Superalgebras with homogeneous structures of Lie type

The objective of this research paper is to broaden the concept of Lie superalgebras into a more generalized framework known as super-Lie superalgebras. It also investigates various super-generalizations of other algebraic structures, including super-associative algebras, left (right) super-Leibniz algebras, and super-left (right)-symmetric superalgebras. The paper presents examples and fundamental results related to these structures. Additionally, it introduces the concept of Rota–Baxter operators with any parity on super-Lie superalgebras and examines representations of these superalgebras along with their associated dual representations. The study also introduces the notion of derivations of super-Lie superalgebras, demonstrating that the inverse of a bijective derivation defines a Rota–Baxter operator. Finally, it offers a generalization of super-Lie superalgebras and other related structures in the ternary case, supported by examples and interesting results.