Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for gl_N

TL;DR

This paper proves the q-difference Noether problem for classical Weyl groups, establishing the quantum Gelfand-Kirillov conjecture for gl_N and sl_N, and provides explicit descriptions of their skew fields of fractions.

Contribution

It introduces a positive solution to the q-difference Noether problem for all classical Weyl groups and links it to the quantum Gelfand-Kirillov conjecture for gl_N.

Findings

Positive solution to the q-difference Noether problem for classical Weyl groups

Quantum Gelfand-Kirillov conjecture for gl_N proven for generic q

Explicit descriptions of skew fields of fractions for quantized gl_N and sl_N

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 gl_N (for a generic q) follows from the positive solution of the q-difference Noether problem for the Weyl group of type D_n. 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 sl_N, thus recovering the result of Fauquant-Millet. Moreover, we provide an explicit description of skew fields of fractions for quantized gl_N and sl_N generalizing Alev and Dumas.