Cohomology and Deformations of Hom-algebras

TL;DR

This paper develops cohomology theories for Hom-associative and Hom-Lie algebras, extending deformation theory by defining brackets and studying obstructions to deformations.

Contribution

It introduces generalized Hochschild and Chevalley-Eilenberg cohomologies for Hom-algebras, including brackets and obstruction analysis, advancing their deformation theory.

Findings

Defined Gerstenhaber and Nijenhuis-Richardson brackets for Hom-algebras.

Extended deformation theory by analyzing obstructions.

Generalized cohomology structures for Hom-associative and Hom-Lie algebras.

Abstract

The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Also we enhance the deformations theory of this Hom-algebras by studying the obstructions.