Split Regular $Hom$-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 $T=\mathcal{U}+\sum _j{I}_j$, with $\mathcal{U}$ a subspace of the $0$-root space $T_0$, and $I_j$ an ideal of $T$ satisfying {for} $j\ne k:$ \[[{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0.\] 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.