Double derivations of $n-$Hom-Lie color algebras

Abstract

We study the double derivation algebra $\mathcal{D}(\mathcal{L})$ of $n-$Hom Lie color algebra $\mathcal{L}$ and describe the relation between $\mathcal{D}(\mathcal{L})$ and the usual derivation Hom-Lie color algebra $Der(\mathcal{L}).$ We prove that the inner derivation algebra $Inn(\mathcal{L})$ is an ideal of the double derivation algebra $\mathcal{D}(\mathcal{L}).$ We also show that if $\mathcal{L}$ is a perfect $n-$Hom Lie color algebra with certain constraints on the base field, then the centralizer of $Inn(\mathcal{L})$ in $\mathcal{D}(\mathcal{L})$ is trivial. In addition, we obtain that for every centerless perfect $n-$Hom Lie color algebra $\mathcal{L}$, the triple derivations of the derivation algebra $Der(\mathcal{L})$ are exactly the derivations of $Der(\mathcal{L}).$