$\mathds{Z}$-graded Hom-Lie superalgebras

In this paper, we introduce the notion of $\mathds{Z}$-graded Hom-Lie superalgebras, and we show that there is a maximal (resp., minimal) $\mathds{Z}$-graded Hom-Lie superalgebra for a given local Hom-Lie superalgebra. Morever, we introduce the invariant bilinear forms on a $\mathds{Z}$-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 $\mathds{Z}$-graded Hom-Lie superalgebra. Furthermore, we check the condition in which the $\mathds{Z}$-graded Hom-Lie superalgebra is simple.