Geometric Models for Noncommutative Algebras

Рубрика: Библиотека, Математика 14 Март 2008

Название: Geometric Models for Noncommutative Algebras
Автор: Ana Cannas, Alan Weinstein2
Аннотация: Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, like the commutative algebras associated to ane algebraic varieties, dierentiable manifolds, topological spaces, and measure spaces. In this book, we discuss several types of geometric objects (in the usual sense of sets with structure) which are closely related to noncommutative algebras. Central to the discussion are symplectic and Poisson manifolds, which arisewhen noncommutative algebras are obtained by deforming commutative algebras. We also make a detailed study of groupoids, whose role in noncommutative geometry has been stressed by Connes, as well as of Lie algebroids, the in nitesimal approximations to dierentiable groupoids.
These notes are based on a topics course, Geometric Models for Noncommutative Algebras,” which one of us (A.W.) taught at Berkeley in the Spring of 1997. We would like to express our appreciation to Kevin Hartshorn for his participation in the early stages of the project { producing typed notes for many of the lectures. Henrique Bursztyn, who read preliminary versions of the notes, has provided us with innumerable suggestions of great value. We are also indebted to Johannes Huebschmann, Kirill Mackenzie, Daniel Markiewicz, Elisa Prato and Olga Radko for several useful commentaries or references. Finally, we would like to dedicate these notes to the memory of four friends and
colleagues who, sadly, passed away in 1998: Moshe Flato, K. Guruprasad, Andr e Lichnerowicz, and Stanis law Zakrzewski.

Ana Cannas da Silva
Alan Weinstein

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