Fusion categories of rank 2
Viktor Ostrik
对复数域上具有两个同构类的单对象的半单刚性幺半范畴进行了分类。在P.Etingof的附录中证明了具有有限个不可约表示的半单霍普夫代数的个数是有限的。
We classify semisimple rigid monoidal categories with two isomorphism classes of simple objects over the field of complex numbers. In the appendix written by P.Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite.
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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 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 $10$.
On the classification of certain fusion categories
我们在两个方向上推进融合范畴的分类。首先，我们完全分类积分融合范畴-因此，半单霍普夫代数-维度$pq^2$，其中$p$和$q$是不同的素数。这种情况特别有趣，因为它是最简单的维类，其中并非所有的积分融合范畴都是群论的。其次，我们对一类$\ZZ/3\ZZ$ -分次融合范畴进行了分类，它们是$\ZZ/2\ZZ$ -分次Tambara-Yamagami范畴的推广。我们的证明是基于最近发展的理论融合范畴的扩展。
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.
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.
Noncommutative near-group fusion categories were completely classified in the previous work of the first named author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group theoretical though the corresponding pointed categories were not identified. In this note we give a purely algebraic construction of the noncommutative near-group fusion categories starting from pointed categories categorically Morita equivalent to them.
Categorical Lagrangian Grassmannians and Brauer-Picard groups of pointed fusion categories
我们分析了点融合范畴的布劳尔-皮卡德群对其中心的拉格朗日子范畴集的作用。利用这个作用，我们计算了与几个经典有限群相关的点融合范畴的Brauer-Picard群.作为应用，我们构造了弱群论融合范畴的新例子。
Weakly group-theoretical and solvable fusion categories
本文引入了两类新的融合范畴，它们是从有限群中通过一定的过程得到的--弱群论范畴和可解范畴。这些融合范畴是森田等价于任意可解有限群的迭代扩展（在融合范畴的世界中）。弱群理论范畴具有整数维，所有已知的整数维融合范畴都是弱群理论范畴。我们的主要结果是：弱群论范畴C具有强Frobenius性质（即，不可分解的C-模范畴中的任何简单对象的维数都能整除C的维数），并且维数至多有两个素除子的任何融合范畴都是可解的（一个与有限群的伯恩赛德定理类似的范畴）。这对融合范畴和给定维数的半单霍普夫代数的分类有很强的应用。特别地，我们证明了任何整数维数小于84的融合范畴都是弱群论的（即来自有限群），并给出了维数为pqr和pq ^2的半单霍普夫代数的完全分类，其中p，q，r是不同的素数。
Extension Theory and Fermionic Strongly Fusion 2-Categories (with an Appendix by Thibault Didier Décoppet and Theo Johnson-Freyd)
本文研究融合2-范畴的群分次扩张，并利用同伦论方法对费米子强融合2-范畴进行分类，同时详细分析了若干具体例子。
On formal codegrees of fusion categories
我们证明了一个一般性的结果，这意味着一个融合范畴的整体和Frobenius-Perron维数生成代数整数代数整数环中的伽罗瓦不变理想。
Constructing TQFTs Using Non-Hermitian Ribbon Fusion Categories
本文提出了一种从非厄米带环融合范畴构造拓扑量子场论的简单方法，并通过证明Müger中心、辫结构和球面结构的相关定理加以验证。
Fusion subcategories of representation categories of twisted quantum doubles of finite groups
我们描述了有限群的扭曲量子偶的表示范畴的所有融合子范畴。鉴于每个群论辫状融合范畴都可以嵌入到有限群的扭曲量子偶的表示范畴中，这就给出了所有群论辫状融合范畴的完整描述.我们描述了格，并给出了有限群的扭曲量子偶的表示范畴的融合子范畴的一些不变量的公式。我们还给出了群论辫融合范畴作为尖范畴的等变化的一个刻画。
We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many isomorphism classes of simple objects.
This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.
A criterion for Müger centralizer of a fusion subcategory of a braided non-degenerate fusion category is given. Along the way we extend some identities on the space of class functions of a fusion category introduced by Shimizu in \cite{scalg}. We also show that in a modular tensor category the product of two conjugacy class sums is a linear combination of conjugacy class sums with rational coefficients.
Let $p\geq5$ be a prime, we show that a non-pointed modular fusion category $\mathcal{C}$ is Grothendieck equivalent to $\mathcal{C}(\mathfrak{sl}_2,2(p-1))_A^0$ if and only if $\dim(\mathcal{C})=p\cdot u$, where $u$ is a certain totally positive algebraic unit and $A$ is the regular algebra of the Tannakian subcategory $\text{Rep}(\mathbb{Z}_2)\subseteq\mathcal{C}(\mathfrak{sl}_2,2(p-1))$. As a direct corollary, we classify non-simple modular fusion categories of global dimensions $p^2$.
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.
In this paper we give a complete classification of unitary fusion categories $\otimes$-generated by an object of dimension $\frac{1 + \sqrt{5}}{2}$. We show that all such categories arise as certain wreath products of either the Fibonacci category, or of the dual even part of the $2D2$ subfactor. As a by-product of proving our main classification result we produce a classification of finite unitarizable quotients of $\operatorname{Fib}^{*N}$ satisfying a certain symmetry condition.
