Solution of a $$q$$ q -difference Noether problem and the quantum Gelfand–Kirillov conjecture for $$\mathfrak gl _N$$ g l 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 (Proc Am Math Soc 19:764–765, 1968) and Miyata (Nagoya Math J 41:69–73, 1971) in the case $$q=1$$ , and $$q$$ -deforming the noncommutative Noether problem for the symmetric group (Futorny et al. in Adv Math 223:773–796, 2010). It is also shown that the quantum Gelfand–Kirillov conjecture for $$\mathfrak gl _N$$ (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 (Futorny and Ovsienko in J Algebra 324:598–630, 2010). From here we obtain a proof of the quantum Gelfand–Kirillov conjecture for $$\mathfrak gl _N$$ , and for a certain extension of $$\mathfrak sl _N$$ . Previously, the case of $$\mathfrak sl _N$$ was shown by Fauquant-Millet (J Algebra 218:93–116, 1999) and by Alev and Dumas (J Algebra 170:229–265, 1994) (for $$N=2,3$$ ). Moreover, we give an explicit description of the skew fields of fractions for $$U_q(\mathfrak gl _N)$$ and $$U_q^\mathrm{ext}(\mathfrak sl _N)$$ which generalizes the results of Alev and Dumas (J Algebra 170:229–265, 1994).