Lie Algebras of Finite and Affine Type

Cover......Page 1
Half-title......Page 2
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
1.1 Elementary properties of Lie algebras......Page 21
1.2 Representations and modules......Page 25
1.3 Abelian, nilpotent and soluble Lie algebras......Page 27
2.1 Representations of soluble Lie algebras......Page 31
2.2 Representations of nilpotent Lie algebras......Page 34
3.1 Existence of Cartan subalgebras......Page 43
3.2 Derivations and automorphisms......Page 45
3.3 Ideas from algebraic geometry......Page 47
3.4 Conjugacy of Cartan subalgebras......Page 53
4.1 Some properties of root spaces......Page 56
4.2 The Killing form......Page 59
4.3 The Cartan decomposition of a semisimple Lie algebra......Page 65
4.4 The Lie algebra.........Page 72
5.1 Positive systems and fundamental systems of roots......Page 76
5.2 The Weyl group......Page 79
5.3 Generators and relations for the Weyl group......Page 85
6.1 The Cartan matrix......Page 89
6.2 The Dynkin diagram......Page 92
6.3 Classification of Dynkin diagrams......Page 94
6.4 Classification of Cartan matrices......Page 100
7.1 Some properties of structure constants......Page 108
7.2 The uniqueness theorem......Page 113
7.3 Some generators and relations in a simple Lie algebra......Page 116
7.4 The Lie algebras L(A) and L(A)......Page 118
7.5 The existence theorem......Page 125
8 The simple Lie algebras......Page 141
8.1 Lie algebras of type A1......Page 142
8.2 Lie algebras of type D1......Page 144
8.3 Lie algebras of type B1......Page 148
8.4 Lie algebras of type C1......Page 152
8.5 Lie algebras of type G2......Page 155
8.6 Lie algebras of type F4......Page 158
8.7 Lie algebras of types E6, E7, E8......Page 160
8.8 Properties of long and short roots......Page 165
9.1 The universal enveloping algebra......Page 172
9.2 The Poincaré–Birkhoff–Witt basis theorem......Page 175
9.3 Free Lie algebras......Page 180
9.4 Lie algebras defined by generators and relations......Page 183
9.5 Graph automorphisms of simple Lie algebras......Page 185
10.1 Verma modules......Page 196
10.2 Finite dimensional irreducible modules......Page 206
10.3 The finite dimensionality criterion......Page 210
11.1 Relations between the enveloping algebra and the symmetric algebra......Page 221
11.2 Invariant polynomial functions......Page 227
11.3 The structure of the ring of polynomial invariants......Page 236
11.4 The Killing isomorphisms......Page 242
11.5 The centre of the enveloping algebra......Page 246
11.6 The Casimir element......Page 258
12.1 Characters of L-modules......Page 261
12.2 Characters of Verma modules......Page 264
12.3 Chambers and roots......Page 266
12.4 Composition factors of Verma modules......Page 275
12.5 Weyl’s character formula......Page 278
12.6 Complete reducibility......Page 282
13.1 An alternative form of Weyl’s dimension formula......Page 287
13.2 Fundamental modules for A1......Page 288
13.3 Exterior powers of modules......Page 290
13.4 Fundamental modules for B1 and D1......Page 294
13.5 Clifford algebras and spin modules......Page 301
13.6 Fundamental modules for C1......Page 312
13.7 Contraction maps......Page 315
13.8 Fundamental modules for exceptional algebras......Page 323
14.1 Realisations of a square matrix......Page 339
14.2 The Lie algebra L(A) associated with a complex matrix......Page 342
14.3 The Kac–Moody algebra L(A)......Page 351
15.1 A trichotomy for indecomposable GCMs......Page 356
15.2 Symmetrisable generalised Cartan matrices......Page 364
15.3 The classification of affine generalised Cartan matrices......Page 371
16.1 The invariant bilinear form......Page 380
16.2 The Weyl group of a Kac–Moody algebra......Page 391
16.3 The roots of a Kac–Moody algebra......Page 397
17.1 Properties of the affine Cartan matrix......Page 406
Summary......Page 412
17.2 The roots of an affine Kac–Moody algebra......Page 414
17.3 The Weyl group of an affine Kac–Moody algebra......Page 424
18.1 Loop algebras and central extensions......Page 436
18.2 Realisations of untwisted affine Kac–Moody algebras......Page 441
18.3 Some graph automorphisms of affine algebras......Page 446
18.4 Realisations of twisted affine algebras......Page 449
Comments on notation......Page 471
19.1 The category of L(A)-modules......Page 472
19.2 The generalised Casimir operator......Page 479
19.3 Kac’ character formula......Page 486
19.4 Generators and relations for symmetrisable algebras......Page 494
20.1 Macdonald’s identities......Page 504
20.2 Specialisations of Macdonald’s identities......Page 511
20.3 Irreducible modules for affine algebras......Page 514
20.4 The fundamental modules for L(Ã 1)......Page 524
20.5 The basic representation......Page 528
21.1 Definition and examples of Borcherds algebras......Page 539
21.2 Representations of Borcherds algebras......Page 544
21.3 The Monster Lie algebra......Page 550
Summary pages – explanation......Page 560
Notation......Page 630
Bibliography of books on Lie algebras......Page 639
Bibliography of articles on Kac–Moody algebras......Page 641
Index......Page 649