On split Regular Hom-Leibniz algebras

TL;DR

This paper introduces split regular Hom-Leibniz algebras, generalizing existing algebra classes, and develops root connection techniques to analyze their structure and simplicity conditions.

Contribution

It defines split regular Hom-Leibniz algebras, describes their decomposition, and characterizes simplicity under certain conditions.

Findings

Algebras decompose into subspaces and ideals with specific commutation properties.

Structural description of split regular Hom-Leibniz algebras.

Conditions for algebra simplicity in maximal length cases.

Abstract

We introduce the class of split regular Hom-Leibniz algebras as the natural generalization of split Leibniz algebras and split regular Hom-Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz algebra $L$ is of the form $L = U + \sum\limits_{[j] \in \Lambda/\sim}I_{[j]}$ with $U$ a subspace of the abelian subalgebra $H$ and any $I_{[j]}$, a well described ideal of $L$, satisfying $[I_{[j]}, I_{[k]}] = 0$ if $[j]\neq [k]$. Under certain conditions, in the case of $L$ being of maximal length, the simplicity of the algebra is characterized.