Gerstenhaber algebra structure on the cohomology of a hom-associative algebra

TL;DR

This paper establishes that the cohomology of a hom-associative algebra naturally forms a Gerstenhaber algebra, extending classical results from associative algebra cohomology to the hom-associative setting.

Contribution

It introduces a cup product and a graded Lie bracket on the cohomology of hom-associative algebras, demonstrating they form a Gerstenhaber algebra structure.

Findings

Cohomology of hom-associative algebras admits a cup product.

A degree -1 graded Lie bracket is defined on the cohomology.

The combined structure forms a Gerstenhaber algebra.

Abstract

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. We show that the cup product together with the degree $-1$ graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology forms a Gerstenhaber algebra. This generalizes a classical fact that the Hochschild cohomology of an associative algebra carries a Gerstenhaber algebra structure.