2011/05/31 by Hanspeter Kraft, Immanuel Stampfli · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Affine transformation #Automorphism #Automorphism group #Automorphisms of the symmetric and alternating groups #Dimension (graph theory) #Field (mathematics) #Group (periodic table) #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #Polynomial #math.AG
paper · pdf · doi:10.5802/aif.2785
published as Annales de l'institut Fourier, 63 no. 3 (2013), p. 1137-1148 · 8 pages, typos corrected, minor changes, improved exposition
openalex publication_date 2013/01/01 · openalex created_date 2016/06/24 · arxiv created 2016/11/23 · arxiv updated 2016/11/24 · openalex updated_date 2026/08/05
We show that every automorphism of the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>𝒢</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>𝔸</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of polynomial automorphisms of complex affine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>𝔸</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>ℂ</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> is inner up to field automorphisms when restricted to the subgroup <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:msub> <mml:mi>𝒢</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> where all automorphisms are tame: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:msub> <mml:mi>𝒢</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>𝒢</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> . The methods are different, based on arguments from algebraic group actions.