Finite-dimensional subalgebras in polynomial Lie algebras of rank one

  • I. V. Arzhantsev
  • E. A. Makedonskii Kyiv Nat. Taras Shevchenko Univ., Ukraine
  • A. P. Petravchuk Kyiv Nat. Taras Shevchenko Univ., Ukraine

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 $K$ 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.
Published
25.05.2011
How to Cite
Arzhantsev, I. V., E. A. Makedonskii, and A. P. Petravchuk. “Finite-Dimensional Subalgebras in Polynomial Lie Algebras of Rank One”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, no. 5, May 2011, pp. 708-12, https://umj.imath.kiev.ua/index.php/umj/article/view/2755.
Section
Short communications