On Automorphisms of the Affine Cremona Group
Annales de l'Institut Fourier, Volume 63 (2013) no. 3, pp. 1137-1148.

We show that every automorphism of the group 𝒢 n :=Aut(𝔸 n ) of polynomial automorphisms of complex affine n-space 𝔸 n = n is inner up to field automorphisms when restricted to the subgroup T𝒢 n of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n=2 where all automorphisms are tame: T𝒢 2 =𝒢 2 . The methods are different, based on arguments from algebraic group actions.

Nous montrons que tous les automorphismes du groupe des automorphismes polynomiaux de l’espace affine n sont des automorphismes intérieurs modulo des automorphismes du corps , si nous nous restreignons au sous-groupe des automorphismes modérés. Ceci généralise un résultat de Julie Déserti traitant le cas de la dimension n=2. Dans ce cas, tous les automorphismes polynomiaux sont modérés. Nos méthodes sont différentes de celles de Julie Déserti et sont basées sur des arguments d’actions de groupes algébriques.

DOI: 10.5802/aif.2785
Classification: 14R10, 14R20, 14L30
Keywords: Polynomial automorphisms, algebraic group actions, ind-varieties, affine n-space
Mot clés : Automorphismes polynomiaux, actions de groupes algébriques, variétés algébriques de dimension infinie, éspace affine
Kraft, Hanspeter 1; Stampfli, Immanuel 1

1 Universität Basel Mathematisches Institut Rheinsprung 21, CH-4051 Basel (Suisse)
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     title = {On {Automorphisms} of the {Affine} {Cremona} {Group}},
     journal = {Annales de l'Institut Fourier},
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Kraft, Hanspeter; Stampfli, Immanuel. On Automorphisms of the Affine Cremona Group. Annales de l'Institut Fourier, Volume 63 (2013) no. 3, pp. 1137-1148. doi : 10.5802/aif.2785. https://aif.centre-mersenne.org/articles/10.5802/aif.2785/

[1] Bass, Hyman; Connell, Edwin H.; Wright, David The Jacobian conjecture: reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. (N.S.), Volume 7 (1982) no. 2, pp. 287-330 | DOI | MR | Zbl

[2] Białynicki-Birula, A. Remarks on the action of an algebraic torus on k n , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., Volume 14 (1966), pp. 177-181 | MR | Zbl

[3] Déserti, Julie Sur le groupe des automorphismes polynomiaux du plan affine, J. Algebra, Volume 297 (2006), pp. 584-599 | DOI | MR | Zbl

[4] Fogarty, John Fixed point schemes, Amer. J. Math., Volume 95 (1973), pp. 35-51 | DOI | MR | Zbl

[5] Kraft, Hanspeter; Popov, Vladimir L. Semisimple group actions on the three-dimensional affine space are linear, Comment. Math. Helv., Volume 60 (1985) no. 3, pp. 466-479 | DOI | MR | Zbl

[6] Kraft, Hanspeter; Russell, Peter Families of group actions, generic isotriviality, and linearization, 2011 (preprint, submitted to Tranformation Groups)

[7] Kraft, Hanspeter; Schwarz, Gerald W. Reductive group actions with one-dimensional quotient, Inst. Hautes Études Sci. Publ. Math. (1992) no. 76, pp. 1-97 | DOI | Numdam | MR | Zbl

[8] Kumar, Shrawan Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204, Birkhäuser Boston Inc., Boston, MA, 2002, xvi+606 pages | DOI | MR | Zbl

[9] Liendo, Alvaro Roots of the affine Cremona group, Transform. Groups, Volume 16 (2011) no. 4, pp. 1137-1142 | DOI | MR | Zbl

[10] Serre, Jean-Pierre How to use finite fields for problems concerning infinite fields, Arithmetic, geometry, cryptography and coding theory (Contemp. Math.), Volume 487, Amer. Math. Soc., Providence, RI, 2009, pp. 183-193 | DOI | MR | Zbl

[11] Smith, P. A. A theorem on fixed points for periodic transformations, Ann. of Math. (2), Volume 35 (1934) no. 3, pp. 572-578 | DOI | MR | Zbl

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