Representation Theory of the Virasoro Algebra (eBook)

Springer Monographs 2
Mathematics 2
Preface 6
Contents 8
Introduction 16
Preliminary 18
1.1 Virasoro Algebra 18
1.1.1 Universal Central Extension 19
1.1.2 Witt Algebra and its Universal Central Extension 20
1.2 Q-graded Lie Algebra 22
1.2.1 G-graded Vector Spaces 23
1.2.2 Definitions 24
1.2.3 Examples 26
1.2.4 Categories of ( g, h)- modules 28
1.2.5 Some Objects of the Category C 30
1.2.6 Simple Objects of the Category Cadm ( the Virasoro Case) 33
1.2.7 Dualising Functors 34
1.2.8 Local Composition Series and Formal Character 35
1.3 (Co)homology of a Q-graded Lie Algebra 41
1.3.1 Preliminaries 42
1.3.2 Frobenius Reciprocity 43
1.3.3 Definitions 45
1.3.4 Some Properties 48
1.4 Bernstein-Gelfand- Gelfand Duality 50
1.4.1 Preliminaries 51
1.4.2 Truncated Category 52
1.4.3 Projective Objects 53
1.4.4 Indecomposable Projective Objects 55
1.4.5 Duality Theorem 58
1.5 Bibliographical Notes and Comments 58
1.A Appendix: Proof of Propositions 1.1, 1.2 and 1.3 59
1.B Appendix: Alternative Proof of Proposition 1.14 62
Classification of Harish-Chandra Modules 64
2.1 Main Result 64
2.1.1 Notations and Conventions 64
2.1.2 Definitions 65
2.2 Partial Lie Algebras 67
2.2.1 Definition and Main Theorems 68
2.2.2 Proof of Theorem 2.2 70
2.2.3 Proof of Theorem 2.3 78
2.3 Z-graded Lie Algebras 81
2.3.1 Z-graded Modules 82
2.3.2 Correspondence between Simple Z-graded Modules and Simple Z/NZ-graded Modules 85
2.3.3 R-forms 89
2.4 Lie p-algebra W(m) 90
2.4.1 Definitions 90
2.4.2 Preliminaries 92
2.4.3 Irreducible Representations of W(m) ( m = 2) 97
2.4.4 Irreducible Representations of W(1) 99
2.4.5 Z/NZ-graded Modules over VirK 103
2.5 Proof of the Classification of Harish-Chandra Modules 107
2.5.1 Structure of Simple Z-graded Modules 107
2.5.2 Semi-continuity Principle 108
2.5.3 Proof of Theorem 2.1 109
2.6 Bibliographical Notes and Comments 113
2.A Appendix: Indecomposable Z-graded Vir-Moduleswith Weight Multiplicities 1 114
2.A.1 Definition of A(a) and B(ß) 114
2.A.2 Classification Theorem 115
The Jantzen Filtration 117
3.1 Motivation 118
3.1.1 Shapovalov Forms 118
3.1.2 Contravariant Forms 119
3.1.3 What is the Jantzen Filtration? 121
3.2 The original Jantzen Filtration 122
3.2.1 Integral Form 122
3.2.2 Definitions 125
3.2.3 Basic Properties 126
3.2.4 Character Sum 128
3.3 The Jantzen Filtration ` a la Feigin and Fuchs I 130
3.3.1 Definitions and Properties 130
3.3.2 Character Sum 132
3.3.3 Duality 134
3.4 The Jantzen Filtration ` a la Feigin and Fuchs II 135
3.4.1 Notation 135
3.4.2 Definitions and Properties 136
3.5 The Jantzen Filtration of Quotient Modules 138
3.6 Bibliographical Notes and Comments 139
Determinant Formulae 141
4.1 Vertex (Super)algebra Structures associated to Bosonic Fock Modules 141
4.1.1 Definitions and Notation 142
4.1.2 Vertex Operator Algebra F 143
4.1.3 Vertex Operator Superalgebra Vv 144
4.2 Isomorphisms among Fock Modules 145
4.2.1 Notation 145
4.2.2 Isomorphisms arising from Automorphisms of H 146
4.2.3 Isomorphisms related to the Contragredient Dual 147
4.3 Intertwining Operators 149
4.3.1 Vertex Operator Vµ( z) 149
4.3.2 Criterion for Non-Triviality 150
4.4 Determinants of Verma Modules 153
4.4.1 Definitions and Formulae 153
4.4.2 Proof of Theorem 4.2 154
4.5 Determinants of Fock Modules 158
4.5.1 Definitions and Formulae 159
4.5.2 Proof of Theorem 4.3 161
4.6 Bibliographical Notes and Comments 164
Verma Modules I: Preliminaries 165
5.1 Classification of Highest Weights 165
