Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

Abstract

Let W n ( $$ {\mathbb K} $$ ) be the Lie algebra of derivations of the polynomial algebra $$ {\mathbb K} $$ [X] := $$ {\mathbb K} $$ [x 1,…,x n ]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 W n ( $$ {\mathbb K} $$ ). 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 finite-dimensional subalgebras in polynomial Lie algebras of rank one.