Skip to main content
SpringerLink
Log in

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

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

An irreducible II1-subfactor \({A\subset B}\) is exactly 1-supertransitive if \({B\ominus A}\) is reducible as an A − A 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Rferences

  1. 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. Alexander, J.W.: A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9(3), 93–95 (1923). http://www.jstor.org/stable/84367

  3. Bisch, D, Haagerup, U.: Composition of subfactors: new examples of infinite depth subfactors. Ann. Sci. École Norm. Sup. (4) 29(3), 329–383 (1996). http://www.numdam.org/item?id=ASENS_1996_4_29_3_329_0

  4. Bisch, D., Jones, V.F.R.: Singly generated planar algebras of small dimension. Duke Math. J. 101(1), 41–75 (2000). MR1733737

  5. 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

    Article  MATH  MathSciNet  Google Scholar 

  6. Bisch, D., Jones, V.F.R., Liu, Z.: Singly generated planar algebras of small dimension, part III (2013, in preparation)

  7. Bhattacharyya, B., Landau, Z.: Intermediate standard invariants and intermediate planar algebras. (2010). http://www.eecs.berkeley.edu/~landau/publications.htm

  8. 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. 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. 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. Brothier, A., Vaes, S.: Families of hyperfinite subfactors with the same standard invariant and prescribed fundamental group (2013). arxiv:1309.5354

  12. 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. 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. 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). MR776477

  15. 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). MR999799

  16. Goldman, M.: On subfactors of factors of type II1. Michigan Math. J. 6, 167–172 (1959). MR0107827

  17. 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. 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. 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). MR1317352

  20. 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. Izumi, M., Morrison, S., Penneys, D.: Fusion categories between \({\mathcal{C} \boxtimes \mathcal{D}}\) and \({\mathcal{C} *\mathcal{D}}\) (2013). arxiv:1308.5723

  22. Izumi, M., Penneys, D., Peters, E., Snyder, N.: Subfactors of index exactly 5. arxiv:1406.2389

  23. 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

    Article  MATH  MathSciNet  Google Scholar 

  24. 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

    Article  MATH  MathSciNet  Google Scholar 

  25. 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. Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983). doi:10.1007/BF01389127

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Jones, V.F.R.: Planar algebras, I (1999). arxiv:math.QA/9909027

  28. 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). MR1865703

  29. 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. 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, Z.: Singly generated planar algebras with small dimensions, part IV (2014, In preparation)

  32. Liu, Z.: Composed inclusions of A 3 and A 4 subfactors (2013). arxiv:1308.5691

  33. Liu, Z.: Planar algebras of small thickness (2013). arxiv:1308.5656

  34. Morrison, S., Penneys, D.: Constructing spoke subfactors using the jellyfish algorithm. Trans. Am. Math. Soc. (2012). arxiv:1208.3637

  35. Morrison, S., Peters, E.: The little desert? Some subfactors with index in the interval \({(5,3+\sqrt{5})}\) (2012). arxiv:1205.2742

  36. 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. 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. 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. 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). MR996454

  40. Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups 8(2), 177–206 (2003). arxiv:math/0111139

  41. Penneys, D.: Chirality and principal graph obstructions (2013). arxiv:1307.5890

  42. Popa S.: Classification of subfactors: the reduction to commuting squares. Invent. Math. 101(1), 19–43 (1990). doi:10.1007/BF01231494

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. Popa S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111(2), 375–405 (1993). doi:10.1007/BF01231293

    Article  ADS  MATH  MathSciNet  Google Scholar 

  44. Popa S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994). doi:10.1007/BF02392646

    Article  MATH  MathSciNet  Google Scholar 

  45. 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

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. Penneys, D., Peters, E.: Calculating two-strand jellyfish relations (2013). arxiv:1308.5197

  47. 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. Szymański W.: Finite index subfactors and Hopf algebra crossed products. Proc. Am. Math. Soc. 120(2), 519–528 (1994). doi:10.2307/2159890

    Article  MATH  Google Scholar 

  49. Wenzl, H.: Quantum groups and subfactors of type B, C, and D. Commun. Math. Phys. 133(2), 383–432 (1990). MR1090432

Download references

Author information

Authors and Affiliations

  1. Vanderbilt University, Nashville, USA

    Zhengwei Liu

  2. Mathematical Sciences Institute, Australian National University, Canberra, Australia

    Scott Morrison

  3. University of California, Los Angeles, USA

    David Penneys

Authors
  1. Zhengwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Scott Morrison
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David Penneys
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott Morrison.

Additional information

Communicated by Y. Kawahigashi

Copyright \({\copyright}\) 2014 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Morrison, S. & Penneys, D. 1-Supertransitive Subfactors with Index at Most \({6\frac{1}{5}}\) . Commun. Math. Phys. 334, 889–922 (2015). https://doi.org/10.1007/s00220-014-2160-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2160-4

Keywords

Search

Navigation