Derivation Hom-Lie 2-algebras and non-abelian extensions of Hom-Lie algebras

TL;DR

This paper introduces derivation Hom-Lie 2-algebras and explores their role in classifying non-abelian extensions of Hom-Lie algebras, establishing a correspondence with homotopy classes of morphisms.

Contribution

It constructs the derivation Hom-Lie 2-algebra and applies it to classify non-abelian extensions of Hom-Lie algebras.

Findings

Derivation Hom-Lie 2-algebras are explicitly constructed.

Non-abelian extensions are classified via homotopy classes of morphisms.

A one-to-one correspondence is established between extension classes and morphism homotopy classes.

Abstract

In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).