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

TL;DR

This paper investigates the structure of double derivations in $n$-Hom-Lie color algebras, revealing their relationship with derivations, inner derivations, and triple derivations, and establishing conditions for their algebraic properties.

Contribution

It characterizes the double derivation algebra of $n$-Hom-Lie color algebras and explores its relation to derivation algebras, including conditions for trivial centralizers and derivation identities.

Findings

Inner derivations form an ideal in the double derivation algebra.

The centralizer of inner derivations is trivial under certain conditions.

Triple derivations coincide with derivations for centerless perfect 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 $Der(\mathcal{L}).$