the fundamental theorem

537

Proof: This follows from Corollary I11 to Theorem VIII, sec. 3.21, in exactly the same way as Theorem VII followed from Corollary 11. Q.E.D. 12.26. Examples. 1. Equal rank pairs: Let ( E , F, i, 8, R, 6,) be a regular operation of an equal rank pair. Then according to Theorem XI sec. 10.22, X # is surjective. But

p,, X j = P. 0

Thus p R is surjective, and so (AE*)iF=O,BF=O is n.c.2. in B F . Moreover P = 0, and Theorem VII, sec. 12.25, provides a linear isomorphism

H(BE) @ H ( E / F )5 H(B,) (cf. also Theorem 11, sec. 12.22). Note that diagram (12.22) reduces to the diagram of Theorem I1 in this case. O n the other hand, diagram (12.23) yields the commutative diagram

This shows that Imeg

=

Imei.

(12.24)

Finally, Proposition VIII, sec. 12.25, yields an algebra isomorphism (H(BE)

@ (vF*)fJ=O)/I-% H(BP'),

where I denotes the ideal which is generated by the elements of the form t @ D @ l - I @ O @ , GEPE. 2. N.c.z. pairs: Let ( E , F , i , 8 , I?, 6,) be a regular operation of a reductive pair such that F is n.c.2. in E. Then, by Theorem IX, sec. 10.18, (%*)+= 0. Thus the isomorphism f of Theorem VII is given by

f = (W 0 i~)*: H(BE @ A P ) 5 H(B,),

538

XII. Operation of a Lie Algebra Pair

and so in this case f is an isomorphism of graded algebras induced from an isomorphism of graded differential algebras. I n particular,

(BE @ Ap,

PB)

7(BF,

dR)*

Moreover, diagrams (12.22) and (12.23) reduce to the commutative diagrams

H(BE @ AP) L A?'

and

'I-

5%

I

8;

12.27. Lie algebra triples. A reductive Lie algebra triple (L, E, F ) is a sequence of Lie algebras L 2 E 3 F such that

(1) L is reductive. (2) E is reductive in L. (3) F is reductive in E. Note that then F is reductive in L (cf. Proposition 111, sec. 4.7). Moreover, the commutative diagram

4. Applications of the fundamental theorem

539

implies that the vertical arrow is an isomorphism, and so (E, F, iE, AL", 6,) is a regular operation of the pair ( E , F ) . With the terminology of sec. 12.1 we have

BE = (AL*)ig=o,eg=o

and

BF

=

OE,

(ALx)~F=o,~F=o.

Thus, Theorem I, sec. 12.10, expresses H(L/F) in terms of H(L/E) and other invariants. Now assume that (E, F ) is a special Cartan pair, and let z be an adapted transgression in W(E)B=o.Then Theorem VII, sec. 12.25, yields a linear isomorphism

f:H((AL")i,=o,eE=o@ A P ) @ I m 1*"-

H(L/F).

Theorem VIII: Let (L, E, F ) be a reductive triple and assume that ( E , F ) is an equal rank pair. Then

def(L, E ) = def(L, F). In particular, (L, E ) is a Cartan pair if and only if (L, F ) is. Proof:

In view of formula (12.24) in sec. 12.26, the inclusions

satisfy I m kg

=

Im k $ .

It follows that the Samelson subspaces PE and Pp for the pairs (L, E ) and (L, F ) coincide. Since by hypothesis dim PE = dim P F , it follows that def(L, E ) = dim PL - dim PE - dim PE = def(L, F).

Q.E.D. Corollary: Let H be a Cartan subalgebra of E. Then (L, E ) is a Cartan pair if and only if (L, H ) is. Proof: Apply Theorem XII, sec. 10.23.

Q.E.D.

85. Bundles with fibre a homogeneous space 12.28. The cohomology diagram. Let 9= (P, n,B, G) be a principal bundle with compact connected structure group G and assume that B is connected. Let K be a closed connected subgroup of G. Restricting the principal action of G on P to K yields an action of K. Let nl:P --t P / K be the canonical projection. Then P1 = ( P I ,nl, P / K , K ) is again a principal bundle (cf. sec. 5.7, volume 11). Moreover, n factors over nl to yield a smooth map n6:P / K B, and E = ( P / K , n6,B, G / K ) is a fibre bundle. Finally, we have the principal bundle 9K = (G, nK 9 G / K ,K ) . These bundles are combined in the commutative diagram --f

(12.25)

6,

+

PIK

n€

B,

where, for z E P,

(1) A, is the inclusion map a H z . a, a E G. (2) A, is the restriction of A , to K. (3) A, is the induced map. Note that A , , a,,and A, are the fibre inclusions for the bundles 9, pl, and 5'. Observe also that A, is a homomorphism of K-principal bundles. The homomorphism Af: H ( P / K )-+H ( G / K ) is independent of the choice of z in P. In fact, if zo E P and z1E P, choose a smooth path zt connecting zo and zl.Then the maps A,t define a homotopy from Azoto AL1,and thus A: = A$. We denote this common homomorphism by ee: H ( P / K ) H(GIK) +

and call it the jibre projection for the bundle 5. 540

5. Bundles with fibre a homogeneous space

T h e diagram (VE”),=,

541

‘sja

H(B)

(12.26)

is called the cohomology diagram corresponding to the diagram (12.25). Here el. = A,# and ,dP = a,”are the fibre projections for the bundles .P and T h e cohomology diagram commutes. In fact, it was shown in sec. 6.27, volume 11, that the upper square commutes. Since A, is a homomorphism of principal bundles, and since the Weil homomorphism is natural it follows that ec 0 hpl = h p K . Finally, the commutativity of the rest of the diagram follows directly from diagram (12.25).

e.

12.29. Induced operation of a pair. Let E and F denote the Lie algebras of G and K. Then we have the associated operations ( E , i, 8, A ( P ) ,6 ) and ( F , i, 8, A ( P ) ,6 ) (cf. sec. 8.22). Since the principal action of K on P is the restriction to K of the principal action of G, it follows that the second operation is the restriction of the first operation. I n view of Theorem I, sec. 4.3, volume 11, the inclusion maps

-

A(P)@,=, A ( P )

and

A(P),,=,

-+

A(P)

induce isomorphisms of cohomology Thus the inclusion map

induces an isomorphism of cohomology, and so (E, F, i, 8, A ( P ) , 6) is an operation of the pair (E, F ) . Now since G is compact, the pair (E, F) is reductive. Thus the “algebraic” fibre projection

PA,,): H ( ’ ( P ) ~ F = O , O F = O )

+

H(’/F)

512

XII. Operation of a Lie Algebra Pair

is defined (cf. sec. 12.4). On the other hand, isomorphism z:

can be regarded as an

7cf

A ( P / K )5 A(P)ip=O,eF=O ,

while in sec. 11.1 we defined an isomorphism

H ( E / F )-% H(G/K).

E & ~ :

Proposition IX: With the hypotheses and notation above, the diagram

I

aH ( E / F )

H(A(P)iF-o,@F-o) ((7I:)-l)*

=

H(PIK)

2

1

e&K

' H(G/K)

et

commutes. Proof: Consider the operation ( E , F, i, 8, A(G), &), where i(h) = i ( X h )and 8(h) = 8(Xh) ( X , is the left invariant vector field generated by h ) (cf. sec. 7.21). Since the map A,: G -+ P is G-equivariant,

A,*:A(G) t A ( P ) is a homomorphism of operations. On the other hand, in sec. 7.21 we defined a homomorphism of operations E ~ ( :E , F , i, 8 , AE*, S,) -+ ( E , F, i, 8, A(G), 6,) inducing an isomorphism in cohomology . Recall from the example of sec. 12.5 that the fibre projection PhE*

-

:H(E/F)

H(E/F)

is just the identity map. Now the naturality of the algebraic fibre projection gives the commutative diagram

connecting the various fibre projections.

5. Bundles with fibre

a homogeneous space

543

It follows that

12.30. The cohomology of P/K. Theorem IX: Let 9= ( P ,n,B, G ) be a principal bundle with compact connected structure group G and connected base. Let K be a closed connected subgroup of G. Then there are c-equivalences

( A ( P / K ) ,4 7 (W) 0(VF*),=llO APE,

(A(P), 6, 7 ( A ( B )@

v,

VB),

and

( A ( G / W ,a) 7 ((Vp')e=o

0APE -K). 9

The induced isomorphism of cohomology algebras determines an isomorphism from the cohomology diagram of sec. 12.10 to the cohomology diagram (12.26) (with BE replaced by A ( B ) and H(B,) replaced by H ( B ) ) . Proof: In view of the canonical isomorphisms

A ( P / K )-5~ ( ~ ) i p = o , f J p = o and

A ( B ) 2A(P)iE-O,eE-O,

the theorem follows from Theorem I, sec. 12.10, together with diagram 8.22, sec. 8.27, diagram 11.1, sec. 11.4, and Proposition IX, sec. 12.29.

Q.E.D.

E, G / K , is called noncohomologous is surjective. I n this case, Theorem IX,

12.31. N.c.z. fibres. The fibre of

to zero in P / K if the map .oc together with Theorem VIII, sec. 3.21, yield the commutative diagram

(12.27)

544

XII. Operation of a Lie Algebra Pair

where f is an isomorphism of graded vector spaces satisfying f(a

0B ) = (.Pa)

*

f(1 08))

a E H(B),

B E H(G/K).

