On the existence of orders in semisimple Hopf algebras

2013/07/11 by Juan Cuadra, Cuadra, Juan, Ehud Meir +1
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.QA #math.RA #math.RT

paper · pdf · doi:10.48550/arxiv.1307.3269

Final version, to appear in Trans. Amer. Math. Soc.; 16 pp

arxiv created 2014/01/25 · arxiv updated 2014/01/28

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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