Intermediate Subfactors with No Extra Structure

If \(N \subset P,Q \subset M\) are type II_1 factors with \(N' \cap M = C id\) and \([M:N]\) finite we show that restrictions on the standard invariants of the elementary inclusions \(N \subset P\), \(N \subset Q\), \(P \subset M\) and \(Q \subset M\) imply drastic restrictions on the indices a...

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Bibliographic Details
Published in:arXiv.org 2005-02
Main Authors: Grossman, Pinhas, Jones, Vaughan F R
Format: Article
Language:English
Subjects:
Inclusions
Invariants
Online Access:Get full text
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Summary:If \(N \subset P,Q \subset M\) are type II_1 factors with \(N' \cap M = C id\) and \([M:N]\) finite we show that restrictions on the standard invariants of the elementary inclusions \(N \subset P\), \(N \subset Q\), \(P \subset M\) and \(Q \subset M\) imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto \(P\) and \(Q\) do not commute, then \([M:N]\) is 6 or \(6 + 4\sqrt 2\). In the former case \(N\) is the fixed point algebra for an outer action of \(S_3\) on \(M\) and the angle is \(\pi/3\), and in the latter case the angle is \(cos^{-1}(\sqrt 2-1)\) and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.
ISSN:2331-8422