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A differential complex for Poisson manifolds. (English) Zbl 0634.58029
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

##### 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)
##### Citations:
Zbl 0635.18010; Zbl 0646.58026
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