On the existence of orders in semisimple Hopf algebras

Cuadra, Juan; Meir, Ehud

doi:10.1090/tran/6380

On the existence of orders in semisimple Hopf algebras
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by Juan Cuadra and Ehud Meir PDF

Trans. Amer. Math. Soc. 368 (2016), 2547-2562 Request permission

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.

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