ELEMENTS OF MATHEMATICS NICOLAS BOURBAKI ELEMENTS OF MATHEMATICS Lie Groups and Lie Algebras Chapters 7–9 1 23 Originally published as GROUPES ET ALGÈBRES DE LIE ©N.Bourbaki Translator Andrew Pressley Department ofMathematics King’s College London Strand,London WC2R 2LS United Kingdom e-mail:[email protected] Mathematics Subject Classification (2000):17B10,17B15,17-01,22C05,22E60,22E46 Library ofCongress Control Number:2004111709 ISBN 3-540-43405-4 Springer Berlin Heidelberg New York This work is subject to copyright.All rights are reserved,whether the whole or part ofthe material is concerned,specifically the rights oftranslation,reprinting,reuse ofillustrations,recitation,broad- casting,reproduction on microfilm or in any other way,and storage in data banks.Duplication ofthis publicationor parts thereofis permitted only under the provisions ofthe German Copyright Law ofSeptember 9,1965,in its current version,and permission for use must always be obtained from Springer.Violations are liable for prosecution under the German Copyright Law. Springer is a part ofSpringer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use ofgeneral descriptive names,registered names,trademarks,etc.in this publication does not imply,even in the absence ofa specific statement,that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the Translator and Frank Herweg,Leutershausen Cover design:design+production GmbH,Heidelberg Printed on acid-free paper 41/3142/XT – 5 4 3 2 1 0 CONTENTS CONTENTS ................................................ V CHAPTER VII CARTAN SUBALGEBRAS AND REGULAR ELEMENTS §1. Primary decomposition of linear representations ........ 1 1. Decomposition of a family of endomorphisms ............. 1 2. The case of a linear family of endomorphisms ............ 6 3. DecompositionofrepresentationsofanilpotentLiealgebra 8 4. Decomposition of a Lie algebra relative to an automorphism ........................................ 11 5. Invariants of a semi-simple Lie algebra relative to a semi-simple action .................................... 11 §2. Cartan subalgebras and regular elements of a Lie algebra ................................................. 12 1. Cartan subalgebras ................................... 13 2. Regular elements of a Lie algebra ....................... 16 3. Cartan subalgebras and regular elements ................ 18 4. Cartan subalgebras of semi-simple Lie algebras ........... 19 §3. Conjugacy theorems .................................... 20 1. Elementary automorphisms ............................ 20 2. Conjugacy of Cartan subalgebras ....................... 22 3. Applications of conjugacy .............................. 24 4. Conjugacy of Cartan subalgebras of solvable Lie algebras .. 25 5. Lie group case ........................................ 26 §4. Regular elements of a Lie group ........................ 27 1. Regular elements for a linear representation .............. 27 2. Regular elements of a Lie group ........................ 29 3. Relations with regular elements of the Lie algebra ........ 31 4. Application to elementary automorphisms ............... 34 VI CONTENTS §5. Decomposable linear Lie algebras ....................... 34 1. Decomposable linear Lie algebras ....................... 34 2. Decomposable envelope ................................ 37 3. Decompositions of decomposable algebras ................ 37 4. Linear Lie algebras of nilpotent endomorphisms .......... 39 5. Characterizations of decomposable Lie algebras ........... 43 Appendix I - Polynomial maps and Zariski topology ........ 45 1. Zariski topology ...................................... 45 2. Dominant polynomial maps ............................ 46 Appendix II - A connectedness property .................... 48 Exercises for §1 .............................................. 51 Exercises for §2 .............................................. 55 Exercises for §3 .............................................. 57 Exercises for §4 .............................................. 63 Exercises for §5 .............................................. 63 Exercises for Appendix I ....................................... 66 Exercises for Appendix II ...................................... 67 CHAPTER VIII SPLIT SEMI-SIMPLE LIE ALGEBRAS §1. The Lie algebra sl(2,k) and its representations .......... 69 1. Canonical basis of sl(2,k) .............................. 69 2. Primitive elements of sl(2,k)-modules ................... 70 3. The simple modules V(m) ............................. 72 4. Linear representations of the group SL(2,k) .............. 74 5. Some elements of SL(2,k) ............................. 76 §2. Root system of a split semi-simple Lie algebra .......... 77 1. Split semi-simple Lie algebras .......................... 77 2. Roots of a split semi-simple Lie algebra .................. 78 3. Invariant bilinear forms ................................ 83 4. The coefficients N .................................. 83 αβ §3. Subalgebras of split semi-simple Lie algebras ............ 86 1. Subalgebras stable under ad h .......................... 