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 $\mathfrak{b}$ by an abelian one $\mathfrak{a}$ and $Z^1(\mathfrak{b},\mathfrak{a}{)}_{\bar{0}}$. We also introduce the notion of $T^{\ast }$-extensions of $n$-ary multiplicative Hom-Nambu-Lie superalgebras and prove that every finite-dimensional nilpotent metric $n$-ary multiplicative Hom-Nambu-Lie superalgebra $(\g,[\cdot,\cdots,\cdot]_{\g},α,\langle ,\rangle_{\g})$ over an algebraically closed field of characteristic not 2 in the case $\alpha$ is a surjection is isometric to a suitable $T^{\ast }$-extension.

32pages. arXiv admin note: substantial text overlap with arXiv:1304.7335

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• Deformations of Lie Algebras using $\sigma$-derivations
• Representations of hom-Lie algebras
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Cited by in corpus (1)

• On the deformations and derivations of $n$-ary multiplicative Hom-Nambu-Lie superalgebras