Poisson Geometry, Deformation Quantisation and Group Representations

Cover......Page 1
Frontmatter......Page 2
Contents......Page 6
Preface......Page 10
Part One - Poisson geometry and morita equivalence......Page 12
1 - Introduction......Page 14
2.1 Poisson manifolds......Page 16
2.2 Dirac structures......Page 18
2.3 Twisted structures......Page 22
2.4 Symplectic leaves and local structure of Poisson manifolds......Page 24
2.5 Presymplectic leaves and Dirac manifolds......Page 26
2.6 Poisson maps......Page 29
2.7 Dirac maps......Page 31
3.1 Ring-theoretic Morita equivalence of algebras......Page 36
3.2 Strong Morita equivalence of C*-algebras......Page 40
3.3 Morita equivalence of deformed algebras......Page 44
4.1 Representations and tensor product......Page 48
4.2 Symplectic groupoids......Page 51
4.3 Morita equivalence for groups and groupoids......Page 58
4.4 Modules over Poisson manifolds and groupoid actions......Page 60
4.5 Morita equivalence and symplectic groupoids......Page 63
4.6 Picard groups......Page 69
4.7 Fibrating Poisson manifolds and Morita invariants......Page 72
4.8 Gauge equivalence of Poisson structures......Page 75
5.1 Symplectic torsors......Page 78
5.2 Symplectic categories......Page 80
5.3 Symplectic categories of representations......Page 81
Bibliography......Page 83
Part Two - Formality and star products......Page 90
1.1 Physical motivation......Page 92
1.2 Historical review of deformation quantization......Page 94
1.3 Plan of the work......Page 96
2 - The star product......Page 98
3 - Rephrasing the main problem: the formality......Page 104
3.1 DGLA\'s, L[INFINITY]- algebras and deformation functors......Page 105
3.2 Multivector fields and multidifferential operators......Page 113
3.3 The first term: U1......Page 122
4 - Digression: what happens in the dual......Page 124
5 - The Kontsevich formula......Page 131
5.1 Admissible graphs, weights and B[GREEK CAPITAL LETTER GAMMA]\'s......Page 132
5.2 The proof: Stokes\' theorem & Vanishing theorems......Page 136
6 - From local to global deformation quantization......Page 145
Bibliography......Page 152
Part Three - Lie groupoids, sheaves and cohomology......Page 156
1 - Introduction......Page 158
2 - Lie groupoids......Page 160
2.1 Lie groupoids and weak equivalences......Page 162
2.2 The monodromy and holonomy groupoids of a foliation......Page 165
2.3 Etale groupoids and foliation groupoids......Page 167
2.4 Some general constructions......Page 170
2.5 Principal bundles as morphisms......Page 175
2.6 The principal bundles category......Page 179
3 - Sheaves on Lie groupoids......Page 186
3.1 Sheaves on groupoids......Page 187
3.2 Functoriality and Morita equivalence......Page 193
3.3 The fundamental group and locally constant sheaves......Page 198
3.4 G-sheaves of R-modules......Page 212
3.5 Derived categories......Page 216
4 - Sheaf cohomology......Page 221
4.1 Sheaf cohomology of foliation groupoids......Page 222
4.2 The bar resolution for étale groupoids......Page 225
4.3 Proper maps and orbifolds......Page 232
4.4 A comparison theorem for foliations......Page 238
4.5 The embedding category of an étale groupoid......Page 243
4.6 Degree one cohomology and the fundamental group......Page 249
5 - Compactly supported cohomology......Page 253
5.1 Sheaves over non-Hausdorff manifolds......Page 254
5.2 Compactly supported cohomology of éetale groupoids......Page 260
5.3 The operation [GREEK SMALL LETTER PHI]!......Page 265
5.4 Leray spectral sequence, and change-of-base......Page 269
5.5 Homology of the embedding category......Page 275
Bibliography......Page 280
Part Four - Geometric methods in representation theory......Page 284
1.1 Basic Definitions and Examples......Page 286
1.2 The Cartan Decomposition......Page 287
1.3 Complexifications of Linear Groups......Page 290
2.1 Maximal Tori, the Unit Lattice, and the Weight Lattice......Page 293
2.2 Weights, Roots, and the Weyl Group......Page 295
2.3 The Theorem of the Highest Weight......Page 297
2.4 Borel Subalgebras and the Flag Variety......Page 300
2.5 The Borel-Weil-Bott Theorem......Page 302
3.1 Continuity, Admissibility, K[DOUBLE-STRUCK CAPITAL R]-finite and C[INFINITY] Vectors......Page 305
3.2 Harish-Chandra Modules......Page 309
4 - Geometric Constructions of Representations......Page 316
Bibliography......Page 332
Part Five - Deformation theory: a powerful tool in physics modelling......Page 336
1.1 It ain\'t necessarily so......Page 338
1.2 Epistemological importance of deformation theory......Page 339
2 - Composite elementary particles in AdS microworld......Page 343
2.1 A qualitative overview......Page 344
2.2 A brief overview of singleton symmetry & field theory......Page 346
3 - Nonlinear covariant field equations......Page 349
4.1 The Gerstenhaber theory of deformations of algebras......Page 351
4.2 The invention of deformation quantisation......Page 353
4.3 Deformation quantisation and its developments......Page 356
Bibliography......Page 359
Index......Page 366