Finite-dimensional subalgebras in polynomial Lie algebras of rank one

TL;DR

This paper classifies finite-dimensional subalgebras within polynomial Lie algebras of rank one, showing that the centralizer of any nonzero element in such subalgebras is abelian, thus revealing their structural properties.

Contribution

It provides a classification of finite-dimensional subalgebras in polynomial Lie algebras of rank one, based on the abelian nature of centralizers of nonzero elements.

Findings

Centralizer of nonzero elements in rank one subalgebras is abelian.

Classification of finite-dimensional subalgebras in polynomial Lie algebras of rank one.

Structural insights into polynomial Lie algebra subalgebras.

Abstract

Let W_n(K) be the Lie algebra of derivations of the polynomial algebra K[X]:=K[x_1,...,x_n] over an algebraically closed field K of characteristic zero. A subalgebra L of W_n(K) is called polynomial if it is a submodule of the K[X]-module W_n(K). We prove that the centralizer of every nonzero element in L is abelian provided L has rank one. This allows to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.