Non-cyclotomic fusion categories

Morrison, Scott; Snyder, Noah

doi:10.1090/S0002-9947-2012-05498-5

Non-cyclotomic fusion categories
HTML articles powered by AMS MathViewer

by Scott Morrison and Noah Snyder PDF

Trans. Amer. Math. Soc. 364 (2012), 4713-4733 Request permission

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