The purpose of this paper is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology.1 of how to construct ternary multiplications from the binary multiplication of a Hom-Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions; and it was shown that this method can be used to construct ternary Hom-Nambu-Lie algebras from Hom-Lie algebras. This construction was generalized to n-Lie algebras and n-Hom-Nambu-Lie algebras in [5].The reference [1] constructed super 3-Lie algebras(that is, 3-ary Lie superalgebras) by super Lie algebras(that is, Lie superalgebras). The reference [14] constructed (n + 1)-Hom-Lie algebras by n-Hom-Lie algebras. Inspired by the references [14] and [1], the paper generalizes them to the case of Hom-superalgebra. Its aim is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We explore the structure properties of objects such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology. In Section 2, we recall the basic notions for Hom-Lie superalgebras and the construction of 3-ary-Hom-Lie superalgebras induced by Hom-Lie superalgebras, we also give a new construction theorem for such algebras and give some basic properties. In Section 3, we define the notion of solvability and nilpotency for 3-ary-Hom-Lie superalgebras and discuss solvability and nilpotency of 3-ary-Hom-Lie superalgebras induced by Hom-Lie superalgebras. In Section 4, we recall the definition and main properties of central extensions of Hom-Lie superalgebras, and then we study central extensions of 3-ary-Hom-Lie superalgebras induced by Hom-Lie superalgebras. The Section 5 is dedicated to study the corresponding cohomology.
The paper studies the structure of restricted hom-Lie algebras. More specifically speaking, we first give the equivalent definition of restricted hom-Lie algebras. Second, we obtain some properties of p-mappings and restrictable hom-Lie algebras. Finally, the cohomology of restricted hom-Lie algebras is researched.
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.
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