Superalgebras with Homogeneous structures of Lie type

TL;DR

This paper generalizes Lie superalgebras to Super-Lie superalgebras, explores their structures, representations, derivations, and extensions to ternary cases, providing new examples and fundamental results.

Contribution

It introduces the concept of Super-Lie superalgebras, studies Rota-Baxter operators, and extends the framework to ternary structures with new examples and results.

Findings

Defined Super-Lie superalgebras and their properties

Established connections between derivations and Rota-Baxter operators

Extended structures to ternary cases with supporting examples

Abstract

In this paper, we extend the concept of Lie superalgebras to a more generalized framework called Super-Lie superalgebras. In addition, they seem to be exploring various supergeneralizations of other algebraic structures, such as Super-associative, left (right) Super-Leibniz, and Super-left(right)-symmetric superalgebras, then we give some examples and related fundamental results. The notion of Rota-Baxter operators with any parity on the Super-Lie superalgebras is given. Moreover, we study a representations of Super-Lie superalgebras and its associate dual representations. The notion of derivations of Super-Lie superalgebras is introduced thus we show that the converse of a bijective derivation defines a Rota-Baxter operator. Finally, we give a generalization of the Super-Lie superalgebras and some other structures in the ternary case which we supported this with some examples and interesting results.