A differential complex for Poisson manifolds.

*(English)*Zbl 0634.58029We 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.

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 |