On the existence of orders in semisimple Hopf algebras

TL;DR

This paper demonstrates that certain complex semisimple Hopf algebras, specifically Drinfel'd twists of group algebras, cannot be defined over any number ring, and links the existence of weak orders to Kaplansky's sixth conjecture.

Contribution

It establishes the non-existence of Hopf orders over number rings for a family of twisted semisimple Hopf algebras and connects weak orders to Kaplansky's sixth conjecture.

Findings

Some semisimple Hopf algebras lack Hopf orders over any number ring.

Weak orders over integers are equivalent to satisfying Kaplansky's sixth conjecture.

Drinfel'd twists with scalar fractions prevent definability over number rings.

Abstract

We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.