Journal of Lie Theory 28 (2018), No. 3, 885--900
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.uni-lj.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
[Abstract-pdf]
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 skew-symmetric 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.
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