Elements of noncommutative geometry:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2001
|
| Series: | Birkhäuser advanced texts
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 685 S. graph. Darst. |
| ISBN: | 0817641246 3764341246 |
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MARC
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| 245 | 1 | 0 | |a Elements of noncommutative geometry |c José M. Gracia-Bondía ; Joseph C. Várilly ; Héctor Figueroa |
| 246 | 1 | 3 | |a Elements of noncommunitative geometry |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2001 | |
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Record in the Search Index
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| adam_text | Contents
Preface xi
I TOPOLOGY 1
1 Noncommutative Topology: Spaces 3
1.1 Continuous functions on a locally compact space ... 4
1.2 Characters and the Gelfand transformation 5
1.3 Trading spaces for algebras 9
1.4 Homotopy in noncommutative language 16
1.5 Exponentials and cohomology 17
1.6 Identifications and attachments 21
LA C* algebra basics 26
1.B Hopf algebras and Tannaka Krein duality 34
2 Noncommutative Topology: Vector Bundles 49
2.1 Vector bundles 49
2.2 The functor I 56
2.3 The Serre Swan theorem 59
2.4 Trading bundles for modules 60
2.5 C* modules 64
2.6 line bundles and the Bott projector 74
2.A Projective modules over unital rings 79
viii Contents
3 Some Aspects of X theory 83
3.1 Endomorphisms of C* modules 83
3.2 The Ko group 92
3.3 The importance of being halfexact 103
3.4 Asymptotic morphisms 110
3.5 The Moyal asymptotic morphism 113
3.6 Bott periodicity and the hexagon 120
3.7 The K functor 127
3.8 K theory of pre C* algebras 133
4 Fredholm Operators on C* modules 141
4.1 Fredholm operators and the Atiyah Janich theorem . . 141
4.2 Fredholm operators on C* modules 145
4.3 The generalized Fredholm index 148
4.4 The noncommutative Atiyah Janich theorem . . . . 156
4.5 Morita equivalence of C* algebras 159
n CALCULUS AND LINEAR ALGEBRA 169
5 Finite dimensional Clifford Algebras and Spinors 171
5.1 The eightfold way 171
5.2 Spin groups 180
5.3 Fock space representations 184
5.4 The exterior algebra viewpoint 195
5.5 Pfaffians and Gaussians 204
5.A Superalgebras 210
6 The Spin Representation 213
6.1 Infinite dimensional Clifford algebras 213
6.2 The infinitesimal spin representation revisited .... 218
6.3 The Shale Stinespring theorem 224
6.4 Charged fields 238
7 The Noncommutative Integral 251
7.1 A rapid course in Riemannian geometry 251
7.2 Laplacians 258
7.3 The Wodzicki residue 264
7.4 Spectral functions 272
7.5 The Dixmier trace 284
7.6 Connes trace theorem 293
7.A Pseudodifferential operators 298
7.B Homogeneous distributions 306
7.C Ideals of compact operators 310
Contents ix
8 Noncommutative Differential Calculi 319
8.1 Universal forms 320
8.2 Cycles and Fredholm modules 326
8.3 Connections and the Chern homomorphism 335
8.4 Hochschild homology and cohomology 343
8.5 The Hochschild Kostant Rosenberg Connes theorem . 355
HI GEOMETRY 367
9 Commutative Geometries 369
9.1 Clifford modules 370
9.2 Spinc structures: the algebraic way 372
9.3 Spin connections and Dirac operators 382
9.4 Analytical aspects of Dirac operators 391
9.5 O cycles and the eightfold way 399
9.A Spin geometry of the Riemann sphere 407
9.B The Hodge Dirac operator 423
10 Spectral Triples 429
10.1 Cyclic cohomology 429
10.2 Chern characters and entire cyclic cocycles 446
10.3 Tameness and regularity of spectral triples 460
10.4 Connes character formula 470
10.5 Terms and conditions for spin geometries 481
11 Connes Spin Manifold Theorem 487
11.1 Commutative spin geometries revisited 487
11.2 The construction of the volume form 491
11.3 The spin structure and the metric 500
11.4 The Dirac operator and the action functional .... 506
11.A The Riemann sphere as a spectral manifold 513
IV TRENDS 517
12 Tori 519
12.1 Crossed products 522
12.2 Structure of NC tori and the Moyal approach 526
12.3 Spin geometries on noncommutative tori 539
12.4 Morita equivalence and crossed products 549
13 Quantum Theory 557
13.1 The Dirac equation and the neutrino paradigm . . . . 558
13.2 Propagators 564
13.3 The classical Dyson expansion in QED 570
x Contents
13.4 The Rules 575
13.5 The quantum Dyson expansion 584
13.A On quantum field theory on noncommutative manifolds 592
14 Kreimer Connes Moscovici Algebras 597
14.1 The Connes Kreimer algebra of rooted trees 598
14.2 The Grossman Larson algebra of rooted trees .... 613
14.3 The Milnor Moore theorem 618
14.4 Duality in Hopf algebras 622
14.5 Hopf algebras of Feynman diagrams 627
14.6 Hopf algebras and diffeomorphism groups 630
14.7 Cyclic cohomology of Hopf algebras 635
References 641
Symbol Index 667
Subject Index 673
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| author | Gracia-Bondía, José M. Várilly, Joseph C. 1952- Figueroa, Héctor |
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| dewey-full | 516.3/5 |
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| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/5 |
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| language | English |
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| spelling | Gracia-Bondía, José M. Verfasser aut Elements of noncommutative geometry José M. Gracia-Bondía ; Joseph C. Várilly ; Héctor Figueroa Elements of noncommunitative geometry Boston [u.a.] Birkhäuser 2001 XVI, 685 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Birkhäuser advanced texts Nichtkommutative Differentialgeometrie Nichtkommutative Geometrie Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Nichtkommutative Differentialgeometrie (DE-588)4311174-9 s Várilly, Joseph C. 1952- Verfasser (DE-588)122620097 aut Figueroa, Héctor Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009204007&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gracia-Bondía, José M. Várilly, Joseph C. 1952- Figueroa, Héctor Elements of noncommutative geometry Nichtkommutative Differentialgeometrie Nichtkommutative Geometrie Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd |
| subject_GND | (DE-588)4311174-9 (DE-588)4171742-9 |
| title | Elements of noncommutative geometry |
| title_alt | Elements of noncommunitative geometry |
| title_auth | Elements of noncommutative geometry |
| title_exact_search | Elements of noncommutative geometry |
| title_full | Elements of noncommutative geometry José M. Gracia-Bondía ; Joseph C. Várilly ; Héctor Figueroa |
| title_fullStr | Elements of noncommutative geometry José M. Gracia-Bondía ; Joseph C. Várilly ; Héctor Figueroa |
| title_full_unstemmed | Elements of noncommutative geometry José M. Gracia-Bondía ; Joseph C. Várilly ; Héctor Figueroa |
| title_short | Elements of noncommutative geometry |
| title_sort | elements of noncommutative geometry |
| topic | Nichtkommutative Differentialgeometrie Nichtkommutative Geometrie Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Nichtkommutative Geometrie (DE-588)4171742-9 gnd |
| topic_facet | Nichtkommutative Differentialgeometrie Nichtkommutative Geometrie |
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