Acta Mathematica

, Volume 209, Issue 1, pp 29-82

Constructing the extended Haagerup planar algebra

  • Stephen BigelowAffiliated withDepartment of Mathematics, University of California, Santa Barbara
  • , Emily PetersAffiliated withDepartment of Mathematics, Massachusetts Institute of Technology Email author 
  • , Scott MorrisonAffiliated withMathematical Sciences Institute, Australian National University
  • , Noah SnyderAffiliated withDepartment of Mathematics, Indiana University, Bloomington

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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].


planar algebras subfactors skein theory principal graphs

2000 Math. Subject Classification

primary 46L37 secondary 18D10