Biderivations and Commuting Linear Maps on Lie Algebras

Matej Bresar¹ ; Kaiming Zhao^2, 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, Tome 28 (2018) no. 3, pp. 885-900

Résumé

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.

Détail

Classification : 17B05, 17B40, 16R60
Mots-clés : Lie algebra, biderivation, commuting linear map, centroid