Poisson cohomology and quantization. (English) Zbl 0699.53037

Huebschmann, Johannes

J. Reine Angew. Math. 408, 57-113 (1990).

Let R be a commutative ring, and let (A,\(\{\), \(\})\) be a Poisson algebra over R. It is shown in the paper that the latter determines a structure of an (R,A)-Lie algebra in the sense of G. S. Rinehart [Trans. Am. Math. Soc. 108, 195-222 (1963; Zbl 0113.262)] on the A-module of Kähler differentials of A depending naturally on A and \(\{\), \(\}\). This gives rise to suitable algebraic notions of Poisson homology and cohomology for an arbitrary Poisson algebra. A smooth 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. For the Poisson algebra of smooth functions on a smooth finite dimensional Poisson manifold the smooth and algebraic notions of Poisson cohomology are isomorphic. For an arbitrary Poisson algebra (A,\(\{\), \(\})\), the Poisson structure determines a closed 2-form \(\pi\) \(\{\), \(\}\) in the complex computing Poisson cohomology. This 2-form generalizes the 2-form \(\sigma\) defining a symplectic structure on a smooth manifold N; moreover, the class of \(\pi\) \(\{\), \(\}\) in Poisson cohomology generalizes the class \([\sigma]\in H^ 2_{deRham}(N,R)\) of a symplectic structure \(\sigma\) on a smooth manifold N 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, formal concepts of connection and curvature generalizing the classical ones are related with extensions of Lie algebras. These results are illustrated with a number of examples of Poisson algebras and with a quantization procedure for a relativistic particle with zero rest mass and spin zero.

Reviewer: J.Huebschmann

MSC:

53C05	Connections (general theory)
17B56	Cohomology of Lie (super)algebras
53D50	Geometric quantization
17B65	Infinite-dimensional Lie (super)algebras
81S10	Geometry and quantization, symplectic methods
53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
17B55	Homological methods in Lie (super)algebras

Keywords:

Poisson algebra; Kähler differentials; Poisson homology; Poisson manifold; symplectic structure; connection; curvature

Citations:

Zbl 0113.262