An introduction to noncommutative geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
European Mathematical Soc.
2006
|
| Series: | EMS series of lectures in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 113 S. |
| ISBN: | 3037190248 9783037190241 |
Staff View
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| 100 | 1 | |a Várilly, Joseph C. |d 1952- |e Verfasser |0 (DE-588)122620097 |4 aut | |
| 245 | 1 | 0 | |a An introduction to noncommutative geometry |c Joseph C. Varilly |
| 264 | 1 | |a Zürich |b European Mathematical Soc. |c 2006 | |
| 300 | |a VIII, 113 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a EMS series of lectures in mathematics | |
| 650 | 7 | |a Globale analyse |2 gtt | |
| 650 | 4 | |a Noncommutative differential geometry | |
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Record in the Search Index
| _version_ | 1804137227050024960 |
|---|---|
| adam_text | Contents
Introduction
..................................
vii
1
Commutative
geometry
from the
noncommutative
point of view
1
1.1
The
Gelfand-Naïmark
cofunctors
................... 2
1.2
The
Γ
functor
.............................. 4
1.3
Hermitian metrics and spinc structures
................. 5
1.4
The Dirac operator and the distance formula
.............. 8
2
Spectral triples on the Riemann sphere
11
2.1
Line bundles and the spinor bundle
................... 11
2.2
The Dirac operator on the sphere S2
.................. 13
2.3
Spinor harmonics and the spectrum of
Џ
................ 15
2.4
Twisted spinor modules
......................... 17
2.5
A reducible spectral triple
........................ 19
3
Real spectral striples: the axiomatic foundation
21
3.1
The data set
............................... 21
3.2
Infinitesimais
and dimension
...................... 23
3.3
The first-order condition
........................ 25
3.4
Smoothness of the algebra
....................... 25
3.5 Hochschild
cycles and orientation
................... 26
3.6
Finiteness of the
К
-cycle
........................ 27
3.7
Poincaré
duality and
К
-theory
..................... 28
3.8
The real structure
............................ 30
4
Geometries on the
noncommutative
torus
32
4.1
Algebras of Weyl operators
....................... 32
4.2
The algebra of the
noncommutative
torus
............... 34
4.3
The skeleton of the
noncommutative
torus
............... 36
4.4
A family of spin geometries on the torus
................ 38
5
The
noncommutative
integral
43
5.1
The Dixmier trace on infinitesimals
.................. 43
5.2
Pseudodifferential operators
...................... 46
5.3
The Wodzicki residue
.......................... 48
5.4
The trace theorem
............................ 49
5.5
Integrals and
zeta
residues
....................... 51
vi
Contents
Quantization and the tangent gronpoid
53
6.1
Moyal quantizers and the Moyal deformation
............. 53
6.2
Smooth groupoids
............................ 56
6.3
The tangent groupoid
.......................... 58
6.4
Moyal quantization as a continuity condition
.............. 60
6.5
The hexagon and the analytical index
.................. 62
6.6
Quantization and the index theorem
.................. 63
Equivalence of geometries
65
7.1
Unitary equivalence of spin geometries
................. 65
7.2
Morita equivalence and connections
.................. 67
7.3
Vector bundles over
noncommutative tori
............... 70
7.4
Morita-equivalent
toral
geometries
................... 72
7.5
Gauge potentials
............................ 74
Action functionals
75
8.1
Algebra automorphisms and the metric
................. 75
8.2
The fermionic action
.......................... 76
8.3
The spectral action principle
...................... 78
8.4
Spectral densities and asymptotics
................... 79
Epilogue: new directions
85
9.1
Noncommutative
field theories
..................... 85
9.2
Isospectral
deformations
........................ 86
9.3
Geometries with quantum group symmetry
.............. 90
9.4
Other developments
........................... 93
Bibliography
................................. 97
Index
..................................... 109
|
| adam_txt |
Contents
Introduction
.
