Geometric models for noncommutative algebras:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
American Math. Soc.
1999
|
| Series: | Berkeley mathematics lecture notes
10 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 184 S. |
| ISBN: | 0821809520 |
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MARC
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| 100 | 1 | |a Silva, Ana Cannas da |d 1968- |e Verfasser |0 (DE-588)122891333 |4 aut | |
| 245 | 1 | 0 | |a Geometric models for noncommutative algebras |c Ana Cannas da Silva ; Alan Weinstein |
| 264 | 1 | |a New York |b American Math. Soc. |c 1999 | |
| 300 | |a XIV, 184 S. | ||
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| 490 | 1 | |a Berkeley mathematics lecture notes |v 10 | |
| 650 | 4 | |a Nichtkommutative Algebra | |
| 650 | 4 | |a Nichtkommutative Geometrie | |
| 650 | 4 | |a Nichtkommutative Algebra - Nichtkommutative Differentialgeometrie | |
| 700 | 1 | |a Weinstein, Alan |d 1943- |e Verfasser |0 (DE-588)113532512 |4 aut | |
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Record in the Search Index
| _version_ | 1804127363402825728 |
|---|---|
| adam_text | Contents
Preface ix
Introduction xi
I Universal Enveloping Algebras 1
1 Algebraic Constructions 1
1.1 Universal Enveloping Algebras 1
1.2 Lie Algebra Deformations 2
1.3 Symmetrization 3
1.4 The Graded Algebra ofU{g) 3
2 The Poincare Birkhoff Witt Theorem 5
2.1 Almost Commutativity of U(g) 5
2.2 Poisson Bracket on 9t W(g) 5
2.3 The Role of the Jacobi Identity 7
2.4 Actions of Lie Algebras 8
2.5 Proof of the Poincare Birkhoff Witt Theorem 9
II Poisson Geometry 11
3 Poisson Structures 11
3.1 Lie Poisson Bracket 11
3.2 Almost Poisson Manifolds 12
3.3 Poisson Manifolds 12
3.4 Structure Functions and Canonical Coordinates 13
3.5 Hamiltonian Vector Fields 14
3.6 Poisson Cohomology 15
4 Normal Forms 17
4.1 Lie s Normal Form 17
4.2 A Faithful Representation of g 17
4.3 The Splitting Theorem 19
4.4 Special Cases of the Splitting Theorem 20
4.5 Almost Symplectic Structures 20
4.6 Incarnations of the Jacobi Identity 21
5 Local Poisson Geometry 23
5.1 Symplectic Foliation 23
5.2 Transverse Structure 24
5.3 The Linearization Problem 25
5.4 The Cases of su(2) ands[(2;K) 27
III Poisson Category 29
V
vi CONTENTS
6 Poisson Maps 29
6.1 Characterization of Poisson Maps 29
6.2 Complete Poisson Maps 31
6.3 Symplectic Realizations 32
6.4 Coisotropic Calculus 34
6.5 Poisson Quotients 34
6.6 Poisson Submanifolds 36
7 Hamiltonian Actions 39
7.1 Momentum Maps 39
7.2 First Obstruction for Momentum Maps 40
7.3 Second Obstruction for Momentum Maps 41
7.4 Killing the Second Obstruction 42
7.5 Obstructions Summarized 43
7.6 Flat Connections for Poisson Maps with Symplectic Target 44
IV Dual Pairs 47
8 Operator Algebras 47
8.1 Norm Topology and C* Algebras 47
8.2 Strong and Weak Topologies 48
8.3 Commutants 49
8.4 Dual Pairs 50
9 Dual Pairs in Poisson Geometry 51
9.1 Commutants in Poisson Geometry 51
9.2 Pairs of Symplectically Complete Foliations 52
9.3 Symplectic Dual Pairs 53
9.4 Morita Equivalence 54
9.5 Representation Equivalence 55
9.6 Topological Restrictions 56
10 Examples of Symplectic Realizations 59
10.1 Injective Realizations of T3 59
10.2 Submersive Realizations of T3 60
10.3 Complex Coordinates in Symplectic Geometry 62
10.4 The Harmonic Oscillator 63
10.5 A Dual Pair from Complex Geometry 65
V Generalized Functions 69
11 Group Algebras 69
11.1 Hopf Algebras 69
11.2 Commutative and Noncommutative Hopf Algebras 72
11.3 Algebras of Measures on Groups 73
11.4 Convolution of Functions 74
11.5 Distribution Group Algebras 76
CONTENTS vii
12 Densities 77
12.1 Densities 77
12.2 Intrinsic Lp Spaces 78
12.3 Generalized Sections 79
12.4 Poincare Birkhoff Witt Revisited 81
VI Groupoids 85
13 Groupoids 85
13.1 Definitions and Notation 85
13.2 Subgroupoids and Orbits 88
13.3 Examples of Groupoids 89
13.4 Groupoids with Structure 92
13.5 The Holonomy Groupoid of a Foliation 93
14 Groupoid Algebras 97
14.1 First Examples 97
14.2 Groupoid Algebras via Haar Systems 98
14.3 Intrinsic Groupoid Algebras 99
14.4 Groupoid Actions 101
14.5 Groupoid Algebra Actions 103
15 Extended Groupoid Algebras 105
15.1 Generalized Sections 105
15.2 Bisections 106
15.3 Actions of Bisections on Groupoids 107
15.4 Sections of the Normal Bundle 109
15.5 Left Invariant Vector Fields 110
