The Noether problem for Hopf algebras
| Autor: | Akira Masuoka, Christian Kassel |
|---|---|
| Přispěvatelé: | Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Institute of Mathematics, University of Tsukuba, University of Tsukuba, Université de Tsukuba = University of Tsukuba |
| Rok vydání: | 2014 |
| Předmět: |
Pure mathematics
Polynomial rationality Field (mathematics) MSC: 16T05 16W22 16R50 Hopf algebra localization polynomial identities Comodule Mathematics::Quantum Algebra Mathematics - Quantum Algebra FOS: Mathematics Quantum Algebra (math.QA) 16T05 16W22 16R50 14E08 (Primary) 13A50 13B30 12F20 (Secondary) Mathematical Physics Mathematics Finite group Algebra and Number Theory Group (mathematics) Subalgebra [MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA] Mathematics::Rings and Algebras Noether problem Mathematics - Rings and Algebras Invariant theory invariant theory Rings and Algebras (math.RA) [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA] Geometry and Topology |
| Zdroj: | Journal of Noncommutative Geometry Journal of Noncommutative Geometry, European Mathematical Society, 2016, 10 (2), pp.405-428. ⟨10.4171/JNCG/237⟩ Journal of Noncommutative Geometry, European Mathematical Society, 2016, 10 (2), pp.405-428 |
| ISSN: | 1661-6952 1661-6960 |
| DOI: | 10.48550/arxiv.1404.4941 |
| Popis: | In previous work, Eli Aljadeff and the first-named author attached an algebra B_H of rational fractions to each Hopf algebra H. The generalized Noether problem is the following: for which finite-dimensional Hopf algebra H is B_H the localization of a polynomial algebra? A positive answer to this question when H is the algebra of functions on a finite group implies a positive answer for the classical Noether problem for the group. We show that the generalized Noether problem has a positive answer for all pointed finite-dimensional Hopf algebras over a field of characteristic zero. We actually give a precise description of B_H for such a Hopf algebra, including a bound on the degrees of the generators. A theory of polynomial identities for comodule algebras over a Hopf algebra H gives rise to a universal comodule algebra whose subalgebra of coinvariants V_H maps injectively into B_H. In the second half of this paper, we show that B_H is a localization of V_H when again H is a pointed finite-dimensional Hopf algebra in characteristic zero. We also report on a result by Uma Iyer showing that the same localization result holds when H is the algebra of functions on a finite group. Comment: 19 pages. Section 4.3 and three references have been added to Version 2 |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|