Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for glN
Abstract
It is shown that the q-difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck and Miyata in the case q=1, and q-deforming the noncommutative Noether problem for the symmetric group. It is also shown that the quantum Gelfand-Kirillov conjecture for glN (for a generic q) follows from the positive solution of the q-difference Noether problem for the Weyl group of type Dn. The proof is based on the theory of Galois rings developed by the first author and Ovsienko. From here we obtain a new proof of the quantum Gelfand-Kirillov conjecture for slN, thus recovering the result of Fauquant-Millet. Moreover, we provide an explicit description of skew fields of fractions for quantized glN and slN generalizing Alev and Dumas.