# alpha-type Chevalley-Eilenberg cohomology of Hom-Lie algebras and bialgebras

## TL;DR

This paper introduces an $ ext{alpha}$-type Chevalley-Eilenberg cohomology for Hom-Lie algebras, relating it to existing cohomology, and explores its applications in deformations and Hom-Lie bialgebras.

## Contribution

It defines a new cohomology theory for Hom-Lie algebras, connects it with known theories, and applies it to study deformations and bialgebra structures.

## Key findings

- Explicit computations of the new cohomology for examples
- Relation between $ ext{alpha}$-type and classical Chevalley-Eilenberg cohomology
- Analysis of formal deformations of Hom-Lie algebras

## Abstract

The purpose of this paper is to define an $\alpha$-type cohomology, which we call $\alpha$-type Chevalley-Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley-Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map $\alpha$ are deformed. Furthermore, we provide a generalization of the Grand Crochet and study, in a particular case, the $\alpha$-type cohomology for Hom-Lie bialgebras.