86 2. Ideals ............................................... 89 3. Borel subalgebras ..................................... 90 4. Parabolic subalgebras ................................. 92 5. Non-split case ........................................ 94 CONTENTS VII §4. Split semi-simple Lie algebra defined by a reduced root system .................................................. 95 1. Framed semi-simple Lie algebras ........................ 95 2. A preliminary construction ............................. 96 3. Existence theorem .................................... 100 4. Uniqueness theorem ................................... 104 §5. Automorphisms of a semi-simple Lie algebra ............ 106 1. Automorphisms of a framed semi-simple Lie algebra ....... 106 2. Automorphisms of a split semi-simple Lie algebra ......... 107 3. Automorphisms of a splittable semi-simple Lie algebra .... 111 4. Zariski topology on Aut(g) ............................. 113 5. Lie group case ........................................ 115 §6. Modules over a split semi-simple Lie algebra ............ 115 1. Weights and primitive elements ......................... 116 2. Simple modules with a highest weight ................... 118 3. Existence and uniqueness theorem ...................... 119 4. Commutant of h in the enveloping algebra of g ........... 122 §7. Finite dimensional modules over a split semi-simple Lie algebra ................................................. 124 1. Weights of a finited imensional simple g-module .......... 124 2. Highest weight of a finite dimensional simple g-module .... 126 3. Minuscule weights .................................... 130 4. Tensor products of g-modules .......................... 132 5. Dual of a g-module ................................... 134 6. Representation ring ................................... 136 7. Characters of g-modules ............................... 139 §8. Symmetric invariants ................................... 141 1. Exponential of a linear form ........................... 141 2. Injection of k[P] into S(h)∗ ............................. 142 3. Invariant polynomial functions ......................... 143 4. Properties of Aut .................................... 148 0 5. Centre of the enveloping algebra ........................ 148 §9. The formula of Hermann Weyl .......................... 152 1. Characters of finite dimensional g-modules ............... 152 2. Dimensions of simple g-modules ........................ 154 3. Multiplicities of weights of simple g-modules ............. 156 4. Decomposition of tensor products of simple g-modules ..... 157 VIII CONTENTS §10. Maximal subalgebras of semi-simple Lie algebras ...... 159 §11. Classes of nilpotent elements and sl2-triplets ........... 163 1. Definition of sl -triplets ............................... 163 2 2. sl -triplets in semi-simple Lie algebras ................... 165 2 3. Simple elements ...................................... 167 4. Principal elements .................................... 170 §12. Chevalley orders ....................................... 173 1. Lattices and orders ................................... 173 2. Divided powers in a bigebra ............................ 173 3. Integral variant of the Poincar´e-Birkhoff-Witt theorem .... 174 4. Example: polynomials with integer values ................ 176 5. Some formulas ....................................... 178 6. Biorders in the enveloping algebra of a split reductive Lie algebra .............................................. 180 7. Chevalley orders ...................................... 185 8. Admissible lattices .................................... 187 §13. Classical splittable simple Lie algebras ................. 189 1. Algebras of type A (l≥1) ............................. 190 l 2. Algebras of type B (l≥1) ............................. 195 l 3. Algebras of type C (l≥1) ............................. 204 l 4. Algebras of type D (l≥2) ............................ 211 l Table 1 ..................................................... 217 Table 2 ..................................................... 218 Exercises for §1 .............................................. 219 Exercises for §2 .............................................. 226 Exercises for §3 .............................................. 229 Exercises for §4 .............................................. 231 Exercises for §5 .............................................. 233 Exercises for §6 .............................................. 238 Exercises for §7 .............................................. 238 Exercises for §8 .............................................. 250 Exercises for §9 .............................................. 253 Exercises for §10 ............................................. 260 Exercises for §11 ............................................. 261 Exercises for §13 ............................................. 266 Summary of some important properties of semi-simple Lie algebras .................................................... 