Colliot-Thélène, Jean-Louis; Kunyavskiĭ, Boris; Popov, Vladimir L.; Reichstein, Zinovy Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action? (English) Zbl 1218.14010 Compos. Math. 147, No. 2, 428-466 (2011). A field extension \(E/F\) is said to be purely transcendental if \(E\) is generated over \(F\) by a finite collection of algebraically independent elements. A slightly weaker condition is to say that \(E/F\) is stably purely transcendental, meaning that \(E\) is contained in a field which is purely transcendental over both \(E\) and \(F\). Let \(G\) be a connected reductive group over a field \(k\) of characteristic zero and \(\mathfrak{g}\) be the Lie algebra of \(G\). Let \(k(G)\) denote the \(k\)-rational functions on \(G\) and \(k(G)^{G}\) denote the subfield of invariants under the conjugation action. Similarly, let \(k(\mathfrak{g})\) denote the field of \(k\)-rational functions on \(\mathfrak{g}\) and \(k(\mathfrak{g})^{G}\) denote the subfield of invariants under the adjoint action. In this paper, the authors investigate the questions of whether or not the field extensions \(k(\mathfrak{g})/k(\mathfrak{g})^{G}\) and \(k(G)/k(G)^{G}\) are (stably) purely transcendental.It is first shown the extension \(k(\mathfrak{g})/k(\mathfrak{g})^{G}\) is (stably) purely transcendental if and only if the extension \(k(G)/k(G)^{G}\) is (stably) purely transcendental. This is done by showing that both situations are equivalent to the (stable) \(K\)-rationality of the homogeneous space \(G_K/T\) where \(T\) is a certain “generic torus” of \(G\) defined over a certain field extension \(K/k\). The authors then reduce the question to the case that \(G\) is semisimple and further simple. The main result is that for \(G\) simple, if \(G\) is split of type \(A_n\) or \(C_n\), then the field extensions are purely transcendental. Conversely, if \(G\) is simple and not of type \(A_n\), \(C_n\), or \(G_2\), then the extensions are not even stably purely transcendental. However, it is shown more generally for a connected split reductive group that the extensions are at least unirational. A field extension \(E/F\) is unirational if there exists a field extension of \(E\) which is purely transcendental over \(F\) but not necessarily over \(E\).In the process of proving the main result, the authors affirmatively answer a question of Grothendieck about the existence of a rational section for the categorical quotient map \(G \to G/\!/G\) for the conjugation action (when \(G\) is a connected quasisplit reductive group). They also obtain a characterization of when the weight lattice can fit into a short exact sequence of permutation lattices for the associated Weyl group. Reviewer: Christopher P. Bendel (Menomonie) Cited in 1 ReviewCited in 8 Documents MSC: 14E08 Rationality questions in algebraic geometry 14L30 Group actions on varieties or schemes (quotients) 14F22 Brauer groups of schemes 17B45 Lie algebras of linear algebraic groups 20C10 Integral representations of finite groups 20G15 Linear algebraic groups over arbitrary fields Keywords:algebraic group; simple Lie algebra; rationality problem; integral representation; algebraic torus; unramified Brauer group; transcendental extensions; conjugation action; adjoint action; special groups; permutation lattices PDFBibTeX XMLCite \textit{J.-L. Colliot-Thélène} et al., Compos. Math. 147, No. 2, 428--466 (2011; Zbl 1218.14010) Full Text: DOI arXiv References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.