Abstract
We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the “quantum subgroups” in the sense of Ocneanu), we find all irreducible subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.
Similar content being viewed by others
References
Asaeda M., Haagerup U.: Exotic subfactors of finite depth with Jones indices \({(5+\sqrt{13})/2}\) and \({(5+\sqrt{17})/2}\) . Commun. Math. Phys. 202(1), 1–63 (1999)
Aschbacher M.: On intervals in subgroup lattices of finite groups. J. Amer. Math. Soc. 21(3), 809–830 (2008)
Böckenhauer J., Evans D.E., Kawahigashi Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210(3), 733–784 (2000)
Bisch D., Jones V.: Algebras associated to intermediate subfactors. Invent. Math. 128(1), 89–157 (1997)
Evans, D.E., Gannon, T.: The exoticness and realisability of twisted Haagerup-Izumi modular data. http://arxiv.org/abs/1006.1326v1 [math.OA], 2010
Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. of Math.(2) 162(2), 581–642 (2005)
Etingof, P., Nikshych, D., Ostrik, V., with an appendix by Meir, E.: Fusion categories and homotopy theory. http://arxiv.org/abs/0909.3140v2 [math.QA], 2009
Etingof P., Ostrik V.: Module categories over representations of SLqq (2) and graphs. Math. Res. Lett. 11(1), 103–114 (2004)
Evans D.E., Pugh M.: Ocneanu cells and Boltzmann weights for the SU(3) ADE graphs. Münster J. Math. 2, 95–142 (2009)
Evans D.E., Pugh M.: SU(3)-Goodman-de la Harpe-Jones subfactors and the realization of su(3) modular invariants. Rev. Math. Phys. 21(7), 877–928 (2009)
Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. Volume 14 of Mathematical Sciences Research Institute Publications. New York: Springer-Verlag, 1989
Grossman P., Izumi M.: Classification of noncommuting quadrilaterals of factors. Internat. J. Math. 19(5), 557–643 (2008)
Grossman P., Jones V.F.R.: Intermediate subfactors with no extra structure. J. Amer. Math. Soc. 20(1), 219–265 (2007) (electronic)
Haagerup, U.: Principal graphs of subfactors in the index range \({4<[M:N] <3 +\sqrt2}\) . In: Subfactors (Kyuzeso, 1993), River Edge, NJ: World Sci. Publ., 1994, pp. 1–38
Izumi M., Kosaki H.: On a subfactor analogue of the second cohomology. Rev. Math. Phys. 14(7-8), 733–757 (2002)
Izumi M.: The structure of sectors associated with Longo-Rehren inclusions. II. Examples. Rev. Math. Phys. 13(5), 603–674 (2001)
Jones V.F.R.: The annular structure of subfactors. In: Essays on geometry and related topics, Vol. 1, 2. Volume 38 of Monogr. Enseign. Math., Geneva: Enseignement Math., 2001, pp. 401–463
Kirillov A. Jr., Ostrik V.: On a q-analogue of the McKay correspondence and the ADE classification of \({\mathfrak{sl}_2}\) conformal field theories. Adv. Math. 171(2), 183–227 (2002)
Liptrap J. : From hypergroups to anyonic twines, Ph.D.dissertation, June 2010. Available at http://www.math.ucsb.edu/~jliptrap/Jesse_Liptrap_dissertation.pdf
Longo R.: A duality for Hopf algebras and for subfactors. I. Commun. Math. Phys. 159(1), 133–150 (1994)
Longo R., Roberts J.E.: A theory of dimension. K-Theory 11(2), 103–159 (1997)
Morrison, S., Snyder, N.: Subfactors of index less than 5, part 1: the principal graph odometer. http://arxiv.org/abs/1007.1730v2 [math.OA], 2011, to appear in Commun. Math. Phys.
Müger M.: From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Alg. 180(1-2), 81–157 (2003)
Ocneanu, A.: Quantized groups, string algebras and Galois theory for algebras. In: Operator algebras and applications, Vol. 2. Volume 136 of London Math. Soc. Lecture Note Ser., Cambridge: Cambridge Univ. Press, 1988, pp. 119–172
Ocneanu, A.: Paths on coxeter diagrams: from platonic solids and singularities to minimal models and subfactors’. In: Rajarama Bhat, B.V., Elliott, G.A., Fillmore, P.A., editors, Lectures on operator theory, Volume 13 of Fields Institute Monographs, Part 5. Providence, RI: Amer. Math. Soc., 1999
Ocneanu, A.: The classification of subgroups of quantum SU(N). In: Quantum symmetries in theoretical physics and mathematics (Bariloche, 2000), Volume 294 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2002, pp. 133–159
Ostrik V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8(2), 177–206 (2003)
Ostrik, V.: Module categories over the Drinfeld double of a finite group. Int. Math. Res. Not. (27), 1507–1520 (2003)
Peters, E.: A planar algebra construction of the Haagerup subfactor. http://arxiv.org/abs/0902.1294v2 [math.OA], 2009, to appear in Internat. J. Math.
Watatani Y.: Lattices of intermediate subfactors. J. Funct. Anal. 140(2), 312–334 (1996)
Xu F.: On representing some lattices as lattices of intermediate subfactors of finite index. Adv. Math. 220(5), 1317–1356 (2009)
Xu F.: On intermediate subfactors of Goodman-de la Harpe-Jones subfactors. Commun. Math. Phys. 298(3), 707–739 (2010)
Yamagami S.: C*-tensor categories and free product bimodules. J. Funct. Anal. 197(2), 323–346 (2003)
Yamagami, S.: Frobenius algebras in tensor categories and bimodule extensions. In: Galois theory, Hopf algebras, and semiabelian categories. Volume 43 of Fields Inst. Commun., Providence, RI: Amer. Math. Soc., 2004, pp. 551–570
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Grossman, P., Snyder, N. Quantum Subgroups of the Haagerup Fusion Categories. Commun. Math. Phys. 311, 617–643 (2012). https://doi.org/10.1007/s00220-012-1427-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1427-x