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

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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 b by an abelian one a and Z¹(b, a)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 (,[·,,·],α,,) over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T*-extension.