2019
Том 71
№ 6

All Issues

Makedonskii E. A.

Articles: 1
Brief Communications (English)

Finite-dimensional subalgebras in polynomial Lie algebras of rank one

Arzhantsev I. V., Makedonskii E. A., Petravchuk A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2011. - 63, № 5. - pp. 708-712

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.