Journal of Noncommutative Geometry

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Volume 11, Issue 3, 2017, pp. 919–955
DOI: 10.4171/JNCG/11-3-5

Published online: 2017-09-26

Orders of Nikshych's Hopf algebra

Juan Cuadra[1] and Ehud Meir[2]

(1) University of Almeria, Spain
(2) Universit├Ąt Hamburg, Germany

Let $p$ be an odd prime number and $K$ a number field having a primitive $p$th root of unity $\zeta_p$. We prove that Nikshych's non group-theoretical Hopf algebra $H_p$, which is defined over $\mathbb Q(\zeta_p)$, 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.

Keywords: Fusion categories, semisimple Hopf algebras, Hopf orders, group schemes, cyclotomic integers

Cuadra Juan, Meir Ehud: Orders of Nikshych's Hopf algebra. J. Noncommut. Geom. 11 (2017), 919-955. doi: 10.4171/JNCG/11-3-5