The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is the skew field of fractions of $A_{n,s}(k)$.
We have the following classical problem which was formulated by I. Gelfand and A. Kirillov in 1966 (in the same wounderful paper where the beloved GK-dimension was introduced):
- Let $\mathfrak{g}$ be a finite dimensional algebraic Lie algebra over an algebraically closed field of zero characteristc. When the skew field of fractions of $U(\mathfrak{g})$ is isomorphic to a Weyl field $F_{n,s}(k)$?
Widely known as the Gelfand-Kirillov Conjecture, it was eventually shown to be false in general, in [ Alev, Jacques; Ooms, Alfons; Van den Bergh, Michel A class of counterexamples to the Gelʹfand-Kirillov conjecture, 1996 ]
An interesting contribution was made by Premet in 2010 in [ Modular Lie algebras and the Gelfand-Kirillov conjecture ]. For $\mathfrak{g}$ a simple Lie algebra, it was known that Gelfand-Kirillov problem had a positive solution for type $A$ algebras since the seminal work of 1966, but the for other simple Lie algebras nothing was known until Premet showed in the above that Gelfand-Kirillov Problem is false for simple Lie algebras of type $B,D,F,E$.
The remaining simple Lie algebras are quite elusive. The last work I know of discussing them is [ Anderson, Dave; Florence, Mathieu; Reichstein, Zinovy The Lie algebra of type G2 is rational over its quotient by the adjoint action, 2013 ].
A couple of years ago, I've met Alev personally, and according to him, nothing is known about the skew field of fractions of $U(\mathfrak{g})$, when $\mathfrak{g}$ is of type $C,G$.
- Has the understanding of these skew field of fractions been clarified by some recent work?
In a certain sense, Gelfand-Kirillov Conjecture fails by a small margin. Namely, in their second seminal work of 1969 [ The structure of the Lie field connected with a split semisimple Lie algebra ] the authors have introduced the following Conjecture
- Let $D(\mathfrak{g})$ be the skew field of fractions of $U(\mathfrak{g})$, $\mathfrak{g}$ an algebraic Lie algebra as above. Let $Z$ be its center. Then there exists a finite field extension of $K$ of $Z$, purely transcedental over $k$, such that the skew field of fractions of $D(\mathfrak{g}) \otimes_Z K$ is a Weyl field $F_{n,s}(k)$.
In their work of 1969, it was shown that this Modified Gelfand-Kirillov Conjecture is true for all semisimple Lie $\mathfrak{g}$. In [ Alev et. al. op. cit ] the authors say they are quite confident that this modified Conjecture is true.
I've not known on any advance on it in the last decades. In the same conversation with Alev a couple of years ago, he pointed that there is a connection between Gelfand and Kirillov 1969 approach to this Conjecture and his work with F. Dumas on the so called Noncommutative Noether's Problem ( [ Opérateurs différentiels invariants et problème de Noether, 2006 ]), but I was unable to figure out precisely the connection.
Anyway,
- Is there any recent progress on the modified Gelfand-Kirillov Conjecture?