VII Algebroids 113
16 Lie Algebroids 113
16.1 Definitions 113
16.2 First Examples of Lie Algebroids 114
16.3 Bundles of Lie Algebras 116
16.4 Integrability and Non Integrability 117
16.5 The Dual of a Lie Algebroid 119
16.6 Complex Lie Algebroids 120
17 Examples of Lie Algebroids 123
17.1 Atiyah Algebras 123
17.2 Connections on Transitive Lie Algebroids 124
17.3 The Lie Algebroid of a Poisson Manifold 125
17.4 Vector Fields Tangent to a Hypersurface 127
17.5 Vector Fields Tangent to the Boundary 128
viii CONTENTS
18 Differential Geometry for Lie Algebroids 131
18.1 The Exterior Differential Algebra of a Lie Algebroid 131
18.2 The Gerstenhaber Algebra of a Lie Algebroid 132
18.3 Poisson Structures on Lie Algebroids 134
18.4 Poisson Cohomology on Lie Algebroids 136
18.5 Infinitesimal Deformations of Poisson Structures 137
18.6 Obstructions to Formal Deformations 138
VIII Deformations of Algebras of Functions 141
19 Algebraic Deformation Theory 141
19.1 The Gerstenhaber Bracket 141
19.2 Hochschild Cohomology 142
19.3 Case of Functions on a Manifold 144
19.4 Deformations of Associative Products 144
19.5 Deformations of the Product of Functions 146
20 Weyl Algebras 149
20.1 The Moyal Weyl Product 149
20.2 The Moyal Weyl Product as an Operator Product 151
20.3 Affine Invariance of the Weyl Product 152
20.4 Derivations of Formal Weyl Algebras 152
20.5 Weyl Algebra Bundles 153
21 Deformation Quantization 155
21.1 Fedosov s Connection 155
21.2 Preparing the Connection 156
21.3 A Derivation and Filtration of the Weyl Algebra 158
21.4 Flattening the Connection 160
21.5 Classification of Deformation Quantizations 161
References 163
Index 175
|
| any_adam_object | 1 |
| author | Silva, Ana Cannas da 1968- Weinstein, Alan 1943- |
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| author_facet | Silva, Ana Cannas da 1968- Weinstein, Alan 1943- |
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| discipline | Mathematik |
| format | Book |
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| indexdate | 2024-07-09T18:32:09Z |
| institution | BVB |
| isbn | 0821809520 |
| language | English |
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| physical | XIV, 184 S. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | American Math. Soc. |
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| series | Berkeley mathematics lecture notes |
| series2 | Berkeley mathematics lecture notes |
| spelling | Silva, Ana Cannas da 1968- Verfasser (DE-588)122891333 aut Geometric models for noncommutative algebras Ana Cannas da Silva ; Alan Weinstein New York American Math. Soc. 1999 XIV, 184 S. txt rdacontent n rdamedia nc rdacarrier Berkeley mathematics lecture notes 10 Nichtkommutative Algebra Nichtkommutative Geometrie Nichtkommutative Algebra - Nichtkommutative Differentialgeometrie Weinstein, Alan 1943- Verfasser (DE-588)113532512 aut Berkeley mathematics lecture notes 10 (DE-604)BV011932272 10 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008633076&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Silva, Ana Cannas da 1968- Weinstein, Alan 1943- Geometric models for noncommutative algebras Berkeley mathematics lecture notes Nichtkommutative Algebra Nichtkommutative Geometrie Nichtkommutative Algebra - Nichtkommutative Differentialgeometrie |
| title | Geometric models for noncommutative algebras |
| title_auth | Geometric models for noncommutative algebras |
| title_exact_search | Geometric models for noncommutative algebras |
| title_full | Geometric models for noncommutative algebras Ana Cannas da Silva ; Alan Weinstein |
| title_fullStr | Geometric models for noncommutative algebras Ana Cannas da Silva ; Alan Weinstein |
| title_full_unstemmed | Geometric models for noncommutative algebras Ana Cannas da Silva ; Alan Weinstein |
| title_short | Geometric models for noncommutative algebras |
| title_sort | geometric models for noncommutative algebras |
| topic | Nichtkommutative Algebra Nichtkommutative Geometrie Nichtkommutative Algebra - Nichtkommutative Differentialgeometrie |
| topic_facet | Nichtkommutative Algebra Nichtkommutative Geometrie Nichtkommutative Algebra - Nichtkommutative Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008633076&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011932272 |
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