In particular, if G and K have the same rank, then the map

h p K :(VF")e,o

-+

H(G/K)

is surjective (cf. sec. 11.7). The cohomology diagram (sec. 12.28) shows that in this case GIK is n.c.2. in PIK. On the other hand, we can apply Theorem IX, sec. 3.23, to obtain Theorem X: Let 9= (P, n,B , G), K , be as in Theorem IX. Then the following conditions are equivalent :

( 1 ) There is a homomorphism of graded algebras W: (VFY)e=o

-+

H ( B ) @ I m hpK

which makes the diagram

commute. (2) There is a c-equivalence

(A(B x GIK), 6) 7 A ( P / K ) such that the induced isomorphism of cohomology makes the diagram (12.27) commute. (3) There is an isomorphism of graded algebras

f:H ( B ) 0H ( G / K )-5H ( P / K ) which makes the diagram (12.27) commute.

5 . Bundles with fibre a homogeneous space

545

Finally, as in Theorem V, sec. 12.23, we have Theorem XI: Let 9=(P, n,B, G), K be as in Theorem IX. Assume the Lie algebra of G is semisimple and that K is a torus in G. Then the following conditions are equivalent:

(1) h& = 0. (2) The conditions of Theorem X hold. Example: Let 9= (P, n,B, G) be a principal bundle as in Theorem XI, and assume h& # 0. Let T be a maximal torus in G. Then rank T = rank G. Hence, as we have just seen, there is a linear isomorphism

f:H ( B ) 0H ( G / T )5 H ( P / T ) such that f(a @ B ) = (@a) .f ( l @ B), a c H ( B ) , @ E H ( G / T ) ,and the diagram (12.27) commutes. On the other hand, since h$ # 0, Theorem XI, together with Theorem X, (3), shows that f cannot be an algebra isomorphism.

Appendix A

Characteristic Coefficients and the Pfaffian I n this chapter all vector spaces are defined over a field I' of characteristic zero.

A.O. The algebra of homogeneous functions. Given a vector space + r is caIled homogeneous of degree p if

F a function f:F

f(aX)

= ~ * j ( ~ ) ,E

F,

a E r.

The functions homogeneous of degree p form a vector space, Multiplication of functions makes the direct sum,

Z(F)=

%'p(F).

fZp(F),

p=0

into a graded commutative algebra. Z ( F ) . Since X ( F ) is a comConsider the inclusion map u: F" mutative algebra, u extends to a homomorphism, -+

a : VF"

-+

Z(F),

of graded algebras. For simplicity, we usually denote a(Y)(x) by Y(x). On the other hand, a homomorphism of graded algebras ,!?: OF*--+%(F) is given by @ E OpF", x E F. /?(@)(x)= @(x, . . . , x),

Let zs: OF"

-+

V F X be the projection (cf. sec. 6.17, volume 11); then

(u 0 ns)(x") = X" = ,!?(x")

X" E

F".

It follows that u 0 zs = p.

This shows that

1

Y(x) = -j- Y(x, .

P.

. ,x),

In particular, u is injective. 546

Y E V p F X , x E F.

§l. Characteristic and trace coefficients A.l. The characteristic algebra of a vector space. Let F be an ndimensional vector space. Define bilinear maps,

xi=,

These bilinear maps make the space L A p F into a graded algebra, C(F). It is called the characteristic algebra for F. On the other hand, make the direct sum d ( F ) = &,(ApF* @ ApF) into a commutative and associative algebra by setting (u* @ u )

- (v"

@ v ) = (u"

A

v*) @ (u A v),

u",

v x E AF", u, v E AF. N

Then the canonical linear isomorphisms ApFX@ ApF 5 L A P F define an algebra isomorphism d ( F )2% C(F). In particular, it follows that C ( F )is commutative and associative. Henceforth we shall identify the algebras d ( F )and C ( F )under the isomorphism above. T h e pth power of an element @ E C ( F ) will be denoted by @m,

@ = @...a@. n ( p factors)

In particular, pm:

P.A 1 p p,

p ELF.

More particularly, if I denotes the identity map of F and the identity map of APF, this formula becomes 1m

=p ! l p . 547

lP

denotes

Appendix A. Characteristic Coefficients and the Pfaffian

548

It follows that

Next, recall the substitution operators i ( x ) : AF" -+ AF" and i ( x w ) : AF + AF determined by vectors x E F and x* E F*. They are the unique antiderivations that satisfy

i ( x ) y x = ( y " , x)

and

i ( x " ) y = (x", y ) ,

y"

E

F", y E F.

An algebra homomorphism i : d ( F )+ L d ( F ) is defined by

i(x#l A

- .- A

x*p

@xi A

. - . AX^) = i ( x p )o - . o i ( x l ) @ i(x*p) . . 0

+

0

i(xW1).

With the aid of the identification above we may regard i as a homomorphism i : C ( F )+ L c : ( F ) . Finally, note that the spaces inner product given by

LAPP

are self-dual with respect to the

(@, Y) = tr(@ o Y) = ( t P , @ o Y) = i(@)!P.

It satisfies

(u"

@I(,

V"

@ V ) = (u", v)(v*, u),

u", V" E APF*,

Moreover i ( @ ) is dual to multiplication by @, @ E

U, v E

ApF.

LAPP.

A.2. Characteristic coefficients. The pth characteristic coeficient for an n-dimensional vector space F is the element C$ E VpLg given by Cf = 1 and qX91,* . ' t Q l p ) = t r ( Q l l n - - - 0 v p ) = < ~ p , f p 1*U -*IJY,,),

p 2 1, Ti E L F . Note that C$ = 0 if p > n. Cg will be denoted by DetF. The homogeneous functions, C g , corresponding to C$ are given by

C$(v)= tr Apv, (cf. sec. A.0). We shall show that

9E

LF

1. Characteristic and trace coefficients

549

In particular, det tp =

~

1 Det"(p, . . . ,tp). n!

To prove formula (A.l) we argue as follows. Let e , , basis of F. Then det(q

+ h )el

= (tp

. . A en + h ) e 1 A . A (p + h ) e ,

1..
C An+

The elements eil A writing

we see that el A . It follows that det(q

.-

+

+ h )el

-.

el A

'

A

yeil A

+ .A yei, A . . A e,

i 1< *

p=0

+

a

A

n,

=

. . . , erz be

*

A

A

eip(il < . .

yeil A

.

A

-

< ip) are a basis for APF.Moreover,

q e i pA

. . . A e,, =

.

.

%,...ige1 A

i211"*t

?l

A

. . A en = C I/-0

il<

C
Af::::I;el

A

... A en. . . . A en

a * .

Relation (A.l) is now established. Relation (A.l) implies that C$ E (VPL$)r;i.e.,

or, equivalently

+

;IL (y E L F , -1 not an eigenvalue of y) and comparing Setting u = y the coefficients of A P - l we obtain

550

Appendix A. Characteristic Coefficients and the Pfaffian

The nonhomogeneous element CF E ( V L $ ) I ,given by

CF = p=o

c;,

is called the characteristic element for F. Next, let H be a second finite-dimensional vector space. T h e inclusion map j : LF @ LH + LFaH, given by j ( p @ y ) = g~ @ y , induces a homomorphism

7 :VL; 0vLE

t

VL$$H.

(Recall that multiplication induces a canonical isomorphism

Proposition I: The characteristic elements of F, H , and F are connected by the relation

0H

j.(CF"") = CF 0CH.

shows that

Let a : V(L$ @ L z ) Z ( L F @ L H ) be the homomorphism of sec. A.O. The relation above yields a(jV(CF@H))= a(CF @ C H ) .

Since u is injective, the proposition follows.

Q.E.D. A.3. Trace coefficients. Let F be a finite-dimensional vector space. The trace coejicients of F are the elements Trp" E V p L $ ,given by

Trf and

= dim

F

1. Characreristic and trace coefficients

551

Evidently, TrpF(u o y1 0 a+,

. .. , u

0

plP

0

u-l) = TrpF(pl,

. . . ,p?,),

uE

GL(F),

and

Let H be a second finite-dimensional vector space. Consider the inclusion maps,

A straightforward computation establishes (cf. sec. A.2) Proposition 11: The trace coefficients of F, H , F connected by the relations

j"(Tr,FeH) = Tr," @ 1 and i'(Tr$@'H)=

,

0H , F 0H

are

+ 1 @ Tr,"

C (f ) Tr? @ Tr;.

%fJ=P

Next, consider the commutative algebra,

v""L; =

fi (VPL;),

p=o

whose elements are the infinite sequences @ = (Qo, GI,

. . . , Q p , . . .),

QP

E

VPLf .

Addition is defined componentwise, while the product is given by

(cf. sec. 6.21, volume 11). Clearly V*"L$ contains VL: as a subalgebra.

552

Appendix A. Characteristic Coefficients and the Pffian

The trace series of F is the element TrF E V**L$, given by

. . . ,Tl T r pF , . . . P. Proposition I1 implies that j”(TrFeH)= TrF @ 1

+ 1 @ TrH

and i‘(TrFeH) = TrF @ TrH. Proposition III: The trace and characteristic coefficients of a finitedimensional vector space F are related by

is a derivation, homogeneous of degree 1. I t satisfies

d Trp” = p Tr,F,, and

dC$ = - ( p

+ l)Cp”+,+ Cp”v TrF,

p 2 0.

