The structure of split regular BiHom-Leibniz color algebras

Abstract

We introduce the class of split regular BiHom-Leibniz color algebras as the natural generalization of split regular Hom-Leibniz algebras. By developing techniques of connections of roots for this kind of algebra, we show that such a split regular BiHom-Leibniz color algebra $\mathcal{ℒ}$ is of the form $\mathcal{ℒ}=\mathcal{𝒰}\oplus {\sum }_{\left[\alpha \right]\in \mathrm{\Pi }∕\sim }{I}_{\left[\alpha \right]}$, with $\mathcal{𝒰}$ a subspace of the abelian subalgebra $\mathcal{\mathscr{H}}$ and any $I_{\left[\alpha \right]}$, a well-described ideal of $\mathcal{ℒ}$, satisfying $\left[I_{\left[\alpha \right]},I_{\left[\beta \right]}\right]=0$ if $\left[\alpha \right]\ne \left[\beta \right]$. Under certain conditions, in the case of $\mathcal{ℒ}$ being of maximal length, the simplicity and the primeness of the algebra is characterized.