Lie algebras: finite and infinite dimensional Lie algebras and applications in physics

E.A. de Kerf, G.G.A. Bäuerle, A.P.E. ten Kroode9780080535463, 9780444828361, 0444828362, 0444887768

This is the long awaited follow-up to Lie Algebras, Part I which covered a major part of the theory of Kac-Moody algebras, stressing primarily their mathematical structure. Part II deals mainly with the representations and applications of Lie Algebras and contains many cross references to Part I.
The theoretical part largely deals with the representation theory of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are prime examples. After setting up the general framework of highest weight representations, the book continues to treat topics as the Casimir operator and the Weyl-Kac character formula, which are specific for Kac-Moody algebras.
The applications have a wide range. First, the book contains an exposition on the role of finite-dimensional semisimple Lie algebras and their representations in the standard and grand unified models of elementary particle physics. A second application is in the realm of soliton equations and their infinite-dimensional symmetry groups and algebras. The book concludes with a chapter on conformal field theory and the importance of the Virasoro and Kac-Moody algebras therein.

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Table of contents :
Cover……Page 1
Contents ……Page 2
Preface from the editors ……Page 5
18 Extensions of Lie algebras ……Page 9
18.1 Generalities ……Page 13
18.2 2-Cocyeles on Lie algebras ……Page 18
18.3 Structure constants and central extensions ……Page 24
18.4 Central extensions of simple Lie algebras ……Page 26
18.5 Central extensions of loop algebras ……Page 30
18.6 The Witt algebra and the Virasoro algebra ……Page 40
18.7 Projective representations and central extensions ……Page 48
19 Explicit construction of affine Kac-Moody algebras ……Page 53
19.1 Main features of affine Kac Moody algebras ……Page 54
19.2 Loop algebras reconsidered ……Page 59
19.3 Chevalley generators of $hat L$ ……Page 64
19.4 Realization of $A_1^{(1)}$ ……Page 73
20 Representations—enveloping algebra techniques ……Page 75
20.1 Poincare-BirkhofT-Will theorem ……Page 80
20.2 Highest weight modules ……Page 86
20.3 Existence of highest weight modules and Verma modules ……Page 92
20.4 More on highest weight modules ……Page 104
20.5 Example – The highest weight representations of sl(2,C) ……Page 110
21 The Weyl group and integrable representations ……Page 118
21.1 The Weyl group revisited ……Page 119
21.2 Weyl chambers and the Tits cone ……Page 125
21.3 Integrable representations ……Page 136
21.4 Integrable highest weight representations ……Page 152
22 More on representations ……Page 160
22.1 Fundamental highest weight modules ……Page 162
22.2 Bilinear forms on scmisimple Lie algebras ……Page 167
22.3 Casimir operators ……Page 179
22.4 Generalized Casimir operators ……Page 187
22.5 Lie algebras with a triangular decomposition ……Page 196
22.6 Lowest weight modules ……Page 199
22.7 Contravariant bilinear form $B_Lambda$ ……Page 209
22.8 Hermitian form $H_Lambda$ on $L(Lambda)$ ……Page 211
23 Characters and multiplicities ……Page 221
23.1 Freudenthal’s formula ……Page 223
23.2 Characters ……Page 228
23.3 Weyl-Kac character formula ……Page 239
23.4 Multiplicities of roots ……Page 244
23.5 Generalized Kostant formula ……Page 249
23.6 Weyl’s dimension formula ……Page 250
23.7 The $q$-dimension ……Page 255
24 Quarks, leptons and gauge fields ……Page 261
24.1 Particle multiplets and symmetries ……Page 263
24.2 Standard model ……Page 281
24.3 Complex and real representations ……Page 285
24.4 Unified models ……Page 288
24.5 Anomalies ……Page 298
25 Lie algebras of infinite matrices ……Page 307
25.1 The algebras sl($infty$,C) and gl($infty$,C) ……Page 309
25.2 Completions ……Page 312
25.3 The fundamental representations of sl($infty$,C) ……Page 322
25.4 The semi-infinite wedge space ……Page 325
25.5 Fermions ……Page 332
25.6 The energy spectrum of $wedge^infty C^infty$ ……Page 339
25.7 Bosons ……Page 345
25.8 Boson-fermion correspondence I ……Page 354
25.9 Boson-fermion correspondence II ……Page 362
26 Representations of loop algebras ……Page 367
26.1 Embedding of loop algebras ……Page 370
26.2 Principal Heisenberg subalgebra ……Page 376
26.3 Automorphisms of finite order ……Page 382
26.4 The principal realization of the fundamental modules ……Page 390
26.5 Other Heistnberg subalgehra ……Page 399
29.6 Realization of type $underline n$, I ……Page 410
26.7 Multicomponent boson-fermion correspondence ……Page 415
26.8 Realization of type $underline n$, II ……Page 423
26.9 Other loop algebras ……Page 429
27 KP-hierarchies ……Page 432
27.1 Finite-dimensional Grassmannians ……Page 435
27.2 Infinite-dimensional Grassmannians ……Page 447
27.3 Completions and extensions ……Page 451
27.4 The KP-hierarchy ……Page 465
27.5 Multicomponent КP-hierarchies ……Page 470
28 Conformal symmetry ……Page 478
28.1 The conformal group ……Page 480
28.2 The energy-momentum tensor ……Page 488
28.3 Conformal lield theory ……Page 493
28.4 Unitary representations of the Virasoro algebra ……Page 506
28.5 The Sugawara construction ……Page 516
28.6 The Coddard-Kent-Olive construction ……Page 532
28.7 The discrete series ……Page 537
Bibliography ……Page 543
Index ……Page 550