BV-differential on Hochschild cohomology of Frobenius algebras

TL;DR

This paper introduces a new BV-algebra structure on the Hochschild cohomology of Frobenius algebras, enabling explicit calculations of algebraic structures for certain self-injective algebras.

Contribution

It defines a novel algebraic structure on Hochschild cohomology for Frobenius algebras and proves it forms a BV-algebra, facilitating detailed structural computations.

Findings

Hochschild cohomology algebra is a BV-algebra under certain conditions

Explicit calculation of Gerstenhaber and BV structures for tree type D_n algebras

The algebra ${ m HH}^*(R)^{ u}$ is isomorphic to Hochschild cohomology when the order of a9 is not divisible by the characteristic of k

Abstract

For a finite-dimensional Frobenius $k$-algebra $R$ with the Nakayama automorphism $\nu$ we define an algebra ${\rm HH}^*(R)^{\nu\uparrow}$. If the order of $\nu$ is not divisible by the characteristic of $k$, this algebra is isomorphic to the Hochschild cohomology algebra of $R$.. We prove that this algebra is a BV-algebra. We use this fact to calculate the Gerstenhaber algebra structure and BV-structure on the Hochschild cohomology algebras of a family of self-injective algebras of tree type $D_n$.