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Cannas da Silva A., Weinstein A. — Geometric Models for Noncommutative Algebra
Cannas da Silva A., Weinstein A. — Geometric Models for Noncommutative Algebra

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Название: Geometric Models for Noncommutative Algebra

Авторы: Cannas da Silva A., Weinstein A.

Аннотация:

The volume is based on a course, "Geometric Models for Noncommutative Algebras" taught by Professor Weinstein at Berkeley. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, for example, the commutative algebras associated to affine algebraic varieties, differentiable manifolds, topological spaces, and measure spaces. In this work, the authors discuss several types of geometric objects (in the usual sense of sets with structure) that are closely related to noncommutative algebras.
Central to the discussion are symplectic and Poisson manifolds, which arise when noncommutative algebras are obtained by deforming commutative algebras. The authors also give a detailed study of groupoids (whose role in noncommutative geometry has been stressed by Connes) as well as of Lie algebroids, the infinitesimal approximations to differentiable groupoids.
Featured are many interesting examples, applications, and exercises. The book starts with basic definitions and builds to (still) open questions. It is suitable for use as a graduate text. An extensive bibliography and index are included.


Язык: en

Рубрика: Математика/Алгебра/Абстрактная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 184

Добавлена в каталог: 28.02.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\alpha$-density      77
$\varphi$-related multivector field      30
$\varphi$-related vector field      29
Action of a groupoid bisection      107
Action of a Lie algebra      8
Action, coadjoint      39
Action, effective      8
Action, free      8
Action, groupoid      90 101
Action, groupoid algebra action      103
Action, Hamiltonian      39
Action, linear action      102
Action, right      8
Adjoint operation      47
Adjoint representation of $C^{\infty}(M)$      14
Admissible section      106
Albert, C.      108
Algebra, center      50
Algebra, dual pair      50
Algebra, factor      50
Algebra, von Neumann      49
Algebra, Weyl      149
Almeida, R.      118
Almost commutativity of the universal enveloping algebra      5
Almost complex structure      120
Almost Lie algebra      7
Almost Poisson structure      12
Almost symplectic manifold      20
Anchor map, definition      113
Anchor map, image      113
Anchor map, injective      115
Anchor map, kernel      113
Anchor map, surjective      123
Antipode      69
Arnold, V.      26
Associativity of the cup product      142
Associativity, associative structure      142
Associativity, coassociativity      70
Atiyah algebra      123
Atiyah sequence      123
Atiyah, M.      121 123
Baer groupoid      91
Baer, A.      91
Berezin, F.A.      81 82
Bisection      106 107
Bivector field      12
Borel groupoid      93
Borel, E.      152
Bracket on alternating multilinear maps      142
Bracket, cobracket      132
Bracket, commutator      150
Bracket, Dirac's quantum Poisson      152 157
Bracket, E-Gerstenhaber bracket      133
Bracket, Gerstenhaber      141
Bracket, Gerstenhaber bracket on Hochschild cohomology      143
Bracket, Lie — Poisson      11
Bracket, Poisson      149
Bracket, properties of $[\cdot, \cdot]_E$      133
Bracket, Schouten — Nijenhuis      12 135 144
Brandt groupoid      87
Brandt, W.      87 89
Bundle of groups      93
Bundle of Lie algebras      114 116
Bundle of Lie groups      116
C*-algebra, definition      47
C*-algebra, groupoid      98
Canonical 1-form      36
Canonical coordinate      14
Canonical Poisson relation      14
Canonical symplectic structure on a cotangent bundle      36 119
Cartan's magic formula      21 126 159
Cartan, E.      9
Casimir function      14 16 136
Cauchy — Riemann structure      121
