Developments and Trends in Infinite-Dimensional Lie Theory (eBook)

Preface 6
Contents 8
Part A Infinite-Dimensional Lie (Super-)Algebras 10
Isotopy for Extended Affine Lie Algebras and Lie Tori 11
1 Introduction 11
2 Extended affine Lie algebras 13
3 Lie tori 14
4 The construction of EALAs from Lie tori 18
5 Isotopy for Lie tori 21
6 Isotopy in the theory of EALAs 22
7 Coordinatization of Lie tori 30
8 Type A1 32
9 Type A2 36
10 Type Ar, r = 3 40
11 Type Cr, r = 4 42
12 Some concluding remarks 48
References 49
Remarks on the Isotriviality of Multiloop Algebras 52
1 Introduction 52
1.1 Notation and conventions 53
1.2 Rn-torsors under finite constant groups 54
2 Bounds on the isotriviality of multiloop algebras 55
References 58
Extended Affine Lie Algebras and Other Generalizations of Affine Lie Algebras – A Survey 59
1 Introduction 59
2 Root systems and other types of reflection systems 64
3 Affine reflection systems 74
4 Graded algebras 82
5 Lie algebras graded by root systems 94
6 Extended affine Lie algebras and generalizations 102
7 Example: slI(A) for A associative 118
References 128
Tensor Representations of Classical Locally Finite Lie Algebras 133
Introduction 133
1 Preliminaries 134
2 Tensor representations of gl8 and sl8 136
3 Tensor representations of sp8 143
4 Tensor representations of so8 146
5 Tensor representations of root-reductive Lie algebras 150
6 Appendix 151
References 155
Lie Algebras, Vertex Algebras, and Automorphic Forms 157
1 Introduction 157
2 Generalized Kac–Moody algebras 158
3 Vertex algebras 161
4 Automorphic forms on orthogonal groups 163
5 Moonshine for Conway’s group 166
7 Construction as strings 171
8 Open problems 172
References 173
Kac–Moody Superalgebras and Integrability 175
1 Introduction 175
2 Basic definitions 177
3 Odd reflections, Weyl groupoid, and principal roots 180
4 Kac–Moody superalgebras 185
5 Regular Kac–Moody superalgebras with two simple roots 188
6 Examples of regular quasisimple superalgebras 188
7 Classification results 194
8 Applications of classification results 195
9 Description of g (B) and g' in examples 197
10 Integrable modules and highest weight modules 200
11 General properties of category O 205
12 Lie superalgebras sl (1|n)(1), osp(2|2n)(1), andS (1, 2 b)
13 Lie superalgebras sl (1|n)(1), osp(2|2n)(1) 214
14 On affine character formulae 217
References 224
Part B Geometry of Infinite-Dimensional Lie (Transformation) Groups 225
Jordan Structures and Non-Associative Geometry 226
Introduction 226
1 Jordan pairs and graded Lie algebras 229
1.1 Z/(2)-graded Lie algebras and Lie triple systems 229
1.2 3-graded Lie algebras and Jordan pairs 230
1.3 Involutive Z-graded Lie algebras 231
1.4 The link with Jordan algebras 232
1.5 Some examples 233
2 The generalized projective geometry of a Jordan pair 234
2.1 The construction 234
2.2 Generalized projective geometries 235
2.3 The geometric Jordan–Lie functor 237
2.4 Examples revisited 238
3 The universal model 240
3.1 Ordinary flag geometries 240
3.2 Filtrations and gradings of Lie algebras 241
4 The geometry of states 242
4.1 Intrinsic subspaces 242
4.2 Examples 244
References 245
Direct Limits of Infinite-Dimensional Lie Groups 247
1 Introduction 247
1.1 Direct limit properties of ascending unions 249
1.2 Existence of direct limit charts: an essential hypothesis 249
1.3 Homotopy groups of ascending unions of Lie groups 250
1.4 Regularity in Milnor’s sense 251
1.5 Subgroups of direct limit groups 252
1.6 Constructions of Lie group structures on ascending unions 252
1.7 Properties of locally convex direct limits 252
1.8 Further comments, and some historical remarks 253
2 Preliminaries, terminology and basic facts 255
3 Direct limits of topological groups 259
4 Non-linear mappings on locally convex direct limits 262
5 Lie group structures on directed unions of Lie groups 266
6 Examples of directed unions of Lie groups 269
