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.

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