Communications in Mathematical Physics

, Volume 334, Issue 2, pp 889–922 | Cite as

1-Supertransitive Subfactors with Index at Most \({6\frac{1}{5}}\)

  • Zhengwei Liu
  • Scott Morrison
  • David Penneys


An irreducible II1-subfactor \({A\subset B}\) is exactly 1-supertransitive if \({B\ominus A}\) is reducible as an AA bimodule. We classify exactly 1-supertransitive subfactors with index at most \({6\frac{1}{5}}\), leaving aside the composite subfactors at index exactly 6 where there are severe difficulties. Previously, such subfactors were only known up to index \({3+\sqrt{5} \approx 5.23}\). Our work is a significant extension, and also shows that index 6 is not an insurmountable barrier.

There are exactly three such subfactors with index in \({(3+\sqrt{5}, 6 \frac{1}{5}]}\), all with index \({3+2\sqrt{2}}\). One of these comes from SO(3) q at a root of unity, while the other two appear to be closely related, and are ‘braided up to a sign’.

This is the published version of arXiv:1310.8566.


Fusion Category Algebraic Integer Reidemeister Move Minimal Projection Principal Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AH99.
    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). arxiv:math.OA/9803044; doi: 10.1007/s002200050574
  2. Ale23.
    Alexander, J.W.: A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9(3), 93–95 (1923).
  3. BH96.
    Bisch, D, Haagerup, U.: Composition of subfactors: new examples of infinite depth subfactors. Ann. Sci. École Norm. Sup. (4) 29(3), 329–383 (1996).
  4. BJ00.
    Bisch, D., Jones, V.F.R.: Singly generated planar algebras of small dimension. Duke Math. J. 101(1), 41–75 (2000). MR1733737Google Scholar
  5. BJ03.
    Bisch D., Jones V.: Singly generated planar algebras of small dimension. II. Adv. Math. 175(2), 297–318 (2003). doi: 10.1016/S0001-8708(02)00060-9 CrossRefzbMATHMathSciNetGoogle Scholar
  6. BJL13.
    Bisch, D., Jones, V.F.R., Liu, Z.: Singly generated planar algebras of small dimension, part III (2013, in preparation)Google Scholar
  7. BL10.
    Bhattacharyya, B., Landau, Z.: Intermediate standard invariants and intermediate planar algebras. (2010).
  8. BMPS12.
    Bigelow, S., Morrison, S., Peters, E., Snyder, N.: Constructing the extended Haagerup planar algebra. Acta Math. 209(1), 29–82 (2012). arxiv:0909.4099; doi: 10.1007/s11511-012-0081-7
  9. BNP07.
    Bisch, D., Nicoara, R., Popa, S.: Continuous families of hyperfinite subfactors with the same standard invariant. Internat. J. Math. 18(3), 255–267 (2007). arxiv:math.OA/0604460; doi: 10.1142/S0129167X07004011
  10. BP14.
    Bigelow, S., Penneys, D.: Principal graph stability and the jellyfish algorithm. Math. Ann. 358(1–2), 1–24 (2014). doi: 10.1007/s00208-013-0941-2; arxiv:1208.1564
  11. BV13.
    Brothier, A., Vaes, S.: Families of hyperfinite subfactors with the same standard invariant and prescribed fundamental group (2013). arxiv:1309.5354
  12. EG98.
    Etingof, P., Gelaki, S.: Semisimple Hopf algebras of dimension pq are trivial. J. Algebra 210(2), 664–669 (1998). doi: 10.1006/jabr.1998.7568; arxiv:math/9801129
  13. EG12.
    Evans, D.E., Gannon, T.: Near-group fusion categories and their doubles. Adv. Math. 255, 586–640 (2014). doi: 10.1016/j.aim.2013.12.014; arxiv:1208.1500
  14. FYH+85.
    Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. (N.S.) 12(2),239–246 (1985). MR776477Google Scholar
  15. GdlHJ89.
    Goodman, F.M., de la Harpe, P., Jones, V.F.R.: Coxeter graphs and towers of algebras. In: Mathematical Sciences Research Institute Publications, vol. 14. Springer, New York (1989). MR999799Google Scholar
  16. Gol59.
    Goldman, M.: On subfactors of factors of type II1. Michigan Math. J. 6, 167–172 (1959). MR0107827Google Scholar
  17. GS12a.
    Grossman, P., Snyder, N.: The Brauer–Picard group of the Asaeda–Haagerup fusion categories. Trans. Am. Math. Soc. (2012, to appear). arxiv:1202.4396
  18. GS12b.
    Grossman, P., Snyder, N.: Quantum subgroups of the Haagerup fusion categories. Commun. Math. Phys. 311(3), 617–643 (2012). arxiv:1102.2631; doi: 10.1007/s00220-012-1427-x
  19. Haa94.
    Haagerup, U.: Principal graphs of subfactors in the index range \({4 \, < \, [M \, : \, N] \, < \, 3 \,+ \,\sqrt2}\). In: Subfactors (Kyuzeso, 1993), pp. 1–38. World Sci. Publ., River Edge (1994). MR1317352Google Scholar
  20. IJMS12.
    Izumi, M., Jones, V.F.R., Morrison, S., Snyder, N.: Subfactors of index less than 5, Part 3: quadruple points. Commun. Math. Phys. 316(2), 531–554 (2012). arxiv:1109.3190; doi: 10.1007/s00220-012-1472-5
  21. IMP13.
    Izumi, M., Morrison, S., Penneys, D.: Fusion categories between \({\mathcal{C} \boxtimes \mathcal{D}}\) and \({\mathcal{C} *\mathcal{D}}\) (2013). arxiv:1308.5723
