On the cohomology and extensions of n-ary multiplicative Hom-Nambu-Lie superalgebras

In this paper, we discuss the representations of $n$-ary multiplicative Hom-Nambu-Lie superalgebras as a generalization of the notion of representations for $n$-ary multiplicative Hom-Nambu-Lie algebras. We also give the cohomology of an $n$-ary multiplicative Hom-Nambu-Lie superalgebra and obtain a relation between extensions of an $n$-ary multiplicative Hom-Nambu-Lie superalgebra $\mathfrak{b}$ by an abelian one $\mathfrak{a}$ and $Z^1(\mathfrak{b}, \mathfrak{a})_{\bar{0}}$. We also introduce the notion of $T^*$-extensions of $n$-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric $n$-ary multiplicative Hom-Nambu-Lie superalgebra $(\g,[\cdot,\cdots,\cdot]_{\g},\alpha,\langle ,\rangle_{\g})$ over an algebraically closed field of characteristic not 2 in the case $\alpha$ is a surjection is isometric to a suitable $T^*$-extension.