Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for gl_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.
Cite
@article{arxiv.1111.6044,
title = {Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for gl_N},
author = {Vyacheslav Futorny and Jonas T. Hartwig},
journal= {arXiv preprint arXiv:1111.6044},
year = {2020}
}