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 these fusion categories, compute the Galois groups, and identify their Galois conjugates.
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…
A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.
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…
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.
Let C be a fusion category which is an extension of a fusion category D by a finite group G. We classify module categories over C in terms of module categories over D and the extension data (c,M,a) of C. We also describe functor categories over C (and in particular the dual categories of C). We use this in order to classify module categories over the Tambara Yamagami fusion categories, and their duals.
We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generali…
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 develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group $G$ as induced from module categories over fusion subcategories associated with the subgroups of $G$. We define invariant $\C_e$-module categories and extensions of $\C_e$-module categories. The construction of module categories over $\C$ is reduced to determine invariant module categories for subgroups of $G$ and the indecomposable extensions of this modules categories. We associate a $G$-crossed product fusion cat…
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…
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 classify braided $\mathbb{Z}_q$-extensions of pointed fusion categories, where $q$ is a prime number. As an application, we classify modular categories of Frobenius-Perron dimension $q^3$.
We describe Lagrangian algebras in twisted Drinfeld centres for finite groups. Using the full centre construction, we establish a 1-1 correspondence between Lagrangian algebras and module categories over pointed fusion categories.
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…
We describe the structure of a generalized near-group fusion category and present an example of this class of fusion categories which arises from the extension of a Fibonacci category. We then classify slightly degenerate generalized near-group fusion categories. We also prove a structure result for braided generalized Tambara-Yamagami fusion categories.
We show that Galois theory of cyclotomic number fields provides a powerful tool to construct systematically integer-valued matrices commuting with the modular matrix S, as well as automorphisms of the fusion rules. Both of these prescriptions allow the construction of modular invariants and offer new insight in the structure of known exceptional invariants.
We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories which have a faithful simple object and show that its universal grading group must be cyclic. As for the converse, we prove that a braided nilpotent fusion category with cyclic universal grading group always has a faithful simple object. We study the universal grading of fusion categories with generalized Tambara-Yamagami fusion rules. As an application, we classify modular categories in this class and describe the modularizations…
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 …
We introduce a weakening of the notion of fusion 2-category given in arXiv:1812.11933. Then, we establish a number of properties of (multi)fusion 2-categories. Finally, we describe the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories.
We describe an approach to classification of fusion categories in terms of Morita equivalence. This is usually achieved by analyzing Drinfeld centers of fusion categories and finding Tannakian subcategories therein.
In this paper, we first show for a slightly degenerate pre-modular fusion category $\mathcal{C}$ that squares of dimensions of simple objects divide half of the dimension of $\mathcal{C}$, and that slightly degenerate fusion categories of FP-dimensions $2p^nd$ and $4p^nd$ are nilpotent, where $p$ is an odd prime and $d$ is an odd square-free integer. Then we classify slightly degenerate generalized Tambara-Yamagami fusion categories and weakly integral slightly degenerate fusion categories of particular dimensions.
We develop methods of computation of the Brauer-Picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension: representation categories of elementary abelian groups with twisted associativity constraint, extra special p-groups, and the Kac-Paljutkin Hopf algebra. We conclude that many finite groups of Lie type occur as composition factors of the Brauer-Picard groups of pointed fusion categories.
In this paper, we study fusion categories which contain a proper fusion subcategory with maximal rank. They can be viewed as generalizations of near-group fusion categories. We first prove that they admit spherical structure. We then classify those which are non-degenerate or symmetric. Finally, we classify such fusion categories of rank 4.
In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two cent…
We give an explicit description, up to gauge equivalence, of group-theoretical quasi-Hopf algebras. We use this description to compute the Frobenius-Schur indicators for group-theoretical fusion categories.
We introduce semisimple 2-categories, fusion 2-categories, and spherical fusion 2-categories. For each spherical fusion 2-category, we construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds.
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module categories.
We first prove an analogue of Lagrange theorem for global dimensions of fusion categories, then we give a complete classifications of pre-modular fusion categories of integer global dimensions less than or equal to $10$.
We classify all fusion categories of rank 3 that admit a pivotal structure over an algebraically closed field of characteristic zero. Also in the Appendix (joint with D.Nikshych) we give some restrictions on Grothendieck rings of near-group categories.
