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 \( {\mathbb K} \) 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 one to classify finite-dimensional subalgebras in polynomial Lie algebras of rank one.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 5, pp. 708–712, May, 2011.
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Arzhantsev, I., Makedonskii, E.A. & Petravchuk, A.P. Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One. Ukr Math J 63, 827–832 (2011). https://doi.org/10.1007/s11253-011-0545-5
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DOI: https://doi.org/10.1007/s11253-011-0545-5