Representations and module-extensions of 3-hom-Lie algebras

In this paper, we study the representations and module-extensions of 3-hom-Lie algebras. We show that a linear map between 3-hom-Lie algebras is a morphism if and only if its graph is a hom subalgebra and show that the set of derivations of a 3-hom-Lie algebra is a Lie algebra. Moreover, we introduce the definition of TθTθ-extensions and Tθ∗-extensions of 3-hom-Lie algebras in terms of modules, providing the necessary and sufficient conditions for a 2k2k-dimensional metric 3-hom-Lie algebra to be isometric to a Tθ∗-extension.