Proof: A simple calculation yields the formula

This implies that d is a derivation. Clearly d is homogeneous of degree 1. The first formula follows at once from the definition of d. T o establish the second formula note that (cf. sec. A.l)

553

1. Characteristic and trace coefficients

and

‘

j
(Yj Pk

+

Vk

Pj)

PI

* * *

@j ‘ ’ ’

0 @k

* * *

Pp*

Combining these relations yields the second formula.

Q.E.D. Proof of the proposition: T h e proposition is trivial for p = 1. In the general case, it follows by induction via the formulae in the lemma and the derivation property of d.

Q.E.D. Corollary I:

Proof:

Apply the proposition and observe that Cr

=

0, p > n. Q.E.D.

Corollary 11: T h e subalgebras of VL; generated respectively, by C$’, . . . , C$ and by Trc, . . . , Tr;, coincide and contain all the trace coefficients and characteristic coefficients. Q.E.D.

92. Inner product spaces In this article F denotes an n-dimensional vector space and ( , ) denotes an inner product in F. I t induces a linear isomorphism F Z F" which we use to identify F with F*. Further, ( , ) extends to an inner product in each space APF. SkF denotes the Lie subalgebra of LF consisting of the linear transformations which are skew with respect to ( , ). N

A.4. Multiplications in AF @AF. In the vector space AF

0AF we

introduce two algebra structures : the first is the canonical tensor product of the algebras AF and AF; the second is the anticommutative tensor product of AF and AF. The first algebra contains d ( F ) as a subalgebra (cf. sec. A.1) and so its multiplication is denoted by 0:

(u 0).

(% 0.l>

=

( u A U l ) 0.(

A v1).

The second algebra is canonically isomorphic to A(F @ F ) , and so multiplication is denoted by A : ( u @ v ) A ( u , @ z ~ ~ () =- l ) q r ( ~ ~ ~ l ) @ v ~€ A~q ~F ,l ,U ~ E A ' F .

The two products are connected by the relation

CD

Y

=

(-t)qr@ AY,

CD

E

AF @AqF, Y E A'F O A F .

This implies that

0 . . 0v,, = (- l)p'P-"/*v,A . . A V P , a

qj E F @ F.

I n particular,

where i p is regarded as an element of ApF 0APF. Now define an inner product in F 0F by (x

0Y,x1 0Yl) = (x, YJ + (Y,31). 554

(A.2)

2. Inner product spaces

555

(This is not the usual inner product!) Extend it to an inner product in

A(F @ F). T h e induced inner product in AF 0AF (via the standard algebra isomorphism (AF @ AF, A) A(F @ F ) ) is given by (ApF @ AqF, A% @ A'F)

= 0,

unless p

and q = r,

=s

and ( u @ ~ , u @ v ) = ( - ~ ) ~ ~ ( u , v ) ( ~ , uu , )v,E A ~ F , ~ , u E A ' F .

Remark: Up to sign, this inner product agrees with the inner product in C ( F ) defined in sec. A.l.

Next, identify F @ F with ( F @ F)* under the above inner product. Let t:F @ F + F @ F be the linear isomorphism given by

.(.,r) Its dual,

t*, is

=

.(

+r,x -r),

X,Y

E

F.

given by y ) = ( y - X, y

Z*(X,

+

X,

X),

Y

E

F.

and t* extend to algebra automorphisms t, and t Aof (AF @ AF, A) which are dual with respect to the inner product defined above. Observe that t

Lemma II:

(l) (2)

p

t

has the following properties:

@ ).

-y

'A('

t"(LP)

=

2((X ' y ) @

-

@ ('

'.Y))'

=2PlP.

Proof: ( 1 ) is immediate from the formula above as is (2) in the case 1. To obtain (2) in general observe that

=

tA(cp)

=

(-l)p(P-l)/z

- ( -1)p(p-1)/2

1

P!

~

1 ~

P!

t"(' A

. .A

(t"L A

.. . A t " l ) = 2 P J P .

6)

Q.E.D.

556

Appendix A. Characteristic Coefficients and the Pfaffian

AS. Characteristic coefficients for F. Let canonical isomorphism given by

Proposition IV: Let C$(p?)are given by

p? E

P: A 2 F Z Skp

be the

Skp. Then the characteristic coefficients

CP"(p?)= 0,

P

odd,

and

Proof: Let

p? E

det(g,

SkF. Then

+

;It) =

det(p?*

+

At)

= det( -p?

+ At).

It follows that C&+l(p?)= -C$+l(p?), whence C$+,(p?) = 0. T o establish the second formula, regard the inclusion j : Skp + LF as a linear map from Skp into F @ F. Then

Now let ( , ) be the inner product in A F @ A F defined in sec. A.4. Then, for q~ E SkF (cf. Lemma I1 and formula (A.2), sec. A.4),

557

2. Inner product spaces

Next, define elements BkE V2kSk$by

Then, as an immediate consequence of Proposition IV, we have Let j : SkF + L F be the inclusion. Then

Proposition V:

j"( Cg,,)

=0

j.(C g ) = Bk.

and

A.6. Pfafiian. Suppose F has even dimension n = 2m and let a E A"F. Then the Pfafian of the pair ( F , u ) is the element, P f i E VmSkg,given by Pfi'(~1,*

* *

v,)

=

(a,P-'(P~)A

*

. . A P-'(Pm)),

~p

E

S~F.

It determines the homogeneous function Pfz given by

The scalar Pff(V) is called the Pfafian of ~1 with respect to a. We extend the definition to odd-dimensional spaces by setting the Pfaffian equal to zero in this case. Proposition VI: Let a

E

A"F and b E AnF. Then

Pfz v Pff where j : Sk,

+ LF

=

(a,b)j"(Det),

denotes the inclusion. In particular,

(Pff(p))2 = ( a , a ) det 9, Proof: I n fact,

~1 E

SkF.

558

Appendix A. Characteristic Coefficients and the P f f i a n

(since a

AnF and b

E

AnF). This shows that

E

Pff v Pff

=

Now apply Proposition V, with k

( a , b)B,.

= m.

Q.E.D. F be an isometry; i.e.,

t: F -+

Next, let

t y ) = (x, y>,

(tX,

If det

t=

1,

t

o

p1 o

(2) If y

E

t-l,

F.

is called proper.

Proposition VII: (1) If

Pf[(t

x, Y E

. . . , t o e),

t

is an isometry of F, then

o t-')

= det t Pff(q1,

. . . ,e),),

yi E Skp.

S k F , then

2 Pf,F(%,

i-1

*

.

* 9

[w, V i l , ' . , p,) *

Ti E SkF

= 0,

Proof: In fact, since

it follows that

pf[(t

0

p1 o

t-1,

. . . , t 0 e),

0

t-l)

. . , v,),

= det z Pf!(e),,

which establishes (1). Similarly, for y E SkF B(YX

"Y

+x

A

VY) = [Y, b ( X * Y ) l ,

* * 9

Tm) = t r y * pfa(P1,

whence

f pf,~(Vl,* . .

i=l

>

[ y , Vi],

F

9

* * 9

vm) =

0.

Q.E.D. Let H be a second inner product space and give F @ H the induced inner product; i.e., (x

0Y , x1 0Yl> = (x,

Xl>

+ ( Y , Yl>.

2. Inner product spaces

559

T h e inclusion map j : SkF 0SkH--t SkFerI induces a homomorphism

v Sk;

Jv:

@ v Sk;

t

v Sk&H.

Moreover, multiplication defines a canonical algebra isomorphism,

AF @ A H %

A(F 0H ) ,

which preserves the inner products. We shall identify the algebras AF @ AH and A(F @ H ) under this isomorphism.

r

Proposition VIE: Let a E AnF and b E A’H, where n dim H . Then, with the identification above,

f(Pf$$bH)

n

= dim

F and

=

=

PfZ @ Pff.

Proof: If n + r is odd both sides are zero. Now + r = 2k. Then we have, for q E SkF and y E S k H ,

(i’Pf::f)(P

0Y ,

* * * 9

P 0Y ) = ( a 0b, (901

+

assume that

0d k )

If n and r are odd, it follows that

i’Pff&H If n

= 2m

=0 =

Pff

0Pff.

and r = 2s, we obtain

A.7. Examples: 1. Oriented inner product spaces: Let F be a real inner product space of dimension n = 2m (note that we do not require the inner product to be positive definite). Let e E AnF be the unique element which represents the orientation and satisfies 1 (e, e ) 1 = 1. Then Pf,F is called the Pfafian of the oriented inner product space F, and is denoted by PfF. Reversing the orientation changes the sign of the

Appendix A. Characteristic Coefficients and the Pfaffian

560

Pfaffian. Proposition VI implies that det 9 = (e, e)(Pfay)2,

'p E

Ska.

Next let F = F+ @ F- be an orthogonaI decomposition of F such that the restriction of the inner product to F+ (respectively, F - ) is positive (respectively, negative) definite. Define a positive definite inner product ( , ) in F by setting (x'

+

X-,

y+

+ y - ) = (x+, y+) - (x-, y-),

x+, y + E F+,

X-,

y- E F-.

