Exchange relation planar algebras of small rank

Zhengwei Liu Department of Mathematics, Vanderbilt University, Nashville TN 37235

Quantum Algebra Spectral Theory and Operator Algebra mathscidoc:2206.29002

Transactions of the American Mathematical Society, 368, (12), 8303–8348, 2017.12
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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@inproceedings{zhengwei2017exchange,
  title={Exchange relation planar algebras of small rank},
  author={Zhengwei Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626171209475276473},
  booktitle={Transactions of the American Mathematical Society},
  volume={368},
  number={12},
  pages={8303–8348},
  year={2017},
}
Zhengwei Liu. Exchange relation planar algebras of small rank. 2017. Vol. 368. In Transactions of the American Mathematical Society. pp.8303–8348. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626171209475276473.
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