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 classify certain $\mathbb{Z}_2 $-graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: $\mathbb{Z}_2 $-graded extensions of $\mathbb{Z}_{2n} $ generalized Haagerup categories for all $n \leq 5 $; $\mathbb{Z}_2 \times \mathbb{Z}_2 $-graded extensions of the Asaeda-Haagerup categories; and extensions of the $\mathbb{Z}_2 \times \mathbb{Z}_2 $ generalized Haagerup category by its outer automorphism group $A_4 $. The construction uses endomorphis…
The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the Z/3Z symmetry is replaced by other finite Abelian groups. The goal of this paper is to give a similarly good description of the Asaeda-Haagerup subfactor which emerged from our study of its Brauer-Picard groupoid. More specifically, we construct a new subfactor S which is a Z/4Z x Z/2Z analogue of the Haagerup subfactor and we show that the even parts…
We prove that the Brauer-Picard group of Morita autoequiv- alences 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 four-group. 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 ma…
We consider generalized Haagerup categories such that $1 \oplus X$ admits a $Q$-system for every non-invertible simple object $X$. We show that in such a category, the group of order two invertible objects has size at most four. We describe the simple objects of the Drinfeld center and give partial formulas for the modular data. We compute the remaining corner of the modular data for several examples and make conjectures about the general case. We also consider several types of equivariantizations and de-equivariantizations of generalized Haagerup categories and describe their Drinfeld centers…
Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $τ$. We prove that if $Γ$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M\rtimesΓ$ has also the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed products von Neumann algebras are $\textrm{II}_1$-factors with the Haagerup proper…
In this paper we construct two new fusion categories and many new subfactors related to the exceptional Extended Haagerup subfactor. The Extended Haagerup subfactor has two even parts EH1 and EH2. These fusion categories are mysterious and are the only known fusion categories which appear to be unrelated to finite groups, quantum groups, or Izumi quadratic categories. One key technique which has previously revealed hidden structure in fusion categories is to study all other fusion categories in the Morita equivalence class, and hope that one of the others is easier to understand. In this paper…
We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some rem…
A fusion category is called transparent if the associator involving any invertible object is the identity map. For the Haagerup-Izumi fusion rings with $G = \mathbb{Z}_{2n+1}$ (the $\mathbb{Z}_3$ case is the Haagerup fusion ring with six simple objects), the transparent ansatz reduces the number of independent $F$-symbols from order $\mathcal{O}(n^6)$ to $\mathcal{O}(n^2)$, rendering the pentagon identity practically solvable. Transparent Haagerup-Izumi fusion categories are thereby constructively classified up to $G = \mathbb{Z}_9$, recovering all known Haagerup-Izumi fusion categories to thi…
Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion category can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely defined over a cyclotomic field. On the other hand, we show that the double of the even part of the Haagerup subfactor is completely defined over a cyclotomic field. We identify the minimal field of definition for each of t…
We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we further develop a Cuntz algebra method to construct $3^n$ subfactors and show that the classification of the $3^n$ subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd $n$.In particular, our method with $n=4$ gives a uniform construction of 4 finite depth subfactors, u…
We develop a representation theory for $λ$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $λ$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with propert…
We numerically study an anyon chain based on the Haagerup fusion category, and find evidence that it leads in the long-distance limit to a conformal field theory whose central charge is $\sim 2$. Fusion categories generalize the concept of finite group symmetries to non-invertible symmetry operations, and the Haagerup fusion category is the simplest one which comes neither from finite groups nor affine Lie algebras. As such, ours is the first example of conformal field theories which have truly exotic generalized symmetries.
We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup-Izumi family of Q-systems. For example, we construct endomorphism realisations of the (non-unitary) Yang-Lee model, and non-unitary analogues of one of the even subsystems of the Haagerup subfactor and of the Grossman-Snyder system. We supplement Izumi's equations for identifying the half-braidings, which were incomplete even in his Q-system setting. We conjecture a remarkably simple form for the modular S and T matric…
We construct a (1+1)$d$ topological field theory (TFT) whose topological defect lines (TDLs) realize the transparent Haagerup $\mathcal{H}_3$ fusion category. This TFT has six vacua, and each of the three non-invertible simple TDLs hosts three defect operators, giving rise to a total of 15 point-like operators. The TFT data, including the three-point coefficients and lasso diagrams, are determined by solving all the sphere four-point crossing equations and torus one-point modular invariance equations. We further verify that the Cardy states furnish a non-negative integer matrix representation …
Given a pair of fusion categories $C$ and $D$, we may form the free product $C * D$ and the tensor product $C \boxtimes D$. It is natural to think of the tensor product as a quotient of the free product. What other quotients are possible? When $C=D=A_2$, there is an infinite family of quotients interpolating between the free product and the tensor product (closely related to the $A_{2n-1}^{(1)}$ and $D_{n+2}^{(1)}$ subfactors at index 4). Bisch and Haagerup discovered one example of such an intermediate quotient when $C=A_2$ and $D=T_2$, and suggested that there might be another family here. W…
We associate a rigid C*-tensor category $C$ to a totally disconnected locally compact group $G$ and a compact open subgroup $K < G$. We characterize when $C$ has the Haagerup property or property (T), and when $C$ is weakly amenable. When $G$ is compactly generated, we prove that $C$ is essentially equivalent to the planar algebra associated by Jones and Burstein to a group acting on a locally finite bipartite graph. We then concretely realize $C$ as the category of bimodules generated by a hyperfinite subfactor.