Let cp be a skew linear transformation of F that stabilizes F+ and F-,

v+: F+

9 = v+ @ v-,

-+

F+,

v-:

F-

-+

F-.

Then q~ is skew with respect to both of the inner products ( , ) and ( , ) and so the Pfaffians Pf?, )(v) and Pff, ,(v) are defined. Proposition M: Suppose cp satisfies the conditions above. Then:

(1) If dim F- is odd, Pff, dv) = 0.

Pft", ,(v) = 0,

(2) If dim F- = 2q. Then Pf?. ,(v)

=

(-1Y Pff, dv).

Proof: The corollary to Proposition VIII, sec. A.6, shows that, for suitable orientations of F+ and F-,

Pftp. ,(d = P f ? 3 v + )

- Pf?:,(v-)

and Pf?,

dv) = PfT,(v+)

*

Pf?,-dv-)*

Since ( , ) and ( , ) coincide in F*, it follows that Pft"C,(F+) = PfiP,',(v+) We are thus reduced to the case that ( , ) is negative definite; i.e., F = F- and 9 = v-. In this case, ( , ) = -( , ) and so the linear isomorphisms ,!I(, ) and ,!I(, are related by &,,= -,!I(*)* If dim F is odd, then, by definition Pft", )

= PfF, ) = 0.

2. Inner product spaces

On the other hand, if dim F

=

561

2q, then

Pf?. ,(d = (e, BTt ,(v) * * A BY! ,(d> = ( - 1)*(e, BTt ,(p) A * . A BTf )(V)>

=

( - 1)* PfiF, ,(p). Q.E.D.

2. Oriented Euclidean spaces: Let F be an oriented 2m-dimensional Euclidean space. Fix p E SkF and choose a positive orthonormal basis xl, . . . , xBmof F so that = AiX2i,

4 % - 1 )

and p ( x Z i )= -Aix2i-l,

i = 1, . . . , m.

-

Then PfF(p) = A, . A,. On the other hand, the characteristic coefficients of p are given by

3. Complex spaces: Let F be an m-dimensional complex space with a Hermitian inner product. Orient the underlying real vector space FR as described in Example 2, sec. 9.17, volume 11, and define a positive definite inner product in FR by

<

9

>w = Re( , >.

Then a skew Hermitian linear transformation p of F may be considered as a skew linear transformation pR of FB. We shall show that imPfFR(pR) = det p,

p E Sk,.

In fact, let xl, . . . , z, be an orthonormal basis of F and let A, E scalars, such that ,u = 1, . . . , m. px, = iA,z,,

R be

--

Then det q = imA, A,. On the other hand, the vectors zl, i x , , . . . , z,, ix, form a positive orthonormal basis of F,. Moreover, cpR(z,)= A,(iz,)

and

p R ( i z p )= -A,(x,),

u , =

It follows that (cf. Example 2)

im PfFR(ppl)= iml, - - . A,

= det

q.

1,

. . . , m.

This Page Intentionally Left Blank

Notes In these notes we attempt to give the original sources for the theorems of this volume, as well as some of the recent applications which have been made. There have been two effectively different techniques applied to the study of the cohomology of principal bundles and homogeneous spaces : E. Cartan's method of invariant differential forms, as extended in [53] by H. Cartan, and the classic methods of algebraic topology. The latter techniques were used by Pontrjagin, Ehresmann, Hopf, Samelson, Leray, and Borel and depend in part on the construction of a topological universal bundle which plays a role analogous to that of the Weil algebra in the first method. Both methods frequently produced the same results at the same time, although the topological method often produced results with coefficients 2 or 2,. Nonetheless this book is an exposition of the first method, and the notes will therefore tend to concentrate on proofs achieved that way. For more details the reader is referred to the two excellent survey articles by Samelson [239] and Borel [25], as well as the Springer Lectures Notes [28] by Borel. Finally, we should like to apologize for any omissions or mistakes in crediting discoveries. We should also like to thank J.-L. Koszul, who gave a series of lectures at Toronto, on which note 18 is based.

1. C-equivalence and minimum models. A minimum model is a graded differential algebra (R, d ) such that (i) R is the tensor product of an exterior algebra (over an oddly graded space) with a symmetric algebra (over an evenly graded space) and (ii) d(R) c R + R + . They were introduced by Sullivan [265], who shows that each c-equivalence class of anticommutative graded differential algebras contains exactly one isomorphism class of minimum models. For example, the Koszul complex of the " associated essential PI-algebra " of a P-algebra is the minimum model for the Koszul complex of the P-algebra (cf. sec. 2.23). Sullivan also shows that if the ground field is Q then there is a natural bijection between isomorphism classes of minimum models and rational homotopy types of C. W. complexes (at least in the simply connected case.)

-

563

564

Notes

2. Koszul complexes of P-algebras and P-dSerential algebras. These were introduced by Koszul in [168], and have become an important homological tool in commutative algebra. They are also used by Bore1 in his thesis [19]. The cohomology of the Koszul complex of a (P, 6) algebra is a function now known as diflerential tor introduced by Eilenberg and Moore (cf. [87], [209], and [140]). Differential tor is defined for two graded differential modules over a graded differential algebra ; the notion has been further extended by Stasheff and Halperin (cf. [253], [212]) to “ strongly homotopy associative modules.” Many of the results of Chapter 11, including the use of the PoincarCKoszul series, are either explicit or implicit in Koszul [168] and explicit in AndrC [9]. A major exception is the proof of Theorem VII, sec. 2.17, which is due to J, C. Moore.

3. Spectral sequences. I n chapter IX we introduce the spectral sequence of an operation and show it is isomorphic with the spectral sequence of the associated ( P E ,6)-algebra. If the operation arises from a principal bundle with compact Lie group, this spectral sequence coincides with the classical Leray-Serre sequence. On the other hand, the ( P E ,6)-algebra determines the lower spectral sequence (cf. sec. 3.5); this was introduced by Koszul in [168]. It coincides with the Eilenberg-Moore sequence for the bundle (cf. [87], [249]) in singular theory as has been observed by So. Further, it has been shown (Halperin, unpublished) that this is the same sequence as that arising from the filtration of the invariant differential forms A(P)B= , by the subspaces F-k= a t y €( A

keri(a,,

A

... A

afk).

+E)e=o

4. Cohomology of Lie groups and Lie algebras. E. Cartan [47l showed that biinvariant forms on a connected compact Lie group coincided with the de Rham cohomology of the group; this was used by Brauer [45] to calculate the PoincarC polynomials of the classical groups. Pontrjagin [221], [222], and [223] constructed the primitive cycles in the classical groups, introduced the Pontrjagin multiplication in homology, and showed that the homology of each group was an exterior algebra over the primitive homology classes. He also obtained the PoincarC polynomials for the Grassmann manifolds and some results on torsion. Ehresmann [82] and [84] calculates the homology of the Grassmann manifolds and classical groups using cell complexes. Hopf [1311 introduced finite-dimensional H-spaces and showed their cohomology was an exterior algebra. Then Samelson [236] gave a topological proof of all the main results of Chapter V for compact Lie groups (the cohomology and homology algebras are dual exterior algebras over the

565

Notes

primitive subspaces) for coefficients a field of characteristic zero. Leray [178] and [187] obtained similar results using fibre bundle methods. Koszul [167] approaches and solves all these problems by working only with the Lie algebra; articles 1 through 6 of Chapter V are an exposition of his work. His starting point is the graded differential algebra (AE", 8,) of E. Cartan (see also Chevalley and Eilenberg [68]). It is E. Cartan's theorem [47] that H ( E )= H(G) when E is.the Lie algebra of the compact connected group G. Stiefel [264] uses the action of the Weyl group on a maximal torus to calculate the PoincarC polynomials of the classical groups. T h e PoincarC polynomials for the exceptional groups were first written down by Yen [294] and Chevalley [65J, although they are implicit in Racah [226] ; for proofs see Borel and Chevalley [29]. 5. Weil algebra and transgression. The Weil algebra was invented by Weil (unpublished) and is introduced by H. Cartan in [53], where most of the major results of Chapter VI are announced; this includes the equalities Im p E = PE and ker p E = ( V E x ) : = of Theorem 11, sec. 6.14 (also proved by Leray [187]). These had been conjectured by Weil, and the first one was partially established by Chevalley, using the methods of Koszul [167]. The map pE has reappeared recently in the work of Chern and Simons (cf. [61], [62], and [120]) under the name "transgression map." T h e fundamental notion of transgression is first defined explicitly by Koszul in his thesis [167] (for the operation in ALx of a reductive subalgebra E of a Lie algebra L), although the idea is clearly present in Hirsch [123] and one example is implicit in Chern [56]. By constructing a specific transgression Koszul shows that primitive elements are transgressive. That same construction in the Weil algebra (used by Chevalley to prove Im p E 3 P E )gives rise to the canonical transgression as described in sec. 6.10. T h e notion of transgression plays a major role in Borel's thesis [19], where he uses it to obtain topological proofs of many of the results in this volume, often with coefficients 2, Z p , or Q rather than R . +

6. The structure of ( V E x ) , = and V E". Let T be a Cartan subalgebra of a reductive Lie algebra E. Then there are isomorphisms (1)

V Q c ( V E")e=o

(2)

(2

(VT")w,=i = V Q as shown in Theorem I, sec. 6.13 and Theorem VIII, sec. 11.9. T h e first isomorphism is given by Koszul [168], the second by Borel [19] and Leray [187], and the third independently by Leray [187] and Chevalley [63]. Chevalley's proof uses only the fact that the action of W, in T" is generated by reflections. In many cases these proofs also

566

Notes

give the PoincarC series for ( v E * ) , = in terms of the PoincarC polynomial for H(E). modules) ( V E*), = 8L g V T", where L The isomorphism (of is the regular representation of W,, is also established by Leray [187] and Chevalley [63]. Borel [19] also obtains analogous results for coefficients 2 or 2, under suitable hypotheses; in this case ( v E+)e = must be replaced by H*(B,) ( E the Lie algebra of a compact connected group G with classifying space B,). In the process he obtains an isomorphism H*(B, ; R ) g ( V E *)e = o . This isomorphism is also constructed in [30]. Suppose E is a reductive Lie algebra. As in the case of AE* the elements a E ( V E ), = determine operators is(a) in V E". Set +

V E*i, =

= as(V

n +

ker is(a).