5.1.1 Strategy of Classification 166
5.1.2 Bijection between ˜D (c, h) and Integral Points on c,h 169
5.1.3 List of Integral Points of c,h 171
5.1.4 Fine Classification of Highest Weights: Class R+ 176
5.1.5 Special Highest Weights 179
5.1.6 Fine classification of Highest Weights: Class R- 180
5.2 Singular Vectors 182
5.2.1 Uniqueness of Singular Vectors 183
5.2.2 Existence of Singular Vectors 186
5.3 Embedding Diagrams of Verma Modules 188
5.3.1 Embedding Diagrams 188
5.3.2 Proof of Propositions 5.4 and 5.5 190
5.4 Singular Vector Formulae 191
5.4.1 Formula I 192
5.4.2 Formula II 193
5.4.3 Proof of Formula I 193
5.5 Character Sums of Jantzen Filtration of Verma Modules 200
5.5.1 Notation 200
5.5.2 Character Sum Formula 201
5.5.3 Explicit Forms 203
5.6 Character Sums of the Jantzen Filtration of Quotient Modules 204
5.6.1 Integral Forms of Quotient Modules 205
5.6.2 Definition 206
5.6.3 Character Sum Formula 208
5.6.4 Explicit Forms 214
5.7 Bibliographical Notes and Comments 215
5.A Appendix: Integral Points on c,h 216
Verma Modules II: Structure Theorem 224
6.1 Structures of Jantzen Filtration 224
6.1.1 Class V and Class I 224
6.1.2 Class R+ 225
6.1.3 Proof of (i) in Theorem 6.3 227
6.1.4 Proof of (ii) and (iv) in Theorem 6.3 229
6.1.5 Proof of (iii) in Theorem 6.3 230
6.1.6 Class R- 231
6.2 Structures of Verma Modules 234
6.2.1 Main Results 234
6.2.2 Proof of Theorems 6.5 and 6.6 235
6.3 Bernstein-Gelfand- Gelfand Type Resolutions 236
6.3.1 Class V and Class I 236
6.3.2 Class R+ 237
6.3.3 Class R- 238
6.4 Characters of Irreducible Highest Weight Representations 239
6.4.1 Normalised Character 239
6.4.2 Characters of the Irreducible Highest Weight Representations 240
6.4.3 Multiplicity 243
6.4.4 Modular Transformation 244
6.4.5 Asymptotic Dimension 247
6.5 Bibliographical Notes and Comments 251
A Duality among Verma Modules 252
7.1 Semi-regular Bimodule 252
7.1.1 Preliminaries 252
7.1.2 Semi-infinite Character 254
7.1.3 Definition 259
7.1.4 Compatibility of Two Actions on S.( g) 261
7.1.5 Isomorphisms 267
7.2 Tilting Equivalence 268
7.2.1 Preliminaries 268
7.2.2 Some Categories 270
7.2.3 Some Functors 272
7.2.4 Equivalence between M and K 275
7.2.5 The Virasoro Case 277
7.3 Bibliographical Notes and Comments 278
Fock Modules 279
8.1 Classification of Weights (., .) 279
8.1.1 Coarse Classification 279
8.1.2 Fine Classification: Class R+ 281
8.1.3 Zeros of det(G.,.)n 283
8.2 The Jantzen (Co)filtrations of Fock Modules 284
8.2.1 Contragredient Dual of gR- Modules 284
8.2.2 Fock Modules over gR 284
8.2.3 The Jantzen (Co)filtrations defined by G.,. andL.,. 285
8.2.4 Character Sum Formula 286
8.2.5 Structures of the Jantzen Filtrations defined byG˜.,˜. and L˜.,˜ 287
8.2.6 Singular Vectors and M(c, .i)(n] 288
8.2.7 Cosingular Vectors and M(c, .i)c[n) 290
8.3 Structure of Fock Modules (Class R+) 292
8.3.1 Main Theorem ( Case 1+) 292
8.3.2 Main Theorem ( Case 2+ and 3+) 296
8.3.3 Main Theorem ( Case 4+) 299
8.3.4 Classification of Singular Vectors 300
8.4 Jack Symmetric Polynomials and Singular Vectors 301
8.4.1 Completion of Fock Modules and Operators 301
8.4.2 Screening Operators 302
8.4.3 Non-Triviality of Screening Operators 304
8.4.4 Jack Symmetric Polynomials 306
8.4.5 Singular Vectors of Fock Modules 308
8.5 Spaces of Semi-infinite Forms and Fock Modules 314
8.5.1 Space of Semi-infinite Forms 314
8.5.2 Clifford Algebra and Fermionic Fock Modules 315
8.5.3 Isomorphism between 82 +•V a,b and FCa,b 318
8.5.4 Boson- Fermion Correspondence 319
