An Invitation to Quantum Groups and Duality (Ems Textbooks in Mathematics)

Preface......Page 7
Contents......Page 9
Introduction......Page 15
I From groups to quantum groups......Page 21
Motivation: Pontrjagin duality......Page 23
Definition......Page 25
Examples related to groups......Page 26
Axiomatics of Hopf algebras......Page 31
Coalgebras and bialgebras......Page 32
Convolution......Page 36
Properties of the antipode......Page 38
Another characterization of Hopf algebras......Page 42
Hopf *-algebras......Page 46
The duality of finite-dimensional Hopf algebras......Page 48
Dual pairings of Hopf algebras......Page 51
The restricted dual of a Hopf algebra......Page 55
Definition of multiplier Hopf algebras......Page 60
Multipliers of algebras......Page 61
Multiplier bialgebras......Page 62
Multiplier Hopf algebras......Page 64
The concept of an integral......Page 67
Existence and uniqueness......Page 72
The modular element of an integral......Page 74
The modular automorphism of an integral......Page 76
The duality of regular multiplier Hopf algebras......Page 78
The duality of algebraic quantum groups......Page 83
3 Algebraic compact quantum groups......Page 85
Definition and examples......Page 86
Reformulation of the concept of a corepresentation......Page 89
Construction of new corepresentations......Page 95
Decomposition into irreducible corepresentations......Page 100
Schur\'s orthogonality relations......Page 103
Characterization of compact quantum groups......Page 106
Characters of corepresentations......Page 107
Modular properties of the Haar state......Page 109
Discrete algebraic quantum groups......Page 111
II Quantum groups and C*-/von Neumann bialgebras......Page 115
C*-bialgebras and von Neumann bialgebras......Page 117
Bialgebras associated to groups......Page 120
Approaches to quantum groups......Page 124
Definition and examples......Page 127
Unitary corepresentations of C*-algebraic compact quantum groups......Page 131
Corepresentation operators of C*-bialgebras......Page 133
Constructions related to corepresentation operators......Page 136
Corepresentation theory and structure theory......Page 141
Decomposition into irreducible corepresentations......Page 142
Schur\'s orthogonality relations......Page 145
Characterization of C*-algebraic compact quantum groups......Page 147
From C*-algebraic to algebraic CQGs......Page 148
From algebraic to C*-algebraic CQGs......Page 150
6 Examples of compact quantum groups......Page 155
Compact matrix quantum groups......Page 156
Definition and first properties......Page 162
Corepresentations and their weights......Page 165
Corepresentations and differential calculi......Page 169
Modular properties of the Haar state......Page 172
Products of compact quantum groups......Page 174
The free unitary and the free orthogonal quantum groups......Page 179
7 Multiplicative unitaries......Page 186
Motivation......Page 187
Definition and examples......Page 188
Definition and first properties......Page 192
Well-behaved multiplicative unitaries......Page 195
Examples......Page 199
The dual pairing, counit, and antipode of the legs......Page 204
Regular multiplicative unitaries......Page 209
Manageable and modular multiplicative unitaries......Page 217
The concept of a locally compact quantum group......Page 223
Weights......Page 224
Locally compact quantum groups in the setting of von Neumann algebras......Page 225
The modular automorphism group of a weight......Page 227
Reduced C*-algebraic quantum groups......Page 230
Additional prerequisites......Page 233
Main properties......Page 236
The multiplicative unitary......Page 237
The antipode and modular properties......Page 238
The duality of locally compact quantum groups......Page 242
Passage between the different levels......Page 245
Examples of locally compact quantum groups......Page 247
C*-algebras generated by unbounded elements......Page 249
The quantum groups E_(2) and _(2)......Page 253
The quantum az+b group......Page 262
III Selected topics......Page 269
9 Coactions on C*-algebras, reduced crossed products, and duality......Page 271
Actions of groups and Takesaki–Takai duality......Page 272
Coactions of C*-bialgebras on C*-algebras......Page 276
Balanced multiplicative unitaries......Page 283
Weak Kac systems......Page 287
Examples of weak Kac systems......Page 288
The reduced crossed product of a coaction of A(V)......Page 294
The dual coaction of a coaction of A(V)......Page 297
The dual coaction of a coaction of (V)......Page 299
Comparison with the reduced crossed product of an action......Page 300
Kac systems......Page 302
The Baaj–Skandalis duality theorem......Page 306
10 Pseudo-multiplicative unitaries on Hilbert spaces......Page 309
Hilbert modules over von Neumann algebras......Page 311
Outline of the construction......Page 313
Bounded elements of a Hilbert module......Page 316
Construction of the relative tensor product......Page 320
Properties of the relative tensor product......Page 322
The fiber product of von Neumann algebras......Page 327
Hopf–von Neumann bimodules......Page 332
Definition......Page 334
The legs of a pseudo-multiplicative unitary......Page 336
The pseudo-multiplicative unitary of a groupoid......Page 343
Pseudo-multiplicative unitaries on C*-modules......Page 348
The flipped internal tensor product of C*-modules......Page 349
Definition and examples......Page 350
Obstructions to the construction of the legs......Page 355
Homogeneous operators and C*-families......Page 357
Homogeneous elements of right C*-bimodules......Page 362
Examples related to groupoids......Page 367
The internal tensor product of C*-families......Page 370
Morphisms of C*-families......Page 374
Hopf C*-families......Page 378
The legs of a decomposable pseudo-multiplicative unitary......Page 379
Coactions of Hopf C*-families......Page 385
C*-algebras......Page 389
C*-modules......Page 393
Von Neumann algebras......Page 395
Slice maps......Page 397
Auxiliary results......Page 401
Index......Page 421
Bibliography......Page 405
Symbol Index......Page 417