Biderivations and commuting linear maps on Hom-Lie algebras

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.