### 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.

Original language | English |
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Pages (from-to) | 2547-2562 |

Number of pages | 16 |

Journal | Transactions of the American Mathematical Society |

Volume | 368 |

Issue number | 4 |

Early online date | 20 Aug 2015 |

DOIs | |

Publication status | Published - Apr 2016 |

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## Cite this

Cuadra, J., & Meir, E. (2016). On the existence of orders in semisimple Hopf algebras.

*Transactions of the American Mathematical Society*,*368*(4), 2547-2562. https://doi.org/10.1090/tran/6380