Journal of Lie Theory

no. 3
Biderivations and Commuting Linear Maps on Lie Algebras
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 
@article{JOLT_2018_28_3_a15,
     author = {M. Bresar and K. Zhao},
     title = {Biderivations and {Commuting} {Linear} {Maps} on {Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {885--900},
     year = {2018},
     volume = {28},
     number = {3},
     doi = {10.5802/jolt.1031},
     zbl = {1422.17006},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1031/}
}
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M. Bresar; K. 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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