Constructing the extended Haagerup planar algebra
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 , 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.
46L37 \secondaryclass 18D10
A subfactor is an inclusion of von Neumann algebras with trivial center. The theory of subfactors can be thought of as a nonabelian version of Galois theory, and has had many applications in operator algebras, quantum algebra, and knot theory. For example, the construction of a new finite depth subfactor, as in this paper, also yields two new fusion categories (by taking the even parts) and a new -dimensional TQFT (via the Ocneanu-Turaev-Viro construction [62, 48, 51]).
A subfactor has three key invariants. From strongest to weakest, they are: the standard invariant (which captures all information about “basic” bimodules over and ), the principal and dual principal graphs (which together describe the fusion rules for these basic bimodules), and the index (which is a real number measuring the “size” of the basic bimodules). We will use the axiomatization of the standard invariant as a subfactor planar algebra, which is due to Jones . Other axiomatizations include Ocneanu’s paragroups  and Popa’s -lattices . (For readers more familiar with tensor categories, these three approaches are analogous to the diagram calculus [51, 58, 35], basic symbols [61, Chapter 5], and towers of endomorphism algebras , respectively.) The standard invariant is a complete invariant of amenable subfactors of the hyperfinite factor [53, 55].
The index of a subfactor must lie in the set
and all numbers in this set can be realized as the index of a subfactor . Early work on classifying subfactors of “small index” concentrated on the case of index less than . The principal graphs of these subfactors are exactly the Dynkin diagrams , , and . Furthermore there is exactly one subfactor planar algebra with principal graph or and there is exactly one pair of complex conjugate subfactor planar algebras with principal graph or . (See  for the outline of this result, and [5, 22, 23, 38] for more details.) The story of the corresponding classification for index equal to is outlined in [55, p. 231]. In this case, the principal graph must be an affine Dynkin diagram. For some principal graphs there are multiple non-conjugate subfactors with the same principal graph, which are distinguished by homological data.
The classification of subfactors of “small index” greater than was initiated by Haagerup . His main result is a list of all possible pairs of principal graphs of irreducible subfactors of index larger than but smaller than . Here we begin to see subfactors whose principal graph is different from its dual principal graph. If refers to a pair of principal graphs and we need to refer to one individually, we will use the notation and . Any subfactor has a dual given by the basic construction . Taking duals reverses the shading on the planar algebra, switches the principal and dual principal graphs, and preserves index. Haagerup’s list is as follows (we list each pair once).
the infinite family
which has , the largest root of , and monotonically increasing with , converging to the real root of , (thus , , and ),
the infinite family
which has , and monotonically increasing in , converging to the real root of , (thus , and ),
one more pair of graphs,
which has index .
Haagerup’s paper announces this result up to index , but only proves it up to index ; this includes all of the graphs , but none of the graphs or . Haagerup’s proof of the full result has not yet appeared. In work in progress, Jones, Morrison, Penneys, Peters, and Snyder have independently confirmed his result (following Haagerup’s outline except at one point using a result from ), and have extended his techniques to give a partial result up to index (see [45, 42, 25, 50]). In this paper, we will only rely on the part of Haagerup’s classification that has appeared in print.
Haagerup’s original result did not specify which of the possible principal graphs are actually realized. Considerable progress has since been made in this direction. Asaeda and Haagerup  proved the existence and uniqueness of a subfactor planar algebra whose principal graphs are (called the Haagerup subfactor), and a subfactor planar algebra for (called the Asaeda-Haagerup subfactor). Izumi  gave an alternate construction of the Haagerup subfactor. Bisch  showed none of the graphs can be principal graphs because they give inconsistent fusion rules. Asaeda  and Asaeda-Yasuda  proved that is not a principal graph for . To do this, they showed that the index is not a cyclotomic integer, and then appealed to a result of Etingof, Nikshych and Ostrik , which in turn is proved by reduction to the case of modular categories, where it was proved in the context of rational conformal field theories by Coste–Gannon  using a result of de Boere–Goeree .
The main result of our paper is
There is a subfactor planar algebra with principal graphs .
In addition, we prove in Theorem 3.9 that this planar algebra is the only one with these principal graphs. This result completes the classification of all subfactor planar algebras up to index :
The only irreducible subfactor planar algebras with index in the range are
the non-amenable Temperley-Lieb planar algebra at every index in this range, with principal graphs ,
the Haagerup planar algebra with principal graphs , and its dual,
the Haagerup-Asaeda planar algebra with principal graphs , and its dual, and
the extended Haagerup planar algebra with principal graphs , and its dual.
By Popa’s classification  the latter three pairs can each be realized uniquely as the standard invariant of a subfactor of the hyperfinite factor. This gives a complete classification of amenable subfactors of the hyperfinite factor with index between . The non-amenable case remains open because it is unknown for which indices Temperley-Lieb can be realized as the standard invariant of the hyperfinite factor, nor in how many ways it can be realized (see [54, 6] for some work in this direction). Furthermore, there remain many interesting questions about small index subfactors of arbitrary factors.
It was already expected that the extended Haagerup subfactor should exist, thanks to approximate numerical evidence coming from computations by Ikeda . We note that although our construction relies on a computation of the traces of a few large matrices, this computation consists of exact arithmetic in a number field, and is a very different calculation from the one Ikeda did numerically.
The search for small index subfactors has so far produced the three pairs of “sporadic” examples: the Haagerup, Asaeda-Haagerup and extended Haagerup subfactors. These are some of the very few known subfactors that do not seem to fit into the frameworks of groups, quantum groups, or conformal field theory . (See also a generalization of the Haagerup subfactor due to Izumi [24, Example 7.2]). You might think of them as analogs of the exceptional simple Lie algebras, or of the sporadic finite simple groups. (Without a good extension theory, it is not yet clear what “simple” should mean in this context.)
