The Lie algebra of type $ {G}_{2}$ is rational over its quotient by the adjoint action

Anderson, Dave; Florence, Mathieu; Reichstein, Zinovy

doi:10.1016/j.crma.2013.10.029

Group theory/Algebraic geometry

The Lie algebra of type

G_{2}

is rational over its quotient by the adjoint action
[Rationalité de lʼalgèbre de Lie de type

G_{2}

sur son quotient par lʼaction adjointe]

Présenté par : Jean-Pierre Serre

Anderson, Dave ¹ ; Florence, Mathieu ² ; Reichstein, Zinovy ³

¹ Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460-320, Brazil
² Institut de mathématiques de Jussieu, Université Paris-6, 4, place Jussieu, 75005 Paris, France
³ Department of Mathematics, University of British Columbia, BC V6T 1Z2, Canada

Comptes Rendus. Mathématique, Tome 351 (2013) no. 23-24, pp. 871-875.

Résumé
Abstract

Soit G un groupe algébrique simple et déployé de type $G_{2}$ sur un corps k. Soit $g$ son algèbre de Lie. On démontre que le corps des fonctions $k (g)$ est transcendant pur sur le corps $k {(g)}^{G}$ des invariants adjoints. Ceci répond par lʼaffirmative à une question posée par J.-L. Colliot-Thélène, B. Kunyavskiĭ, V.L. Popov et Z. Reichstein.

Let G be a split simple group of type $G_{2}$ over a field k, and let $g$ be its Lie algebra. Answering a question of J.-L. Colliot-Thélène, B. Kunyavskiĭ, V.L. Popov, and Z. Reichstein, we show that the function field $k (g)$ is generated by algebraically independent elements over the field of adjoint invariants $k {(g)}^{G}$ .

Reçu le : 2013-08-27
Accepté le : 2013-10-24
Publié le : 2013-11-21

DOI : 10.1016/j.crma.2013.10.029

Affiliations des auteurs :

Anderson, Dave ¹ ; Florence, Mathieu ² ; Reichstein, Zinovy ³

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Anderson, Dave; Florence, Mathieu; Reichstein, Zinovy. The Lie algebra of type $ {G}_{2}$ is rational over its quotient by the adjoint action. Comptes Rendus. Mathématique, Tome 351 (2013) no. 23-24, pp. 871-875. doi : 10.1016/j.crma.2013.10.029. http://www.numdam.org/articles/10.1016/j.crma.2013.10.029/

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Cité par Sources :

^☆ D.A. was partially supported by NSF Grant DMS-0902967. Z.R. was partially supported by National Sciences and Engineering Research Council of Canada Grant No. 250217-2012.