E)B=O

Then Kostant [161] shows that multiplication defines a linear isomorphism ( V E*), = @ ( V E"),, = 5 V E". A major simplification in the proof is given by Johnson [143].

7. Operations and algebraic connections. The notions of an operation of a Lie algebra and of an algebraic connection were introduced by H. Cartan in [53], together with the example of differential forms on a principal bundle. The results of the latter part of Chapter VIII (construction of the classifying homomorphism, construction of the Weil homomorphism, independence of connection) as well as the idea of the proofs are all in [53] ; details can be found in [9]. These ideas derive from the earlier work of Chern [56], who first expressed characteristic classes as polynomials in the curvature; Ehresmann, who introduced the notion of connection in a principal bundle (C.R. Acad. Sci. Paris, 1938, p. 1433-1434); and Koszul [167], who analyzed the special case of the principal bundle associated with a homogeneous space. In the subsequent development of the general theory Weil (unpublished) played a major role; in particular, he constructed the Weil algebra and the Weil homomorphism and proved the latter independent of connection. 8. Cohomology of an operation. T h e fundamental theorem of Chapter IX, which identifies the cohomology of an operation with the cohomology of a Koszul complex, is due to Chevalley (cf. [53]) as an application of the theory of Hirsch-Koszul (exposited in [9]). T h e proof we give is essentially the same. A topological proof for bundles is given by Borel [19]. T h e simpler Koszul complex ( H ( B )0PE, V) for the case A(B) is c-split is announced by Koszul [168] and established by Borel [19].

Notes

567

The specific application to n-connected operations (Theorem IV, sec. 9.8.) is also given by H. Cartan [53]. The definition of the filtration of an operation, its spectral sequence, and the first three terms of the spectral sequence are all given earlier by Kozul in [167] for the operation of a subalgebra E of L in AL"; this is apparently the first time Leray's spectral sequence is interpreted as a sequence of differential spaces ( E i , di) with E,,, g H ( E , ) .

9. Cohomology of homogeneous spaces. Most of the results of Chapters X and XI are due to H. Cartan [53] (although many of the proofs are only given in AndrC [9] or Rashevskii [227]). These include Theorems I1 through V and Theorem VII of Chapter X and the corresponding results in Chapter XI; at least part of Theorem V is apparently also due to Weil. Theorem VI is due to Koszul [168]; it generalizes a conjecture of Hirsch (see below) and is frequently referred to as a "Hirsch formula." Most of the results in article 1 (operation of a subalgebra), article 5 (n.c.2. subalgebras), and article 8 (relative PoincarC duality) in Chapter X are in Koszul [167]. In particular the identification of the algebra ( A E"iF= o, = o , 6,) as the differential algebra of invariant forms on G / K is made by Chevalley and Eilenberg in [68] and is a starting point for Koszul; the cohomology is sometimes called the relative cohomology and written HX(E,F ) rather than our notation of H ( E / F ) . The proofs given in this book in Chapter X are based on the ideas discovered by H. Cartan and Koszul. N.c.2. subgroups were studied earlier by Samelson [236] and by Kudo [173]. The formula for the PoincarC polynomial of H(G/K)(Theorem VI) was originally conjectured by Hirsch for equal rank pairs, and it is established by Leray [I871 in that case. Homogeneous spaces in which the subgroup has the same rank were studied extensively by Leray [185], [187] and Borel [19], who establish all of the results of article 6, Chapter X as well as those in article 3, Chapter XI. Many of the results of article 6 are also at least implicit in H. Cartan [531* Theorem XII, sec. 10.23, is equivalent to dim H ( E ) = Z', 2 the dimension of a Cartan subalgebra, and in this form it was known to E. Cartan, at least for compact Lie groups. As stated, it is proved explicitly by Hopf [131] again for compact groups. The algebraic proof given here is a slight modification of the argument due to Borel [19]. The main idea is to prove that if K c G has the same rank (G, K compact and connected), then H(G/K)is evenly graded ; this result remains true for integer coefficients, as was shown by Bott and Samelson [43].

568

Notes

The example in sec. 11.15 is due to Borel (unpublished). H. Cartan’s main theorem, H(G/K)= H( V F:= @ APE),is also in Borel [19], again for coefficients a field of characteristic zero and with H(B,) replacing ( V F*),= o . The same theorem has recently been established for coefficients or 2, (under suitable homological restrictions on G and K ) independently by Husemoller et al. [140], May [202], Munkholm [213], and Wolf [290] ; partial results had been obtained earlier by Baum [12].

10. Symmetric spaces. If G/K is a symmetric space with G compact and K connected, then H(G/K)= ( AE*),, =,,, = ,,, as was proved by E. Cartan [47]. This enabled Ehresmann, and Iwamoto [141], to calculate the cohomology in certain cases. This same fact is used by Koszul [168] to show that symmetric spaces satisfy the Cartan condition (cf. sec. 10.13). Theorem VIII, sec. 10.17, is a simple generalization of Koszul’s result. 11. The Samelson and reduction theorems. The “ Samelson ” theorem (Theorem IV, sec. 2.13; Theorem IV, sec. 3.13; Theorem I, sec. 7.13; Theorem 111, sec. 7.23; Theorem I, sec. 10.4; Theorem I, sec. 11.2) is in each case effectively the same theorem proved the same way. The original version is due to Samelson [236]; the algebraic form (and proof) which we give is due to Koszul [167]. The reduction theorem (Theorem V, sec. 2.15; Theorem V, sec. 3.15; Theorem 11, sec. 7.14; Theorem 3, sec. 7.23; Theorem IV, sec. 10.12; Theorem 111, sec. 11.5) is again effectively a single theorem. The version in sec. 2.15 is established in AndrC [9] (and hence also those in sec. 10.12 and sec. 11.5). The one in sec. 10.12, however, is announced earlier by H. Cartan in [53]. Dynkin [76] gives a topological proof of the Samelson theorem; Leray [183] and [184] does the same for both theorems. Borel [28] extends these results to actions of a Hopf space. These proofs work in any characteristic with suitable homological restrictions on the group or Hopf space. 12. Operation of a Lie algebra pair. The main theorem of Chapter XI1 (sec. 12.10) was inspired by a theorem of Baum and Smith [13], which is essentially an additive version of Corollary IV, sec. 12.21. A weaker version of part (1) of the main theorem was established independently by Kamber and Tondeur [151]. They also considered special Cartan pairs and obtained many of the results of sec. 12.24. Diagram (12.27) was established by Leray [187] for homogeneous spaces GIK with rank K = rank G. 13. Tensor difference. Suppose {A(&), 6,; Ti} (i= 1, 2) are the (PE, 6) algebras arising from principal bundles Pi 5 Bi . Then the Koszul

Notes

569

complex of the tensor difference is c-equivalent to the algebra of differential forms on P, x GP, if G is compact and connected (Halperin, unpublished); this follows easily from the main theorem of Chapter XI1 given the identification of P, X GP, as a bundle over B , x B, with fibre (G x G)/G.

14. Associated bundles. Let G be a compact connected Lie group with Lie algebra E . The canonical transgression T : P, --f ( v E*), = extends to a canonical linear map r : A P E 4 ( V E *)@ = o , homogeneous of degree 1. Moreover, let (P, T , B, G) be a smooth principal bundle and suppose G acts smoothly on a manifold F. Then H(P x GF) H(A(B) @ A,(F), D), where

D =6 ,

-

Cp(@") @ (a,).

