$\alpha$-type Hochschild cohomology of Hom-associative algebras and Hom-bialgebras

TL;DR

This paper introduces a new cohomology theory for Hom-associative algebras and Hom-bialgebras, extending classical Hochschild and Gerstenhaber-Schack cohomologies to include the structure map , facilitating deformation analysis.

Contribution

It defines a novel cohomology framework for Hom-associative algebras and Hom-bialgebras, generalizing existing Hochschild and Gerstenhaber-Schack cohomologies.

Findings

New cohomology for Hom-associative algebras incorporating .

Extension of Hochschild cohomology to Hom-associative structures.

Extension of Gerstenhaber-Schack cohomology to Hom-bialgebras.

Abstract

In this paper we define a new cohomology for multiplicative Hom-associative algebras, which generalize Hochschild cohomology and fits with deformations of Hom-associative algebras including the structure map $\alpha$. It is a generalization of the known Hochschild-type cohomology for Hom-associative algebras defined by Amar, Ejbehi and Makhlouf. Moreover, we provide various observations and similarly a new cohomology of Hom-bialgebras extending the Gerstenhaber-Schack cohomology for Hom-bialgebras given by Dekkar and Makhlouf.