Abelian extensions and crossed modules of Hom-Lie algebras

TL;DR

This paper explores the cohomology of Hom-Lie algebras, introducing new concepts like $eta$-abelian extensions and $eta$-crossed modules, and establishes their relationships with derivations and exact sequences.

Contribution

It introduces $eta$-abelian extensions and $eta$-crossed modules of Hom-Lie algebras, linking them to cohomology and providing new structural insights.

Findings

Established a five-term exact sequence in cohomology

Defined $eta$-abelian extensions and $eta$-crossed modules

Proved the equivalence between crossed modules and cat$^1$-Hom-Lie algebras

Abstract

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of $\alpha$-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we introduce crossed modules of Hom-Lie algebras showing their equivalence with cat$^1$-Hom-Lie algebras, and we introduce $\alpha$-crossed modules to have a better understanding of the third cohomology group.