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2010

Non-cyclotomic fusion categories

Morrison, Scott, Snyder, Noah

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Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion category can be completely defined over a cyclotomic field.

We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely defined over a cyclotomic field.
On the other hand, we show that the double of the even part of the Haagerup subfactor is completely defined over a cyclotomic field.
We identify the minimal field of definition for each of these fusion categories, compute the Galois groups, and identify their Galois conjugates.

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Roger Penrose
1971
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Jean-Pierre Serre
Original
1977
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Nathan Jacobson
1989
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N Yu Reshetikhin
1990
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Jan de Boer
1991
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André Joyal
1991
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Dietmar Bisch
1997
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Marta Asaeda
1999
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