Solution of a \(q\)-difference Noether problem and the quantum Gelfand-Kirillov conjecture for \(\mathfrak{gl}_N\) (Q2636969)

The main result of this paper asserts that the skew field of fractions of the quantum algebra \(U_q(\mathfrak{gl}_N)\) for \(q\in\mathbb{C}\) not a root of unity is isomorphic to a very explicitly given quantum Weyl field over a purely transcendental field extension of \(\mathbb{C}\). The proof is based on reduction to a \(q\)-difference analogue of the so-called noncommutative Noether problem which asks for an explicit description of the algebra of invariants in the skew field of fractions of a quantum affine space under the action of a Weyl group. Some additional results on various types of non-commutative invariants are obtained.