7 Direct limit properties of ascending unions 272
8 Regularity in Milnor’s sense 273
9 Homotopy groups of ascending unions of Lie groups 276
References 279
Lie Groups of Bundle Automorphisms and Their Extensions 285
Introduction 285
Notation and basic concepts 289
1 Lie group structures on mapping groups and automorphism groups of bundles 290
1.1 Automorphism groups of bundles 290
1.2 Mapping groups on non-compact manifolds 292
2 Central extensions of mapping groups 293
2.1 Central extensions of C8(M, t) 294
2.2 Covariance of the Lie algebra cocycles 298
2.3 Corresponding Lie group extensions 301
3 Twists and the cohomology of vector fields 305
3.1 Some cohomology of the Lie algebra of vector fields 306
3.2 Abelian extensions of diffeomorphism groups 310
4 Central extensions of gauge groups 318
4.1 Central extensions of gau(P) 318
4.2 Covariance of the Lie algebra cocycles 321
4.3 Corresponding Lie group extensions 324
5 Multiloop algebras 327
5.1 The algebraic picture 327
5.2 Geometric realization of multiloop algebras 328
5.3 A generalization of multiloop algebras 329
5.4 Connections to forms of Lie algebras over rings 330
6 Concluding remarks 331
7 Appendix A. Abelian extensions of Lie groups 332
8 Appendix B. Abelian extensions of semidirect sums 335
9 Appendix C. Triviality of the group action on Lie algebra cohomology 338
References 339
Gerbes and Lie Groups 343
Introduction 343
1 Bundle gerbes 344
2 Connections on bundle gerbes and holonomy 348
3 Bundle gerbes over compact Lie groups 353
4 Structure on loop spaces from bundle gerbes 359
5 Algebraic structures for gerbes 361
5.1 Bundle gerbe modules 361
5.2 Bundle Gerbe bimodules 362
6 Applications to conformal field theory 365
7 Open questions 366
References 367
Part C Representation Theory of Infinite-Dimensional Lie Groups 369
Functional Analytic Background for a Theory of Infinite-Dimensional Reductive Lie Groups 370
1 Introduction: What a reductive Lie group is supposed to be 370
2 Triangular integrals and factorizations 372
3 Invariant means on groups 378
4 Lifting group decompositions to covering groups 383
5 What a reductive Banach–Lie group could be 387
References 393
Heat Kernel Measures and Critical Limits 396
1 Introduction 396
2 General constructions 400
2.1 Abstract Wiener spaces and Gaussian measures 400
2.2 Abstract Wiener groups and heat kernel measures 402
3 Invariance questions 404
4 The Example of Map(X, F) (see [M]) 405
5 The critical Sobolev exponent and X = S1 408
6 2D quantum field theory 414
References 417
Coadjoint Orbits and the Beginnings of a Geometric Representation Theory 419
1 Introduction 419
2 Banach Poisson manifolds 421
2.1 The definition 421
2.2 Banach symplectic manifolds 422
3 Banach–Lie–Poisson spaces 423
3.1 Characterization 423
3.2 Examples 424
4 Symplectic leaves 426
4.1 Attempt at constructing symplectic leaves 427
4.2 Coadjoint orbits in Banach–Lie–Poisson spaces 429
5 Coadjoint orbits in operator spaces 431
5.1 Symplectic leaves in preduals of W*-algebras 432
5.2 Symplectic leaves in C*-algebras 434
5.3 Symplectic leaves in preduals of operator ideals 435
5.4 The restricted unitary algebra and its central extension 437
5.5 The restricted Grassmannian 443
5.6 Hilbert–Schmidt skew-Hermitian operators 444
6 Geometric representation theory 446
6.1 GNS unital *-representation 446
6.2 The fundamental construction 447
6.3 Reproducing kernels 448
6.4 Reproducing kernels and GNS-representations 449
6.5 Example 451
References 453
Infinite-Dimensional Multiplicity-Free Spaces I: Limits of Compact Commutative Spaces 460
1 Introduction 460
2 Direct limit groups and representations 462
3 Limit theorem for symmetric spaces 463
4 Gelfand pairs and defining representations 469
5 Function algebras 471
6 Pairs related to spheres and Grassmann manifolds 474
7 Limits related to spheres and Grassmann manifolds 477
8 Conclusions 480
References 482
Index 483