Exchange relation planar algebras of small rank
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- by Zhengwei Liu PDF
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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.References
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Additional Information
- Zhengwei Liu
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 1095405
- Email: zhengwei.liu@vanderbilt.edu
- 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.
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8303-8348
- MSC (2010): Primary 46L37, 46L10, 20C05
- DOI: https://doi.org/10.1090/tran/6582
- MathSciNet review: 3551573