We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\ZZ/3\ZZ$-graded fusion categories, which are generalizations of the $\ZZ/2\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory o…
We prove a version of the Jordan-H\" older theorem in the context of weakly group-theoretical fusion categories. This allows us to introduce the composition factors and the length of such a fusion category C, which are in fact Morita invariants of C.
We prove some results on the structure of certain classes of integral fusion categories and semisimple Hopf algebras under restrictions on the set of its irreducible degrees.
A $G$-graded extension of a fusion category $\mathcal{C}$ yields a categorical action of $G$ on the center $Z(\mathcal C)$. If the extension admits a spherical structure, we provide a method for recovering its fusion rules in terms of the action. We then apply this to find closed formulas for the fusion rules of extensions of some group theoretical categories and of cyclic permutation crossed extensions of modular categories.
We prove several results in the theory of fusion categories using the product (norm) and sum (trace) of Galois conjugates of formal codegrees. First, we prove that finitely-many fusion categories exist up to equivalence whose global dimension has a fixed norm. Furthermore, with two exceptions, all formal codegrees of spherical fusion categories with square-free norm are rational integers. This implies, with three exceptions, that every spherical braided fusion category whose global dimension has prime norm is pointed. The reason exceptions occur is related to the classical Schur-Siegel-Smyth p…
We collate information about the fusion categories with $A_n$ fusion rules. This note includes the classification of these categories, a realisation via the Temperley-Lieb categories, the auto-equivalence groups (both braided and tensor), identifications of the subcategories of invertible objects, and explicit descriptions of the Drinfeld centres. The first section describes the classification of these categories (as monoidal, dagger, pivotal, and braided categories). The second section describes the properties of these categories.
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 show that braidings on a fusion category $\mathcal{C}$ correspond to certain fusion subcategories of the center of $\mathcal{C}$ transversal to the canonical Lagrangian algebra. This allows to classify braidings on non-degenerate and group-theoretical fusion categories.
We provide a general formula for Mueger's centralizer of any fusion subcategory of a braided fusion category containing a tannakian subcategory. This entails a description for Mueger's centralizer of all fusion subcategories of a group theoretical braided fusion category.
We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius-Perron dimension and the Frobenius-Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braid…
We study a class of strictly weakly integral fusion categories $\mathfrak{I}_{N, ζ}$, where $N \geq 1$ is a natural number and $ζ$ is a $2^N$th root of unity, that we call $N$-Ising fusion categories. An $N$-Ising fusion category has Frobenius-Perron dimension $2^{N+1}$ and is a graded extension of a pointed fusion category of rank 2 by the cyclic group of order $\mathbb Z_{2^N}$. We show that every braided $N$-Ising fusion category is prime and also that there exists a slightly degenerate $N$-Ising braided fusion category for all $N > 2$. We also prove a structure result for braided extension…
In this paper, we show that non-degenerate fusion categories of FP-dimensions $p^2q^3d$ and $p^3q^3d$ are group-theoretical, where $p, q$ are odd primes, $d$ is a square-free integer such that $(pq,d) = 1$.
We introduce the notion of a $\textit{reflection fusion category}$, which is a type of a $G$-crossed category generated by objects of Frobenius-Perron dimension $1$ and $\sqrt{p}$, where $p$ is an odd prime. We show that such categories correspond to orthogonal reflection groups over $\mathbb{F}_p$. This allows us to use the known classification of irreducible reflection groups over finite fields to classify irreducible reflection fusion categories.
In this paper we prove an analogue of Brauer's theorem for faithful objects in fusion categories. Other notions, such as the order and the index associated to faithful objects of fusion categories are also discussed. We show that the index of a faithful simple object of a fusion categories coincides with the order of the universal grading group of the fusion category.
We show that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between such categories. We prove analogous results for unitarizable braided fusion categories and module categories.
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…
Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any modular category (not necessarily hermitian) is unitary. We also show that the category of module functors between two module categories over a fusion category is semisimple, and that fusion categories and tensor functors between them are undeformable (generalized Ocneanu rigidity). In particular the num…
The problem of determining gauge and monoidal equivalence classes of fusion categories is considered from the perspective of geometric invariant theory. It is shown that the gauge (or monoidal) class of a fusion category is determined by the evaluation of a finite set of gauge (or monoidal) invariant functions. In the multiplicity free case this leads to a fast algorithm for computing a classifying set of functions.
