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Noncommutative Noether's problem for complex reflection groups


Authors: Farkhod Eshmatov, Vyacheslav Futorny, Sergiy Ovsienko and Joao Fernando Schwarz
Journal: Proc. Amer. Math. Soc. 145 (2017), 5043-5052
MSC (2010): Primary 16Z05, 16R30
DOI: https://doi.org/10.1090/proc/13646
Published electronically: August 29, 2017
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Abstract | References | Similar Articles | Additional Information

Abstract: We solve some noncommutative analogue of the Noether's problem for the reflection groups by showing that the skew field of fractions of the invariant subalgebra of the Weyl algebra under the action of any finite complex reflection group is a Weyl field, that is, isomorphic to the skew field of fractions of some Weyl algebra. We also extend this result to the invariants of the ring of differential operators on any finite dimensional torus. The results are applied to obtain analogs of the Gelfand-Kirillov conjecture for Cherednik algebras and Galois algebras.


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Additional Information

Farkhod Eshmatov
Affiliation: Department of Mathematics, University of Sichuan, Chengdu, People’s Republic of China
Email: olimjon55@hotmail.com

Vyacheslav Futorny
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil
Email: futorny@ime.usp.br

Sergiy Ovsienko
Affiliation: Faculty of Mechanics and Mathematics, Taras Shevchenko Kiev University Kiev, Ukraine
Email: ovsiyenko.sergiy@gmail.com

Joao Fernando Schwarz
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brazil
Email: jfschwarz.0791@gmail.com

DOI: https://doi.org/10.1090/proc/13646
Received by editor(s): July 3, 2015
Received by editor(s) in revised form: November 27, 2016
Published electronically: August 29, 2017
Additional Notes: The first author was supported in part by Fapesp (2013/22068-6)
The second author was supported in part by CNPq (301320/2013-6) and by Fapesp (2014/09310-5)
The fourth author was supported in part by Fapesp (2014/25612-1)
Communicated by: Harm Derksen
Article copyright: © Copyright 2017 American Mathematical Society

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