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Exchange relation planar algebras of small rank


Author: Zhengwei Liu
Journal: Trans. Amer. Math. Soc. 368 (2016), 8303-8348
MSC (2010): Primary 46L37, 46L10, 20C05
DOI: https://doi.org/10.1090/tran/6582
Published electronically: March 1, 2016
MathSciNet review: 3551573
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Abstract: The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.


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Additional Information

Zhengwei Liu
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: zhengwei.liu@vanderbilt.edu

DOI: https://doi.org/10.1090/tran/6582
Received by editor(s): March 25, 2014
Received by editor(s) in revised form: September 5, 2014
Published electronically: March 1, 2016
Additional Notes: The author was supported by DOD-DARPA grant HR0011-12-1-0009.
Article copyright: © Copyright 2016 American Mathematical Society