Biderivations and Commuting Linear Maps on Lie Algebras

摘要

Let L be a Lie algebra over a commutative unital ring F containing 1/2. If L is perfect and centerless, then every skew-symmetric biderivation δ: L x L → L is of the form δ(x,y) = γ([x, y]) for all x,y ∈ L, where γ ∈ 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 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.