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In this paper, we study hom-Lie superalgebras. We give the definition of hom-Nijienhuis operators of regualr hom-Lie superalgebras and show that the deformation generated by a hom-Nijienhuis operator is trivial. Moreover, we introduce the definition of T * -extensions of Hom-Lie superalgebras and show that T * -extensions preserve many properties such as nilpotency, solvability and decomposition in some sense. We also investigate the equivalence of T * -extensions.

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We introduce the class of split regular Hom-Lie color algebras as the natural generalization of split Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Lie color algebra L is of the form L = U + [j]∈Λ/∼ I [j] with U a subspace of the abelian graded subalgebra H and any I [j] , a well described ideal of L, satisfying [I [j] , I [k] ] = 0 if [j] = [k]. Under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized.

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