(Here a, is a basis for A P E with dual basis Y", and@" is a closed form representing x # (TYv).) This result (Halperin, unpublished) uses the result on the tensor difference stated above

15. Cohomology of an operation with coefficients in a module. A linear connection in a vector bundle 4 determines a covariant exterior derivative V in the space A(B; 6) of bundle-valued forms over the base (cf. Chapter VII, volume 11). If R is the curvature of V, then V2(@)= R(@) and so (A(B;E)/R(A(B;t)),0)is a chain complex. T h e cohomology of this chain complex was introduced by Vaisman [272], who used a different complex to calculate it. Later it was observed by Halperin and Lehmann [115] that the chain complex described above could be constructed abstractly from the following data: an operation of a Lie algebra E, a finite dimensional representation of E, and an algebraic connection for the operation. Moreover the cohomology contains a characteristic submodule which is a finitely generated submodule over the characteristic subalgebra of H(B). 16. Foliations. Let 5 be an involutive distribution on a manifold n/! and let (P, T , M , GL(q))be the principal bundle associated with the ~ f tangent bundle). Every linear connection V quotient bundle ~ ~( T / the in T ~ / (determines a principal connection and hence an algebraic connection in A(P). Bott [37] shows that for suitable linear connections x,x v(C, > V I E * ) = 0 ( E the Lie algebra of GL(q)).Thus the classifying homomorphism becomes a homomorphism

570

Notes

Moreover, it is clear that the operators O,(x), i(x), and d , induce operators in ( V E"/I) 0 A E". The resulting operation of E is called the truncated WeiE a2gebp.a. T h e fundamental theorem of Chapter IX shows that H { ( V E " / I ) O AE"}rH(W,, V), where W,={V(c,,..., c,)lJ} 0 A(xl, x2, x3, x *, . . . , x,), deg ci= 2i, deg xi = 2i- 1, J is the ideal of deterelements of degree > Zq, and V is given by Vx, = E , . Thus mines a homomorphism H( W, , V) -+ H(P). Now let F be the Lie algebra of O(q). Write

zw

{( v

"/I)0 A E*}O(q)-basic = {( V

E " / I ) OAE"}ip=~ n {( V E*/I) 0 AE+}O
and Then

zw restricts to a homomorphism

A EW}O(q)-basic +A(P)O(qLbaslc Applying the main theorem of Chapter XI1 (slightly modified because O(q) is not connected), we find that H(( VE"/I) @ A E " ) O ( q ) - b s s i c = H(WO,, V), where WO, is the sub-differential algebra of W, given by XO

: {( V E"/I)

WO, = ( v (c19 ' * * C J U ) 0 A (XI, x3 , x5 . . .). (Explicit bases for H(W,) and H(W0,) have been given by Vey; cf. [1191.) On the other hand A ( P ) O ( q ) - b a s i c = A(P/O(q)) and the projection P/O(q)+M induces a cohomology isomorphism. Thus xo induces a homomorphism H( WO,) + H ( M ) . 9

This clearly extends the classic characteristic homomorphism of P ; the new elements are called secondary characteristic classes. The first construction of such a class is due to Godbillon and Vey [loo]; Roussarie [loo] gave an example of a foliation for which the class did not vanish. The main results listed above seem to have been discovered independently by a number of people (see for example Bott [39], Bott-Haefliger [41], Bernstein-Rosenfeld [14], Kamber-Tondeur [15 11).

17. Gelf'and-Fuks cohomology. Let E be the Lie algebra of compactly supported vector fields on a manifold M . Let P ( E ) denote the p-linear skew symmetric functions E x ... x E+ 62, which are continuous with respect to the C" topology of E. Then formula (5.7), sec. 5.3, defines an antiderivation 6, in C"(E); the cohomology algebra H"(E) is called the Gelf'and-Fuks cohomology of M (cf. [95]). They show that if H ( M ) is finite dimensional, then each Hp(E) is finite dimensional.

571

Notes

18. Formal vector fields. An important step in the study of Gelf'and-Fuks cohomology is the study of the Lie algebra of formal vector fields in R q . An account is given in Godbillon [99]. A polynomial vector Jcield in R q of degree i p is a vector field of the form Cp= ff alax,, where each ft is a polynomial of degree s p l(p = -1, 0,. . .); these form a Lie algebra L, . There are obvious projections -+ L, + -+ L,+ +L - 1, and the inverse limit L = lim L, is called the Lie algebra of formal vector fields on Rq. Note that Lo:sthe Lie algebra of the Lie group of affine isomorphisms of R Q . Now consider the induced directed system + A L," +A L; + -+ -.. of graded differential algebras and set {C*(L), S} = 1iT (ALP*, 8). Its cohomology is called the cohomology algebra of L and is written H(L). Next set E = L( R Q ;RQ). Observe that L, = C j g p + l V'( Rq)" 0 Rq,and the inclusions

+

E=(Rq)+@ RQ+L,

are Lie algebra homomorphisms and define a Lie algebra homomorphism of E into L. This defines an operation of E in {C"(L), S}, which admits a natural algebraic connection x. Moreover the homomorphism xw : W(E),,= o+ V fE"). C"(L),,,, maps the ideal ( I @ r\E"),,=, into zero (I Thus xw induces a homomorphism

{( v E"/ I) 0 AE"),,

= 0 --f

C"(L),, = 0 '

It has been shown that the induced homomorphism (cf. note 16) H( W,)-+ H ( L ) is an isomorphism; this is a main step in the proof of the Gelf 'and-Fuks theorem (cf. note 17).

19. Bundles over any space. Let (P, 7r, B, G) be a principal bundle over any topological space B, with G a compact connected Lie group. Then the methods of Chapter IX can be applied as follows to calculate H(P), as has been shown by Watkiss ([283]). Case I : B is a simplicia1 complex. For each simplex a of B, set Po= Then a smooth structure can be assigned to each P, so that the inclusions P, c, f,(when T < u) are smooth. Let A(P)be the subalgebra = a, whenever of nu A(P,) defined by (@,) E A(P) if and only if @,I T
Case I I : 3 is arbitrary. Replace B by the associated singular complex (subdivided twice).

572

Notes

20. The bicomplex of an open cover. Let % = {U,}, I be an open cover of a manifold B. For each ordered ( p 1)-tuple u = (i, ,. . . , i,) (iv E I ) write p = I uI and (i, . . . lv.. . i,) = a, U. Then set U , = U,, and define AS(%,) = II,,,=, A“(U,){note that A(+)= O!}. Set A p * q ( % ) =Aq(%,) and A(%)= Ap.q(%). Operators d and 8 of bidegrees (1, 0) and (0, 1) are defined in A(%)by

+

P+1

=

C (-1)Wav0l

and

(a@),

= (-

0,”

l)p8(@,), @ E

v=o

+

Set D = d 6; then D 2= 0 and H(A, 0) g H(B). (A(%), D ) is called the bicomplex of the open cover. Next observe that a ‘ I piecewise smooth ” space 1% I can be constructed by glueing together the pieces u x U , via the inclusions avu x U,+ avu x U,,,. There is an algebra of piecewise differential forms on I%] (defined as described above in note 19) and a chain equivalence J : {A( % I ), 6} -+{A(%),D } obtained by “ fibre integration over the simplices.’’ Suppose now that B is the base of a principal bundle and that connections cot are given for the restriction of the bundle to each U , . Then a piecewise smooth bundle is determined over % 1, and the w , determine a canonical connection for the corresponding operation (which is constructed as in note 19). Thus we obtain the sequence

I

I

( V ,E*)o=o

yv-B=o,

A( p q ) -5. A(%)’

which provides representatives in the bicomplex for the characteristic classes. T h e composite J 0 x v , e = o was first constructed by Bott [39]. Shulman [247] provides a “universal construction’’ of this composite map in the case that the bundle is trivial over each U , . T h e approach described above is due to Watkiss [283]. Another approach is given by Kamber and Tondeur [150]. They construct a semisimplicial Weil algebra W , and show that H{(Wl)basic} ( V E*)e =,. T h e w , then determine a chain map (Wl)b,,,, +A(%). Given representatives of the characteristic classes in A(%), one would like to write down a Koszul complex {A(%)@ A P E , V) whose cohomology is isomorphic with the cohomology of the total space of the bundle (in analogy with the fundamental theorem of Chapter IX). This problem is completely solved by Watkiss [283]. T h e operator V = V, V1 0, ; V, carries A(%)@ APPEinto A(%)@ I\p-IPE. V, and V, are the operators defined in Chapter 111, while the other V, are needed because A(%)is not anticommutative. These constructions apply generally to the bicomplex of a simplicia1 graded commutative differential algebra, and most of the results of Chap-

=

+

+ +

Notes

573

ter I11 carry over to this case. Applied to bundles over a simplicia1 complex K , it gives an operator in C*(K)@ APE whose cohomology is isomorphic with that of the total space (C*(K)is the algebra of simplicia1 cochains). It is conjectured that this remains true when the coefficients are Z or

z?J.

Finally, an earlier construction of Toledo and Tong (cf. [270] and [270a]), in some ways analogous to that of Watkiss, makes use of local Koszul complexes to prove the Riemann-Roch theorem.

References 1. N. Bourbaki, “J?lbrnents de Mathbmatique, Groupes et Algebres de Lie I,” Hermann, Paris, 1960. 2. H. Cartan and S. Eilenberg, “Homological Algebra,” Princeton Univ. Press, Princeton, New Jersey, 1956. 3. W. Feit, “Characters of Finite Groups,” Benjamin, New York, 1967. 4. W. H. Greub, “Linear Algebra,” 4th edition, Springer-Verlag, Berlin and New York, 1975. 5. W. H. Greub, “Multilinear Algebra,” Springer-Verlag, Berlin and New York, 1967. 6. N. Jacobson, “Lie .4lgebras,” Wiley (Interscience), New York, 1966.

574

Bibliography The reader is also referred to the bibliographies of Volumes I and 11.

