Biderivations and commuting linear maps on Hom-Lie algebras

TL;DR

This paper characterizes skew-symmetric biderivations and commuting linear maps on Hom-Lie algebras, linking them to the centroid, and provides algorithms and examples for their description.

Contribution

It establishes a clear relationship between biderivations, commuting maps, and the centroid in Hom-Lie algebras, including explicit formulas and algorithms.

Findings

Every skew-symmetric biderivation is expressed via the centroid.

Commuting linear maps coincide with the centroid.

Algorithms for describing these maps are provided.

Abstract

The purpose of this paper is to determine skew-symmetric biderivations $\text{Bider}_{\text{s}}(L, V)$ and commuting linear maps $\text{Com}(L, V)$ on a Hom-Lie algebra $(L,\alpha)$ having their ranges in an $(L,\alpha)$-module $(V, \rho, \beta)$, which are both closely related to $\text{Cent} (L, V)$, the centroid of $(V, \rho, \beta)$. Specifically, under appropriate assumptions, every $\delta\in\text{Bider}_{\text{s}}(L, V)$ is of the form $\delta(x,y)=\beta^{-1}\gamma([x,y])$ for some $\gamma\in \text{Cent} (L, V)$, and $\text{Com}(L, V)$ coincides with $\text{Cent} (L, V)$. Besides, we give the algorithm for describing $\text{Bider}_{\text{s}}(L, V)$ and $\text{Com}(L, V)$ respectively, and provide several examples.