Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

@article{Arzhantsev2010FiniteDimensionalSI,
title={Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One},
author={Ivan Arzhantsev and E. A. Makedonskii and Anatoliy P. Petravchuk},
journal={Ukrainian Mathematical Journal},
year={2010},
volume={63},
pages={827-832}
}
• Published 9 May 2010
• Mathematics
• Ukrainian Mathematical Journal
Let Wn($${\mathbb K}$$) be the Lie algebra of derivations of the polynomial algebra $${\mathbb K}$$[X] :=$${\mathbb K}$$[x1,…,xn]over an algebraically closed field $${\mathbb K}$$ of characteristic zero. A subalgebra $$L \subseteq {W_n}(\mathbb{K})$$ is called polynomial if it is a submodule of the $${\mathbb K}$$[X]-module Wn($${\mathbb K}$$). We prove that the centralizer of every nonzero element in L is abelian, provided that L is of rank one. This fact allows one to classify…
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