Biderivations and commuting linear maps on Lie algebras

TL;DR

This paper characterizes skew-symmetric biderivations and commuting linear maps on certain Lie algebras, showing they are closely related to the centroid, under specific algebraic conditions.

Contribution

It generalizes known results by describing biderivations and commuting maps with ranges in modules, under milder assumptions on the Lie algebra.

Findings

Skew-symmetric biderivations are of the form $ heta([x,y])$ with $ heta$ in the centroid.

Commuting linear maps are in the centroid under a mild condition.

The results include and extend several existing theorems in the literature.

Abstract

Let $L$ be a Lie algebra over a field of characteristic different from $2$. If $L$ is perfect and centerless, then every skew-symmetric biderivation $\delta:L\times L\to L$ is of the form $\delta(x,y)=\gamma([x,y])$ for all $x,y\in L$, where $\gamma\in{\rm Cent}(L)$, the centroid of $L$. Under a milder assumption that $[c,[L,L]]=\{0\}$ implies $c=0$, every commuting linear map from $L$ to $L$ lies in ${\rm Cent}(L)$. These two results are special cases of our main theorems which concern biderivations and commuting linear maps having their ranges in an $L$-module. We provide a variety of examples, some of them showing the necessity of our assumptions and some of them showing that our results cover several results from the literature.