7. J. F. Adams, On the cobar construction, Proc. Nut. Acad. Sci. U.S.A.42 (1956), 409-412. 8. J. F. Adams, “ Lectures on Lie Groups,” Benjamin, New York, 1969. 9. M. Andre, Cohomologie des alghbres diffkrentielles bu ophre une alghbre de Lie, Tohoku Math. J. 14 (1962), 263-311. 10. V. I. Arnold, Characteristic class entering in quantization conditions, Funct. Anal. Appl. 1 (1967), 1-13. 11. P. Baum, Cohomology of homogeneous spaces, Thesis, Princeton Univ., Princeton, New Jersey, 1963. 12. P. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15-38. 13. P. F. Baum and L. Smith, The real cohomology of differentiable fibre bundles, Comm. Math. Helv. 42 (1967), 171-179. 14. I. N. Bernstein and B. I. Rosenfeld, Characteristic classes of foliations, Funct. Anal. Appl. 6 (1972), 68-69. 15. A. Borel, Le plan projectif des octaves et les spheres comme espaces homogenes, C . R. Acad. Sci. Paris 230 (1950), 1378-1380. 16. A. Borel, Sur la cohornologie des variktds de Stiefel et de certaines groupes de Lie, C . R . Acad. Sci. Paris 232 (1951), 1628-1630. 17. A. Borel, La transgression dans les espaces fibres principaux, C . R. Acad. Sci. Pards 232 (1951), 2392-2394. 18. A. Borel, Sur la cohornologie des espaces homoghnes des groupes de Lie compacts, C . R. Acad. Sci. Paris 233 (1951), 569-571. 19. A. Borel, Sur la cohomologie des espaces fibrCs principaux et des espaces homoghnes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-207. 20. A. Borel, La cohomologie mod 2 de certains espaces homogenes, Comment. Math. Helv. 27 (1953), 165-197. 21. A. Borel, Les bouts des espaces homogenes de groupes de Lie, Ann. of Math. 58 (1953), 443-457. 22. A. Borel, Sur l’homologie et la cohomologie des groupes de Lie compacts connexes, Amer. J. Math. 76 (1954), 273-342. 23. A. Borel, Kahlerian coset spaces of semi-simple Lie groups, Proc. Nut. Acad. Sci. U.S.A. 40 (1954), 1147-1151. 24. A. Borel, Sur la torsion des groupes de Lie, J. Math. Pures AppZ. 35 (1956), 127-139. 25. A. Borel, Topology of Lie groups and characteristic classes, Bull. Amer. Math. SOC. 61 (1955), 397-432. 26. A. Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tohoku Math. J. 13 (1961), 216-240. 575

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Index Numbers in parentheses refer to pages in Volume 11; e.g., (999). Numbers in brackets refer to pages in Volume I ; e.g., [999].

A Abelian Lie algebra, 158, 182 Action of a Lie group, 278, 307, (109) Adjoint representation, 159, 163, 439, 457, (43) Admissible subspace, 279 Admissible vector, 280 Algebra, 4 differential, 12ff., 45, 332, 343, 363, (10, 155) graded, 4, 159, (3, 4) graded differential, 12, 169, 273, 498, (10) graded filtered differential, 31, 45 isomorphism theorem, 137 P-, 58 Algebraic connection, 314, 415, 504, (298) and principal bundles, Chapter VIII, 314 Alternating graded algebra, 5 Anticommutative graded algebra, 5 Anticommutative tensor product, 5, 554 Antiderivation, 5, 207, 225, 226, 322 Associated cohomology space, 55 essential P-algebra, 92 graded algebra, 45 graded space, 19 P-algebra, 98 semisimple operation, 279, 281, 499

B B-sequence, 106 Base, 31, 37, 95

degree, 32 diagram, 421 inclusion, 96 space, 36, (15), [38] Basic factor, 295, 296 Basic subalgebra, 276, (241) Betti numbers, 179, 203, 376, (19), [178, 2051 Bidegree, 3, 55 Bigradation, 3, 55, (17) Bigraded differential space, 33 Bigraded vector space, 3 Bundle frame, 402, 405, (194, 220, 404) principal, 352, 390, 397, (193, 403) Bundle diagram, 418 Bundle with fibre a homogeneous space, 540

C c-connected, c-equivalent, c-related, c-split, see Cohomologically Canonical map, 231 Canonical tensor product, 5, 554, (4) Cartan map, 232 Cartan pair, 431, 448, 532 Cartan subalgebra, 163 Centre of a Lie algebra, 158, (69, 96) Characteristic algebra, 547 coefficients, 546ff., 556 element, 550 factor, 427 homomorphism, 442, (372, 391, 400) subalgebra, 341, 427, (265, 391, 426)

587

588

Index

zero, field of, 9 Chevalley homomorphism, 363ff., 423ff., 515, 521 Classification Theorem, 382, 387 Classifying homomorphism, 341 Closed symplectic metric, 454 Coboundary space, 10 Cocycle space, 10 Cohomologically (c-) connected, 13, 97, 115 equivalent, 14, 147, 387 related, 14, 148, 387 split, 15, 151, 391, 435 Cohomology algebra, 96, 179, 543, (11, 19, 155), [I761 associated with a P-space, 55 of the classical Lie algebras, 253, 258 classification theorem, 382, 387 with coefficients in a graded Lie module, 210 diagram, 126, 128, 129, 506, 519, 526, 540 invariant, 213 of Lie algebras, 179, 543, (155) of Lie groups, 174ff., 474 of operations and principal bundles, Chapter IX, 359 for principal bundle, 352, 390, 459 sequence, 72, 115, 350, 357, 371, 417, 460 space, 10, 169 structure homomorphism, 289, 297 of the tensor difference, 109, 126, 129, 135 of the Weil algebra, 228 Collapse of a spectral sequence, 25 Compact Lie algebra, 162 Compact Lie group, 222, 264, 457, 459, (48, 149) Comparison theorem, 39 Compatible filtration, 27, 31 Complexification of a real Lie algebra, 264 Complex spaces, 561 Comultiplication, 199, 219, 291 Conjugate of a quaternion, 270 Connecting homomorphism, 11, (lo), D811 Connection algebraic, 314, 415, 504, (298) form, 352, (250)

invariant, 459, (280) Contravariant Koszul formula, 177 Convergence of spectral sequences, 35 Covariant derivative, 322, 354, (355) Curvature, 323, 354, 416, (257)

D Decomposable curvature, 402 Decomposition theorem, 83, 118 Decreasing filtration, 19 Deficiency number, 431 Degree, 33 of homogeneous functions, 546 involution, 3 of root, 443 total, 33 Derivation, 4, (3) Derived algebra, 158, (96) Diagonal map, 183 Differential algebras, 12ff., 45, 243, 263, 332, (10, 155) filtered, 45, 241 graded, llff., 169, 273, 498, (10, 155) graded filtered, 45 Differential couple, 48, 51 Differential operator, 10, 48, 96, (9), P331 Differential spaces, 10, 19, 31, 169, (9, 41) Direct sum of Lie algebras, 182 of P-spaces, 57 Distinguished transgression, 239, 378 Double complex, 52 Dual basis, 1 Dual graded spaces, 3 Duality theorems, 206, 249 Dual pair of vector spaces, 1

E E-linear, E-stable, 158 Equal rank pair, 442,492ff., 537 Equivalent basic factors, 296 Equivalent P-differential algebras, 147 Essential P-algebra, 90, 141 Essential subspace, 90 Exact sequence of P-spaces, 54 Euclidean space, oriented, 561

Index Euler class, 401, (23, 477), [316, 334, 3911 Euler-PoincarC characteristic, 3, 43, 67, 125, 134, 401, 434, 492ff., 529, (19, 182), [178, 186, 205, 3911 Euler-PoincarC formula, 11 Evenly graded space, 3, 8 Even root, 443 Exterior algebra, 6, 10

F Factor algebra, 157 Factor space, 54 Faithful representation, 159 Fibre degree, 33 Fibre a homogeneous space, 540 Fibre integral, 394, (22, 83, 242), [300] Fibre n.c.z., 392, 543 Fibre projection, 292, 310, 316, 357, 413, 501, 503, 540 Filtered differential algebra, 45, 241 Filtered differential space, 20, 31 Filtration, 19, 45, 241 compatible with gradation, 27, 31 induced by gradation, 27 induced by operation, 359, 361 spectral sequence of, 25 Finite type gradation, 3 Frame bundle, 402, 405, (194, 220, 404) Fundamental vector field, 352, (121)

G Generators, 4 Gradation, 3, 13, 20, 55 Graded algebra, 4, 159, 213, 300 Graded differential algebra, 12, 169, 273, 498, (10) couple, 51 space, 11, 169, (10) Graded filtered differential algebra, 45 space, 31 Graded space, 3, 19, 20, 159, 300 Gysin sequence/triangle, 101, [320, 2821

H Homogeneous differential couple, 50 Homogeneous element, 3, 55

589

Homogeneous function, 546 Homogeneous linear map, 3 Homogeneous spaces, Chapter XI, 457, 474, 540, (77ff., 205) Homology of a Lie algebra, 179 Homomorphism of algebras, 4 characteristic, 442, (372, 391, 400) Chevalley, 363ff., 423ff., 515, 521 classifying, 341 cohomology structure, 289, 297 connecting, 11, (lo), [181] of differential couples, 50 of differential spaces, 11 of filtered spaces, 19, 23 of graded algebras, 5 of graded differential algebras, 13, 228, 515 of graded differential couples, 52 of graded filtered differential algebras, 47 of graded filtered differential spaces, 38 of graded filtered spaces, 32 of graded P-algebras, 59 of graded P-spaces, 54 of Lie algebras, 180 n-regular, 13, 39 of operations, 274, 327, 370, 507 of P-spaces, 54, 57 of spectral sequences, 25 structure, 288, 297, 328 Horizontal projection, 322, 353, (253, 299) Horizontal subalgebra, 276 1

