Representations Of Generalized Hom-Lie Algebras

In this paper, we study hom-Nijienhuis operators and T*-extensions of hom-Lie superalgebras and hom-Lie color algebras. We show that a linear map between hom-Lie superalgebras(resp. hom-Lie color algebras) is a morphism if and only if its graph is a hom-subalgebra. We also prove that the infinitesimal deformation gener-ated by a hom-Nijienhuis operator is trivial. Moreover, we introduce the definition of T*-extensions of hom-Lie superalgebras and hom-Lie color algebras and show that T*-extensions preserve many properties such as nilpotency, solvability and decom-position in some sense. In particular, we discuss the equivalence of T*-extensions using cohomology.In the last part, we study the representations and module-extensions of hom3-Lie algebras. We show that the set of derivations of a hom3-Lie algebra is a Lie algebra. Moreover, we introduce the definition of Tθ-extensions and Tθ*-extensions of hom3-Lie algebras in terms of modules, and provide a necessary and sufficient condition for a2K-dimensional metric hom3-Lie algebra to be isometric to a Tθ*-extension.