# Orders of Nikshych's Hopf algebra

### Juan Cuadra

University of Almeria, Spain### Ehud Meir

Universität Hamburg, Germany

## Abstract

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.

## Cite this article

Juan Cuadra, Ehud Meir, Orders of Nikshych's Hopf algebra. J. Noncommut. Geom. 11 (2017), no. 3, pp. 919–955

DOI 10.4171/JNCG/11-3-5