[Paper Review] Singly generated planar algebras of small dimension, Part III

Dietmar Bisch, Vaughan F. R. Jones+1 more|arXiv (Cornell University)|Oct 10, 2014

Algebraic structures and combinatorial models38 references3 citations

TL;DR

This paper classifies singly generated subfactor planar algebras with 14-dimensional 3-boxes, proving they are all BMW-type planar algebras—specifically, either the depth-3 algebra from quantum $SO(3)$ or a one-parameter family from quantum $Sp(4)$. Using dimension constraints and the Yang-Baxter relation, the authors show that the entire structure is determined by 2-box operations, extending prior classifications up to dimension 13.

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 $\mathbb{Z}_2\subset \mathbb{Z}_5 times \mathbb{Z}_2$. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all BMW. Precisely it contains a depth 3 one from quantum $SO(3)$, and a one-parameter family from quantum $Sp(4)$.

Motivation & Objective

• To extend the classification of singly generated subfactor planar algebras beyond dimension 13 to include 14-dimensional 3-boxes.
• To determine whether such planar algebras are governed by the BMW algebra structure rather than exchange relations.
• To prove that the planar algebra structure is fully determined by 2-box operations under dimension 14 constraints.
• To identify all such algebras as either the depth-3 quantum $SO(3)$ algebra or a one-parameter family from quantum $Sp(4)$.

Proposed method

• Use dimension constraints on 3-boxes to limit possible relations among generators in the planar algebra.
• Apply the Yang-Baxter equation and Reidemeister moves to constrain the algebraic structure of the generator.
• Leverage Lemma 2.11 to show that the entire planar algebra is determined by 2-box operations when the Yang-Baxter relation holds.
• Use the existence of a biprojection under specific conditions (from Liu [Liu]) to simplify computations and verify structure.
• Derive parameters of 2-box operations via direct computation, including adjoints, products, and coproducts.
• Establish rational function dependence of parameters on deformation variables, proving the generator satisfies BMW relations universally.

Experimental results

Research questions

• RQ1 Are all subfactor planar algebras generated by a single 2-box with 14-dimensional 3-boxes isomorphic to a BMW algebra?
• RQ2 Does the Yang-Baxter relation suffice to determine the full structure of such planar algebras when the 3-box dimension is 14?
• RQ3 What are the complete list of such algebras, and how do they relate to quantum groups like $SO(3)$ and $Sp(4)$?
• RQ4 Can the structure of 2-boxes alone determine the entire planar algebra in the absence of exchange relations at dimension 14?
• RQ5 How do the parameters of the 2-box operations relate to quantum group deformation parameters?

Key findings

• All subfactor planar algebras generated by a single 2-box with 14-dimensional 3-boxes are isomorphic to BMW algebras.
• The classification includes the depth-3 planar algebra from quantum $SO(3)$ and a one-parameter family from quantum $Sp(4)$.
• The Yang-Baxter relation is sufficient to determine the full planar algebra structure when dim(3-boxes) = 14.
• The generator $U$ satisfies the BMW quadratic relation and the Yang-Baxter equation, confirming its BMW nature.
• The parameters $a$, $b$, and $ ilde{q}$ are rational functions of the deformation parameter $q$, ensuring consistency across all values.
• The structure of 2-boxes uniquely determines the entire planar algebra via Lemma 2.11, confirming the classification is complete.