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On the existence of orders in semisimple Hopf algebras


Authors: Juan Cuadra and Ehud Meir
Journal: Trans. Amer. Math. Soc. 368 (2016), 2547-2562
MSC (2010): Primary 16T05
DOI: https://doi.org/10.1090/tran/6380
Published electronically: August 20, 2015
MathSciNet review: 3449248
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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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Additional Information

Juan Cuadra
Affiliation: Departamento de Matemáticas, Universidad de Almería, E-04120 Almería, Spain
Email: jcdiaz@ual.es

Ehud Meir
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitets- parken 5, DK-2100, Denmark
Email: meirehud@gmail.com

DOI: https://doi.org/10.1090/tran/6380
Received by editor(s): September 10, 2013
Received by editor(s) in revised form: December 24, 2013, and January 21, 2014
Published electronically: August 20, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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