Geometric models for noncommutative algebras. (English) Zbl 1135.58300

Berkeley Mathematics Lecture Notes 10. Providence, RI: American Mathematical Society (AMS); Berkekey, CA: Berkeley Center for Pure and Applied Mathematics (ISBN 0-8218-0952-0/pbk). xiv, 184 p. (1999).
Publisher’s description: 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.


58B34 Noncommutative geometry (à la Connes)
58B32 Geometry of quantum groups
16S80 Deformations of associative rings
17B63 Poisson algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D55 Deformation quantization, star products