Ukrainian Mathematical Journal

, Volume 63, Issue 5, pp 827–832 | Cite as

Finite-Dimensional Subalgebras In Polynomial Lie Algebras Of Rank One

  • I.V. Arzhantsev
  • E. A. Makedonskii
  • A. P. Petravchuk
Brief Communications

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.


Nonzero Element Characteristic Zero Polynomial Algebra Real Plane Borel Subalgebra 
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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • I.V. Arzhantsev
    • 1
  • E. A. Makedonskii
    • 2
  • A. P. Petravchuk
    • 2
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Shevchenko Kyiv National UniversityKyivUkraine

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