Poisson geometry, deformation quantisation and group representations:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2005
|
| Series: | London Mathematical Society lecture note series
323 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | X, 359 S. graph. Darst. |
| ISBN: | 9780521615051 0521615054 |
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MARC
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| 245 | 1 | 0 | |a Poisson geometry, deformation quantisation and group representations |c ed. by Simone Gutt ... |
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2005 | |
| 300 | |a X, 359 S. |b graph. Darst. | ||
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| 490 | 1 | |a London Mathematical Society lecture note series |v 323 | |
| 500 | |a Literaturangaben | ||
| 650 | 4 | |a Darstellungstheorie | |
| 650 | 4 | |a Deformationsquantisierung | |
| 650 | 4 | |a Differentialgeometrie | |
| 650 | 4 | |a Poisson algebras | |
| 650 | 4 | |a Poisson manifolds | |
| 650 | 4 | |a Poisson-Mannigfaltigkeit | |
| 650 | 4 | |a Representations of groups | |
| 655 | 4 | |a Kongress - Brüssel <2003> - Darstellungstheorie - Poisson-Mannigfaltigkeit - Deformationsquantisierung | |
| 700 | 1 | |a Gutt, Simone |e Sonstige |4 oth | |
| 830 | 0 | |a London Mathematical Society lecture note series |v 323 |w (DE-604)BV000000130 |9 323 | |
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Record in the Search Index
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| adam_text | Contents
Preface page ix
Part One: Poisson geometry and morita equivalence 1
1 Introduction 3
2 Poisson geometry and some generalizations 5
2.1 Poisson manifolds 5
2.2 Dirac structures 7
2.3 Twisted structures 11
2.4 Symplectic leaves and local structure of Poisson
manifolds 13
2.5 Presymplectic leaves and Dirac manifolds 15
2.6 Poisson maps 18
2.7 Dirac maps 20
3 Algebraic Morita equivalence 25
3.1 Ring theoretic Morita equivalence of algebras 25
3.2 Strong Morita equivalence of C* algebras 29
3.3 Morita equivalence of deformed algebras 33
4 Geometric Morita equivalence 37
4.1 Representations and tensor product 37
4.2 Symplectic groupoids 40
4.3 Morita equivalence for groups and groupoids 47
4.4 Modules over Poisson manifolds and groupoid actions 49
4.5 Morita equivalence and symplectic groupoids 52
4.6 Picard groups 58
4.7 Fibrating Poisson manifolds and Morita invariants 61
4.8 Gauge equivalence of Poisson structures 64
5 Geometric representation equivalence 67
5.1 Symplectic torsors 67
5.2 Symplectic categories 69
v
vi Contents
5.3 Symplectic categories of representations 70
Bibliography 72
Part Two: Formality and star products 79
1 Introduction 81
1.1 Physical motivation 81
1.2 Historical review of deformation quantization 83
1.3 Plan of the work 85
2 The star product 87
3 Rephrasing the main problem: the formality 93
3.1 DGLA s, Lqo algebras and deformation functors 94
3.2 Multivector fields and multidifferential operators 102
3.2.1 The DGLA V 103
3.2.2 The DGLA V 106
3.3 The first term: Ui 111
4 Digression: what happens in the dual 113
5 The Kontsevich formula 120
5.1 Admissible graphs, weights and _Bp s 121
5.2 The proof: Stokes theorem Vanishing theorems 125
6 Prom local to global deformation quantization 134
Bibliography 141
Part Three: Lie groupoids, sheaves and cohomology 145
1 Introduction 147
2 Lie groupoids 149
2.1 Lie groupoids and weak equivalences 151
2.2 The monodromy and holonomy groupoids
of a foliation 154
2.3 Etale groupoids and foliation groupoids 156
2.4 Some general constructions 159
2.5 Principal bundles as morphisms 164
2.6 The principal bundles category 168
3 Sheaves on Lie groupoids 175
3.1 Sheaves on groupoids 176
3.2 Functoriality and Morita equivalence 182
3.3 The fundamental group and locally constant sheaves 187
3.4 G sheaves of fl modules 201
3.5 Derived categories 205
4 Sheaf cohomology 210
4.1 Sheaf cohomology of foliation groupoids 211
4.2 The bar resolution for etale groupoids 214
4.3 Proper maps and orbifolds 221
Contents vii
4.4 A comparison theorem for foliations 227
4.5 The embedding category of an etale groupoid 232
4.6 Degree one cohomology and the fundamental group 238
5 Compactly supported cohomology 242
5.1 Sheaves over non Hausdorff manifolds 243
5.2 Compactly supported cohomology of etale groupoids 249
5.3 The operation cj 254
5.4 Leray spectral sequence, and change of base 258
5.5 Homology of the embedding category 264
Bibliography 269
