@article{Arzhantsev_Makedonskii_Petravchuk_2011, title={Finite-dimensional subalgebras in polynomial Lie algebras of rank one}, volume={63}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/2755}, abstractNote={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.}, number={5}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Arzhantsev, I. V. and Makedonskii, E. A. and Petravchuk, A. P.}, year={2011}, month={May}, pages={708-712} }