Mathematische Zeitschrift

, Volume 276, Issue 1–2, pp 1–37 | Cite as

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

Article

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).

Keywords

Gelfand–Kirillov conjecture Quantum group  Noncommutative invariant theory 

Mathematics Subject Classification (2000)

16T20 (16K40 16W22 ) 

Notes

Acknowledgments

The authors are grateful to Michel Van den Bergh, Fedor Malikov, Eugene Mukhin and Alan Weinstein for encouraging discussions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of São PauloSão PauloBrazil
  2. 2.Max Planck Institute for MathematicsBonnGermany
  3. 3.Department of MathematicsUniversity of CaliforniaRiversideUSA

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