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

TL;DR

This paper proves that for a split simple group of type G_2 over a field, its Lie algebra's function field is purely transcendental over the field of invariants, answering a longstanding question.

Contribution

It establishes that the function field of the Lie algebra of type G_2 is generated by algebraically independent elements over the invariants, confirming a conjecture by Colliot-Thélène et al.

Findings

The function field k(g) is purely transcendental over k(g)^G.

The result confirms the rationality of the Lie algebra over its quotient by the adjoint action.

It resolves an open question in the theory of algebraic groups and Lie algebras.

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.