A bicommutant category is a higher categorical analog of a von Neumann algebra. We study the bicommutant categories which arise as the commutant $\mathcal{C}'$ of a fully faithful representation $\mathcal{C}\to\operatorname{Bim}(R)$ of a unitary fusion category $\mathcal{C}$. Using results of Izumi, Popa, and Tomatsu about existence and uniqueness of representations of unitary (multi)fusion categories, we prove that if $\mathcal{C}$ and $\mathcal{D}$ are Morita equivalent unitary fusion categories, then their commutant categories $\mathcal{C}'$ and $\mathcal{D}'$ are equivalent as bicommutant categories. In particular, they are equivalent as tensor categories: \[ \Big(\,\,\mathcal{C}\,\,\simeq_{\text{Morita}}\,\,\mathcal{D}\,\,\Big) \qquad\Longrightarrow\qquad \Big(\,\,\mathcal{C}'\,\,\simeq_{\text{tensor}}\,\,\mathcal{D}'\,\,\Big). \] This categorifies the well-known result according to which the commutants (in some representations) of Morita equivalent finite dimensional $\rm C^*$-algebras are isomorphic von Neumann algebras, provided the representations are `big enough'.
We also introduce a notion of positivity for bi-involutive tensor categories. For dagger categories, positivity is a property (the property of being a $\rm C^*$-category). But for bi-involutive tensor categories, positivity is extra structure. We show that unitary fusion categories and $\operatorname{Bim}(R)$ admit distinguished positive structures, and that fully faithful representations $\mathcal{C}\to\operatorname{Bim}(R)$ automatically respect these positive structures.
We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect to the fusion subcategory, and minimal central projections in the relative tube algebra. Based on this, we explicitly compute certain relative Drinfeld commutants of fusion categories arising from $α$-induction for braided subfactors. We present examples arising from chiral conformal field theory.
Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well known result according to which a finite dimensional *-algebra that can be faithfully represented on a Hilbert space is in fact a von Neumann algebra.
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…
A unitary fusion category is called $\mathbb{Z}/2\mathbb{Z}$-quadratic if it has a $\mathbb{Z}/2\mathbb{Z}$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $\mathbb{Z}/2\mathbb{Z}$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostrik's results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larson's rank-finiteness bound for $\mathbb{Z}/3\mathbb{Z}$-near group pseudounitary fusio…
We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that $\cup_{k\geq 1}M_k(D)$ admits a factor trace if and only if $D\in \{2\cos(π/n)+1|n\geq 3\}\cup [3,\infty)$ and higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over the factors are constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.
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 equiva…
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 classify fusion categories which are Morita equivalent to even parts of subfactors with index $3+\sqrt{5} $, and module categories over these fusion categories. For the fusion category $\mathcal{C} $ which is the even part of the self-dual $3^{\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} } $ subfactor, we show that there are $30$ simple module categories over $ \mathcal{C}$; there are no other fusion categories in the Morita equivalence class; and the order of the Brauer-Picard group is $360$. The proof proceeds indirectly by first describing the Brauer-Picard groupoid of a $ \mathb…
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 …
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…
We construct analogs of the embedding of orthogonal and symplectic groups into unitary groups in the context of fusion categories. At least some of the resulting module categories also appear in boundary conformal field theory. We determine when these categories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors.
We compute the group of Morita auto-equivalences of the even parts of the $ADE$ subfactors, and Galois conjugates. To achieve this we study the braided auto-equivalences of the Drinfeld centres of these categories. We give planar algebra presentations for each of these Drinfeld centres, which we leverage to obtain information about the braided auto-equivalences of the corresponding categories. We also perform the same calculations for the fusion categories constructed from the full $ADE$ subfactors. Of particular interest, the even part of the $D_{10}$ subfactor is shown to have Brauer-Picard …
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…
A near-group fusion category is a fusion category C where all but 1 simple objects are invertible. Examples of these include the Tambara-Yamagami categories and the even sectors of the E6 and affine-D5 subfactors, though there are infinitely many others. We classify the near-group fusion categories, and compute their doubles and the modular data relevant to conformal field theory. Among other things, we explicitly construct over 40 new finite depth subfactors, with Jones index ranging from around 6.85 to around 14.93. We expect all of these doubles to be realised by rational conformal field th…
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…
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 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 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…
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 Frobenius-Perron 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 t…
Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is…
