Lectures on the geometry of Poisson manifolds
type: Item
instance of: scholarly article
title: Lectures on the geometry of Poisson manifolds
zbMATH Open document ID: 0810.53019
author: Izu Vaisman
published in: Progress in Mathematics
publication date: 1994-04-14T00:00:00Z
review text: A long overdue systematic self-contained presentation of the subject. The book presents the basic theory including most results of the fundamental article by \textit{A. Weinstein} [J. Differ. Geom. 18, 523-557 (1983; Zbl 0524.58011)] and outlines some of the further developments. In principle the only prerequisite is calculus on manifolds, however for understanding background and purpose of the theory some familiarity with symplectic geometry and its application is necessary. Beginning with an invariant definition of the Schouten-Nijenhuis bracket fundamental properties of Poisson manifolds are developed: the symplectic foliation, the fundamental splitting, regularity and Poisson connections. The standard examples linear Poisson structure on \(\mathbb{R}^ n\) and Lie Poisson structure on the dual Lie algebra of Lie groups and coadjoint orbits and the embedding into an almost symplectic manifold with Dirac brackets are presented. A chapter on Poisson calculus includes contravariant exterior differentiation and graded Lie algebra structures on Poisson manifolds. Chapters on Poisson cohomology and its spectral sequence, quantization and deformations, a discussion of Poisson morphisms, group actions, momenta and reduction of Poisson manifolds follow. The final three chapters deal with symplectic realizations, realizations by symplectic groupoids and with Poisson-Lie groups. All major results include complete proofs, however no physical applications are given. The book serves well as an introduction and an overview of the subject and a long list of references helps with further studies.
reviewed by: Christian Günther
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 53-02
zbMATH DE Number: 551044
zbMATH Keywords: Schouten-Nijenhuis bracket
zbMATH Keywords: Poisson manifolds
zbMATH Keywords: symplectic foliation
zbMATH Keywords: fundamental splitting
zbMATH Keywords: Poisson connections
zbMATH Keywords: graded Lie algebra structures
zbMATH Keywords: Poisson cohomology
zbMATH Keywords: quantization
zbMATH Keywords: Poisson morphisms
zbMATH Keywords: symplectic groupoids
zbMATH Keywords: Poisson-Lie groups
zbMATH Keywords: bibliography
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 551044