Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Group actions on graphs related to Krishnan-Sunder subfactors


Author: Bina Bhattacharyya
Journal: Trans. Amer. Math. Soc. 355 (2003), 433-463
MSC (2000): Primary 46L37
Published electronically: October 8, 2002
MathSciNet review: 1932707
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We describe the principal graphs of the subfactors studied by Krishnan and Sunder in terms of group actions on Cayley-type graphs. This leads to the construction of a tower of tree algebras, for every positive integer $k$, which are symmetries of the Krishnan-Sunder subfactors of index $k^2$. Using our theory, we prove that the principal graph of the irreducible infinite depth subfactor of index 9 constructed by Krishnan and Sunder is not a tree, contrary to their expectations. We also show that the principal graphs of the Krishnan-Sunder subfactors of index 4 are the affine A and D Coxeter graphs.


References [Enhancements On Off] (What's this?)

  • 1. The GAP Group, Aachen, St. Andrews, GAP - Groups, Algorithms, and Programming, Version 4, 1998, (http://www-gap.dcs.st-and.ac.uk/~gap).
  • 2. B. Bhattacharyya, Krishnan-Sunder subfactors and a new countable family of subfactors related to trees, Ph.D. thesis, UC Berkeley, 1998.
  • 3. Chris Godsil and Gordon Royle, Algebraic graph theory, Springer-Verlag, New York, 2001.
  • 4. F. Goodman, P. de la Harpe, and V. F. R. Jones, Coxeter graphs and towers of algebras, MSRI Publications, vol. 14, Springer, 1989. MR 91c:46082
  • 5. Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981, with a foreword by P. M. Cohn, with an introduction by Gilbert de B. Robinson. MR 83k:20003
  • 6. Jonathan L. Gross and Thomas W. Tucker, Topological graph theory, John Wiley & Sons Inc., New York, 1987. MR 88h:05034
  • 7. U. Haagerup and J. Schou, Some new subfactors of the hyperfinite $II_1$ subfactor, 1989, preprint.
  • 8. V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 1-25. MR 84d:46097
  • 9. V. F. R. Jones and V. S. Sunder, Introduction to subfactors, London Mathematical Society Lecture Note Series, vol. 234, Cambridge University Press, 1997. MR 98h:46067
  • 10. U. Krishnan and V. S. Sunder, On biunitary permutation matrices and some subfactors of index $9$, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4691-4736. MR 97c:46077
  • 11. W. S. Massey, Algebraic topology: An introduction, ch. 6, Springer-Verlag, 1977. MR 56:6638
  • 12. A. Ocneanu (Lecture Notes by Y. Kawahigashi), Quantum symmetry, differential geometry of finite graphs and classification of subfactors, 1990, University of Tokyo Seminar Notes.
  • 13. Adrian Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119-172. MR 91k:46068
  • 14. S. Popa, Orthogonal pairs of $\ast $-subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253-268. MR 84h:46077
  • 15. -, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no. 1, 19-43. MR 91h:46109
  • 16. -, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163-255. MR 95f:46105
  • 17. -, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), 427-445. MR 96g:46051
  • 18. V. S. Sunder, A model for AF algebras and a representation of the Jones projections, J. Operator Theory 18 (1987), 289-301. MR 89e:46079
  • 19. Hans Wenzl, Hecke algebras of type ${A}\sb n$ and subfactors, Invent. Math. 92 (1988), no. 2, 349-383. MR 90b:46118

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L37

Retrieve articles in all journals with MSC (2000): 46L37


Additional Information

Bina Bhattacharyya
Affiliation: Elance, 820A Kifer Rd., Sunnyvale, California 94086
Email: Bina_Bhattacharyya_91@post.harvard.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02986-0
PII: S 0002-9947(02)02986-0
Received by editor(s): March 8, 1999
Received by editor(s) in revised form: December 17, 2001
Published electronically: October 8, 2002
Article copyright: © Copyright 2002 American Mathematical Society