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Planar algebras and the Ocneanu-Szymanski theorem

Author(s): Paramita Das; Vijay Kodiyalam
Journal: Proc. Amer. Math. Soc. 133 (2005), 2751-2759.
MSC (1991): Primary 54C40, 14E20; Secondary 46E25, 20C20
Posted: April 19, 2005
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Abstract: We give a very simple `planar algebra' proof of the part of the Ocneanu-Szymanski theorem which asserts that for a finite index, depth two, irreducible $II_1$-subfactor $N \subset M$, the relative commutants $N^\prime \cap M_1$ and $M^\prime \cap M_2$ admit mutually dual Kac algebra structures. In the hyperfinite case, the same techniques also prove the other part, which asserts that $N^\prime \cap M_1$ acts on $M$ with invariants $N$.

References:

[BhmNllSzl]
G.Bohm, F.Nill and K.Szlachanyi, Weak Hopf algebras. I. Integral theory and $C^*$-structure, J. Algebra 221 (1999), no. 2, 385-438 MR 1726707 (2001a:16059)

[Das]
Paramita Das, Weak Hopf $C^*$-algebras and depth two subfactors, J. Funct. Anal. 214 (2004), no. 1, 74-105 MR 2079886

[Jns]
V. F. R. Jones, Planar Algebras I, New Zealand Journal of Mathematics, To appear.

[KdyLndSnd]
V. Kodiyalam, Zeph Landau and V. S. Sunder, The planar algebra associated to a Kac algebra, Proc. Ind. Acad. Sciences, 113 (2003), no. 1, 15-51 MR 1971553 (2004d:46075)

[NksVnr]
D.Nikshych, L.Vainerman, A characterization of depth 2 subfactors of $II_1$ factors, J. Funct. Anal. 171 (2000), no. 2, 278-307 MR 1745634 (2000m:46129)

[Cnn]
A.Ocneanu, Quantized groups, string algebras and Galois theory for algebras, Operator algebras and applications, Vol. 2, 119-172, London Math. Soc. Lecture Note Ser., 136, Cambridge Univ. Press, Cambridge, 1988. MR 0996454 (91k:46068)

[Ppa]
S. Popa, Classification of amenable subfactors of type $II$, Acta Math. 172 (1994), 163-255. MR 1278111 (95f:46105)

[Szy]
W.Szymanski, Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994), no. 2, 519-528. MR 1186139 (94d:46061)

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Additional Information:

Paramita Das
Affiliation: The Institute of Mathematical Sciences, Taramani, Chennai, India 600113
Address at time of publication: Department of Mathematics and Statistics, University of New Hampshire, Durham, New Hampshire 03824
Email: pdas@imsc.res.in, pnt2@unh.edu

Vijay Kodiyalam
Affiliation: The Institute of Mathematical Sciences, Taramani, Chennai, India 600113
Email: vijay@imsc.res.in

DOI: 10.1090/S0002-9939-05-07789-0
PII: S 0002-9939(05)07789-0
Keywords: Planar algebra, subfactor, Kac algebra, Ocneanu-Szyma\'{n}ski theorem
Received by editor(s): December 2, 2002
Received by editor(s) in revised form: June 25, 2003
Posted: April 19, 2005
Communicated by: David R. Larson
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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