Biderivations and commuting linear maps on Lie algebras

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 $\gamma \in \mathrm{Cent}(L)$, 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 $\mathrm{Cent}(L)$. 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.

Cited by in corpus (4)

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• Biderivations and commuting linear maps on Hom-Lie algebras
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