In this paper, we study the extended Haagerup planar algebra. We construct the extended Haagerup planar algebra by locating it inside the graph planar algebra  of its principal graph. By a result of Jones–Penneys  (generalized in ) every subfactor planar algebra occurs in this way. We find the right planar subalgebra by following a recipe outlined by Jones [29, 31] and further developed by Peters , who applied it to the Haagerup planar algebra.
We also give a presentation of the extended Haagerup planar algebra using a single planar generator and explicit relations. We prove that the subalgebra of the graph planar algebra contains an element also satisfying these relations. This is convenient because different properties become more apparent in different descriptions of the planar algebra. For example, the subalgebra of the graph planar algebra is clearly non-trivial, which would be difficult to prove directly from the generators and relations. In the other direction, in §5 we prove that our relations result in a space of closed diagrams that is at most one dimensional, which would be difficult to prove in the graph planar algebra setting.
In §2 we recall the definitions of planar algebras and graph planar algebras [26, 29]. We also set some notation for the graph planar algebra of . In §3 we prove our two main theorems, Theorems 3.9 and 3.10. Theorem 3.9, the uniqueness theorem, says that for each there is at most one subfactor planar algebra with principal graphs . Furthermore we give a skein theoretic description by generators and relations of the unique candidate planar algebra. Theorem 3.10, the existence theorem, constructs a subfactor planar algebra with principal graphs by realizing the skein theoretic planar algebra as a subalgebra of the graph planar algebra. Proofs of several key results needed for the main existence and uniqueness arguments are deferred to §4, §5, and §6. In particular, §4 describes an evaluation algorithm that uses the skein theory to evaluate any closed diagram (Theorem 3.8). This is crucial to our proofs of both existence and uniqueness and may be of broader interest in quantum topology. This section can be read independently of the rest of the paper. Section 5 consists of calculations of inner products using generators and relations. Section 6 gives the description of the generator of our subfactor planar algebra inside the graph planar algebra and verifies its properties. Appendix A gives the tensor product rules for the two fusion categories associated to the extended Haagerup subfactor.
Part of this work was done while Stephen Bigelow and Emily Peters were visiting the University of Melbourne. Scott Morrison was at Microsoft Station Q and the Miller Institute for Basic Research during this work. Emily Peters was supported in part by NSF Grant DMS0401734 and a fellowship from Soroptimist International and Noah Snyder was supported in part by RTG grant DMS-0354321 and in part by an NSF Postdoctoral Fellowship. We would like to thank Vaughan Jones for many useful discussions, and Yossi Farjoun for lessons on Newton’s method.
2.1 Planar algebras
Planar algebras were defined in  and . More general definitions have since appeared elsewhere, but we only need the original notion of a shaded planar algebra, which we sketch here. For further details see [29, §2], [26, §0], or .
A (shaded) planar tangle has an outer disk, a finite number of inner disks, and a finite number of non-intersecting strings. A string can be either a closed loop or an edge with endpoints on boundary circles. We require that there be an even number of endpoints on each boundary circle, and a checkerboard shading of the regions in the complement of the interior disks. We further require that there be a marked point on the boundary of each disk, and that the inner disks are ordered.
Two planar tangles are considered equal if they are isotopic (not necessarily rel boundary).
Here is an example of a planar tangle.
Planar tangles can be composed by placing one planar tangle inside an interior disk of another, lining up the marked points, and connecting endpoints of strands. The numbers of endpoints and the shadings must match up appropriately. This composition turns the collection of planar tangles into a colored operad.
A (shaded) planar algebra consists of
A family of vector spaces , called the positive and negative -box spaces.
For each planar tangle, a multilinear map where is half the number of endpoints on the th interior boundary circle, is half the number of endpoints on the outer boundary circle, and the signs are positive (respectively negative) when the marked point on the corresponding boundary circle is in an unshaded region (respectively shaded region).
For example, the planar tangle above gives a map
The linear map associated to a ‘radial’ tangle (with one inner disc, radial strings, and matching marked points) must be the identity. We require that the action of planar tangles be compatible with composition of planar tangles. In other words, composition of planar tangles must correspond to the obvious composition of multilinear maps. In operadic language this says that a planar algebra is an algebra over the operad of planar tangles.
We will refer to an element of (and specifically , unless otherwise stated) as an “-box.”
We make frequent use of three families of planar tangles called multiplication, trace, and tensor product, which are shown in Figure 1. “Multiplication” gives an associative product . “Trace” gives a map . “Tensor product” gives an associative product if is even, or if is odd.
The (shaded or unshaded) empty diagrams can be thought of as elements of , since the ‘empty tangle’ induces a map from the empty tensor product to the space . If the space is one dimensional then we can identify it with by sending the empty diagram to one. In many other cases, we can make do with the following.
A partition function is a pair of linear maps
that send the empty diagrams to .
In a planar algebra with a partition function, let
denote the composition of the trace tangle with .
Sometimes we will need to refer simply to the action of the trace tangle, which we denote .
Notice that the above trace tangle is the “right trace.” There is also a “left trace” where all the strands are connected around the left side.
A planar algebra with a partition function can be:
Positive definite: There is an antilinear adjoint operation on each , compatible with the adjoint operation on planar tangles given by reflection. The sesquilinear form is positive definite.
Spherical: The left trace
and the right trace
The spherical property implies that the left and right traces are equal on every . Since every planar algebra we consider is spherical, we will usually ignore the distinction between left and right trace.
A subfactor planar algebra is a positive definite spherical planar algebra such that and .
As a consequence of being spherical and having -dimensional -box spaces, subfactor planar algebras always have a well-defined modulus, as described below.
We say that the planar algebra has modulus if the following relations hold.