On the (co)homology of Frobenius Poisson algebras

TL;DR

This paper explores the duality between Poisson homology and cohomology in Frobenius Poisson algebras and constructs a Batalin-Vilkovisky structure on their cohomology rings.

Contribution

It establishes a duality between Poisson homology and cohomology for Frobenius Poisson algebras and introduces a Batalin-Vilkovisky structure on their cohomology.

Findings

Duality between Poisson homology and cohomology established.

Batalin-Vilkovisky structure constructed on Poisson cohomology.

Framework extends known dualities in Frobenius algebras to Poisson setting.

Abstract

In this paper, we study the Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between the Poisson homology and the Poisson cohomology, similar to the duality between the Hochschild homology and the Hochschild cohomology of a Frobenius algebra. Using the non-degenerated bilinear form on a Frobenius algebra we construct a Batalin-Vilkovisky structure on the Poisson cohomology ring of a class of Frobenius Poisson algebras.