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.

45 pages (final version; improved introduction)

References in corpus (2)

• Skein theory for the D_{2n} planar algebras
• On Haagerup's list of potential principal graphs of subfactors

Cited by in corpus (16)

• The classification of subfactors of index at most 5
• The two-color Soergel calculus
• Conformal Field Theory, Tensor Categories and Operator Algebras
• Planar Para Algebras, Reflection Positivity
• Exchange relation planar algebras of small rank
• Composed inclusions of $A_3$ and $A_4$ subfactors
• 1-supertransitive subfactors with index at most 6+1/5
• 2-supertransitive subfactors at index $3+\sqrt{5}$
• The little desert? Some subfactors with index in the interval (5,3+\sqrt{5})
• Calculating two-strand jellyfish relations
• Constructing spoke subfactors using the jellyfish algorithm
• Modular data for the extended Haagerup subfactor
• Subfactors of index exactly 5
• An obstruction to subfactor principal graphs from the graph planar algebra embedding theorem
• Infinite families of potential modular data related to quadratic categories
• Interpolation categories for Conformal Embeddings