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 $\zeta.$ 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.