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Some Finiteness Results on Triangular Automorphisms

Results in Mathematics volume 77, Article number: 75 (2022) Cite this article

Abstract

In this note we prove that every finite collection of connected algebraic subgroups of the group of triangular automorphisms of the affine space generates a connected solvable algebraic subgroup.

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Notes

  1. The results of [3] are obtained under the assumption that the ground field has characteristic zero. But it is clear that over any field the first derived subgroup of the group of triangular automorphisms is contained in the group of unitriangular automorphisms, while the \((s+1)\)th derived subgroup fixes the variables \(x_1,\ldots , x_s\); see [3] for details. In particular, the \((n+1)\)th derived subgroup is trivial.

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Acknowledgements

We thank the referee for useful comments and corrections. Also the first author is grateful to Mikhail Zaidenberg for fruitful discussions and suggestions.

Funding

The research was supported by Russian Science Foundation, Grant 19-11-00172.

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Authors and Affiliations

  1. Faculty of Computer Science, National Research University Higher School of Economics, Pokrovsky Boulevard 11, Moscow, Russia, 109028

    Ivan Arzhantsev & Kirill Shakhmatov

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  1. Ivan Arzhantsev
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  2. Kirill Shakhmatov
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Correspondence to Ivan Arzhantsev.

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The research was supported by Russian Science Foundation, Grant 19-11-00172.

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Arzhantsev, I., Shakhmatov, K. Some Finiteness Results on Triangular Automorphisms. Results Math 77, 75 (2022). https://doi.org/10.1007/s00025-022-01612-9

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  • DOI: https://doi.org/10.1007/s00025-022-01612-9

Keywords

  • Affine space
  • triangular automorphism
  • algebraic group
  • regular action

Mathematics Subject Classification

  • Primary 14R10
  • 14R20
  • Secondary 13A50
  • 13N15