Geometric Models for Noncommutative Algebras

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Brand new. We distribute directly for the publisher. 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. Read more Show Less

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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.

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Product Details

Table of Contents

Preface
Introduction
I Universal Enveloping Algebras 1
1 Algebraic Constructions 1
2 The Poincare-Birkhoff-Witt Theorem 5
II Poisson Geometry 11
3 Poisson Structures 11
4 Normal Forms 17
5 Local Poisson Geometry 23
III Poisson Category 29
6 Poisson Maps 29
7 Hamiltonian Actions 39
IV Dual Pairs 47
8 Operator Algebras 47
9 Dual Pairs in Poisson Geometry 51
10 Examples of Symplectic Realizations 59
V Generalized Functions 69
11 Group Algebras 69
12 Densities 77
VI Groupoids 85
13 Groupoids 85
14 Groupoid Algebras 97
15 Extended Groupoid Algebras 105
VII Algebroids 113
16 Lie Algebroids 113
17 Examples of Lie Algebroids 123
18 Differential Geometry for Lie Algebroids 131
VIII Deformations of Algebras of Functions 141
19 Algebraic Deformation Theory 141
20 Weyl Algebras 149
21 Deformation Quantization 155
References 163
Index 175
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