
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 be an odd prime number and a number field having a primitive -th root of unity We prove that Nikshych's non-group theoretical Hopf algebra , which is defined over , admits a Hopf order over the ring of integers if and only if there is an ideal of such that . 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 exists, it is unique and we describe it explicitly.
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