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Izv. RAN. Ser. Mat., 2001, Volume 65, Issue 3, Pages 153–174 (Mi izv340)  

This article is cited in 5 scientific papers (total in 5 papers)

On polynomial automorphisms of affine spaces

V. L. Popov

Moscow State Institute of Electronics and Mathematics

Abstract: In the first part of this paper we prove some general results on the linearizability of algebraic group actions on $\mathbb A^n$. As an application, we get a method of construction and concrete examples of non-linearizable algebraic actions of infinite non-reductive insoluble algebraic groups on $\mathbb A^n$ with a fixed point. In the second part we use these general results to prove that every effective algebraic action of a connected reductive algebraic group $G$ on the $n$-dimensional affine space $\mathbb A^n$ over an algebraically closed field $k$ of characteristic zero is linearizable in each of the following cases: 1) $n=3$; 2) $n=4$ and $G$ is not a one- or two-dimensional torus. In particular, this means that $\operatorname{GL}_3(k)$ is the unique (up to conjugacy) maximal connected reductive subgroup of the automorphism group of the algebra of polynomials in three variables over $k$.

DOI: https://doi.org/10.4213/im340

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English version:
Izvestiya: Mathematics, 2001, 65:3, 569–587

Bibliographic databases:

MSC: 14L17, 14L30
Received: 06.03.2000

Citation: V. L. Popov, “On polynomial automorphisms of affine spaces”, Izv. RAN. Ser. Mat., 65:3 (2001), 153–174; Izv. Math., 65:3 (2001), 569–587

Citation in format AMSBIB
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\paper On polynomial automorphisms of affine spaces
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\transl
\jour Izv. Math.
\yr 2001
\vol 65
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Flenner H., Zaidenberg M., “On the uniqueness of C*-actions on affine surfaces”, Affine Algebraic Geometry, Contemporary Mathematics Series, 369, 2005, 97–111  crossref  mathscinet  zmath  isi
    2. Kaliman Sh., “Actions of C* and C+ on affine algebraic varieties”, Algebraic Geometry, Proceedings of Symposia in Pure Mathematics, 80, no. 1- 2, 2009, 629–654  crossref  mathscinet  zmath  isi
    3. Donzelli F., Dvorsky A., Kaliman S., “Algebraic Density Property of Homogeneous Spaces”, Transformation Groups, 15:3 (2010), 551–576  crossref  mathscinet  zmath  isi  scopus
    4. Faber E., Hauser H., “Today'S Menu: Geometry and Resolution of Singular Algebraic Surfaces”, Bulletin of the American Mathematical Society, 47:3 (2010), 373–417  crossref  mathscinet  zmath  isi  scopus
    5. Arzhantsev I. Zaidenberg M., “Acyclic Curves and Group Actions on Affine Toric Surfaces”, Affine Algebraic Geometry, ed. Masuda K. Kojima H. Kishimoto T., World Scientific Publ Co Pte Ltd, 2013, 1–41  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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