Poisson cohomology and quantization

Abstract

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 Kähler 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.