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.

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.

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If N ⊆ P , Q ⊆ M N\subseteq P,Q\subseteq M are type II 1 _1 factors with N ′ ∩ M = C i d N’\cap M =\mathbb C id and [ M : N ] > ∞ [M:N]>\infty we show that restrictions on the standard invariants of the elementary inclusions N ⊆ P N\subseteq P , N ⊆ Q N\subseteq Q , P ⊆ M P\subseteq M and Q ⊆ M Q\subseteq M imply drastic restrictions on the indices and angles between the subfactors. In particular we show that if these standard invariants are trivial and the conditional expectations onto P P and Q Q do not commute, then [ M : N ] [M:N] is 6 6 or 6 + 4 2 6+4\sqrt 2 . In the former case N N is the fixed point algebra for an outer action of S 3 S_3 on M M and the angle is π / 3 \pi /3 , and in the latter case the angle is cos − 1 ⁡ ( 2 − 1 ) \cos ^{-1}(\sqrt 2-1) and an example may be found in the GHJ subfactor family. The techniques of proof rely heavily on planar algebras.

A quadrilateral of factors is an irreducible inclusion of factors N ⊂ M with intermediate subfactors P and Q such that P and Q generate M and the intersection of P and Q is N. We investigate the structure of a noncommuting quadrilateral of factors with all the elementary inclusions P ⊂ M, Q ⊂ M, N ⊂ P, and N ⊂ Q 2-supertransitive. In particular, we classify noncommuting quadrilaterals with the indices of the elementary subfactors less than or equal to 4. We also compute the angles between P and Q for quadrilaterals coming from α-induction and asymptotic inclusions.