Part Four: Geometric methods in representation
theory 273
1 Reductive Lie Groups: Definitions and
Basic Properties 275
1.1 Basic Definitions and Examples 275
1.2 The Cartan Decomposition 276
1.3 Complexifications of Linear Groups 279
2 Compact Lie Groups 282
2.1 Maximal Tori, the Unit Lattice, and the Weight
Lattice 282
2.2 Weights, Roots, and the Weyl Group 284
2.3 The Theorem of the Highest Weight 286
2.4 Borel Subalgebras and the Flag Variety 289
2.5 The Borel Weil Bott Theorem 291
3 Representations of Reductive Lie Groups 294
3.1 Continuity, Admissibility. A R nnite and C°° Vectors 294
3.2 Harish Chandra Modules 298
4 Geometric Constructions of Representations 305
4.1 The Universal Cartan Algebra and Infinitesimal
Characters 306
4.2 Twisted P modules 307
4.3 Construction of Harish Chaudra Modules 311
4.4 Construction of ds representations 314
4.5 Matsuki Correspondence 317
Bibliography 321
Part Five: Deformation theory: a powerful tool in
physics modelling 325
1 Introduction 327
1.1 It ain t necessarily so 327
1.2 Epistemological importance of deformation theory 328
viii Contents
2 Composite elementary particles in AdS microworld 332
2.1 A qualitative overview 333
2.2 A brief overview of singleton symmetry field theory 335
3 Nonlinear covariant field equations 338
4 Quantisation is a deformation 340
4.1 The Gerstenhaber theory of deformations of algebras 340
4.2 The invention of deformation quantisation 342
4.3 Deformation quantisation and its developments 345
Bibliography 348
Index 355
|
| adam_txt |
Contents
Preface page ix
Part One: Poisson geometry and morita equivalence 1
1 Introduction 3
2 Poisson geometry and some generalizations 5
2.1 Poisson manifolds 5
2.2 Dirac structures 7
2.3 Twisted structures 11
2.4 Symplectic leaves and local structure of Poisson
manifolds 13
2.5 Presymplectic leaves and Dirac manifolds 15
2.6 Poisson maps 18
2.7 Dirac maps 20
3 Algebraic Morita equivalence 25
3.1 Ring theoretic Morita equivalence of algebras 25
3.2 Strong Morita equivalence of C* algebras 29
3.3 Morita equivalence of deformed algebras 33
4 Geometric Morita equivalence 37
4.1 Representations and tensor product 37
4.2 Symplectic groupoids 40
4.3 Morita equivalence for groups and groupoids 47
4.4 Modules over Poisson manifolds and groupoid actions 49
4.5 Morita equivalence and symplectic groupoids 52
4.6 Picard groups 58
4.7 Fibrating Poisson manifolds and Morita invariants 61
4.8 Gauge equivalence of Poisson structures 64
5 Geometric representation equivalence 67
5.1 Symplectic torsors 67
5.2 Symplectic categories 69
v
vi Contents
5.3 Symplectic categories of representations 70
Bibliography 72
Part Two: Formality and star products 79
1 Introduction 81
1.1 Physical motivation 81
1.2 Historical review of deformation quantization 83
1.3 Plan of the work 85
2 The star product 87
3 Rephrasing the main problem: the formality 93
3.1 DGLA's, Lqo algebras and deformation functors 94
3.2 Multivector fields and multidifferential operators 102
3.2.1 The DGLA V 103
3.2.2 The DGLA V 106
3.3 The first term: Ui 111
4 Digression: what happens in the dual 113
5 The Kontsevich formula 120
5.1 Admissible graphs, weights and _Bp's 121
5.2 The proof: Stokes' theorem Vanishing theorems 125
6 Prom local to global deformation quantization 134
Bibliography 141
Part Three: Lie groupoids, sheaves and cohomology 145
1 Introduction 147
2 Lie groupoids 149
2.1 Lie groupoids and weak equivalences 151
2.2 The monodromy and holonomy groupoids
of a foliation 154
2.3 Etale groupoids and foliation groupoids 156
2.4 Some general constructions 159
2.5 Principal bundles as morphisms 164
2.6 The principal bundles category 168
3 Sheaves on Lie groupoids 175
3.1 Sheaves on groupoids 176
3.2 Functoriality and Morita equivalence 182
3.3 The fundamental group and locally constant sheaves 187
3.4 G sheaves of fl modules 201
3.5 Derived categories 205
4 Sheaf cohomology 210
4.1 Sheaf cohomology of foliation groupoids 211
4.2 The bar resolution for etale groupoids 214
4.3 Proper maps and orbifolds 221
Contents vii
4.4 A comparison theorem for foliations 227
4.5 The embedding category of an etale groupoid 232
4.6 Degree one cohomology and the fundamental group 238
5 Compactly supported cohomology 242
5.1 Sheaves over non Hausdorff manifolds 243
5.2 Compactly supported cohomology of etale groupoids 249
5.3 The operation cj \ 254
5.4 Leray spectral sequence, and change of base 258
5.5 Homology of the embedding category 264
Bibliography 269
