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Graph planar algebra embeddings and infinite depth subfactors

Dietmar Bisch, Julio Cáceres

Abstract

Subfactors of the hyperfinite II$_1$ factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index $\frac{5+\sqrt{13}}{2}$, i.e. the index of the Haagerup subfactor. We are led to a conjecture pertaining to Jones indices of irreducible, hyperfinite subfactors.

Graph planar algebra embeddings and infinite depth subfactors

Abstract

Subfactors of the hyperfinite II factor with ''exotic'' properties can be constructed from nondegenerate commuting squares of multi-matrix algebras. We show that the subfactor planar algebra of these commuting square subfactors necessarily embeds into Jones' graph planar algebra associated to one of the inclusion graphs in the commuting square. This leads to a powerful obstruction for the standard invariant of the subfactor, and we use it to give an example of a hyperfinite subfactor with Temperley-Lieb-Jones standard invariant and index , i.e. the index of the Haagerup subfactor. We are led to a conjecture pertaining to Jones indices of irreducible, hyperfinite subfactors.

Paper Structure

This paper contains 11 sections, 18 theorems, 76 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Subfactors and commuting squares
  3. Finite-dimensional inclusions and loops
  4. Loop algebras
  5. Isomorphism of towers
  6. Relative commutants of loop algebras
  7. Planar algebras
  8. Subfactor planar algebra
  9. Graph planar algebra
  10. The embedding theorem
  11. A hyperfinite $A_\infty$ subfactor with index $\frac{5+\sqrt{13}}{2}$

Key Result

Theorem 1.1

Let $P_\bullet$ be the subfactor planar algebra associated to the commuting square subfactor $A_{0,\infty}\subset A_{1,\infty}$ and let $G_\bullet$ the graph planar algebra associated to the inclusion graph of $A_{0,0}\subset A_{1,0}$. There is an embedding of planar algebras $\psi:P_\bullet\rightar

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 60 more