Finite-dimensional subalgebras in polynomial Lie algebras of rank one - Arzhantsev, I. V.; Makedonskii, E. A.; Petravchuk, A. P. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
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 $K$ 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 to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.