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.

Keywords

Nonzero Element Characteristic Zero Polynomial Algebra Real Plane Borel Subalgebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    V. M. Buchstaber and D. V. Leykin, “Polynomial Lie algebras,” Funkts. Anal. Prilozhen., 36, No. 4, 18–34 (2002); English translation: Funct. Anal. Appl., 36, No. 4, 267–280 (2002).Google Scholar
  2. 2.
    A. González-López, N. Kamran, and P. J. Olver, “Lie algebras of vector fields in the real plane,” Proc. London Math. Soc., Third Ser., 64, No. 2, 339–368 (1992).MATHCrossRefGoogle Scholar
  3. 3.
    J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, New York (1972).MATHCrossRefGoogle Scholar
  4. 4.
    D. A. Jordan, “On the ideals of a Lie algebra of derivations,” J. London Math. Soc., 33, No. 1, 33–39 (1986).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    S. Lie, Theorie der Transformationsgruppen, Vols. 1–3, Leipzig (1888, 1890, 1893).Google Scholar
  6. 6.
    A. Nowicki and M. Nagata, “Rings of constants for k-derivations of k[x 1,…,x n],” J. Math. Kyoto Univ., 28, No. 1, 111–118 (1988).MathSciNetMATHGoogle Scholar

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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