Split Regular $Hom$-Leibniz Color $3$-Algebras

TL;DR

This paper introduces split regular Hom-Leibniz color 3-algebras, extending various algebra classes, and characterizes their structure and simplicity based on root connectivity.

Contribution

It defines and analyzes the structure of split regular Hom-Leibniz color 3-algebras, providing a decomposition and simplicity criteria.

Findings

Decomposition of split regular Hom-Leibniz color 3-algebras into subspaces and ideals.

Characterization of simplicity via root connectivity.

Extension of algebra classes to Hom-Leibniz color 3-algebras.

Abstract

We introduce and describe the class of split regular $Hom$-Leibniz color $3$-algebras as the natural extension of the class of split Lie algebras, split Leibniz algebras, split Lie $3$-algebras, split Lie triple systems, split Leibniz $3$-algebras, and some other algebras. More precisely, we show that any of such split regular $Hom$-Leibniz color $3$-algebras $T$ is of the form ${T}={\mathcal U} +\sum\limits_{j}I_{j}$, with $\mathcal U$ a subspace of the $0$-root space ${T}_0$, and $I_{j}$ an ideal of $T$ satisfying {for} $j\neq k:$ \[[{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0.\] Moreover, if $T$ is of maximal length, we characterize the simplicity of $T$ in terms of a connectivity property in its set of non-zero roots.