The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Dave Anderson; Mathieu Florence; Zinovy Reichstein

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arXiv · 1308.5940

The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Dave Anderson·Mathieu Florence·Zinovy Reichstein

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Abstract

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 Colliot-Th\'el\`ene, Kunyavski\u{i}, Popov, and Reichstein, we show that the function field k(g) is generated by algebraically independent elements over the field of adjoint invariants k(g)^G. Soit G un groupe alg\'ebrique simple et d\'eploy\'e de type G_2 sur un corps k. Soit g son alg\`ebre de Lie. On d\'emontre que le corps des fonctions k(g) est transcendant pur sur le corps k(g)^G des invariants adjoints. Ceci r\'epond par l'affirmative \`a une question pos\'ee par Colliot-Th\'el\`ene, Kunyavski\u{i}, Popov et Reichstein.

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Dave Anderson, Mathieu Florence, Zinovy Reichstein. 2013-08-27. The Lie algebra of type G_2 is rational over its quotient by the adjoint action. https://arxiv.org/abs/1308.5940

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