  Let R be a commutative ring, and let A be a Poisson algebra over R. We
construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the
A-module of K\"ahler differentials of A depending naturally on A and the
Poisson bracket. This gives rise to suitable algebraic notions of Poisson
homology and cohomology for an arbitrary Poisson algebra. A geometric version
thereof includes the canonical homology and Poisson cohomology of a Poisson
manifold introduced by Brylinski, Koszul, and Lichnerowicz, and absorbes the
latter in standard homological algebra by expressing them as Tor and Ext
groups, respectively, over a suitable algebra of differential operators.
Furthermore, the Poisson structure determines a closed 2-form in the complex
computing Poisson cohomology. This 2-form generalizes the 2-form defining a
symplectic structure on a smooth manifold; moreover, the class of that 2-form
in Poisson cohomology generalizes the class in de Rham cohomology of a
symplectic structure on a smooth manifold and appears as a crucial ingredient
for the construction of suitable linear representations of A, viewed as a Lie
algebra; representations of this kind occur in quantum theory. To describe this
class and to construct the representations, we relate formal concepts of
connection and curvature generalizing the classical ones with extensions of Lie
algebras. We illustrate our results with a number of examples of Poisson
algebras and with a quantization procedure for a relativistic particle with
zero rest mass and spin zero.
