Group-type subfactors and Hadamard matrices

Burstein, Richard

doi:10.1090/tran/5314

Group-type subfactors and Hadamard matrices

Author: Richard D. Burstein
Journal: Trans. Amer. Math. Soc. 367 (2015), 6783-6807
MSC (2010): Primary 46L37
DOI: https://doi.org/10.1090/tran/5314
Published electronically: June 11, 2015
MathSciNet review: 3378814
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A hyperfinite $\textup {II}_1$ subfactor may be obtained from a symmetric commuting square via iteration of the basic construction. For certain commuting squares constructed from Hadamard matrices, we describe this subfactor as a group-type inclusion $R^H \subset R \rtimes K$ , where and are finite groups with outer actions on the hyperfinite $\textup {II}_1$ factor . We find the group of outer automorphisms generated by and and use the method of Bisch and Haagerup to determine the principal and dual principal graphs. In some cases a complete classification is obtained by examining the element of $H^3(H \ast K / \textup {Int} R)$ associated with the action.

[1] Dietmar Bisch, Paramita Das, and Shamindra Kumar Ghosh, The planar algebra of group-type subfactors, J. Funct. Anal. 257 (2009), no. 1, 20–46. MR 2523334, https://doi.org/10.1016/j.jfa.2009.03.014
[2] D. Bisch, P. Das, and S. Ghosh, The planar algebra of group-type subfactors with cocycle, Work in progress.
[3] Dietmar Bisch and Uffe Haagerup, Composition of subfactors: new examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 3, 329–383. MR 1386923
[4] Dietmar Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), no. 2, 201–216. MR 1262294
[5] Dietmar Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, Operator algebras and their applications (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 13–63. MR 1424954
[6] Ola Bratteli, Inductive limits of finite dimensional 𝐶*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. MR 0312282, https://doi.org/10.1090/S0002-9947-1972-0312282-2
[7] Dietmar Bisch, Remus Nicoara, and Sorin Popa, Continuous families of hyperfinite subfactors with the same standard invariant, Internat. J. Math. 18 (2007), no. 3, 255–267. MR 2314611, https://doi.org/10.1142/S0129167X07004011
[8] R. D. Burstein, Hadamard Subfactors of Bisch-Haagerup Type, PhD dissertation, University of California, Berkeley, Department of Mathematics, 2008.
[9] Remus Nicoara, Subfactors and Hadamard matrices, J. Operator Theory 64 (2010), no. 2, 453–468. MR 2718953
[10] David E. Evans and Yasuyuki Kawahigashi, Quantum symmetries on operator algebras, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1998. Oxford Science Publications. MR 1642584
[11] Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones, Coxeter graphs and towers of algebras, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR 999799
[12] K. J. Horadam, Hadamard matrices and their applications, Princeton University Press, Princeton, NJ, 2007. MR 2265694
[13] Masaki Izumi and Yasuyuki Kawahigashi, Classification of subfactors with the principal graph 𝐷⁽¹⁾_{𝑛}, J. Funct. Anal. 112 (1993), no. 2, 257–286. MR 1213139, https://doi.org/10.1006/jfan.1993.1033
[14] Vaughan F. R. Jones, Actions of finite groups on the hyperfinite type 𝐼𝐼₁ factor, Mem. Amer. Math. Soc. 28 (1980), no. 237, v+70. MR 587749, https://doi.org/10.1090/memo/0237
[15] V. F. R. Jones, Index for subfactors, Invent. Math. 72 (1983), no. 1, 1–25. MR 696688, https://doi.org/10.1007/BF01389127
[16] V. F. R. Jones, Planar algebras. I, arxiv.org, arXiv:9909027 [math.OA]:122 pp, 1999.
[17] V. Jones and V. S. Sunder, Introduction to subfactors, London Mathematical Society Lecture Note Series, vol. 234, Cambridge University Press, Cambridge, 1997. MR 1473221
[18] Uma 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 1360226, https://doi.org/10.1090/S0002-9947-96-01669-8
[19] Zeph A. Landau, Intermediate subfactors, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of California, Berkeley. MR 2698865
[20] Phan H. Loi, On automorphisms of subfactors, J. Funct. Anal. 141 (1996), no. 2, 275–293. MR 1418506, https://doi.org/10.1006/jfan.1996.0128
[21] 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 996454
[22] S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), no. 1, 19–43. MR 1055708, https://doi.org/10.1007/BF01231494
[23] Sorin Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), no. 2, 163–255. MR 1278111, https://doi.org/10.1007/BF02392646
[24] Anne Louise Svendsen, Automorphisms of subfactors from commuting squares, Trans. Amer. Math. Soc. 356 (2004), no. 6, 2515–2543. MR 2048528, https://doi.org/10.1090/S0002-9947-04-03447-6