vii
1
Commutative
geometry
from the
noncommutative
point of view
1
1.1
The
Gelfand-Naïmark
cofunctors
. 2
1.2
The
Γ
functor
. 4
1.3
Hermitian metrics and spinc structures
. 5
1.4
The Dirac operator and the distance formula
. 8
2
Spectral triples on the Riemann sphere
11
2.1
Line bundles and the spinor bundle
. 11
2.2
The Dirac operator on the sphere S2
. 13
2.3
Spinor harmonics and the spectrum of
Џ
. 15
2.4
Twisted spinor modules
. 17
2.5
A reducible spectral triple
. 19
3
Real spectral striples: the axiomatic foundation
21
3.1
The data set
. 21
3.2
Infinitesimais
and dimension
. 23
3.3
The first-order condition
. 25
3.4
Smoothness of the algebra
. 25
3.5 Hochschild
cycles and orientation
. 26
3.6
Finiteness of the
К
-cycle
. 27
3.7
Poincaré
duality and
К
-theory
. 28
3.8
The real structure
. 30
4
Geometries on the
noncommutative
torus
32
4.1
Algebras of Weyl operators
. 32
4.2
The algebra of the
noncommutative
torus
. 34
4.3
The skeleton of the
noncommutative
torus
. 36
4.4
A family of spin geometries on the torus
. 38
5
The
noncommutative
integral
43
5.1
The Dixmier trace on infinitesimals
. 43
5.2
Pseudodifferential operators
. 46
5.3
The Wodzicki residue
. 48
5.4
The trace theorem
. 49
5.5
Integrals and
zeta
residues
. 51
vi
Contents
Quantization and the tangent gronpoid
53
6.1
Moyal quantizers and the Moyal deformation
. 53
6.2
Smooth groupoids
. 56
6.3
The tangent groupoid
. 58
6.4
Moyal quantization as a continuity condition
. 60
6.5
The hexagon and the analytical index
. 62
6.6
Quantization and the index theorem
. 63
Equivalence of geometries
65
7.1
Unitary equivalence of spin geometries
. 65
7.2
Morita equivalence and connections
. 67
7.3
Vector bundles over
noncommutative tori
. 70
7.4
Morita-equivalent
toral
geometries
. 72
7.5
Gauge potentials
. 74
Action functionals
75
8.1
Algebra automorphisms and the metric
. 75
8.2
The fermionic action
. 76
8.3
The spectral action principle
. 78
8.4
Spectral densities and asymptotics
. 79
Epilogue: new directions
85
9.1
Noncommutative
field theories
. 85
9.2
Isospectral
deformations
. 86
9.3
Geometries with quantum group symmetry
. 90
9.4
Other developments
. 93
Bibliography
. 97
Index
. 109 |
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| author | Várilly, Joseph C. 1952- |
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| callnumber-subject | QC - Physics |
| classification_rvk | SK 240 |
| ctrlnum | (OCoLC)255573700 (DE-599)BVBBV023012079 |
| dewey-full | 516.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 |
| dewey-search | 516.36 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| spelling | Várilly, Joseph C. 1952- Verfasser (DE-588)122620097 aut An introduction to noncommutative geometry Joseph C. Varilly Zürich European Mathematical Soc. 2006 VIII, 113 S. txt rdacontent n rdamedia nc rdacarrier EMS series of lectures in mathematics Globale analyse gtt Noncommutative differential geometry Nichtkommutative Geometrie (DE-588)4171742-9 gnd rswk-swf 1\p (DE-588)1071861417 Konferenzschrift 1997 Monsaraz ; Lisboa gnd-content Nichtkommutative Geometrie (DE-588)4171742-9 s DE-604 Erscheint auch als Online-Ausgabe Várilly, Joseph C. An introduction to noncommutative geometry 978-3-03719-524-6 (DE-604)BV036706024 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016216285&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Várilly, Joseph C. 1952- An introduction to noncommutative geometry Globale analyse gtt Noncommutative differential geometry Nichtkommutative Geometrie (DE-588)4171742-9 gnd |
| subject_GND | (DE-588)4171742-9 (DE-588)1071861417 |
| title | An introduction to noncommutative geometry |
| title_auth | An introduction to noncommutative geometry |
| title_exact_search | An introduction to noncommutative geometry |
| title_exact_search_txtP | An introduction to noncommutative geometry |
| title_full | An introduction to noncommutative geometry Joseph C. Varilly |
| title_fullStr | An introduction to noncommutative geometry Joseph C. Varilly |
| title_full_unstemmed | An introduction to noncommutative geometry Joseph C. Varilly |
| title_short | An introduction to noncommutative geometry |
| title_sort | an introduction to noncommutative geometry |
| topic | Globale analyse gtt Noncommutative differential geometry Nichtkommutative Geometrie (DE-588)4171742-9 gnd |
| topic_facet | Globale analyse Noncommutative differential geometry Nichtkommutative Geometrie Konferenzschrift 1997 Monsaraz ; Lisboa |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016216285&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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