# Cohomology of BiHom-associative algebras

## TL;DR

This paper develops a Hochschild cohomology theory for Bihom-associative algebras, establishing its algebraic structure, deformation control, and connections to homotopy versions, advancing the algebraic understanding of these structures.

## Contribution

It introduces a Hochschild cohomology framework for Bihom-associative algebras, including operad and Gerstenhaber structures, and explores their homotopy counterparts.

## Key findings

- Cohomology inherits a Gerstenhaber algebra structure.
- Cohomology controls formal deformations of Bihom-associative algebras.
- Connections established between homotopy Bihom-associative algebras and Hochschild cohomology.

## Abstract

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.