Brylinski, Jean-Luc A differential complex for Poisson manifolds. (English) Zbl 0634.58029 J. Differ. Geom. 28, No. 1, 93-114 (1988). 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 M. Wodzicki, Duke Math. J. 54, 641–647 (1987; Zbl 0635.18010), and the article by 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. Reviewer: J.-L. Brylinski Cited in 22 ReviewsCited in 128 Documents MSC: 58J10 Differential complexes 17B55 Homological methods in Lie (super)algebras 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B63 Poisson algebras 53D05 Symplectic manifolds (general theory) Keywords:differential operators; cyclic homology; Hochschild homology; Poisson brackets; symplectic manifold Citations:Zbl 0635.18010; Zbl 0646.58026 PDFBibTeX XMLCite \textit{J.-L. Brylinski}, J. Differ. Geom. 28, No. 1, 93--114 (1988; Zbl 0634.58029) Full Text: DOI Euclid