### Abstract

Original language | English |
---|---|

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

*Transactions of the American Mathematical Society*,

*368*(4), 2547-2562. https://doi.org/10.1090/tran/6380

**On the existence of orders in semisimple Hopf algebras.** / Cuadra, Juan; Meir, Ehud.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 368, no. 4, pp. 2547-2562. https://doi.org/10.1090/tran/6380

}

TY - JOUR

T1 - On the existence of orders in semisimple Hopf algebras

AU - Cuadra, Juan

AU - Meir, Ehud

N1 - The first author was supported by the projects MTM2011-27090 from MICINN and FEDER and by the research group FQM0211 from Junta de Andalucıa. The second author was supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation. The authors would like to thank Dmitriy Rumynin and Martin Lorenz for bring- ing to their attention the notion of weak order, Cesar Galindo for pointing out that these examples were already discussed in [8], and Yevgenia Kashina and Sonia Na-tale for useful discussions on the classification of low dimensional semisimple Hopf algebras.

PY - 2016/4

Y1 - 2016/4

N2 - 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.

AB - 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.

UR - http://www.ams.org/publications/authors/ctp

U2 - 10.1090/tran/6380

DO - 10.1090/tran/6380

M3 - Article

VL - 368

SP - 2547

EP - 2562

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -