On Poisson structures of differentiable manifold and its non-commutative properties

Abstract

Our main aim is to analyse geometric application for non-commutative Poission structure on Manifold. There exists a relationship between Poisson geometry and deformation theory which was initiated by Ezra Getzler. We derive properties of deformation quantization of a noncommutative Poisson structure with objective to compute infinitesimal deformation on Poisson manifold. J. Block and Ping Xu introduced the notion of non-commutative Poisson structure on an associative algebra. The main focus here is to reformulate its impact on symplectic reflection algebra. It is closely related to Horchs-child algebra based on computations of Gerstenhaber bracket. M. Gertenheber established its connection with the deformation theory of an associative algebra A as well as the Hochschild cohomology. A. Gerstenhaber studied deformation theory on a topological algebra where Gerstenhaber bracket is used in defining non-commutative Poisson structures compatible with differential forms on Hochs child cohomology X. Tang introduced the notion of Lie bracket on the Hochs child cohomology. We show the set of noncommutative Poisson structures on an algebra A has one to one correspondence with the set of infinitesimal deformations of A. We integrate the infinitesimal deformation associated to a noncommutative Poisson structure to a real one which is closely related to the notion of deformation quantization in mathematical physics.