Abstract:
The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.

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 $T^*$-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.

Abstract:
In this paper, we define $\omega$-derivations, and study some properties of $\omega$-derivations, with its properties we can structure a new $n$-ary Hom-Nambu algebra from an $n$-ary Hom-Nambu algebra. In addition, we also give derivations and representations of $n$-ary Hom-Nambu algebras.

Abstract:
The aim of this paper is to provide cohomologies of $n$-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to $n$-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of $n$-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.

Abstract:
The purpose of this paper is define the representation and the cohomology of n-ary-Nambu-Lie superalgebras. Morever we study extensions and provide the computation of the derivations and second cohomology group of super w_1 3-algebra

Abstract:
Hom-alternative and Hom-Jordan algebras are shown to give rise to Hom-Nambu algebras of arities 2^{k+1} + 1. The class of n-ary Hom-Maltsev algebras is studied. Multiplicative n-ary Hom-Nambu-Lie algebras are shown to be n-ary Hom-Maltsev algebras. Examples of ternary Hom-Maltsev algebras that are not ternary Hom-Nambu-Lie algebras are given. Ternary Hom-Maltsev algebras are shown to arise from composition algebras.

Abstract:
The purpose of this paper is to introduce and study quadratic $n$-ary Hom-Nambu algebras, which are $n$-ary Hom-Nambu algebras with an invariant, nondegenerate and symmetric bilinear forms that are also $\alpha$-symmetric and $\beta$-invariant where $\alpha$ and $\beta$ are twisting maps. We provide constructions of these $n$-ary algebras by using twisting principles, tensor product and T*-extension. Also is discussed their connections with representation theory and centroids. Moreover we show that one may derive from quadratic $n$-ary Hom-Nambu algebra ones of increasingly higher arities and that under suitable assumptions it reduces to a quadratic $(n-1)$-ary Hom-Nambu algebra.

Abstract:
It is observed that the category of n-ary Hom-Nambu(-Lie) algebras is closed under twisting by self-weak morphisms. Constructions of ternary Hom-Nambu algebras from Hom-associative algebras, Hom-Lie algebras, ternary totally Hom-associative algebras, and Hom-Jordan triple systems are given. Every multiplicative n-ary Hom-Nambu algebra gives rise to a sequence of Hom-Nambu algebras of exponentially higher arities. Under some conditions, an n-ary Hom-Nambu(-Lie) algebra gives rise to an (n-1)-ary Hom-Nambu(-Lie) algebra.

Abstract:
It is shown that every n-ary totally Hom-associative algebra with equal twisting maps yields an n-ary Hom-Nambu algebra via an n-ary version of the commutator bracket. The class of n-ary totally Hom-associative algebras is shown to be closed under twisting by self-weak morphisms. Every multiplicative n-ary totally Hom-associative algebra yields a sequence of multiplicative totally Hom-associative algebras of exponentially higher arities. Under suitable conditions, an n-ary totally Hom-associative algebra gives an (n-k)-ary totally Hom-associative algebra.

Abstract:
In this paper, we introduce the concepts of Rota-Baxter operators and differential operators with weights on a multiplicative $n$-ary Hom-algebra. We then focus on Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras and show that they can be derived from Rota-Baxter Hom-Lie algebras, Hom-preLie algebras and Rota-Baxter commutative Hom-associative algebras. We also explore the connections between these Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras.