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Bulletin of the American Mathematical Society

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The classification of subfactors of index at most 5

Authors: Vaughan F. R. Jones, Scott Morrison and Noah Snyder
Journal: Bull. Amer. Math. Soc. 51 (2014), 277-327
MSC (2010): Primary 46L37
Published electronically: December 24, 2013
MathSciNet review: 3166042
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Abstract: A subfactor is an inclusion $ N \subset M$ of von Neumann algebras with trivial centers. The simplest example comes from the fixed points of a group action $ M^G \subset M$, and subfactors can be thought of as fixed points of more general group-like algebraic structures. These algebraic structures are closely related to tensor categories and have played important roles in knot theory, quantum groups, statistical mechanics, and topological quantum field theory. There is a measure of size of a subfactor, called the index. Remarkably, the values of the index below 4 are quantized, which suggests that it may be possible to classify subfactors of small index. Subfactors of index at most 4 were classified in the 1980s and early 1990s. The possible index values above 4 are not quantized, but once you exclude a certain family, it turns out that again the possibilities are quantized. Recently, the classification of subfactors has been extended up to index 5, and (outside of the infinite families) there are only 10 subfactors of index between 4 and 5. We give a summary of the key ideas in this classification and discuss what is known about these special small subfactors.

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Additional Information

Vaughan F. R. Jones
Affiliation: Department of Mathematics, University of California, Berkeley, California

Scott Morrison
Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, Australia

Noah Snyder
Affiliation: Department of Mathematics, Columbia University, New York

Received by editor(s): June 18, 2013
Published electronically: December 24, 2013
Additional Notes: The first author was supported by the NSF under Grant No. DMS-0301173
The second author was supported by the Australian Research Council under the Discovery Early Career Researcher Award DE120100232, and Discovery Project DP140100732
The third author was supported by a NSF Postdoctoral Fellowship at Columbia University.
All authors were supported by DARPA grants HR0011-11-1-0001 and HR0011-12-1-0009.
Article copyright: © Copyright 2013 American Mathematical Society