Abstract
Let p be an odd prime number and K a number field having a primitive pth root of unity ζp. We prove that Nikshych's non group-theoretical Hopf algebra Hp, which is defined over Q(ζp), admits a Hopf order over the ring of integers OK if and only if there is an ideal I of OK such that I2(p−1)=(p). This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over OK exists, it is unique and we describe it explicitly.
| Original language | English |
|---|---|
| Pages (from-to) | 919-955 |
| Number of pages | 37 |
| Journal | Journal of Noncommutative Geometry |
| Volume | 11 |
| Issue number | 3 |
| Early online date | 26 Sept 2017 |
| DOIs | |
| Publication status | Published - Sept 2017 |
Bibliographical note
The first author was supported by grant MTM2014-54439-P from MICINN and FEDER and by the research group FQM0211 from Junta de Andalucía. The second author was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. The authors are grateful to Dror Speiser for doing the previous computer calculation and to Bjorn Poonen for a conversation about the number theoretical condition in Theorem 7.1. The authors are finally indebted to the referee for his/her comments and suggestions, which helped to improve substantially the presentation of the results.Keywords
- fusion categories
- semisimple Hopf algebras
- Hopf orders
- group schemes
- cyclotomic integers