Modular Lie algebras and the Gelfand-Kirillov conjecture. (English) Zbl 1209.17012
The famous Gelfand-Kirillov (G-K) conjecture states that the skew-field of fractions of the universal enveloping algebra \(U(\mathfrak{g})\) of a Lie algebra over a field \(k\) of characteristic \(0\) is the Weyl skew-field over the field of fractions of the centre of \(U(\mathfrak{g})\). The conjecture was formulated in 1966 and was verified for \(\mathfrak{gl}_{n}\), \(\mathfrak{sl}_{n}\) and every nilpotent Lie algebra by Gelfand and Kirillov. In the sequel in 1973, the conjecture was confirmed in the solvable case independently by Borho, Joseph and McConnell. In 1979 Nghiem proved the conjecture for the semi-direct products of \(\mathfrak{sl}_{n}\), \(\mathfrak{sp}_{2n}\), and \(\mathfrak{so}_n\) with their standard modules. A breakthrough in the general case came in 1996 when Alev, Ooms and Van den Bergh constructed a series of counterexamples to the conjecture for \(\mathfrak{g}\) a semi-direct product of \(\text{Lie}(H)\) and \(V\), with \(H\) a simple algebraic group and \(V\) a rational \(H\)-module admitting a trivial generic stabiliser.
It is a question for which types of simple Lie algebras the G-K conjecture is true and for which it isn’t. In this paper, the author gives an almost complete answer, showing that the G-K conjecture does not hold for simple Lie algebras of type \(B_n\) \((n\geq 3)\), \(D_n\) \((n\geq 4)\), \(E_6\), \(E_7\), \(E_8\), and \(F_4\). Let \(\mathfrak{g}_{\mathbb{K}}=\mathfrak{g}_{\mathbb{Z}} \otimes_{\mathbb{Z}}\mathbb{K}\), where \(\mathfrak{g}_{\mathbb{Z}}\) is a Chevalley \(\mathbb{Z}\)-form of \(\mathfrak{g}\) and \(\mathbb{K}\) is the algebraic closure of \(\mathbb{F}_{p}\). The author applies a result of R. Tange on the fraction field of the center of \(U(\mathfrak{g}_{\mathbb{K}})\) to show that if the G-K conjecture holds for \(\mathfrak{g}\), then for all \(p\gg 0\) the field of rational functions \(\mathbb{K}(\mathfrak{g}_{\mathbb{K}})\) is purely transcendental over its subfield \(\mathbb{K}(\mathfrak{g}_{\mathbb{K}})^{G_{\mathbb{K}}}\) (\(G_\mathbb{K}\) simple, simply connected algebraic Lie group with Lie (\(G_\mathbb{K})=\mathfrak{g}_\mathbb{K}\)). Completing the proof, he shows a modular version (for \(p\gg 0\)) of a resent result of Colliot-Thélène, Kunyavskii, Popov and Reichstein that the field of rational fractions \(k(\mathfrak{g})\) is not purely transcendental over its subfield \(k(\mathfrak{g})^{\mathfrak{g}}\) for the Lie algebras of type type \(B_n\) \((n\geq 2)\), \(D_n\) \((n\geq 4)\), \(E_6\), \(E_7\), \(E_8\), and \(F_4\).
It is a question for which types of simple Lie algebras the G-K conjecture is true and for which it isn’t. In this paper, the author gives an almost complete answer, showing that the G-K conjecture does not hold for simple Lie algebras of type \(B_n\) \((n\geq 3)\), \(D_n\) \((n\geq 4)\), \(E_6\), \(E_7\), \(E_8\), and \(F_4\). Let \(\mathfrak{g}_{\mathbb{K}}=\mathfrak{g}_{\mathbb{Z}} \otimes_{\mathbb{Z}}\mathbb{K}\), where \(\mathfrak{g}_{\mathbb{Z}}\) is a Chevalley \(\mathbb{Z}\)-form of \(\mathfrak{g}\) and \(\mathbb{K}\) is the algebraic closure of \(\mathbb{F}_{p}\). The author applies a result of R. Tange on the fraction field of the center of \(U(\mathfrak{g}_{\mathbb{K}})\) to show that if the G-K conjecture holds for \(\mathfrak{g}\), then for all \(p\gg 0\) the field of rational functions \(\mathbb{K}(\mathfrak{g}_{\mathbb{K}})\) is purely transcendental over its subfield \(\mathbb{K}(\mathfrak{g}_{\mathbb{K}})^{G_{\mathbb{K}}}\) (\(G_\mathbb{K}\) simple, simply connected algebraic Lie group with Lie (\(G_\mathbb{K})=\mathfrak{g}_\mathbb{K}\)). Completing the proof, he shows a modular version (for \(p\gg 0\)) of a resent result of Colliot-Thélène, Kunyavskii, Popov and Reichstein that the field of rational fractions \(k(\mathfrak{g})\) is not purely transcendental over its subfield \(k(\mathfrak{g})^{\mathfrak{g}}\) for the Lie algebras of type type \(B_n\) \((n\geq 2)\), \(D_n\) \((n\geq 4)\), \(E_6\), \(E_7\), \(E_8\), and \(F_4\).
Reviewer: Nikolaos Papalexiou (Karlovassi)
MSC:
| 17B35 | Universal enveloping (super)algebras |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B50 | Modular Lie (super)algebras |
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