Quantum Gelfand-Kirillov Conjecture for gl(n)

The classical Gelfand-Kirillov Conjecture states that the
skew field of fractions of the enveloping algebra of an algebraic Lie
algebra is a Weyl skew field.  The conjecture holds for gl(n) (hence
sl(n)), nilpotent and solvable Lie algebras and for all Lie algebras
of dimension less than 9. On the other hand there are counter-examples
for "mixed" Lie algebras.  In the quantum setting the conjecture is
known to hold for Borel subalgebras and for gl(2) and  gl(3).   We
will discuss the advances and the state of the Gelfand-Kirillov
Conjecture for the quantized enveloping algebra of gl(n) with n>3.
The talk is based on recent joint results with J.Hartwig.  The
technique uses the theory of Gelfand-Tsetlin modules for a class of
Galois algebras which are certain invariant subalgebras in skew group
rings.  This is a joint talk  with J.Hartwig.