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Constructing the extended Haagerup planar algebra. (English) Zbl 1270.46058
A subfactor is an inclusion $$N \subset M$$ of von Neumann algebras with trivial centers. There are three invariants associated to a subfactor. The weakest is the index $$[M:N]$$, which measures the “relative size” of $$N$$ in $$M$$. The second is the pair of principal and dual principal graphs, which describe the fusion rules of “basic” bimodules over $$N$$ and $$M$$. The strongest is the standard invariant, which encodes all information about these bimodules. This article uses the planar algebra formalism of Jones for the standard invariant, other axiomatizations are the paragroups of Ocneanu and the $$\lambda$$-lattices of Popa.
A fundamental result of V. F. R. Jones [Invent. Math. 72, 1–25 (1983; Zbl 0508.46040)] is that the index lies in the set $$\{4\cos^2(\frac{\pi}{n}):n \geq 3\} \cup [4,\infty)$$. For “small” index values above $$4$$, the possible pairs of principal and dual principal graphs remain quantized. In fact, U. Haagerup [Subfactors. Proceedings of the Taniguchi symposium on operator algebras, Kyuzeso, Japan, July 6–10, 1993. Singapore: World Scientific. 1–38 (1994; Zbl 0933.46058)] produced a list of all potential pairs of principal and dual principal graphs in the index range $$(4,3 + \sqrt{3})$$. Asaeda and Haagerup constructed subfactors with principal and dual principal graphs equal to two of these candidate pairs. Other candidates were ruled out by Bisch, Asaeda and Yasuda. The only case which remained was the extended Haagerup principal graph.
The main result of this article is the construction of a planar algebra with extended Haagerup principal graph. This completes the classification of irreducible subfactors with index below $$3 + \sqrt{3}$$ which was initiated by Haagerup in the early 1990’s, and is a breakthrough result in the area. This is also a key component of the recent classification of subfactors of index up to $$5$$ due to the authors and others in a series of articles (see, e.g., [S. Morrison and N. Snyder, Commun. Math. Phys. 312, No. 1, 1–35 (2012; Zbl 1246.46055)]). The method of proof uses skein theoretic techniques to find the subfactor planar algebra inside the graph planar algebra of the extended Haagerup principal graph. This idea was outlined by V. F. R. Jones [Ser. Knots Everything 24, 94–117 (2000; Zbl 1021.46047)], and was applied to the Haagerup principal graph by E. Peters [Int. J. Math. 21, No. 8, 987–1045 (2010; Zbl 1203.46039)]. A key technique developed in the article is the “jellyfish” algorithm for evaluating closed diagrams given certain skein relations, which is likely to have further applications in quantum topology.

##### MSC:
 46L37 Subfactors and their classification 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
##### Keywords:
planar algebras; subfactors; skein theory; principal graphs
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##### References:
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