Advertisement

Communications in Mathematical Physics

, Volume 312, Issue 1, pp 1–35 | Cite as

Subfactors of Index less than 5, Part 1: The Principal Graph Odometer

  • Scott Morrison
  • Noah Snyder
Article

Abstract

In this series of papers we show that there are exactly ten subfactors, other than A subfactors, of index between 4 and 5. Previously this classification was known up to index \({3+\sqrt{3}}\). In the first paper we give an analogue of Haagerup’s initial classification of subfactors of index less than \({3+\sqrt{3}}\), showing that any subfactor of index less than 5 must appear in one of a large list of families. These families will be considered separately in the three subsequent papers in this series.

Keywords

Triple Point Fusion Category Graph Norm Root Vertex Quadruple Point 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. Asa07.
    Asaeda M.: Galois groups and an obstruction to principal graphs of subfactors. Internat. J. Math. 18(2), 191–202 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. AY09.
    Asaeda M., Yasuda S.: On Haagerup’s list of potential principal graphs of subfactors. Commun. Math. Phys. 286(3), 1141–1157 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. Bis94.
    Bisch D.: An example of an irreducible subfactor of the hyperfinite II1 factor with rational, noninteger index. J. Reine Angew. Math. 455, 21–34 (1994)MathSciNetzbMATHGoogle Scholar
  5. Bis98.
    Bisch D.: Principal graphs of subfactors with small Jones index. Math. Ann. 311(2), 223–231 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  6. BJ97.
    Bisch D., Jones V.: Algebras associated to intermediate subfactors. Invent. Math. 128(1), 89–157 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. BMPS09.
    Bigelow, S., Morrison, S., Peters, E., Snyder, N.: Constructing the extended Haagerup planar algebra. http://arxiv.org/abs/0909.4099 [math.OA], 2011 to appear Acta Mathematica
  8. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  9. BW99.
    Barrett J.W., Westbury B.W.: Spherical categories. Adv. Math. 143(2), 357–375 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  10. CMS11.
    Calegari F., Morrison S., Snyder N.: Cyclotomic integers, fusion categories, and subfactors. Commun. Math. Phys. 303(3), 845–896 (2011) With an appendix by V. OstrikMathSciNetADSzbMATHCrossRefGoogle Scholar
  11. GdlHJ89.
    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, 1989Google Scholar
  12. GL98.
    Graham J.J., Lehrer G.I.: The representation theory of affine Temperley-Lieb algebras. Enseign. Math. (2) 44(3-4), 173–218 (1998)MathSciNetzbMATHGoogle Scholar
  13. Haa94.
    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. 1–38Google Scholar
  14. Han10.
    Han, R.: A Construction of the −2221+ Planar Algebra. PhD thesis, University of California, Riverside, 2010. http://arxiv.org/abs/1102.2052 [math.OA], 2011
  15. IJMS.
    Izumi, M., Jones, V.F.R., Morrison, S., Snyder, N. Classification of subfactors of index less than 5, Part 3: Quadruple points. Commun. Math. Phys. http://arxiv.org/abs/1109.3190 (2011)
  16. Izu91.
    Izumi M.: Application of fusion rules to classification of subfactors. Publ. Res. Inst. Math. Sci. 27(6), 953–994 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  17. Izu97.
    Izumi, M.: Goldman’s type theorems in index theory. In Operator algebras and quantum field theory (Rome, 1996). Cambridge, MA: Int. Press, 1997, pp. 249–269Google Scholar
  18. Izu01.
    Izumi M.: The structure of sectors associated with Longo-Rehren inclusions. II. Examples. Rev. Math. Phys. 13(5), 603–674 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. Jon.
    Jones, V.F.R.: Planar algebras, I. http://arxiv.org/abs/math.QA/9909027 [math.OA], 1999
  20. Jon80.
    Jones, V.F.R.: Actions of finite groups on the hyperfinite type II1 factor. Mem. Amer. Math. Soc. 28(237), v+70 1980Google Scholar
  21. Jon83.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72(1), 1–25 (1983)MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. Jon01.
    Jones, V.F.R.: The annular structure of subfactors. In: Essays on geometry and related topics, Vol. 1, 2, Vol. 38 of Monogr. Enseign. Math., Geneva: Enseignement Math., 2001, pp. 401–463Google Scholar
  23. Jon03.
    Jones, V.F.R.: Quadratic tangles in planar algebras, 2003 available at http://arxiv.org/abs/:1007.1158 [math.OA], 2010
  24. JR06.
    Jones V.F.R., Reznikoff S.A.: Hilbert space representations of the annular Temperley-Lieb algebra. Pacific J. Math. 228(2), 219–249 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. Kaw95.
    Kawahigashi Y.: Classification of paragroup actions in subfactors. Publ. Res. Inst. Math. Sci. 31(3), 481–517 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  26. MPPS10.
    Morrison, S., Penneys, D., Peters, E., Snyder, N.: Classification of subfactors of index less than 5, Part 2: Triple points. Int. J. Math. doi: 10.1142/S0129167X11007586. http://arxiv.org/abs/1007.2240 [math.OA] 2011
  27. Ocn80.
    Ocneanu A.: Actions des groupes moyennables sur les algèbres de von Neumann. C. R. Acad. Sci. Paris Sér. A-B 291(6), A399–A401 (1980)MathSciNetGoogle Scholar
  28. Ocn88.
    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. 119–172Google Scholar
  29. Ocn94.
    Ocneanu, A.: Chirality for operator algebras. In: Subfactors (Kyuzeso, 1993), River Edge, NJ: World Sci. Publ., 1994, pp. 39–63Google Scholar
  30. Pop90.
    Popa S.: Classification of subfactors: the reduction to commuting squares. Invent. Math. 101(1), 19–43 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. Pop91.
    Popa, S.: Subfactors and classification in von Neumann algebras. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Tokyo: Math. Soc. Japan., pp. 987–996Google Scholar
  32. Pop93.
    Popa S.: Markov traces on universal Jones algebras and subfactors of finite index. Invent. Math. 111(2), 375–405 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. Pop94.
    Popa S.: Classification of amenable subfactors of type II. Acta Math. 172(2), 163–255 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Pop95.
    Popa S.: An axiomatization of the lattice of higher relative commutants of a subfactor. Invent. Math. 120(3), 427–445 (1995)MathSciNetADSzbMATHCrossRefGoogle Scholar
  35. PT10.
    Penneys, D., Tener, J.: Classification of subfactors of index less than 5, Part 4: Cyclotomicity. Int. J. Math. doi: 10.1142/S0129167X11007641. http://arxiv.org/abs/1010.3797 [math.OA], 2011
  36. TV92.
    Turaev V.G., Viro O.Ya.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of CaliforniaBerkeleyUSA
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA

Personalised recommendations