Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

  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},
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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Arzhantsev: Department of Higher Algebra, Faculty of Mechanics and Mathematics

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