Contragredient hom-Lie superalgebras

In this paper we introduce the notions of Z-graded hom-Lie superalgebras and contragredient hom-Lie superalgebras. We introduce the invariant bilinear forms on a Z-graded hom-Lie superalgebra and we prove that a consistent supersymmetric alpha-invariant form on the local part can be extended uniquely to a bilinear form with the same property on the whole Z-graded hom-Lie superalgebra. Furthermore, we check the condition in which hom-Lie superalgebra is simple.