273 CONTENTS IX CHAPTER IX COMPACT REAL LIE GROUPS §1. Compact Lie algebras ................................... 281 1. Invariant hermitian forms .............................. 281 2. Connected commutative real Lie groups ................. 282 3. Compact Lie algebras ................................. 283 4. Groups whose Lie algebra is compact .................... 284 §2. Maximal tori of compact Lie groups .................... 287 1. Cartan subalgebras of compact algebras ................. 287 2. Maximal tori ......................................... 288 3. Maximal tori of subgroups and quotient groups ........... 291 4. Subgroups of maximal rank ............................ 292 5. Weyl group .......................................... 293 6. Maximal tori and covering of homomorphisms ............ 295 §3. Compact forms of complex semi-simple Lie algebras .... 296 1. Real forms ........................................... 296 2. Real forms associated to a Chevalley system ............. 297 3. Conjugacy of compact forms ........................... 299 4. Example I: compact algebras of type A ................. 300 n 5. Example II: compact algebras of type B and D ......... 301 n n 6. Compact groups of rank 1 ............................. 302 §4. Root system associated to a compact group ............. 304 1. The group X(H) ...................................... 304 2. Nodal group of a torus ................................ 305 3. Weights of a linear representation ....................... 307 4. Roots ............................................... 309 5. Nodal vectors and inverse roots ......................... 311 6. Fundamental group ................................... 314 7. Subgroups of maximum rank ........................... 316 8. Root diagrams ....................................... 317 9. Compact Lie groups and root systems ................... 319 10. Automorphisms of a connected compact Lie group ........ 322 §5. Conjugacy classes ....................................... 324 1. Regular elements ..................................... 324 2. Chambers and alcoves ................................. 325 3. Automorphisms and regular elements .................... 327 4. The maps (G/T)×T→G and (G/T)×A→G ......... 331 r X CONTENTS §6. Integration on compact Lie groups ...................... 333 1. Product of alternating multilinear forms ................. 333 2. Integration formula of H. Weyl ......................... 334 3. Integration on Lie algebras ............................. 339 4. Integration of sections of a vector bundle ................ 341 5. Invariant differential forms ............................. 344 §7. Irreducible representations of connected compact Lie groups .................................................. 347 1. Dominant characters .................................. 347 2. Highest weight of an irreducible representation ........... 348 3. The ring R(G) ....................................... 351 4. Character formula .................................... 353 5. Degree of irreducible representations .................... 356 6. Casimir elements ..................................... 358 §8. Fourier transform ....................................... 359 1. Fourier transforms of integrable functions ................ 360 2. Fourier transforms of infinitely-differentiable functions ..... 362 3. Fourier transforms of central functions .................. 366 4. Central functions on G and functions on T ............... 368 §9. Compact Lie groups operating on manifolds ............. 369 1. Embedding of a manifold in the neighbourhood of a compact set .......................................... 369 2. Equivariant embedding theorem ........................ 373 3. Tubes and transversals ................................ 375 4. Orbit types .......................................... 377 Appendix I - Structure of compact groups .................. 381 1. Embedding a compact group in a product of Lie groups ... 381 2. Projective limits of Lie groups .......................... 382 3. Structure of connected compact groups .................. 384 Appendix II - Representations of real, complex or quaternionic type ........................................... 385 1. Representations of real algebras ........................ 385 2. Representations of compact groups ...................... 387 Exercises for §1 .............................................. 389 Exercises for §2 .............................................. 391 Exercises for §3 .............................................. 394 Exercises for §4 .............................................. 396 Exercises for §5 .............................................. 405 CONTENTS XI Exercises for §6 .............................................. 409 Exercises for §7 .............................................. 414 Exercises for §8 .............................................. 417 Exercises for §9 .............................................. 419 Exercises for Appendix I ....................................... 424 INDEX OF NOTATION .................................... 427 INDEX OF TERMINOLOGY .............................. 431