Z-graded Hom-Lie superalgebras

In this paper, we introduce the notion of ?Z-graded Hom-Lie sup eralge-bras, and we show that there is a maximal (resp., minimal) ?Z-graded Hom-Lie sup eralge-bra for a given local Hom-Lie superalgebra. Morever, we introduce the invariant bilinear forms on a ?Z-graded Hom-Lie superalgebra and 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 the ?Z-graded Hom-Lie superalgebra is simple.