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Journal of Lie Theory 31 (2021), No. 1, 119--126
Copyright Heldermann Verlag 2021

Biderivations and Commuting Linear Maps on Current Lie Algebras

Daniel Eremita
Dept. of Mathematics and Computer Science, Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor 2000, Slovenia


Let $L$ be a Lie algebra and let $A$ be an associative commutative algebra with unity, both over the same field $F$. We consider the following two questions. Is every skew-symmetric biderivation on the current Lie algebra $L\otimes A$ of the form $(x,y) \mapsto \lambda([x,y])$ for some $\gamma \in {\rm Cent}(L\otimes A)$, if the same holds true for $L$? Does every commuting linear map of $L\otimes A$ belong to ${\rm Cent}(L\otimes A)$, if the same holds true for $L$?

Keywords: Lie algebra, current Lie algebra, tensor product of algebras, biderivation, commuting linear map, centroid.

MSC: 17B05, 17B40, 15A69, 16R60.

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