Finite-dimensional subalgebras in polynomial Lie algebras of rank one
Abstract
Let Wn(K) be the Lie algebra of derivations of the polynomial algebra K[X]:=K[x1,...,xn] over an algebraically closed field K of characteristic zero. A subalgebra L of Wn(K) is called polynomial if it is a submodule of the K[X]-module Wn(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.
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