Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

Let W _n ( \mathbb K {\mathbb K} ) be the Lie algebra of derivations of the polynomial algebra \mathbb K {\mathbb K} X] := \mathbb K {\mathbb K} x ₁,…,x _n ]over an algebraically closed field \mathbb K {\mathbb K} of characteristic zero. A subalgebra L í W_n(\mathbbK) L \subseteq {W_n}(\mathbb{K}) is called polynomial if it is a submodule of the \mathbb K {\mathbb K} X]-module W _n ( \mathbb K {\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.