We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0. The other terms in the sequence can be computed directly from the fusion ring of C_0. We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as G-extensions. The most striking of these is a Z/2Z-extension of one of the Asaeda-Haagerup fusion categories, which is one of only two known 3-supertransitive fusion categories outside the ADE series. In another direction, we show that our long exac…
Let $(Γ,μ)$ be a bipartite graph together with a weight on its vertices. Assume that $μ$ is an eigenvector for the adjacency matrix of $Γ$. Let Aut$(Γ, μ)$ be the automorphism group of the bipartite graph $Γ$ that scales the weight $μ$. It is a locally compact totally disconnected group that acts on the bipartite graph planar algebra $P$ associated to $(Γ,μ)$. Consider a subgroup G < Aut$(Γ, μ)$ and the set of fixed points $P^G \subset P$ that we assume to be a subfactor planar algebra. If the closure of G inside Aut$(Γ, μ)$ satisfies an approximation property such as amenability, the Haagerup…
Continuing our previous work on graded twisting of Hopf algebras and monoidal categories, we introduce a graded twisting construction for equivariant comodule algebras and module categories. As an example we study actions of quantum subgroups of $G\subset SL_{-1}(2)$ on $K_{-1}[x,y]$ and show that in most cases the corresponding invariant rings $K_{-1}[x,y]^G$ are invariant rings $K[x,y]^{G'}$ for the action of a classical subgroup $G'\subset SL(2)$. As another example we study Poisson boundaries of graded twisted categories and show that under the assumption of weak amenability they are grade…
This paper is the first of a pair that aims to classify a large number of the type $II$ quantum subgroups of the categories $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. In this work we classify the braided auto-equivalences of the categories of local modules for all known type $I$ quantum subgroups of $\mathcal{C}(\mathfrak{sl}_{r+1},k)$. We find that the symmetries are all non-exceptional except for four cases (up to level-rank duality). These exceptional cases are the orbifolds $\mathcal{C}( \mathfrak{sl}_{2},16)_{\operatorname{Rep}(\mathbb{Z}_2)}$, $\mathcal{C}( \mathfrak{sl}_{3},9)_{\operatorname…
We define new compact matrix quantum groups whose intertwiner spaces are dual to tensor categories of three-dimensional set partitions -- which we call spatial partitions. This extends substantially Banica and Speicher's approach of the so called easy quantum groups: It enables us to find new examples of quantum subgroups of Wang's free orthogonal quantum group $O_n^+$ which do not contain the symmetric group $S_n$; we may define new kinds of products of quantum groups coming from new products of categories of partitions; and we give a quantum group interpretation of certain categories of part…
Given a Hopf algebra $A$ graded by a discrete group together with an action of the same group preserving the grading, we define a new Hopf algebra, which we call the graded twisting of $A$. If the action is adjoint, this new Hopf algebra is a twist of $A$ by a pseudo-$2$-cocycle. Analogous construction can be carried out for monoidal categories. As examples we consider graded twistings of the Hopf algebras of nondegenerate bilinear forms, their free products, hyperoctahedral quantum groups and $q$-deformations of compact semisimple Lie groups. As applications, we show that the analogues of the…
The goal of this paper is to classify fusion categories $\otimes$-generated by a $K$-normal object (defined in this paper) of Frobenius-Perron dimension less than 2. This classification has recently become accessible due to a result of Morrison and Snyder, showing that any such category must be a cyclic extension of a category of adjoint $ADE$ type. Our main tools in this classification are the results of Etingof, Ostrik, and Nikshych, classifying cyclic extensions of a given category in terms of data computed from the Brauer-Picard group, and Drinfeld centre of that category, and the results …
