
Journal of Lie Theory 28 (2018), No. 3, 885900 Copyright Heldermann Verlag 2018 Biderivations and Commuting Linear Maps on Lie Algebras Matej Bresar Faculty of Mathematics and Physics, University of Ljubljana, and: Fac. of Nat. Sci. and Mathematics, University of Maribor, Slovenia matej.bresar@fmf.unilj.si Kaiming Zhao Dept. of Mathematics, Wilfrid Laurier University, Waterloo, Canada and: Coll. of Mathematics and Inf. Science, Hebei Normal University, Shijiazhuang, P. R. China kzhao@wlu.ca [Abstractpdf] Let \,$L$ \,be a Lie algebra over a commutative unital ring $F$ contai\ning $\frac{1}{2}$. If $L$ is perfect and centerless, then every skewsymmetric biderivation $\delta\colon L\times L\to L$ 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. Keywords: Lie algebra, biderivation, commuting linear map, centroid. MSC: 17B05, 17B40, 16R60. [ Fulltextpdf (137 KB)] for subscribers only. 