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Cannas da Silva A., Weinstein A. — Geometric Models for Noncommutative Algebra
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Название: Geometric Models for Noncommutative AlgebraАвторы: 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.
Язык: Рубрика: Математика /Алгебра /Абстрактная алгебра /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1998Количество страниц: 184Добавлена в каталог: 28.02.2005Операции: Положить на полку |
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Groupoid, set of composable pairs 85 Groupoid, source 85 Groupoid, subgroupoid 88 Groupoid, symplectic groupoid 127 Groupoid, target 85 Groupoid, topological 92 Groupoid, transformation groupoid 90 Groupoid, transitive 89 Groupoid, trivial 87 Groupoid, Weyl groupoid 91 Groupoid, wide subgroupoid 88 Haar measure 74 Haar system 92 98 Haefliger, A. 95 Hamiltonian action 39 44 Hamiltonian function 40 Hamiltonian set of hamiltonian vector fields 40 Hamiltonian vector field 14 17 20 Hamiltonian vector field on a Lie algebroid 136 Hamiltonian, strongly 44 Hamiltonian, weakly 44 harmonic oscillator 63 Harrison cohomology 142 Heisenberg algebra 150 Hochschild cohomology, action of symmetric groups 143 Hochschild cohomology, algebraic structure 143 Hochschild cohomology, cup product 143 Hochschild cohomology, decomposition 143 Hochschild cohomology, definition 142 Hochschild cohomology, Gerstenhaber 134 Hochschild cohomology, Gerstenhaber bracket 143 Hochschild cohomology, squaring map 143 Hochschild complex 142 Hodge decomposition 156 Holonomy on a regular Poisson manifold 24 Holonomy, definition 45 93 Holonomy, description 93 Holonomy, equivalence relation 93 Holonomy, flat connection 104 Holonomy, groupoid 115 Holonomy, groupoid of a foliation 93 94 Holonomy, one-sided 95 Homogeneous, E-differential form 131 Homogeneous, E-multivector field 132 Hopf algebra, antipode 69 Hopf algebra, associativity of multiplication 70 Hopf algebra, co-unit or coidentity 69 Hopf algebra, coassociativity of comultiplication 70 Hopf algebra, commutative 72 Hopf algebra, comultiplication 69 Hopf algebra, definition 69 Hopf algebra, examples 69 Hopf algebra, multiplication 69 Hopf algebra, noncommutative 72 Hopf algebra, Poisson 72 Hopf algebra, quantum group 72 Hopf algebra, relation to groups 72 Hopf algebra, unit or identity 69 Infinitesimal deformation of a Poisson structure 137 Infinitesimal deformation, obstructions to continuing 138 Infinitesimal deformation, trivial 138 Infinitesimal neighborhood xiii 155 Inner derivation 15 157 Integrability conditions 21 Integrability Jacobi identity 15 Integrability Newlander — Nirenberg theorem 120 Integrability of Lie algebroids 114 117 Intrinsic 78 Intrinsic groupoid algebra 99 Irrational foliation 59 Isotropic 34 Isotropy, algebroid 113 Isotropy, subgroup 89 Isotropy, subgroupoid 89 Jacobi identity for elements of 142 Jacobi identity, definition 6 Jacobi identity, deformation of products 145 146 Jacobi identity, jacobiator 7 13 145 Jacobi identity, Poisson structure 12 Jacobi identity, super-Jacobi identity 133 141 142 Jacobi, C. 15 Jacobiator 7 13 145 Karasev, M. 33 Keel, S. 89 Kirillov, A. 23 Kontsevich, M. 144 Kostant, B. 42 Lakoff, G. xiii Lazard, M. 118 Leaf, breaking the leaves 27 Leaf, definition 93 Left invariant measure 74 Left invariant vector field 111 Leibniz identity for abstract products 145 Leibniz identity in the Weyl algebra 152 Leibniz identity, definition 6 Leibniz identity, deformation of products 146 Leibniz identity, Lie algebroid 113 Leibniz identity, Lie algebroid of a Poisson manifold 126 Leibniz identity, super-Leibniz identity 133 Lie algebra, action 8 Lie algebra, almost 7 Lie algebra, bundle of Lie algebras 114 Lie algebra, cohomology 132 142 Lie algebra, deformation 2 Lie algebra, degenerate 26 Lie algebra, non-degenerate 26 Lie algebra, representation 17 Lie algebra, structure constant 8 Lie algebra, super-Lie algebra 133 Lie algebra, transverse 24 Lie algebroid as a supermanifold 131 Lie algebroid of a Lie groupoid 114 Lie algebroid of a Poisson manifold 125 Lie algebroid of a symplectic manifold 125 Lie algebroid, cohomology 132 Lie algebroid, complex Lie algebroid 120 Lie algebroid, connection 124 Lie algebroid, curvature 124 Lie algebroid, definition 113 Lie algebroid, degree of an E-form 131 Lie algebroid, differential complex 136 Lie algebroid, differential geometry 131 Lie algebroid, dual 119 Lie algebroid, E- 136 Lie algebroid, E-differential form 131 Lie algebroid, E-Gerstenhaber bracket 133 Lie algebroid, E-k-form 131 Lie algebroid, E-Lie derivative 113 133 137 Lie algebroid, E-Poisson bivector field 135 Lie algebroid, E-symplectic form 135 Lie algebroid, E-symplectic structure 135 Lie algebroid, examples 114 123 Lie algebroid, exterior differential algebra 131 Lie algebroid, Gerstenhaber algebra 132 133 Lie algebroid, hamiltonian vector field 136 Lie algebroid, history 115 Lie algebroid, homogeneous E-form 131 Lie