Split regular Hom-Leibniz color 3-algebras
Volume 157 / 2019
Colloquium Mathematicum 157 (2019), 251-277
MSC: Primary 17A60, 17A32, 17A40.
DOI: 10.4064/cm7671-9-2018
Published online: 24 May 2019
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 and superalgebras.
More precisely, we show that any such split regular Hom-Leibniz color $3$-algebra $T$ 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 $T$ such that for $j\not =k$, \[ [{ T},I_j,I_k]+[I_j,{ T},I_k]+[I_j,I_k,T]=0. \] 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.