  22. IMP+14.
    Izumi, M., Penneys, D., Peters, E., Snyder, N.: Subfactors of index exactly 5. arxiv:1406.2389
  23. Izu91.
    Izumi M.: Application of fusion rules to classification of subfactors. Publ. Res. Inst. Math. Sci. 27(6), 953–994 (1991). doi: 10.2977/prims/1195169007 CrossRefzbMATHMathSciNetGoogle Scholar
  24. Izu01.
    Izumi M.: The structure of sectors associated with Longo–Rehren inclusions. II. Examples. Rev. Math. Phys. 13(5), 603–674 (2001). doi: 10.1142/S0129055X01000818 CrossRefzbMATHMathSciNetGoogle Scholar
  25. JMS13.
    Jones V.F.R., Morrison S., Snyder N.: The classification of subfactors of index at most 5. Bull. Am. Math. Soc. 51, 277–327 (2014). arxiv:1304.6141
  26. Jon83.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983). doi: 10.1007/BF01389127 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. Jon99.
    Jones, V.F.R.: Planar algebras, I (1999). arxiv:math.QA/9909027
  28. Jon00.
    Jones, V.F.R.: The planar algebra of a bipartite graph. In: Knots in Hellas ’98 (Delphi). Ser. Knots Everything, vol. 24, pp. 94–117. World Sci. Publ., River Edge (2000). MR1865703Google Scholar
  29. Jon12.
    Jones, V.F.R.: Quadratic tangles in planar algebras. Duke Math. J. 161(12), 2257–2295 (2012). arxiv:1007.1158; doi: 10.1215/00127094-1723608
  30. JP11.
    Jones, V.F.R., Penneys, D.: The embedding theorem for finite depth subfactor planar algebras. Quantum Topol. 2(3), 301–337 (2011). arxiv:1007.3173; doi: 10.4171/QT/23
  31. Liu.
    Liu, Z.: Singly generated planar algebras with small dimensions, part IV (2014, In preparation)Google Scholar
  32. Liu13a.
    Liu, Z.: Composed inclusions of A 3 and A 4 subfactors (2013). arxiv:1308.5691
  33. Liu13b.
    Liu, Z.: Planar algebras of small thickness (2013). arxiv:1308.5656
  34. MP12a.
    Morrison, S., Penneys, D.: Constructing spoke subfactors using the jellyfish algorithm. Trans. Am. Math. Soc. (2012). arxiv:1208.3637
  35. MP12b.
    Morrison, S., Peters, E.: The little desert? Some subfactors with index in the interval \({(5,3+\sqrt{5})}\) (2012). arxiv:1205.2742
  36. MPPS12.
    Morrison, S., Penneys, D., Peters, E., Snyder, N.: Subfactors of index less than 5, Part 2: triple points. Internat. J. Math. 23(3), 1250016, 33 (2012). arxiv:1007.2240; doi: 10.1142/S0129167X11007586
  37. MS12.
    Morrison, S., Snyder, N.: Subfactors of index less than 5, part 1: the principal graph odometer. Commun. Math. Phys. 312(1), 1–35 (2012). arxiv:1007.1730; doi: 10.1007/s00220-012-1426-y
  38. Mg03.
    Müger, M.: From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180(1–2), 81–157 (2003). doi: 10.1016/S0022-4049(02)00247-5: arxiv:math.CT/0111204
  39. Ocn88.
    Ocneanu, A.: Quantized groups, string algebras and Galois theory for algebras. In: Operator algebras and applications, vol. 2. In: London Math. Soc. Lecture Note Ser., vol. 136, pp. 119–172. Cambridge Univ. Press, Cambridge (1988). MR996454Google Scholar
  40. Ost03.
    Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8(2), 177–206 (2003). arxiv:math/0111139
  41. Pen13.
    Penneys, D.: Chirality and principal graph obstructions (2013). arxiv:1307.5890
  42. Pop90.
    Popa S.: Classification of subfactors: the reduction to commuting squares. Invent. Math. 101(1), 19–43 (1990). doi: 10.1007/BF01231494 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  43. Pop93.
    Popa S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111(2), 375–405 (1993). doi: 10.1007/BF01231293 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  44. Pop94.
    Popa S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994). doi: 10.1007/BF02392646 CrossRefzbMATHMathSciNetGoogle Scholar
  45. Pop95.
    Popa S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120(3), 427–445 (1995). doi: 10.1007/BF01241137 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  46. PP13.
    Penneys, D., Peters, E.: Calculating two-strand jellyfish relations (2013). arxiv:1308.5197
  47. PT12.
    Penneys, D., Tener, J.: Classification of subfactors of index less than 5, part 4: vines. Int. J. Math. 23(3), 1250017 (2012). arxiv:1010.3797; doi: 10.1142/S0129167X11007641
  48. Szy94.
    Szymański W.: Finite index subfactors and Hopf algebra crossed products. Proc. Am. Math. Soc. 120(2), 519–528 (1994). doi: 10.2307/2159890 CrossRefzbMATHGoogle Scholar
  49. Wen90.
    Wenzl, H.: Quantum groups and subfactors of type B, C, and D. Commun. Math. Phys. 133(2), 383–432 (1990). MR1090432Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Vanderbilt UniversityNashvilleUSA
  2. 2.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  3. 3.University of CaliforniaLos AngelesUSA

Personalised recommendations