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

Author(s)

Futorny, Vyacheslav ^{[1, 2]} ; Hartwig, Jonas T. ^[3]

Affiliation

^[1] Univ Sao Paulo, Dept Math, Sao Paulo - Brazil

^[2] Max Planck Inst Math, D-53111 Bonn - Germany

^[3] Univ Calif Riverside, Dept Math, Riverside, CA 92521 - USA

• Document type: Journal article
• Source: MATHEMATISCHE ZEITSCHRIFT; v. 276, n. 1-2, p. 1-37, FEB 2014.
• Web of Science Citations: 5

Abstract

It is shown that the -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 , and -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 (for a generic ) follows from the positive solution of the -difference Noether problem for the Weyl group of type . 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 , and for a certain extension of . Previously, the case of was shown by Fauquant-Millet (J Algebra 218:93-116, 1999) and by Alev and Dumas (J Algebra 170:229-265, 1994) (for ). Moreover, we give an explicit description of the skew fields of fractions for and which generalizes the results of Alev and Dumas (J Algebra 170:229-265, 1994). (AU)

• FAPESP's process: 10/50347-9 - Algebras, representations e applications
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants