On split regular Hom-Lie color algebras

TL;DR

This paper introduces split regular Hom-Lie color algebras, generalizing split Lie color algebras, and characterizes their structure and simplicity conditions using root connection techniques.

Contribution

It develops a framework for understanding the structure of split regular Hom-Lie color algebras and characterizes their simplicity in maximal length cases.

Findings

Decomposition of split regular Hom-Lie color algebras into subalgebras and ideals.

Conditions under which the algebra is simple.

Structural description using root connection techniques.

Abstract

We introduce the class of split regular Hom-Lie color algebras as the natural generalization of split Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Lie color algebra $L$ is of the form $L = U + \sum\limits_{[j] \in \Lambda/\sim}I_{[j]}$ with $U$ a subspace of the abelian graded 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.