algebroid, integrability 117 Lie algebroid, Leibniz identity 113 Lie algebroid, Lie-Poisson bracket 119 Lie algebroid, morphism 120 Lie algebroid, multivector field 132 Lie algebroid, orbits 113 Lie algebroid, Poisson bracket 136 Lie algebroid, Poisson cohomology 136 Lie algebroid, Poisson structure 134 Lie algebroid, Poisson vector field 137 Lie algebroid, properties of 131 Lie algebroid, squaring map 139 Lie bracket 6 Lie derivative, Cartan's magic formula 21 126 159 Lie derivative, Lie algebroid 113 133 137 Lie group, modular character 75 Lie group, modular function 75 Lie group, unimodular 75 Lie groupoid, definition 93 Lie groupoid, Lie algebroid of a 114 Lie — Poisson bracket, definition 11 Lie — Poisson bracket, dual of a Lie algebroid 119 Lie — Poisson manifold, definition 11 Lie — Poisson manifold, hamiltonian action 39 Lie — Poisson manifold, Jacobi identity 13 Lie — Poisson manifold, Lie — Poisson bracket 11 Lie — Poisson manifold, normal form 20 Lie — Poisson manifold, rank 17 Lie's theorem 17 Lie, S. 8 9 17 40 Linear Poisson structure 14 Linearizable Poisson structure 25 Linearized Poisson structure 24 Liouville vector field 137 Local bisection 107 Lu, J.-H. 57 Mackenzie, K. 118 Mackey, G. 89 Maximal torus 91 Measurable groupoid 93 Measure, algebras of measures on groups 73 Measure, class 93 Measure, group algebra 73 Measure, Haar measure 74 Measure, left-invariant 74 Measure, quasi-invariant 74 Melrose, R. 127 Modular character 75 Modular function 75 Moebius band 94 Molino, P. 118 Moment map vs. momentum map 101 Moment map, groupoid action 101 Momentum map for a group action 42 Momentum map vs. moment map 101 Momentum map, definition 39 40 Momentum map, equivariance 42 Momentum map, first obstruction 40 43 Momentum map, second obstruction 41—43 Momentum, phase space xv Mori, S. 89 Morita equivalence 55 56 Morphism of groupoids 88 Moyal — Weyl product 149—151 Multilinear maps, brackets 142 Multilinear maps, symmetric 142 Multivector field, 30 Multivector field, definition 12 Multivector field, Lie algebroid 132 Naimark, M. 48 Newlander — Nirenberg theorem 120 Newton's method 156 Non-degenerate Lie algebra 26 Norm topology 47 Novikov, S. 95 Nunez, R. xiii Obstruction to a holomorphic connection 121 Obstruction to a momentum map 40—43 Obstruction to deformation of a Poisson structure 138 Obstruction to the Jacobi identity 7 Odd, differential forms 78 Odd, vector field xv One-sided holonomy 95 Operator, bounded 47 Operator, compact 48 Operator, product 151 Orbit of a Lie algebroid 113 Orbit, coadjoint 39 Orbit, groupoid 89 Outer derivation 15 Pair groupoid 87 94 Palais, R. 118 Permutation group 143 Phase space xv Planck's constant 146 Poincare — Birkhoff — Witt theorem and group algebras 81 Poincare — Birkhoff — Witt theorem, discussion 7 Poincare — Birkhoff — Witt theorem, proof 9 Poincare — Birkhoff — Witt theorem, statement 5 Poisson algebra 6 Poisson automorphism, definition 29 Poisson automorphism, group of Poisson automorphisms 29 Poisson bivector field on a Lie algebra 135 Poisson bivector field, definition 135 Poisson bivector field, E-Poisson bivector field 135 Poisson bivector field, exact 137 Poisson bracket, differential operators 149 Poisson bracket, Lie algebroid 136 Poisson bracket, universal enveloping algebra 5 Poisson cohomology on a Lie algebroid 136 Poisson cohomology, 0-th 16 Poisson cohomology, first 16 Poisson cohomology, symplectic case 23 Poisson Hopf algebra 72 Poisson Lie group, definition 72 Poisson Lie group, non-linearizability 26 Poisson manifold, almost symplectic 20 Poisson manifold, coisotropic 34 Poisson manifold, definition 12 Poisson manifold, Lie algebroid of a 125 Poisson manifold, regular 17 Poisson manifold, symplectic 20 Poisson map, complete 31 Poisson map, definition 29 Poisson quotient 34 Poisson relation 34 Poisson structure on a Lie algebroid 134 Poisson structure, almost 12 Poisson structure, canonical coordinates 13 Poisson structure, definition 12 Poisson structure, formal deformation 137 138 Poisson structure, infinitesimal deformation 137 Poisson structure, Lie's theorem 17 Poisson structure, linear 14 Poisson structure, linearization 25 Poisson structure, linearized 24 Poisson structure, normal form 17 Poisson structure, obstructions to deformation 138 Poisson structure, structure functions 13 Poisson structure, transverse 24 Poisson submanifold 36 Poisson tensor 12 Poisson vector field, definition 15 Poisson vector field, Lie algebroid 137 Poisson vector field, set of hamiltonian vector fields 40 Poisson's theorem 15 19 Poisson, S.-D. 14 15 Poisson-algebra homomorphism 29 Pradines, J. 115 Principal groupoid 90 Product of groupoids 87 Product, coproduct 69 Product, star 151 Product, von Neumann 151 Quantization, classification 161 Quantization, deformation 155 Quantization, Fedosov 161 Quantization, patching from local 155 Quantum group 72 Quantum operator 151 Quasi-invariant measure 74 Rank of a Lie algebra 18
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