Orders of Nikshych's Hopf algebra
| Autor: | Juan Cuadra, Ehud Meir |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
Algebra and Number Theory
Group (mathematics) Mathematics::Number Theory Mathematics::Rings and Algebras Mathematics - Rings and Algebras Hopf algebra Rings and Algebras (math.RA) Mathematics::Quantum Algebra Mathematics - Quantum Algebra FOS: Mathematics Quantum Algebra (math.QA) Geometry and Topology Representation Theory (math.RT) Mathematical economics Mathematics - Representation Theory Mathematical Physics 16T05 18D10 16G30 14L15 Mathematics |
| Zdroj: | Journal of Noncommutative Geometry. 11:919-955 |
| ISSN: | 1661-6952 |
| DOI: | 10.4171/jncg/11-3-5 |
| Popis: | Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $\zeta.$ We prove that Nikshych's non-group theoretical Hopf algebra $H_p$, which is defined over $\mathbb{Q}(\zeta)$, admits a Hopf order over the ring of integers $\mathcal{O}_K$ if and only if there is an ideal $I$ of $\mathcal{O}_K$ such that $I^{2(p-1)} = (p)$. 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 $\mathcal{O}_K$ exists, it is unique and we describe it explicitly. Comment: 33 pages. Major changes in the presentation |
| Databáze: | OpenAIRE |
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