Non-cyclotomic fusion categories
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- by Scott Morrison and Noah Snyder PDF
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Abstract:
Etingof, Nikshych and Ostrik asked 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 and one coming from the newly constructed extended Haagerup subfactor cannot be completely defined over a cyclotomic field. On the other hand, we show that the Drinfel’d center 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.References
- 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
- Stephen Bigelow, Scott Morrison, Emily Peters, and Noah Snyder, Constructing the extended Haagerup planar algebra, 2009, arXiv:0909.4099, to appear Acta Mathematica.
- 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
- D. Bisch, Subfactors and planar algebras, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 775–785. MR 1957084
- A. Coste and T. Gannon, Remarks on Galois symmetry in rational conformal field theories, Phys. Lett. B 323 (1994), no. 3-4, 316–321. MR 1266785, DOI 10.1016/0370-2693(94)91226-2
- Jan de Boer and Jacob Goeree, Markov traces and $\textrm {II}_1$ factors in conformal field theory, Comm. Math. Phys. 139 (1991), no. 2, 267–304. MR 1120140
- 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
- Phùng Hô Hai, Tannaka-Krein duality for Hopf algebroids, Israel J. Math. 167 (2008), 193–225. MR 2448024, DOI 10.1007/s11856-008-1047-5
- Takahiro Hayashi, A canonical Tannaka duality for finite semiisimple tensor categories, 1999, arXiv:9904073.
- Seung-Moon Hong, Eric Rowell, and Zhenghan Wang, On exotic modular tensor categories, Commun. Contemp. Math. 10 (2008), no. suppl. 1, 1049–1074. MR 2468378, DOI 10.1142/S0219199708003162
- Masaki Izumi, The structure of sectors associated with Longo-Rehren inclusions. II. Examples, Rev. Math. Phys. 13 (2001), no. 5, 603–674. MR 1832764, DOI 10.1142/S0129055X01000818
- Nathan Jacobson, Basic algebra. II, 2nd ed., W. H. Freeman and Company, New York, 1989. MR 1009787
- Vaughan F. R. Jones, Planar algebras, I, arXiv:math.QA/9909027.
- Vaughan F. R. Jones, The planar algebra of a bipartite graph, Knots in Hellas ’98 (Delphi), Ser. Knots Everything, vol. 24, World Sci. Publ., River Edge, NJ, 2000, pp. 94–117. MR 1865703, DOI 10.1142/9789812792679_{0}008
- Vaughan F. R. Jones, Quadratic tangles in planar algebras, 2003, arXiv:1007.1158.
- 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 Benjamin Balsam, Turaev-viro invariants as an extended tqft, arXiv:1004.1533.
- Greg Kuperberg, Finite, connected, semisimple, rigid tensor categories are linear, Math. Res. Lett. 10 (2003), no. 4, 411–421. MR 1995781, DOI 10.4310/MRL.2003.v10.n4.a1
- 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
- Viktor Ostrik, Module categories over the Drinfeld double of a finite group, Int. Math. Res. Not. 27 (2003), 1507–1520. MR 1976233, DOI 10.1155/S1073792803205079
- 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
- Emily Peters, A planar algebra construction of the Haagerup subfactor, Internat. J. Math. 21 (2010), no. 8, 987–1045. MR 2679382, DOI 10.1142/S0129167X10006380
- Hendryk Pfeiffer, Finitely semisimple spherical categories and modular categories are self-dual, Adv. Math. 221 (2009), no. 5, 1608–1652. MR 2522429, DOI 10.1016/j.aim.2009.03.002
- N. Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26. MR 1036112
- Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
Additional Information
- Scott Morrison
- Affiliation: Miller Institute for Basic Research, University of California at Berkeley, Berkeley, California 94720
- MR Author ID: 788724
- Email: scott@tqft.net
- Noah Snyder
- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 667772
- Email: nsnyder@math.columbia.edu
- Received by editor(s): October 1, 2010
- Published electronically: April 17, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 4713-4733
- MSC (2010): Primary 18D10; Secondary 46L37
- DOI: https://doi.org/10.1090/S0002-9947-2012-05498-5
- MathSciNet review: 2922607