Constructing the extended Haagerup planar algebra

Abstract

We construct a new subfactor planar algebra, and as a corollary a new subfactor, with the `extended Haagerup' principal graph pair. This completes the classification of irreducible amenable subfactors with index in the range $(4,3+\sqrt{3})$, which was initiated by Haagerup in 1993. We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. In the skein theoretic description there is an explicit algorithm for evaluating closed diagrams. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.Comment: 45 pages (final version; improved introduction

A Planar Calculus for Infinite Index Subfactors

“…The annular Temperley-Lieb category, especially the rotation, played an important role in the construction of certain exotic finite index subfactors [Pet10,BMPS09]. In a future paper with Jones, we will incorporate the odd Jones projections for infinite index (see [Bur03]) into the planar calculus, and we will give the analog of the annular Tempeley-Lieb category for infinite index.…”

Singly generated planar algebras of small dimension, Part III

“…Indeed subfactors of index at most 4 were classified [Ocn88,GdlHJ89,Pop94,Izu91]. This approach has been extremely successful in work of Haagerup [Haa94] and others [AH99,Bis98,Izu91,SV93,BMPS12]. Recently the classification has been extended up to index 5, and even beyond.…”

“…See [Wen90,MPS10,Pet10,BMPS12] for examples. A planar algebra S consists of vector spaces S n,± for n ∈ N ∪ {0}.…”

Subfactors of Index Less Than 5, Part 3: Quadruple Points

“…Theorem 1.1. There are exactly ten subfactor planar algebras other than Temperley-Lieb with index between 4 and 5: the Haagerup planar algebra and its dual [AH99], the extended Haagerup planar algebra and its dual [BMPS09], the Asaeda-Haagerup planar algebra [AH99] and its dual, the 3311 Goodman-de la Harpe-Jones planar algebra [GdlHJ89] and its dual, and Izumi's self-dual 2221 planar algebra [Izu01] and its complex conjugate.…”

“…We prove Theorem 1.3 using planar algebra techniques based on [Jon01,Jon03,BMPS09]. In particular, we show that any subfactor planar algebra whose principal graph is a translated extension of Q has a proper planar subalgebra which begins with a triple point at the same depth.…”