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The Brauer-Picard group of the Asaeda-Haagerup fusion categories


Authors: Pinhas Grossman and Noah Snyder
Journal: Trans. Amer. Math. Soc. 368 (2016), 2289-2331
MSC (2010): Primary 46L37; Secondary 18D10
DOI: https://doi.org/10.1090/tran/6364
Published electronically: August 18, 2015
MathSciNet review: 3449240
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Abstract: We prove that the Brauer-Picard group of Morita autoequivalences of each of the three fusion categories which arise as an even part of the Asaeda-Haagerup subfactor or of its index 2 extension is the Klein four-group. We describe the 36 bimodule categories which occur in the full subgroupoid of the Brauer-Picard groupoid on these three fusion categories. We also classify all irreducible subfactors both of whose even parts are among these categories, of which there are 111 up to isomorphism of the planar algebra (76 up to duality). Although we identify the entire Brauer-Picard group, there may be additional fusion categories in the groupoid. We prove a partial classification of possible additional fusion categories Morita equivalent to the Asaeda-Haagerup fusion categories and make some conjectures about their existence.


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Additional Information

Pinhas Grossman
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: p.grossman@unsw.edu.au

Noah Snyder
Affiliation: Department of Mathematics, Indiana University, 831 E. Third Street, Bloomington, Indiana 47405
Email: nsnyder@math.columbia.edu

DOI: https://doi.org/10.1090/tran/6364
Keywords: Fusion categories, subfactors, quantum subgroups
Received by editor(s): October 4, 2012
Received by editor(s) in revised form: January 2, 2014
Published electronically: August 18, 2015
Article copyright: © Copyright 2015 American Mathematical Society