Center      50
Chevalley cohomology      132 142
Chevalley complex      142
Chevalley — Eilenberg cohomology      142
Classical observables      151
Classical Yang — Baxter equation      135
Co-commutativity of the coproduct      82
Co-unit or coidentity      69
Coadjoint action      39
Coadjoint orbit      39
Coarse groupoid      87
Coboundary      43
Cobracket      132
Cochain      43
Cocycle      43
Cohomology, Chevalley      132 142
Cohomology, Chevalley — Eilenberg      142
Cohomology, de Rham      23
Cohomology, E-$\Pi$-cohomology      136
Cohomology, E-cohomology      132
Cohomology, Gel'fand — Fuks      162
Cohomology, Harrison      142
Cohomology, Hochschild      142
Cohomology, Lie algebra      132 142
Cohomology, Lie algebroid      132 136
Cohomology, Poisson      16
Cohomology, Poisson cohomology on a Lie algebroid      136
Cohomology, squaring map      139
Coisotropic      34
Collective function      66
Commutant, definition      49
Commutant, double      50
Commutant, double commutant theorem      50
Commutant, Poisson geometry      51
Commutative Hopf algebra      72
Compact operator      48
Compatible equivalence relation      34
Complete Poisson map      31
Complete symplectically complete foliation      53
Complex coordinates in symplectic geometry      62
Complex Lie algebroid      120
Complex structure      120
Complexes, Chevalley      142
Complexes, Hochschild      142
Complexified tangent bundle      62
Connection form      124
Connection on a Lie algebroid      124
Connection on a transitive Lie algebroid      124
Connection, Ehresmann      45
Connection, Fedosov's      155
Connection, flat      44 102
Connection, flattening      156 160
Connection, iterative construction      156 160
Connection, torsionless flat Poisson      155
Connes, A.      89
Conormal space      25
Convolution of functions      75
Convolution of measures      73 98
Coproduct, co-commutativity      82
Coproduct, definition      69
CR-functions      121
CR-structure      121
Cup product of multilinear maps      142
Cup product on Hochschild cohomology      143
Cup product, associativity      142
Cup product, supercommutativity      143
Curvature, effective      160
Curvature, form      124
Curvature, Lie algebroid      124
Curvature, Weyl      160
Darboux's theorem      20 21
Dazord, P.      108 117
Deformation of products      144
Deformation of products of functions      144 146
Deformation, Lie algebra      2
Deformation, obstructions      143
Deformation, quantization      6 144 146 155
Deformation, quantization of $R^{2n}$      151
Deformation, theory      141
Degenerate Lie algebra      26
Degree of $d_E$      131
Degree of $[\codt, \cdot]_E$      133
Degree of an E-differential form      131
Degree of an E-multivector field      132
Degree, multilinear map      141
Density, $\alpha$-density      77
Density, definition      77
Density, generalized      80
Derivation of a Poisson algebra      15
Derivation of a superalgebra      xv
Derivation, inner      15 157
Derivation, law for Dirac's bracket      152
Derivation, outer      15
Diagonal subgroupoid      89
Diffeological space      118
Differentiable groupoid      93
Dirac's quantum Poisson bracket      152 157
Dirac, P.      151 152 157 158
Distribution, compactly supported      79
Distribution, group algebra      76
Distributional section      79
Douady, A.      118
Dual of a Lie algebroid      119
Dual pair from complex geometry      65
Dual pair in algebra      50
Dual pair in Poisson geometry      51
Dual pair, symplectic      53
E-$\Pi$-cohomology      136
E-cohomology      132
E-differential form, definition      131
E-differential form, degree      131
E-differential form, E-k-form      131
E-differential form, homogeneous      131
E-differential form, properties of $d_E$      131
E-Gerstenhaber bracket      133
E-Lie derivative      113 133 137
E-multivector field      132
E-Poisson bivector field      135
E-symplectic form      135
