Orders of Nikshych's Hopf algebra

TL;DR

This paper investigates conditions under which Nikshych's non-group theoretical Hopf algebra admits a Hopf order over rings of integers in number fields, revealing specific ideal conditions and uniqueness results.

Contribution

It establishes a precise criterion for the existence of Hopf orders of Nikshych's algebra over number field rings of integers, including explicit descriptions and uniqueness.

Findings

Hopf order exists iff an ideal I satisfies I^{2(p-1)} = (p)

No Hopf order over cyclotomic rings of integers in certain cases

Hopf orders, when they exist, are unique and explicitly described

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.