Communications in Mathematical Physics

, Volume 286, Issue 3, pp 1141–1157

On Haagerup’s List of Potential Principal Graphs of Subfactors



We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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, 1–63 (1999)MATHCrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Asaeda M.: Galois groups and an obstruction to principal graphs of subfactors. Int. J. Math. 18, 191–202 (2007)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Banica T., Bisch D.: Spectral measures of small index principal graphs. Commun. Math. Phys. 260, 259–281 (2007)MathSciNetADSGoogle Scholar
  4. 4.
    Bion-Nadal J.: Subfactor of the hyperfinite II1 factor with Coxeter graph E 6 as invariant. J. Op. Th. 28, 27–50 (1992)MATHMathSciNetGoogle Scholar
  5. 5.
    Bisch D.: Principal graphs of subfactors with small Jones index. Math. Ann. 311, 223–231 (1998)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Coste A., Gannon T.: Remarks on Galois symmetry in rational conformal field theories. Phys. Lett. B 323, 316–321 (1994)CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. Math. 162, 581–642 (2005)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Evans D.E., Kawahigashi Y.: Quantum symmetries on operator algebras. Oxford University Press, Oxford (1998)MATHGoogle Scholar
  9. 9.
    Ghorpade, S.R.: Lectures on Topics in Algebraic Number Theory., 2000
  10. 10.
    Haagerup, U.: Principal graphs of subfactors in the index range \({4 < 3+\sqrt2}\). In: Subfactors—Proceedings of the Taniguchi Symposium, Katata —, ed. H. Araki, et al., Singapore: World Scientific, 1994, pp. 1–38Google Scholar
  11. 11.
    Haagerup, U.: Private communications. 2006Google Scholar
  12. 12.
    Hungerford T.W.: Algebra, GTM 73. Springer Verlag, Berlin-Heidelberg-New York (1974)Google Scholar
  13. 13.
    Ikeda K.: Numerical evidence for flatness of Haagerup’s connections. J. Math. Sci. Univ. Tokyo 5, 257–272 (1998)MATHMathSciNetGoogle Scholar
  14. 14.
    Izumi M.: Application of fusion rules to classification of subfactors. Publications of the RIMS, Kyoto University 27, 953–994 (1991)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Izumi M., Kawahigashi Y.: Classification of subfactors with the principal graph \({D^{(1)}_n}\). J. Funct. Anal. 112, 257–286 (1993)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Izumi M.: On flatness of the Coxeter graph E 8. Pac. J. Math. 166, 305–327 (1994)MATHMathSciNetGoogle Scholar
  17. 17.
    Jones V.F.R.: Index for subfactors. Invent. Math. 72, 1–25 (1983)MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Jones, V.F.R.: Annular structure of subfactors. L’Enseignement Mathématique, in pressGoogle Scholar
  19. 19.
    Kawahigashi Y.: On flatness of Ocneanu’s connections on the Dynkin diagrams and classification of subfactors. J. Funct. Ana. 127, 63–107 (1995)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Komatsu K.: Square-free discriminants and affect-free equations. Tokyo J. Math 14(1), 57–60 (1991)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Kondo T.: Algebraic number fields with the discriminant equal to that of a quadratic number field. J. Math. Soc. Japan 47, 31–36 (1995)MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Lang S.: Algebraic Number Theory. GTM 110. Springer Verlag, Berlin-Heidelberg-NewYork (1994)Google Scholar
  23. 23.
    Milne, J.S.: Fields and Galois Theory., 2008
  24. 24.
    Narkiewitcz W.: Elementary and Analytic Theory of Algebraic Numbers, Third Edition. Springer Verlag, Berlin- Heidelberg-NewYork (2004)Google Scholar
  25. 25.
    Ocneanu, A.: Quantized group, string algebras and Galois theory for algebras. In: Operator algebras and applications, Vol. 2 (Warwick, 1987), ed. D.E. Evans, M. Takesaki, London Mathematical Society Lecture Note Series Vol. 136, Cambridge: Cambridge University Press, 1988, pp. 119–172Google Scholar
  26. 26.
    Popa S.: Classification of amenable subfactors of type II. Acta Math. 172, 163–255 (1994)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Sunder V.S., Vijayarajan A.K.: On the non-occurrence of the Coxeter graphs β 2n+1, E 7, D 2n+1 as principal graphs of an inclusion of II1 factors. Pac. J. Math. 161, 185–200 (1993)MATHMathSciNetGoogle Scholar
  28. 28.
    van der Waerden B.L.: Modern algebra (English). Frederick Ungar Publishing Co., New York (1949)Google Scholar
  29. 29.
    Washington L.: Introduction to Cyclotomic Fields. GTM 83. Springer Verlag, Berlin-Heidelberg-New York (1996)Google Scholar
  30. 30.
    Wenzl H.: Hecke algebras of type A n and subfactors. Invent Math. 92, 345–383 (1988)CrossRefADSMathSciNetGoogle Scholar
  31. 31.

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California RiversideRiversideUSA
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations