Orders of Nikshych's Hopf algebra

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.