Gerstenhaber algebra of an associative conformal algebra

TL;DR

This paper establishes a Gerstenhaber algebra structure on the Hochschild cohomology of associative conformal algebras, including the cup product and graded Lie bracket, and explores extensions with bimodules.

Contribution

It introduces a Gerstenhaber algebra structure on Hochschild cohomology for associative conformal algebras and analyzes the cohomology of split extensions with bimodules.

Findings

Cup product is graded commutative on Hochschild cohomology.

A graded Lie bracket of degree -1 is defined, forming a Gerstenhaber algebra.

Existence of an algebra homomorphism from the cohomology of split extensions to that of the base algebra.

Abstract

We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology $\HH^{\ast}(A)$ of an associative conformal algebra $A$, and show that the Lie bracket together with the cup product is a Gerstenhaber algebra on the Hochschild cohomology of an associative conformal algebra. Moreover, we consider the Hochschild cohomology of split extension conformal algebra $A\hat{\oplus}M$ of $A$ with a conformal bimodule $M$, and show that there exist an algebra homomorphism from $\HH^{\ast}(A\hat{\oplus}M)$ to $\HH^{\ast}(A)$.