Finite-dimensional subalgebras in polynomial Lie algebras of rank one - Arzhantsev, I. V.; Makedonskii, E. A.; Petravchuk, A. P. (Ukrains’kyi Matematychnyi Zhurnal, 2011)
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 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 that L is of rank one. This fact allows to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.