On quasi-Poisson Cohomology

TL;DR

This paper introduces a new framework for computing quasi-Poisson cohomology of Poisson algebras using a projective resolution and a specialized complex, facilitating calculations in specific cases.

Contribution

It constructs a projective resolution for the quasi-Poisson enveloping algebra and introduces a quasi-Poisson complex for easier cohomology computation.

Findings

Established a standard method for computing quasi-Poisson cohomology.

Derived explicit quasi-Poisson cohomologies in particular cases.

Abstract

Let $A$ be a Poisson algebra and $\Q(A)$ its quasi-Poisson enveloping algebra. In this paper, the Yoneda-Ext algebra $\Ext^*_{\Q(A)}(A, A)$, which we call the quasi-Poisson cohomology algebra of $A$, is investigated. We construct a projective resolution of $A$ as $\Q(A)$-modules, which enables to compute the quasi-Poisson cohomologies in a standard way. To simplify calculation, we also introduce the quasi-Poisson complex and apply to obtain quasi-Poisson cohomologies in some special cases.