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

Abstract

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 Z 1 (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 (g, [•, • • • , •] g , α, , g ) over an algebraically closed field of characteristic not 2 in the case α is a surjection is isometric to a suitable T * -extension.

On the Deformations and Derivations ofn-Ary Multiplicative Hom-Nambu-Lie Superalgebras

“…3 Deformations of n-ary multiplicative Hom-Nambu-Lie superalgebras Definition 3.1. [16] For m ≥ 1, we call m-coboundary operator of the n-ary multiplicative Hom-Nambu-Lie superalgebra (g, [•,…”

“…In [16], it also points out that, the map f ∈ C m (g, V ) is called an m-supercocycle if δ m f = 0. We denote by Z m (g, V ) the graded subspace spanned by m-supercocycles.…”