Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras

TL;DR

This paper extends the theory of formal deformations to Hom-associative and Hom-Lie algebras, constructing deformation cohomology groups and providing examples including deformations of $sl_2$ and q-deformed Witt algebra.

Contribution

It introduces deformation cohomology for Hom-algebras and presents new examples of deformations, including families of Hom-Lie algebras and q-deformations.

Findings

Constructed the first deformation cohomology groups for Hom-algebras

Provided explicit examples of deformations of $sl_2$ and Witt algebra

Described formal deformations including q-deformed structures

Abstract

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.