8.6 Bibliographical Notes and Comments 322
8.A Appendix: Another Proof of Theorem 8.8 322
8.B Appendix: List of the Integral Points on ±.,. 331
Rational Vertex Operator Algebras 333
9.1 Vertex Operator Algebra Structure 333
9.2 The Zhu Algebra of Vc 335
9.2.1 Preliminary 335
9.2.2 A( Vc) 337
9.3 Rationality and the Fusion Algebra of BPZ Series 341
9.3.1 Coinvariants I 341
9.3.2 Rationality of L( c, 0) 344
9.3.3 Fusion Algebra 347
9.4 Characterisations of BPZ Series 352
9.4.1 Coinvariants II 352
9.4.2 Lisse Modules 353
9.4.3 Finiteness Condition 354
9.5 Bibliographical Notes and Comments 356
9.A Appendix: Associated Variety 356
9.A.1 Filtration 356
9.A.2 Associated Variety 358
9.A.3 Involutivity 359
9.B Appendix: Tauberian Theorem 360
Coset Constructions for ˆ sl2 362
10.1 Admissible Representations 362
10.1.1 Affine Lie Algebra ˆ sl2 362
10.1.2 Admissible Representations 365
10.2 Sugawara Construction 368
10.2.1 Vertex Algebra Structure of Vacuum ˆ sl2- modules 368
10.2.2 Segal- Sugawara Operator 369
10.3 Coset Constructions 370
10.3.1 Fundamental Characters of ˆ sl2 371
10.3.2 Coset ( ˆ sl2)1 × ( ˆ sl2)k/( ˆ sl2)k+1 373
10.3.3 Properties of the Theta Function and Proof of Lemma 10.5 376
10.3.4 Level 1 ˆ sl2- modules as Vir . sl2- module 380
10.4 Unitarisable Vir-modules 381
10.4.1 Unitarisable Representations of ˆ sl2 381
10.4.2 Unitarisable Representations of Vir 382
10.5 Bibliographical Notes and Comments 383
Unitarisable Harish-Chandra Modules 384
11.1 Definition of Unitarisable Representations 384
11.2 Anti-linear Anti-involutions of Vir 386
11.2.1 The Classification 386
11.2.2 Anti-linear Anti-involutions admitting Unitarisable Vir- Modules 388
11.3 Hermitian Form on Harish-Chandra Modules 389
11.3.1 Intermediate Series 389
11.3.2 Verma Modules 391
11.4 Main Results 392
11.5 Proof of Main Results 393
11.5.1 Determinant Formulae 394
11.5.2 Proof of Theorem 11.1 395
11.6 Bibliographical Notes and Comments 410
Homological Algebras 411
A.1 Categories and Functors 411
A.1.1 Categories 411
A.1.2 Functors 412
A.1.3 Additive Categories 413
A.1.4 Abelian Categories 414
A.2 Derived Functors 415
A.2.1 Definition 416
A.2.2 Extension of Modules 418
A.3 Lie Algebra Homology and Cohomology 420
A.3.1 Chevalley- Eilenberg ( Co) complex and Lie Algebra ( Co) homology 421
A.3.2 Koszul Complex 423
A.3.3 Tensor Identity 426
Lie p-algebras 428
B.1 Basic Objects 428
B.1.1 Definition of a Lie p-algebra 428
B.1.2 Restricted Enveloping Algebra 431
B.1.3 Central Character 433
B.1.4 Induced Representations 434
B.2 Completely Solvable Lie Algebras 436
B.2.1 Definition 436
B.2.2 Polarisation 436
B.2.3 Completely Solvable Lie p-algebras 440
B.3 Irreducible Representations of a Completely Solvable Lie p- algebra 442
B.3.1 Simplicity of Induced Representations 442
B.3.2 Dimension of Irreducible Representations over a Completely Solvable Lie Algebra 446
Vertex Operator Algebras 449
C.1 Basic Objects 449
C.1.1 Notation 449
C.1.2 Definition of a Vertex Operator Algebra 450
C.1.3 Strong Reconstruction Theorem 452
C.1.4 Operator Product Expansion 453
C.2 Rationality 455
C.2.1 Modules 455
C.2.2 The Zhu Algebra 456
C.2.3 A Theorem of Y. Zhu 457
C.3 Fusion Rule 458
C.3.1 Intertwining Operators 458
C.3.2 A( V )- Bimodule associated to a V - Module 459
C.3.3 Fusion Rule 460
C.4 Vertex Superalgebras 461
C.4.1 Notations 461
C.4.2 Definition of a Vertex Superalgebra 462
C.4.3 Operator Product Expansion 463
Further Topics 465
References 467
List of Symbols 476
Index 479