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On the structure of split involutive Hom-Lie color algebras
Volume 60, no. 1 (2019), pp. 61–77

Abstract

In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra $\mathcal{L}$ is of the form $\mathcal{L}=\mathcal{U}\oplus\sum_{[\alpha]\in\Pi/\sim} I_{[\alpha]}$, with $\mathcal{U}$ a subspace of the involutive abelian subalgebra $\mathcal{H}$ and any $I_{[\alpha]}$, a well-described involutive ideal of $\mathcal{L}$, satisfying $[I_{[\alpha]}, I_{[\beta]}]=0$ if $[\alpha]\neq[\beta]$. Under certain conditions, in the case of $\mathcal{L}$ being of maximal length, the simplicity of the algebra is characterized and it is shown that $\mathcal{L}$ is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras.