Constructing the extended Haagerup planar algebra

By Stephen Bigelow, E. Peters, S. Morrison, Noah Snyder. 2009

111 citations recorded by the source corpus.

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].

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eaa3547253893bee1b20c0261c695dd8a96d05fb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. Zhengwei Liu
2. Vaughan F. R. Jones
3. Noah Snyder
4. S. Morrison
5. Ben Elias
6. Geordie Williamson
7. C. Blanchet
8. David E. V. Rose

In Geometry and Topology

1. Zhengwei Liu
2. Noah Snyder
3. S. Morrison
4. Vaughan F. R. Jones
5. E. Peters
6. Ben Elias
7. David Penneys
8. David E. V. Rose

In Mathematical Physics

1. Zhengwei Liu
2. Noah Snyder
3. S. Morrison
4. Vaughan F. R. Jones
5. E. Peters
6. Ben Elias
7. David Penneys
8. David E. V. Rose

Citations in the corpus

• Quadratic Tangles in Planar Algebras (2010, background): “rm as early as 2003. It was one of the contributing factors to a project to classify all subfactors of index ≤ 5 which involves several people including Bigelow, Morrison, Peters and Snyder (see [18],[2]). This project in turn influenced more recent versions of this paper and changed its emphasis som”
• Exchange relation planar algebras of small rank (2013, methodology): “This approach has been extremely successful in the hands of Haagerup [16] and others [1, 3, 7, 17, 41].”
• The classification of subfactors of index at most 5 (2013, methodology): “hundred new subfactors whose even parts are Morita equivalent to the even parts of the Asaeda-Haagerup subfactor. 4.2.3. The extended Haagerup subfactor. In 2009 Bigelow, Morrison, Peters, and Snyder [BMPS12] gave a construction of a long suspected [Ike98] missing case: the extended Haagerup subfact”
• Categories generated by a trivalent vertex (2015, background): “this paper. There is a well-known theorem about nondegeneracy and negligibles which we will need. Proposition 2.6. An evaluable pivotal category has a unique maximal ideal, the negligible ideal. (cf. [BMPS12]) Corollary 2.7. Given a collection of relations, such that any closed diagram can be reduce”
• The two-color Soergel calculus (2013, background): “These maps can be defined in any extension of Z[δ] where the quantum numbers [2], [3], .”
• Cyclotomic Integers, Fusion Categories, and Subfactors (2010, unknown): “(4) Γ1,14 corresponds to the extended Haagerup subfactor [5], and ‖Γ‖ = λ(2) = 3 + ζ + ζ−1 + ζ(3) + ζ−3 + ζ(4) + ζ−4, with ζ(13) = 1.”