E-symplectic structure      135
Effective curvature      160
Ehresmann, C.      45 92
Exact Poisson bivector field      137
Examples of a groupoid algebra      97
Examples of groupoid      89
Examples of Lie algebroids, Atiyah algebra      123
Examples of Lie algebroids, Atiyah sequence      123
Examples of Lie algebroids, bundle of Lie algebras      114
Examples of Lie algebroids, Lie algebra      114
Examples of Lie algebroids, Lie algebroid of a Lie groupoid      114
Examples of Lie algebroids, Lie algebroid of a Poisson manifold      125
Examples of Lie algebroids, Lie algebroid of a symplectic manifold      125
Examples of Lie algebroids, tangent bundle      114
Examples of Lie algebroids, transformation Lie algebroid      114
Examples of Lie algebroids, transitive Lie algebroid      123
Examples of Lie algebroids, Y-tangent bundle      127 128
Extended groupoid algebra      105
F-homotopic      94
Factor      50
Fedosov quantization      161
Fedosov, B.      154
Flat connection      45 102
Foliation, F-homotopy      94
Foliation, graph      94
Foliation, holonomy groupoid      93
Foliation, irrational      59
Foliation, leaf      93
Foliation, one-sided holonomy      95
Foliation, Reeb foliation      94
Foliation, symplectic      23
Foliation, symplectically complete      53
Formal adjoint      80
Formal Weyl algebra      150 158
Formality conjecture      144
Frobenius theorem      8 19 45
Fuchssteiner, B.      125
Function, Casimir      14 16 136
Function, collective      66
Function, Hamiltonian      40
Fundamental groupoid      89 94
Gel'fand, I.      48
General linear groupoid      102
Generalized density      80
Generalized section      79 105
Gerstenhaber algebra of a Lie algebroid      133
Gerstenhaber algebra, definition      133
Gerstenhaber algebra, E-Gerstenhaber algebra      133
Gerstenhaber algebra, E-Gerstenhaber bracket      133
Gerstenhaber, M.      134 141
Ginzburg, V.      57
Grading formal Weyl algebra      158
Grading universal enveloping algebra      3
Graph      94
Grothendieck, A.      89
Group algebra, distribution      76
Group algebra, distribution group algebra      76
Group algebra, group(-element)-like      74
Group algebra, list of structures      73
Group algebra, measure      73
Group, definition      xvi 87
Group, isotropy subgroup      89
Groupoid action      90 101
Groupoid action of a bisection      107
Groupoid as a category      xvi
Groupoid with structure      92
Groupoid, Baer groupoid      91
Groupoid, Borel      93
Groupoid, Brandt groupoid      87
Groupoid, bundle of groups      93
Groupoid, bundle of Lie algebras      116
Groupoid, bundle of Lie groups      116
Groupoid, C* -algebra      98
Groupoid, coarse groupoid      87
Groupoid, comparison with group      xvi
Groupoid, definition      85
Groupoid, diagonal subgroupoid      89
Groupoid, differentiable      93
Groupoid, ergodic theory      89
Groupoid, example of a groupoid algebra      97
Groupoid, examples      89
Groupoid, extended groupoid algebra      105
Groupoid, fundamental      89
Groupoid, fundamental groupoid      94
Groupoid, Galois theory      89
Groupoid, general linear groupoid      102
Groupoid, groupoid algebra      98
Groupoid, groupoid algebra action      103
Groupoid, groupoid algebra with coeficients in a vector bundle      103
Groupoid, Haar system      98
Groupoid, holonomy of a foliation      93
Groupoid, identity section      85
Groupoid, intrinsic groupoid algebra      99
Groupoid, inversion      86
Groupoid, isotropy subgroupoid      89
Groupoid, left invariant vector field      111
Groupoid, Lie      93
Groupoid, measurable      93
Groupoid, morphism      88
Groupoid, normal bundle of G(0)      100
Groupoid, orbit      89
Groupoid, pair groupoid      87 94
Groupoid, principal groupoid      90
Groupoid, product      85
Groupoid, product of groupoids      87
Groupoid, relation      88
Groupoid, representation      102
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