Group actions on graphs related to KrishnanSunder subfactors
Author:
Bina Bhattacharyya
Journal:
Trans. Amer. Math. Soc. 355 (2003), 433463
MSC (2000):
Primary 46L37
Published electronically:
October 8, 2002
MathSciNet review:
1932707
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Abstract: We describe the principal graphs of the subfactors studied by Krishnan and Sunder in terms of group actions on Cayleytype graphs. This leads to the construction of a tower of tree algebras, for every positive integer , which are symmetries of the KrishnanSunder subfactors of index . 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 KrishnanSunder subfactors of index 4 are the affine A and D Coxeter graphs.
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V. S. Sunder, A model for AF algebras and a representation of the Jones projections, J. Operator Theory 18 (1987), 289301. MR 89e:46079
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 1.
 The GAP Group, Aachen, St. Andrews, GAP  Groups, Algorithms, and Programming, Version 4, 1998,
(http://wwwgap.dcs.stand.ac.uk/~gap) .
 2.
 B. Bhattacharyya, KrishnanSunder 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, SpringerVerlag, 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, AddisonWesley 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 subfactor, 1989, preprint.
 8.
 V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), 125. 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 , Trans. Amer. Math. Soc. 348 (1996), no. 12, 46914736. MR 97c:46077
 11.
 W. S. Massey, Algebraic topology: An introduction, ch. 6, SpringerVerlag, 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. 119172. MR 91k:46068
 14.
 S. Popa, Orthogonal pairs of subalgebras in finite von Neumann algebras, J. Operator Theory 9 (1983), no. 2, 253268. MR 84h:46077
 15.
 , Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no. 1, 1943. MR 91h:46109
 16.
 , Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163255. MR 95f:46105
 17.
 , An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995), 427445. MR 96g:46051
 18.
 V. S. Sunder, A model for AF algebras and a representation of the Jones projections, J. Operator Theory 18 (1987), 289301. MR 89e:46079
 19.
 Hans Wenzl, Hecke algebras of type and subfactors, Invent. Math. 92 (1988), no. 2, 349383. MR 90b:46118
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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/S0002994702029860
PII:
S 00029947(02)029860
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
