Journal of Lie Theory

no. 3
Biderivations and Commuting Linear Maps on Lie Algebras
Matej Bresar1; Kaiming Zhao2, 3
1Faculty of Mathematics and Physics, University of Ljubljana, Fac. of Nat. Sci. and Mathematics, University of Maribor, Slovenia
2Dept. of Mathematics, Wilfrid Laurier University, Waterloo, Canada
3Coll. of Mathematics and Inf. Science, Hebei Normal University, Shijiazhuang, P. R. China
Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 885-900
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.
DOI: 10.5802/jolt.1031
Classification: 17B05, 17B40, 16R60
Keywords: Lie algebra, biderivation, commuting linear map, centroid

Matej Bresar  1 ; Kaiming Zhao  2 , 3

1 Faculty of Mathematics and Physics, University of Ljubljana, Fac. of Nat. Sci. and Mathematics, University of Maribor, Slovenia
2 Dept. of Mathematics, Wilfrid Laurier University, Waterloo, Canada
3 Coll. of Mathematics and Inf. Science, Hebei Normal University, Shijiazhuang, P. R. China 
Matej Bresar; Kaiming Zhao. Biderivations and Commuting Linear Maps on Lie Algebras. Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 885-900. doi: 10.5802/jolt.1031
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     title = {Biderivations and {Commuting} {Linear} {Maps} on {Lie} {Algebras}},
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