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Transactions of the American Mathematical Society

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Singly generated planar algebras of small dimension, Part III


Authors: Dietmar Bisch, Vaughan F. R. Jones and Zhengwei Liu
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 46L37; Secondary 46L10
Published electronically: July 20, 2016
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Abstract: The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for $ \mathbb{Z}_3$, the Fuss-Catalan planar algebra, and the group/subgroup planar algebra for $ \mathbb{Z}_2\subset \mathbb{Z}_5\rtimes \mathbb{Z}_2$. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all Birman-Murakami-Wenzl algebras. Precisely it contains a depth 3 one from quantum $ O(3)$, and a one-parameter family from quantum $ Sp(4)$.


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Additional Information

Dietmar Bisch
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: dietmar.bisch@vanderbilt.edu

Vaughan F. R. Jones
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: vaughan.f.jones@vanderbilt.edu

Zhengwei Liu
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: zhengweiliu@fas.harvard.edu

DOI: https://doi.org/10.1090/tran/6719
Received by editor(s): November 12, 2014
Received by editor(s) in revised form: April 8, 2015
Published electronically: July 20, 2016
Article copyright: © Copyright 2016 American Mathematical Society