Representations and module-extensions of hom 3-Lie algebras

TL;DR

This paper explores the structure and extensions of hom 3-Lie algebras, including their representations, derivations, and specific module-extensions, providing new insights into their algebraic properties.

Contribution

It introduces the concepts of $T_{ heta}$-extensions and $T^{*}_{ heta}$-extensions for hom 3-Lie algebras and characterizes metric extensions in terms of these modules.

Findings

Derivations of hom 3-Lie algebras form a Lie algebra.

Graph of a linear map is a hom 3-Lie subalgebra if and only if the map is a morphism.

Necessary and sufficient conditions for metric hom 3-Lie algebras to be $T^{*}_{ heta}$-extensions.

Abstract

In this paper, we study the representations and module-extensions of hom 3-Lie algebras. We show that a linear map between hom 3-Lie algebras is a morphism if and only if its graph is a hom 3-Lie subalgebra and show that the derivations of a hom 3-Lie algebra is a Lie algebra. Derivation extension of hom 3-Lie algebras are also studied as an application. Moreover, we introduce the definition of $T_{\theta}$-extensions and $T^{*}_{\theta}$-extensions of hom 3-Lie sub-algebras in terms of modules, provide the necessary and sufficient conditions for $2k$-dimensional metric hom 3-Lie algebra to be isomorphic to a $T^{*}_{\theta}$-extensions.