Part Four: Geometric methods in representation
theory 273
1 Reductive Lie Groups: Definitions and
Basic Properties 275
1.1 Basic Definitions and Examples 275
1.2 The Cartan Decomposition 276
1.3 Complexifications of Linear Groups 279
2 Compact Lie Groups 282
2.1 Maximal Tori, the Unit Lattice, and the Weight
Lattice 282
2.2 Weights, Roots, and the Weyl Group 284
2.3 The Theorem of the Highest Weight 286
2.4 Borel Subalgebras and the Flag Variety 289
2.5 The Borel Weil Bott Theorem 291
3 Representations of Reductive Lie Groups 294
3.1 Continuity, Admissibility. A"R nnite and C°° Vectors 294
3.2 Harish Chandra Modules 298
4 Geometric Constructions of Representations 305
4.1 The Universal Cartan Algebra and Infinitesimal
Characters 306
4.2 Twisted P modules 307
4.3 Construction of Harish Chaudra Modules 311
4.4 Construction of ds representations 314
4.5 Matsuki Correspondence 317
Bibliography 321
Part Five: Deformation theory: a powerful tool in
physics modelling 325
1 Introduction 327
1.1 It ain't necessarily so 327
1.2 Epistemological importance of deformation theory 328
viii Contents
2 Composite elementary particles in AdS microworld 332
2.1 A qualitative overview 333
2.2 A brief overview of singleton symmetry field theory 335
3 Nonlinear covariant field equations 338
4 Quantisation is a deformation 340
4.1 The Gerstenhaber theory of deformations of algebras 340
4.2 The invention of deformation quantisation 342
4.3 Deformation quantisation and its developments 345
Bibliography 348
Index 355 |
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| genre_facet | Kongress - Brüssel <2003> - Darstellungstheorie - Poisson-Mannigfaltigkeit - Deformationsquantisierung |
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| illustrated | Illustrated |
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| language | English |
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| physical | X, 359 S. graph. Darst. |
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| series | London Mathematical Society lecture note series |
| series2 | London Mathematical Society lecture note series |
| spelling | Poisson geometry, deformation quantisation and group representations ed. by Simone Gutt ... Cambridge [u.a.] Cambridge Univ. Press 2005 X, 359 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society lecture note series 323 Literaturangaben Darstellungstheorie Deformationsquantisierung Differentialgeometrie Poisson algebras Poisson manifolds Poisson-Mannigfaltigkeit Representations of groups Kongress - Brüssel <2003> - Darstellungstheorie - Poisson-Mannigfaltigkeit - Deformationsquantisierung Gutt, Simone Sonstige oth London Mathematical Society lecture note series 323 (DE-604)BV000000130 323 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014189321&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Poisson geometry, deformation quantisation and group representations London Mathematical Society lecture note series Darstellungstheorie Deformationsquantisierung Differentialgeometrie Poisson algebras Poisson manifolds Poisson-Mannigfaltigkeit Representations of groups |
| title | Poisson geometry, deformation quantisation and group representations |
| title_auth | Poisson geometry, deformation quantisation and group representations |
| title_exact_search | Poisson geometry, deformation quantisation and group representations |
| title_exact_search_txtP | Poisson geometry, deformation quantisation and group representations |
| title_full | Poisson geometry, deformation quantisation and group representations ed. by Simone Gutt ... |
| title_fullStr | Poisson geometry, deformation quantisation and group representations ed. by Simone Gutt ... |
| title_full_unstemmed | Poisson geometry, deformation quantisation and group representations ed. by Simone Gutt ... |
| title_short | Poisson geometry, deformation quantisation and group representations |
| title_sort | poisson geometry deformation quantisation and group representations |
| topic | Darstellungstheorie Deformationsquantisierung Differentialgeometrie Poisson algebras Poisson manifolds Poisson-Mannigfaltigkeit Representations of groups |
| topic_facet | Darstellungstheorie Deformationsquantisierung Differentialgeometrie Poisson algebras Poisson manifolds Poisson-Mannigfaltigkeit Representations of groups Kongress - Brüssel <2003> - Darstellungstheorie - Poisson-Mannigfaltigkeit - Deformationsquantisierung |
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