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.
Bibliographical noteThe 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 , and Yevgenia Kashina and Sonia Na-tale for useful discussions on the classification of low dimensional semisimple Hopf algebras.