Cohomology characterizations of non-abelian extensions of Hom-Lie algebras

TL;DR

This paper explores the classification of non-abelian extensions of Hom-Lie algebras using cohomology, establishing a correspondence with certain morphisms and identifying obstructions to their existence.

Contribution

It provides a cohomological framework for characterizing non-abelian extensions of Hom-Lie algebras, linking extensions to morphisms and obstructions.

Findings

Diagonal non-abelian extensions correspond to Hom-Lie algebra morphisms when the center is zero.

A cohomology class acts as an obstruction to the existence of extensions for general morphisms.

The paper establishes a one-to-one correspondence under specific conditions.

Abstract

In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out(h). Then for a general Hom-Lie algebra morphism from g to Out(h), we construct a cohomology class as the obstruction of existence of a non-abelian extension that induce the given Hom-Lie algebra morphism.