Acta Mathematica

Constructing the extended Haagerup planar algebra

Stephen Bigelow, Emily Peters, Scott Morrison, and Noah Snyder

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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. This is the published version of arXiv:0909.4099 [math.OA].

Article information

Acta Math., Volume 209, Number 1 (2012), 29-82.

Received: 31 January 2010
First available in Project Euclid: 31 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L37: Subfactors and their classification
Secondary: 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories [See also 19D23]

planar algebras subfactors skein theory principal graphs

2012 © Institut Mittag-Leffler


Bigelow, Stephen; Peters, Emily; Morrison, Scott; Snyder, Noah. Constructing the extended Haagerup planar algebra. Acta Math. 209 (2012), no. 1, 29--82. doi:10.1007/s11511-012-0081-7.

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