Cohomology and deformations of O-operators on Hom-associative algebras

TL;DR

This paper develops a cohomology theory for O-operators on Hom-associative algebras, linking it to Hochschild cohomology, and explores their deformations and triviality conditions.

Contribution

It introduces the cohomology framework for O-operators on Hom-associative algebras and analyzes their deformations and Nijenhuis elements.

Findings

Cohomology of O-operators is established and related to Hochschild cohomology.

Deformations of O-operators are governed by the newly defined cohomology.

Nijenhuis elements characterize trivial infinitesimal deformations.

Abstract

In this paper, we introduce the cohomology theory of $\mathcal{O}$-operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an $\mathcal{O}$-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an $\mathcal{O}$-operator is introduced to characterize trivial infinitesimal deformations.