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Geometric Models for Noncommutative Algebras
Ana Cannas da Silva and Alan Weinstein, University of California, Berkeley, CA
A co-publication of the AMS and Center for Pure and Applied Mathematics at University of California, Berkeley.
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Berkeley Mathematical Lecture Notes
1999; 184 pp; softcover
Volume: 10
Reprint/Revision History:
reprinted 2000
ISBN-10: 0-8218-0952-0
ISBN-13: 978-0-8218-0952-5
List Price: US$23
Member Price: US$18.40
Order Code: BMLN/10
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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.

This series is jointly published between the AMS and the Center for Pure and Applied Mathematics at the University of California at Berkeley (UCB CPAM).

Readership

Graduate students and research mathematicians working in symplectic and Poisson geometry and its applications to classical and quantum physics and to noncommutative algebra.

Reviews

"The only available monograph on the topic, it is readable and stimulating and contains material which otherwise is only spread over the literature ... The book is well done ... It should be an essential purchase for mathematics libraries and is likely to be a standard reference for years to come, providing an introduction to an attractive area of further research."

-- Mathematical Reviews

Table of Contents

Universal enveloping algebras
Poisson Geometry
Poisson Category
Dual Pairs
Generalized Functions
Algebroids
Deformations of Algebras of Functions
  • References
  • Index
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