Ideal in an algebra, 4, 45 Ideal in a Lie algebra, 157 Induced gradation, 3, 7, 8, 20 Induced isomorphism, 149 Induced operation, 382, 541 Induced representation, 169 Inner product space, 2, 554 oriented, 559 Invariant cohomology, 213 connection, 459, (280) operation, 308 subalgebra, 276, 308

590

Index

subspace, 158, 204 vector, 158 Involution, 166 degree, 3 Isometry, proper, 558 Isomorphism induced by c-equivalence, 149

J Jacobi identity, 157, 160

K Killing form, 160, 161, 166, 272, (97, 186) Koszul complex, 54, 390 for the pair, 420 of P-differential algebras, Chapter 111, 96 of P-spaces and P-algebras, Chapter 11, 59 Koszul formula, 177, 185, (184) Kunneth formula, 61 Kunneth isomorphism, 12, (11)

L Left invariant operation, 308 Left regular representation, 469 Levi decomposition, 440 Lie algebra, (4) abelian, 158, 182 cohomology of, 179, 543, (155) compact, 162 connected, 218 derived algebra of, 158, (96) and differential space, Chapter IV, 157 direct sum of, 182 n.c.z., 436 operation of, 273, 307 pair, 411, 417, 420, 422, 427, 453, 455, 457, 460, 498 real, 264 reductive, 162, 186, 188, 213, 239, 245, 457, (164) representation of, 158 semisimple/simple, 161 Sk(n), 256, 269, 474, 493 S d m ) , 260 triple, 538 unimodular, 185

Lie group, (24) action of, 278, 307, (109) cohomology of, 174fF., 474 compact, 222, 264, 457, 474, (48, 149) connected, 457, 459 n.c.z., 392 Q(n), 270, 474, 483, 493 SO(n), 269, 397, 398, 474, 477, 493, 494 SU(n), 268, 474 U ( m ) , 399, 474, 492 Linear connection, 402, (318ff,, 353) Linear groups, 265ff., (70ff.) see also specific types under Lie group Lower gradation, 55 Lower spectral sequence, 99

M Manifolds with vanishing Pontrjagin class, 40 1 Maximal torus, 222, 466, 469, (87ff.) Multiplication operator, 7, (6), [209]

N n.c.z., see Noncohomologous to zero n-regular, 13, 39 Nilpotent, 163 Non-Cartan pair, 486 Noncohomologous to zero, 121, 129 fibres, 392, 543 operations, 376, 377, 528 pairs, 436, 537 subalgebras, 436 subgroups, 465, 492ff. 0 Oddly graded space, 3 Odd root, 443 Operation(s) associated with a pair, 413 with the principal bundle, 352 semisimple, 279, 281, 499 classifying homomorphism of, 341 cohornoiogicaily equivalentlrelated, 387 cohomology classification theorem for, 382, 387 differential algebra associated with, 363 fibre projection for, 292

591

Index filtration induced by, 359, 361 geometric definition of, 331 of a graded vector space, 300 homomorphism of, 274, 327, 370, 507 induced, 382, 541 left invariant, 308 of a Lie algebra, Chapter VII, 273, 307 of a Lie algebra pair, Chapter VII, 498 n.c.z., 376, 377, 528 projectable, 292, 503, 522 regular, 350, 376, 377, 506 restriction of, 278 of a special Cartan pair, 534 spectral sequence of, 361 structure, 284 tensor product, 278, 281 in the Weil algebra, 279 Oriented Euclidean space, 561 Oriented inner product space, 559 Orthogonal complement, 1

P P-algebra, 58 associated with a pair, 420 cohomology sequence of, 72, 115 essential, 90, 141 symmetric, 78 P-differential algebra, 95 equivalent, 147 n.c.z., 121 spectral sequence of, 99 P-homomorphism, P-linear map, P-space, P-subspace, P-quotient space, 53, 54, 55, 57 Pair Cartan, 431, 448, 532ff. cohomology sequence for, 460 c-split, 435 equal rank, 442, 492ff., 537 n.c.z., 436, 537 operation of, 498ff., 507, 541 Pfaffian of, 557 special Cartan, 532ff. split, 435 symmetric, 447, 492ff. Pairs of Lie algebras, 411, 417, 420, 422, 427, 453, 455, 457 Permanent, 9, (5) Pfaffian, 257, 546, 557, 559

Poincarb duality, 186, (20), [194, 201, 2491 algebras, 9 relative, 450 Poincarb inner product, 186 PoincarC isomorphism, 10, 186, (20) relative, 451 Poincarb-Koszul series, 67 Poincarb polynomial, 3, 8, 180, 203, 244, 255, 260, 269, 369, 376, 402, 404, 408, 410, 433, 446, 465, 467, 488, 492ff., (19, 163, 176, 186, 224, 227), [178, 186, 215, 3451 Poincarb series, 3, 8, 42, 86, 124, 255, 259, 260, 369 Pontrjagin algebra, 190 Pontrjagin,class, 410, (426, 430) Pontrjagin number, 467, (428) Positively graded space, 3, 8 , 13 Primitive subspace, 193, 201, 221 Principal action, 352 Principal bundle, 352, 390, 397 and algebraic connections, Chapter VIII, 314 cohomology for, 358 Principal connection, 352, 390, 459 Product, in an algebra, 4 Projectable operation, 292, 503, 522 Projection, 310, 316 fibre, 292, 357, 413, 501, 503, 540 horizontal, 322, 353 Samelson, 70, 115 surjective fibre, 316 Proper isometry, 558

Q Quasi-sernisimple, 160 Quarternion, 270 €7

Radical, 440 Rank, 194 Reduction theorem, 73, 116, 296 Reductive Lie algebra, 162, 163, 186, 188ff., 213, 239, 245, 457, (164) pair, 411, 417, 427, 442, 453, 455, 457, 498 triple, 538

592

Index

Regular homomorphism, 13,39 operation, 350,376,377,506 representation, 469 Relative Poincarh duality isomorphism,

450,451 Representation, 158 in a differential space, 169 faithful, 159 in a graded algebra, 159, 213 in a graded space, 159 induced, 169 left regular, 469 of a Lie algebra, 158 semisimple, 160,166,169,457 in a tensor product, 172 Restriction of an operation, 278 Root, 164,443,(106) Root space, 164

S S-sequence, 107 Samelson complement, 71, 83,115, 118, 414,

427,459

projection, 70, 115 subspace, 70,83, 115, 118, 144,295,

414,421,427,459,474 theorem, 71, 114,414 Scalar product, 1 tensor product of, 2 Semidirect sum, 439 Semimorphism, 95

Semisimple associated operation, 281, 499 Semisimple Lie algebra, 160,161 Semisimple representation, 160,166,

169,457

Simple Lie algebra, 161 Simplification theorem, 76 Skew Pfaffian, 257 Skew symmetric, 2 Skew symplectic, 2, 260 Spectral sequences, 25ff. convergence of, 35 lower, 99 of an operation, 361 of a P-differential algebra, 99 of a tensor difference, 106 Split pairs, 435 Structure homomorphism, 288, 297, 328

Structure map, 59 Structure operation, 284 Structure theorems, 70,88,114,193,196,

241,249

Subalgebra, Chapter X, 411 basic, 276 characteristic, 341,427,(262,391,426) horizontal, 276 invariant, 276,308 of a Lie algebra, 157, 315 n.c.z., 436 Subgroup, n.c.z., 465, 492ff. Substitution opetator, 7,9,208 Surjective fibre projection, 316 Symmetric algebra, 8 Symmetric connection, 447 Symmetric P-algebra, 78, 135, 152 Symmetric pair, 447,492ff. Symmetric spaces, 465 Symplectic metric, 2,454 skew, 2, 260 space, 2 System of generators, 4 \

T Tensor difference, 103 cohomology diagram of, 126,129 cohomology of, 109, 135 spectral sequence of, 106 with a symmetric P-algebra, 135 Tensor product canonical, 5, 554,(4) of graded algebras, 5, 10 of graded differential spaces, 12 operation, 278, 281 of P-spaces, 57 representation in, 172 of scalar products, 2 Tor, 61 Total degree, 33 Total differential operator, 48 Trace coefficient, 550 Trace form, 159 Trace series, 552 Transformation groups, 307, (109) Transgression, 241, 378 adapted to specid Cartan pairs, 533 distinguished, 239,378 Triple of Lie algebras, 538

Index

U Unimodular, 185, (48) W Weil algebra, Chapter VI, 223 ff.,279

A 6 8 7

c a D 9

E O F 1 6 H 1 J

2 3 4 5

593

of a Lie algebra, 227, 327 of a pair, 422 Weil homomorphism, 340, 348, 355, 368, 375, 390,467,472, (26off., 286, 295, 409) Weyl group, 469, (88)

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