(Bi)Hom-Leibniz algebras

TL;DR

This paper introduces symmetric (Bi)Hom-Leibniz algebras, explores their derivations and cohomology, and establishes a hierarchy of BiHom-Lie algebra types, expanding the theoretical framework of Leibniz and Hom-Lie structures.

Contribution

It defines symmetric (Bi)Hom-Leibniz algebras, studies derivations and cohomology, and introduces a hierarchy of BiHom-Lie algebra types.

Findings

Defined symmetric (Bi)Hom-Leibniz algebras.

Established hierarchy of BiHom-Lie algebra types.

Developed representations and cohomology for these algebras.

Abstract

The first aim of this paper is to introduce and study symmetric (Bi)Hom-Leibniz algebras, which are left and right Leibniz algebras. We discuss $\alpha^k\beta^l$-generalized derivations, $\alpha^k\beta^l$ -quasi-derivations and $\alpha^k\beta^l$-quasi-centroid of (Bi)Hom-Leibniz algebras and colour BiHom-Leibniz algebras. The second aim is to define a new type of BiHom-Lie algebras satisfies the following hierarchy \begin{equation*} \displaystyle \{ \text{ BiHom-Lie type $B_1$} \}\supseteq_{\beta=id} \{ \text{ Hom-Lie} \}\supseteq_{\alpha=id} \{ \text{ Lie} \}. \end{equation*} Moreover, define representations and a cohomology of symmetric BiHom- Leibniz algebras of type $B_1$.