The Brauer-Picard group of the Asaeda-Haagerup fusion categories
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- by Pinhas Grossman and Noah Snyder PDF
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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.References
- Marta Asaeda and Pinhas Grossman, A quadrilateral in the Asaeda-Haagerup category, Quantum Topol. 2 (2011), no. 3, 269–300. MR 2812458, DOI 10.4171/QT/22
- M. Asaeda and U. Haagerup, Exotic subfactors of finite depth with Jones indices $(5+\sqrt {13})/2$ and $(5+\sqrt {17})/2$, Comm. Math. Phys. 202 (1999), no. 1, 1–63. MR 1686551, DOI 10.1007/s002200050574
- Jens Böckenhauer, David E. Evans, and Yasuyuki Kawahigashi, Chiral structure of modular invariants for subfactors, Comm. Math. Phys. 210 (2000), no. 3, 733–784. MR 1777347, DOI 10.1007/s002200050798
- Dietmar Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, Operator algebras and their applications (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 13–63. MR 1424954, DOI 10.1007/s002220050137
- Alexei Davydov, Michael Müger, Dmitri Nikshych, and Victor Ostrik, The Witt group of non-degenerate braided fusion categories, J. Reine Angew. Math. 677 (2013), 135–177. MR 3039775, DOI 10.1515/crelle.2012.014
- Sergio Doplicher and John E. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989), no. 1, 157–218. MR 1010160, DOI 10.1007/BF01388849
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, online.
- David E. Evans and Yasuyuki Kawahigashi, Quantum symmetries on operator algebras, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1642584
- Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI 10.4007/annals.2005.162.581
- Pavel Etingof, Dmitri Nikshych, and Victor Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010), no. 3, 209–273. With an appendix by Ehud Meir. MR 2677836, DOI 10.4171/QT/6
- Pavel Etingof and Viktor Ostrik, Module categories over representations of $\textrm {SL}_q(2)$ and graphs, Math. Res. Lett. 11 (2004), no. 1, 103–114. MR 2046203, DOI 10.4310/MRL.2004.v11.n1.a10
- David E. Evans and Mathew Pugh, Ocneanu cells and Boltzmann weights for the $\rm SU(3)$ $\scr {ADE}$ graphs, Münster J. Math. 2 (2009), 95–142. MR 2545609
- David E. Evans and Mathew Pugh, SU(3)-Goodman-de la Harpe-Jones subfactors and the realization of SU(3) modular invariants, Rev. Math. Phys. 21 (2009), no. 7, 877–928. MR 2553429, DOI 10.1142/S0129055X09003761
- Jürgen Fuchs and Carl Stigner, On Frobenius algebras in rigid monoidal categories, Arab. J. Sci. Eng. Sect. C Theme Issues 33 (2008), no. 2, 175–191 (English, with English and Arabic summaries). MR 2500035
- Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799, DOI 10.1007/978-1-4613-9641-3
- Pinhas Grossman and Masaki Izumi, Classification of noncommuting quadrilaterals of factors, Internat. J. Math. 19 (2008), no. 5, 557–643. MR 2418197, DOI 10.1142/S0129167X08004807
- Pinhas Grossman and Noah Snyder, Quantum subgroups of the Haagerup fusion categories, Comm. Math. Phys. 311 (2012), no. 3, 617–643. MR 2909758, DOI 10.1007/s00220-012-1427-x
- Masaki Izumi, Application of fusion rules to classification of subfactors, Publ. Res. Inst. Math. Sci. 27 (1991), no. 6, 953–994. MR 1145672, DOI 10.2977/prims/1195169007
- Vaughan F. R. Jones, Planar algebras, I, arXiv:math.QA/9909027.
- V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, DOI 10.1007/BF01389127
- Vaughan F. R. Jones, Quadratic tangles in planar algebras, Duke Math. J. 161 (2012), no. 12, 2257–2295. MR 2972458, DOI 10.1215/00127094-1723608
- André Joyal and Ross Street, The geometry of tensor calculus. I, Adv. Math. 88 (1991), no. 1, 55–112. MR 1113284, DOI 10.1016/0001-8708(91)90003-P
- Alexander Kirillov Jr. and Viktor Ostrik, On a $q$-analogue of the McKay correspondence and the ADE classification of $\mathfrak {sl}_2$ conformal field theories, Adv. Math. 171 (2002), no. 2, 183–227. MR 1936496, DOI 10.1006/aima.2002.2072
- David Kazhdan and Hans Wenzl, Reconstructing monoidal categories, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 111–136. MR 1237835
- Roberto Longo, A duality for Hopf algebras and for subfactors. I, Comm. Math. Phys. 159 (1994), no. 1, 133–150. MR 1257245
- R. Longo and J. E. Roberts, A theory of dimension, $K$-Theory 11 (1997), no. 2, 103–159. MR 1444286, DOI 10.1023/A:1007714415067
- Scott Morrison, Emily Peters, and Noah Snyder, Knot polynomial identities and quantum group coincidences, Quantum Topol. 2 (2011), no. 2, 101–156. MR 2783128, DOI 10.4171/QT/16
- Michael Müger, From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories, J. Pure Appl. Algebra 180 (2003), no. 1-2, 81–157. MR 1966524, DOI 10.1016/S0022-4049(02)00247-5
- Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. MR 996454
- B. V. Rajarama Bhat, George A. Elliott, and Peter A. Fillmore (eds.), Lectures on operator theory, Fields Institute Monographs, vol. 13, American Mathematical Society, Providence, RI, 1999. MR 1743202, DOI 10.1090/fim/013
- Adrian Ocneanu, Operator algebras, topology and subgroups of quantum symmetry—construction of subgroups of quantum groups, Taniguchi Conference on Mathematics Nara ’98, Adv. Stud. Pure Math., vol. 31, Math. Soc. Japan, Tokyo, 2001, pp. 235–263. MR 1865095, DOI 10.2969/aspm/03110235
- Adrian Ocneanu, The classification of subgroups of quantum $\textrm {SU}(N)$, Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000) Contemp. Math., vol. 294, Amer. Math. Soc., Providence, RI, 2002, pp. 133–159. MR 1907188, DOI 10.1090/conm/294/04972
- Victor Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), no. 2, 177–206. MR 1976459, DOI 10.1007/s00031-003-0515-6
- Roger Penrose, Applications of negative dimensional tensors, Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) Academic Press, London, 1971, pp. 221–244. MR 0281657
- Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
- Imre Tuba and Hans Wenzl, On braided tensor categories of type $BCD$, J. Reine Angew. Math. 581 (2005), 31–69. MR 2132671, DOI 10.1515/crll.2005.2005.581.31
- Shigeru Yamagami, $C^\ast$-tensor categories and free product bimodules, J. Funct. Anal. 197 (2003), no. 2, 323–346. MR 1960417, DOI 10.1016/S0022-1236(02)00036-8
- Shigeru Yamagami, Frobenius algebras in tensor categories and bimodule extensions, Galois theory, Hopf algebras, and semiabelian categories, Fields Inst. Commun., vol. 43, Amer. Math. Soc., Providence, RI, 2004, pp. 551–570. MR 2075605, DOI 10.1090/fic/043
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
- MR Author ID: 667772
- Email: nsnyder@math.columbia.edu
- Received by editor(s): October 4, 2012
- Received by editor(s) in revised form: January 2, 2014
- Published electronically: August 18, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 2289-2331
- MSC (2010): Primary 46L37; Secondary 18D10
- DOI: https://doi.org/10.1090/tran/6364
- MathSciNet review: 3449240