3-ary Hom-Lie Superalgebras Induced By Hom-Lie Superalgebras

Abstract

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.

Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras

“…In [47], the authors introduced a construction of a 3-Hom-Lie superalgebra from a Hom-Lie superalgebra. It is called 3-Hom-Lie superalgebra induced by Hom-Lie superalgebra.…”

“…, y n ) = 0. In the following, we generalise the notion of supertrace introduced in [47] for the even multilinear form. Proof.…”

On the Cohomology and Extensions of n-ary Multiplicative Hom-Nambu-Lie Superalgebras

“…These quasideformations lead to quasi-Lie algebras, a generalized Lie algebra structure in which the skew symmetry and Jacobi conditions are twisted. Hom-Lie algebras, Hom-Lie superalgebras, Hom-Lie bialgebras, Hom-Lie 2-algebras, and quasi-Hom-Lie algebras are discussed in [4][5][6][7][8][9][10][11][12][13][14]. The n-ary Hom-Nambu-Lie algebras have been introduced in [15].…”

3-Lie Superalgebras Induced by Lie Superalgebras

“…Particularly it was proved that Lie superalgebra gl(m, n) of (m, n)-block matrices induces the 3-Lie superalgebra if we endow it with a graded triple bracket, constructed with the help of binary graded commutator of gl(m, n) and the supertrace of (m, n)-block matrix. In [9] this approach was extended to 3-ary Hom-Lie superalgebras and the authors showed that given a Hom-Lie superalgebra and its representation one can construct the 3-ary Hom-Lie superalgebra, whose graded triple Lie bracket is constructed by means of binary graded Lie bracket of Hom-Lie superalgebra and the supertrace. The authors of [9] explored the structures of induced 3-ary Hom-Lie superalgebras such that ideals, center, derived series and central extensions.…”