Geometric Models for Noncommutative AlgebrasThe 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. |
Contents
Universal Enveloping Algebras 1 Algebraic Constructions | 1 |
2 The PoincareBirkhoffWitt Theorem | 5 |
Poisson Geometry 3 Poisson Structures | 11 |
4 Normal Forms | 17 |
5 Local Poisson Geometry | 23 |
Poisson Category 6 Poisson Maps | 29 |
7 Hamiltonian Actions | 39 |
Dual Pairs 8 Operator Algebras | 47 |
12 Densities | 77 |
Groupoids 13 Groupoids | 85 |
14 Groupoid Algebras | 97 |
15 Extended Groupoid Algebras | 105 |
Algebroids 16 Lie Algebroids | 113 |
17 Examples of Lie Algebroids | 123 |
18 Differential Geometry for Lie Algebroids | 131 |
Deformations of Algebras of Functions | 141 |
9 Dual Pairs in Poisson Geometry | 51 |
10 Examples of Symplectic Realizations | 59 |
Generalized Functions 11 Group Algebras | 69 |
20 Weyl Algebras | 149 |
21 Deformation Quantization | 155 |
Common terms and phrases
action of G algebra g algebra structure an+1 bisection bivector bracket operation bundle of Lie called canonical coordinates commutative compactly supported complete Poisson map corresponding defined deformation quantization denote derivation dual pair element equation Example Exercise fiber flat connection foliation formal Gerstenhaber group algebra groupoid algebra groupoid G hamiltonian vector fields Hausdorff hence holonomy Hopf algebra induced integrable invariant isomorphism Jacobi identity Leibniz identity Lie algebra homomorphism Lie algebroid Lie group Lie groupoid linear m₁ Math momentum map Morita equivalence morphism multiplication noncommutative orbits Poincaré-Birkhoff-Witt theorem pointwise Poisson bracket Poisson cohomology Poisson geometry Poisson manifold Poisson structure Poisson vector fields polynomial quantum quotient Remark representation Section smooth subalgebra subgroupoid subset symmetric symplectic leaves symplectic manifold symplectic realization symplectic structure tangent bundle tensor topology trivial vector bundle vector space Weinstein Weyl algebra მ მ
Popular passages
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Page 164 - Cahen, M, Gutt, S. and Rawnsley, J., Tangential star products for the coadjoint Poisson structure, Comm. Math. Phys. 180 (1996), 99-108.
Page 166 - Emmrich, C., and Weinstein, A., Geometry of the transport equation in multicomponent WKB approximations, Comm. Math. Phys. 176 (1996), 701-711.
Page 165 - Dazord, P., and Weinstein, A., eds., Symplectic Geometry, Groupoids, and Integrable Systems, Seminaire Sud- Rhodanien de Geometric a Berkeley (1989), Springer-MSRI Series, 1991.
Page 165 - Espaces fibres en algebres de Lie et en groupes, Invent. Math. 1 (1966), 133-151.
Page 165 - Lance, translated from the second French edition by F. Jellett, North-Holland Mathematical Library 27, North-Holland Publishing Co., Amsterdam-New York, 1981.