On Automorphisms of the Affine Cremona Group

Hanspeter Kraft[1]; Immanuel Stampfli[1]

  • [1] Universität Basel Mathematisches Institut Rheinsprung 21, CH-4051 Basel (Suisse)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 3, page 1137-1148
  • ISSN: 0373-0956

Abstract

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

How to cite

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Kraft, Hanspeter, and Stampfli, Immanuel. "On Automorphisms of the Affine Cremona Group." Annales de l’institut Fourier 63.3 (2013): 1137-1148. <http://eudml.org/doc/275671>.

@article{Kraft2013,
abstract = {We show that every automorphism of the group $\{\mathcal\{G\}\}_\{n\}:=Aut (\{\mathbb\{A\}\}^\{n\})$ of polynomial automorphisms of complex affine $n$-space $\{\mathbb\{A\}\}^\{n\}=\{\mathbb\{C\}\}^\{n\}$ is inner up to field automorphisms when restricted to the subgroup $T\{\mathcal\{G\}\}_\{n\}$ of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension $n=2$ where all automorphisms are tame: $T\{\mathcal\{G\}\}_\{2\} = \{\mathcal\{G\}\}_\{2\}$. The methods are different, based on arguments from algebraic group actions.},
affiliation = {Universität Basel Mathematisches Institut Rheinsprung 21, CH-4051 Basel (Suisse); Universität Basel Mathematisches Institut Rheinsprung 21, CH-4051 Basel (Suisse)},
author = {Kraft, Hanspeter, Stampfli, Immanuel},
journal = {Annales de l’institut Fourier},
keywords = {Polynomial automorphisms; algebraic group actions; ind-varieties; affine n-space; polynomial automorphisms; affine -space},
language = {eng},
number = {3},
pages = {1137-1148},
publisher = {Association des Annales de l’institut Fourier},
title = {On Automorphisms of the Affine Cremona Group},
url = {http://eudml.org/doc/275671},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Kraft, Hanspeter
AU - Stampfli, Immanuel
TI - On Automorphisms of the Affine Cremona Group
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 3
SP - 1137
EP - 1148
AB - We show that every automorphism of the group ${\mathcal{G}}_{n}:=Aut ({\mathbb{A}}^{n})$ of polynomial automorphisms of complex affine $n$-space ${\mathbb{A}}^{n}={\mathbb{C}}^{n}$ is inner up to field automorphisms when restricted to the subgroup $T{\mathcal{G}}_{n}$ of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension $n=2$ where all automorphisms are tame: $T{\mathcal{G}}_{2} = {\mathcal{G}}_{2}$. The methods are different, based on arguments from algebraic group actions.
LA - eng
KW - Polynomial automorphisms; algebraic group actions; ind-varieties; affine n-space; polynomial automorphisms; affine -space
UR - http://eudml.org/doc/275671
ER -

References

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  6. Hanspeter Kraft, Peter Russell, Families of group actions, generic isotriviality, and linearization, (2011) Zbl1317.14104
  7. Hanspeter Kraft, Gerald W. Schwarz, Reductive group actions with one-dimensional quotient, Inst. Hautes Études Sci. Publ. Math. (1992), 1-97 Zbl0783.14026MR1215592
  8. Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, 204 (2002), Birkhäuser Boston Inc., Boston, MA Zbl1026.17030MR1923198
  9. Alvaro Liendo, Roots of the affine Cremona group, Transform. Groups 16 (2011), 1137-1142 Zbl1255.14036MR2852493
  10. Jean-Pierre Serre, How to use finite fields for problems concerning infinite fields, Arithmetic, geometry, cryptography and coding theory 487 (2009), 183-193, Amer. Math. Soc., Providence, RI Zbl1199.14007MR2555994
  11. P. A. Smith, A theorem on fixed points for periodic transformations, Ann. of Math. (2) 35 (1934), 572-578 Zbl0009.41101MR1503180

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