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.Comment: 45 pages (final version; improved introduction

We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the "quantum subgroups" in the sense of Ocneanu), we find all subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the FrobeniusPerron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the Appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A n or D n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less than 5.

The balanced tensor product M ⊗ A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M × N . The balanced tensor product M C N of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M × N . We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.Tensor categories are a higher dimensional analogue of algebras. Just as modules and bimodules play a key role in the theory of algebras, the analogous notions of module categories and bimodule categories play a key role in the study of tensor categories (as pioneered by Ostrik [Ost03]). One of the key constructions in the theory of modules and bimodules is the relative tensor product M ⊗ A N . As first recognized by Tambara [Tam01], a similarly important role is played by the balanced tensor product of module categories over a monoidal category M C N (for example, see [Müg03, ENO10, JL09, GS12b, GS12a, DNO13, FSV13]). For C = Vect, this agrees with Deligne's [Del90] tensor product K L of finite linear categories. Etingof, Nikshych, and Ostrik [ENO10] established the existence of a balanced tensor product M C N of finite semisimple module categories over a fusion category, and Davydov and Nikshych [DN13, §2.7] outlined how to generalize this construction to module categories over a finite tensor category. 1 We give a new construction of the balanced tensor product over a finite tensor category as a category of bimodule objects.Recall that the balanced tensor product M ⊗ A N of modules is, by definition, the vector space corepresenting A-balanced bilinear functions out of M × N . In other words, giving a map M ⊗ A N → X is the same as giving a billinear map f : M × N → X with the property that f (ma, n) = f (m, an). If the balanced tensor product exists, it is certainly unique (up to unique isomorphism), but the universal property does not guarantee existence. Instead existence is typically established by an explicit construction as a quotient of a free abelian group on the product M ×N . We now describe the balanced tensor product M C N of module categories over a tensor category. Again this should be universal for certain bilinear functors, however when passing from algebras to tensor categories, the analogue of the equality f (ma, n) = f (m, an) is a natural system of isomorphisms η m,a,n : F(m ⊗ a, n) → F(m, a ⊗ n) satisfying some natural coherence properties. A bilinear functor F together with a natural coherent system of isomorphisms η m,a,n is called a C-balanced functor. Thus the balanced tensor product M C N is defined to be the linear category corepresenting C-balanced right-exact bilinear functors out of M × N . In other words, giving a right-exact functor M C N → X is the same as giving a right-exact bilinear functor M × N → X together with isomorphisms η m,a,n : F(m ⊗ a, n) → F(m, a ⊗ n) sa...

We prove that the Brauer-Picard group of Morita autoequivalences of each of the three fusion categories which arise as an even part of the Asaeda-Haagerup subfactor or of its index 2 extension is the Klein fourgroup. We describe the 36 bimodule categories which occur in the full subgroupoid of the Brauer-Picard groupoid on these three fusion categories. We also classify all irreducible subfactors both of whose even parts are among these categories, of which there are 111 up to isomorphism of the planar algebra (76 up to duality). Although we identify the entire Brauer-Picard group, there may be additional fusion categories in the groupoid. We prove a partial classification of possible additional fusion categories Morita equivalent to the Asaeda-Haagerup fusion categories and make some conjectures about their existence. This is the submitted version of arXiv:1202.4396. √ 13 2 and 7+ √ 17 2; we call these subfactors H + 1 and AH + 1. In the Haagerup case H + 1 just implements the trivial autoequivalence of H 1 , so it does not give any new information about the groupoid (and indeed, in [GS12] we gave a "trivial" construction of H + 1 exploiting this fact). However in the Asaeda-Haagerup case, AH + 1 gives a second Morita equivalence between A H 1 and A H 3 , which immediately implies that the group of autoequivalences of each of the Asaeda-Haagerup fusion categories is non-trivial.Moreover, it was conjectured in [AG11] that the "plus one" construction can be iterated once more in the Asaeda-Haagerup case to find a subfactor AH + 2 with index 9+ √ 17 2. We verify the existence of AH + 2 and show that it gives a new autoequivalence of A H 1 which is not in the groupoid generated by AH and AH + 1. Finally, we find the full group of Morita autoequivalences of each of the three Asaeda-Haagerup fusion categories: Theorem 1.1 The Brauer-Picard group of Morita autoequivalences of each of the Asaeda-Haagerup fusion categories is Z/2Z × Z/2Z.