Cohomology Structures of A Poisson Algebra: II

TL;DR

This paper develops a new bicomplex framework for Poisson algebra cohomology, linking it to derived functors and cohomological tools like Lie and Hochschild cohomology for computational purposes.

Contribution

It introduces a bicomplex of free Poisson modules and relates Poisson cohomology to derived functors, enhancing computational methods.

Findings

Established a bicomplex for Poisson modules

Connected Poisson cohomology with Yoneda-extension groups

Provided a method to compute Poisson cohomology via Lie and Hochschild cohomology

Abstract

We introduce for any Poisson algebra a bicomplex of free Poisson modules, and use it to show that the Poisson cohomology theory introduced in the paper "[M. Flato, M. Gerstenhaber and A. A. Voronov, Cohomology and Deformation of Leibniz Pairs, Lett. Math. Phys. 34 (1995) 77--90]" is given by certain derived functor. Moreover, by constructing a long exact sequence connecting Poisson cohomology groups and Yoneda-extension groups of certain quasi-Poisson modules, we provide a way to compute this Poisson cohomology via the Lie algebra cohomology and the Hochschild cohomology.