Modular Lie algebras and the Gelfand–Kirillov conjecture

Abstract

Abstract. Let g be a finite dimensional simple Lie algebra over and algebraically closed field K of characteristic 0. Let g Z be a Chevalley Z-form of g and g k = g Z ⊗ Z k, where k is the algebraic closure of F p . Let G k be a simple, simply connected algebraic k-group with Lie(G k ) = g k . In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U (g k ) to show that if the Gelfand-Kirillov conjecture (from 1966) holds for g, then for all p ≫ 0 the function field k(g k ) on the dual space g k is purely transcendental over its subfield k(g k ) G k . Very recently, it was proved by Colliot-Thélène-Kunyavskiȋ-Popov-Reichstein that the function field K(g) is not purely transcendental over its subfield K(g) g provided that g is of type B n , n ≥ 3, D n , n ≥ 4, E 6 , E 7 , E 8 or F 4 . We prove a modular version of this result (valid for p ≫ 0) and use it to show that, in characteristic 0, the Gelfand-Kirilov conjecture fails for the simple Lie algebras of the above types. In other words, if g of type B n , n ≥ 3, D n , n ≥ 4, E 6 , E 7 , E 8 or F 4 , then the Lie field of g is more complicated than expected.

On the Gelfand–Kirillov conjecture for the W-algebras attached to the minimal nilpotent orbits

“…to show that the quotient field of U(g) is isomorphic to the quotient field of the tensor product of the same W-algebra with a suitable Weyl algebra. This, together with results of [Pr2], implies that the Gelfand-Kirillov conjecture fails for some W-algebras. It worth mentioning that such W-algebras are deeply studied in [Pr1] and explicit generators and relations are known for them.…”

“…In his paper [Pr2], A. Premet shows that the Gelfand-Kirillov conjecture fails for U(g) if g is simple and g is not of type A n , C n or G 2 . Another result of the same author [Pr1] shows that for a simple Lie algebra g we have that U(g) is "almost equal" to the tensor product of some W-algebra with a suitable Weyl algebra.…”

Some aspects of noncommutative invariant theory and the Noether’s problem

“…The Gelfand-Kirillov conjecture [21] asks: is this skew-field of fractions isomorphic to D n (K ), where K is a purely transcendental field extension of k, of finite transcendence degree? The answer is known to be true for L solvable [7,30,36], and is almost settled for the simple Lie algebras (over an algebraically closed field): it is true for sl n (k) (and gl n (k)), the one with root system G 2 , open when the root system is C n , and false for all other cases (see [44] for details).…”

On enveloping skew fields of some Lie superalgebras

“…We show here that the same is true for any n describing explicitely some classes of skew fields arising from this context. The even part g 0 of g = osp(1, 2n) is the Lie algebra sp(2n) of the symplectic group, for which the Gelfand-Kirillov property remains an open question (see [16]). Therefore we concentrate in this exploratory paper on the case of osp(1, 2) and on some significant subsuperalgebras of osp (1,4).…”