J reine angew Math 408(1990) 57^113 Joumal fur die геше und

angewandte Mathematik

© Walter de Gruyter Berlin New York 1990

Poisson cohomology and quantization

By Johannes Huebschmann*) at Heidelberg

0 . Introduction

The concept of a Poisson manifold is currently of much interest, see e g Berger [7], Bhaskara-Viswanath [8], Braconnier [10], [11], Brylinski [12], Coste-Dazord- Weinstein [17], Conn [18], [19], De Wilde-Le Compte [21], Gelfand-Dorfman [27]—[29], Karasev [44], Kassel [45], Kosmann-Schwarzbach and Magn [111], Koszul [52], Lichnerowicz [56]—[62], Magri-Morosi [66], Magri-Morosi-Ragnisco [67], Mikami-Weinstein [74], Stasheff [95], [96], Tulczyjew [97], [98], Vinogradov- Krasil'shchik [100], Weinstein [102]—[106] A Poisson structure on a smooth manifold iV IS a Lie bracket { , } on the (multiplicative) algebra of smooth functions on N satisfying the additional condition

{ fg , h } =J { g , h } + {f,h}g

More generally, an algebra A over a commutative ring R together with a Lie bracket { , } on v4 satisfying the formal analogue of the above additional condition is called a Poisson algebra The significance of Poisson structures in physics is classical, see Lie [63] and Dirac [22], [23]

For a symplectic manifold {N, a), the rule {/, g} = ö-(Xj, Z^), where Xf is the Hamiltoman vector field corresponding to /, defines a Poisson structure on N However, this is not the only way in which a Poisson structure on a manifold arises, see e g Weinstein [103]

In [58] Lichnerowicz introduced what he called "Poisson cohomology" of a Poisson manifold, when the Poisson structure comes from a symplectic one, this Poisson cohomology coincides with de Rham cohomology [58], cf 3 15 below In [52] Koszul introduced a notion of homology for a Poisson manifold which was christened "canonical homology" by Brylinski [12] In the present paper we introduce sponding notions of Poisson homology and cohomology for an arbitrary Poisson algebra {A, { , }) We now explain briefly and informally our approach

* ) Supported by the Deutsche Forschungsgemeinschaft under a Heisenberg grant