A differential complex for Poisson manifolds
type: Item
instance of: scholarly article
title: A differential complex for Poisson manifolds
zbMATH Open document ID: 0634.58029
published in: Journal of Differential Geometry
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jdg/1214442161
review text: We study a differential complex defined on any manifold for which Poisson brackets exist, for instance a symplectic manifold. The differential decreases the degree of differential forms by one. Such a complex was first introduced by J.-L. Koszul. We prove that for a symplectic manifold, this differential is \({}^*d^*\) (up to sign), where \({}^*\) is the duality involution on differential forms. It follows that the homology of the complex (which we call canonical homology) is equal, up to a shift of degrees, to the de Rham cohomology. We conjecture that, on compact symplectic manifolds, every de Rham cohomology class has a (nonunqiue) representative which is also closed for \({}^*d^*\). Our main application here is to the computation of the Hochschild and cyclic homologies of the algebra of P.D.O. on a \(C^{\infty}\), or complex-analytic manifold [see also \textit{M. Wodzicki}, Duke Math. J. 54, 641--647 (1987; Zbl 0635.18010), and the article by \textit{E. Getzler} and the author, K-Theory 1, 385--403 (1987; Zbl 0646.58026)]. It is hoped that these ideas will have further applications to the topology of symplectic manifolds, the study of infinitesimal Lie algebra actions, and Lie algebra homology.
Mathematics Subject Classification ID: 58J10
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 17B63
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 4032666
zbMATH Keywords: differential operators
zbMATH Keywords: cyclic homology
zbMATH Keywords: Hochschild homology
zbMATH Keywords: Poisson brackets
zbMATH Keywords: symplectic manifold
author: Jean-Luc Brylinski
MaRDI profile type: Publication
OpenAlex ID: W1547281537
Wikidata QID: Q115183643
DOI: 10.4310/JDG/1214442161
description: scientific article; zbMATH DE number 4032666