Rigid and Separable Algebras in Fusion 2-Categories
本文推广刚性幺半1-范畴概念，研究融合2-范畴中的刚性代数与可分代数，给出模与双模2-范畴伴随存在的判别准则，证明可分代数的模、双模2-范畴有限半单，且连通刚性代数可分等价于维数非零。
The Morita Theory of Fusion 2-Categories
本文建立融合 2-范畴的 Morita 理论，证明可分代数模的相对张量积存在，构造并等价刻画相关 Morita 3-范畴与可分模 2-范畴，还给出 Morita 等价的三种表述。
Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories
本研究围绕融合2-范畴的Morita等价展开，证明其Drinfeld中心在Morita等价下不变，建立了融合2-范畴的结构分解与可分性等性质，定义了其非零维数并证明Drinfeld中心为有限半单2-范畴。
Module categories over graded fusion categories
设C是一个融合范畴，它是一个融合范畴D通过一个有限群G的扩张.利用D上的模范畴和C的扩张数据（c，M，a）对C上的模范畴进行分类.我们还描述了C上的函子范畴（特别是C的对偶范畴）。我们使用它来对Tambara Yamagami融合类别及其子类别进行分类。
Compact Semisimple Tensor 2-Categories are Morita Connected
本文证明紧致半单张量2-范畴在任意特征零域上具有Morita连通性，推广了融合2-范畴的相应结果，并由此构造了由第四Galois上同调群索引的辫融合1-范畴。
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.
We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.
Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual fusion category of $\mathcal{C}$ with respect to $\mathcal{M}$. We describe tensor functors between two arbitrary duals $\mathcal{C}^*_\mathcal{M}$ and $\mathcal{D}^*_\mathcal{N}$ in terms of data associated to $\mathcal{C}$ and $\mathcal{D}$. We apply the results to $G$-equivariantizations of fusion categories and group-theoretical fusion categories. We describe…
Ocneanu rigidity implies that there are finitely many (braided) fusion categories with a given set of fusion rules. While there is no method for determining all such categories up to equivalence, there are a few cases for which can. For example, Kazhdan and Wenzl described all fusion categories with fusion rules isomorphic to those of $SU(N)_k$. In this paper we extend their results to a statement about braided fusion categories, and obtain similar results for certain subcategories of $SU(N)_k$.
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.
For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary fusion category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$ of $\mathcal{D}$ which are compatible with the enrichment of $\mathcal{C}$. We show that G-crossed extensions of a braided fusion category …
In this paper we classify all semisimple tensor categories with the same fusion rules as $\operatorname{Rep}(SO(4))$, or one of the associated truncations. We show that such categories are explicitly classified by two non-zero complex numbers. Furthermore we show these tensor categories are always braided, and there exist exactly 8 braidings.
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…
The Reduced Tensor Product of Braided Tensor Categories Containing a Symmetric Fusion Category
本研究针对含固定对称融合子范畴的辫子张量范畴，借助富集程序构造约化张量积对称幺模结构，并建立其与Drinfeld中心交叉辫子张量范畴2-范畴的等价性，还给出其在极小模扩张配对中的应用。
On normal tensor functors and coset decompositions for fusion categories
我们引入了相对于融合范畴的两个融合子范畴的双陪集的概念。给定融合范畴之间的张量函子$F : \C \to \D$，在$\C$的简单对象的同构类的集合$Λ_\C$上引入等价关系$\approx^F$，当$F$占优时，在$Λ_\D$上引入等价关系$\approx_F$.证明了$\approx^F$的等价类是陪集.给出了$F$为正规张量函子时的象的描述，证明了$F$正规的充要条件是$Λ_\C$的$\approx^F$等价元的象共线.研究了两个张量函子$F=F'F"$的合成正规的情形，给出了$F"$正规的一个判别准则，并应用于等变分.最后，我们引入了fusion子范畴的根，并将它与正规子范畴的换位子作了比较。我们也给出了一个正常的张量函子之间的任何两个融合类别的图像的描述。
On some integral properties of dimensions in Isaacs fusion categories
本文证明了融合范畴中生成Isaacs融合子范畴的简单对象维数的整性性质，并针对模融合范畴得到更强整除性结果，从而蕴含Ito-Michler型结果的逆方向。
We describe how the study of superfusion categories (roughly speaking, fusion categories enriched over the category of super vector spaces) reduces to that of fusion categories over sVect, in the sense of Drinfeld, Gelaki, Nikshych, and Ostrik. Following Brundan and Ellis, we give the construction of the underlying fusion category of a superfusion category, and give an explicit formula for the associator in this category in terms of 6$j$-symbols. We give a definition of the $π$-Grothendieck ring of a superfusion category, and prove a version of Ocneanu rigidity for superfusion categories.
Clifford theory for graded fusion categories
我们对分次融合范畴上的强分次环发展了Clifford理论的范畴模拟。我们描述了模块类别的融合类别分级的一组$G$诱导模块类别的融合子类别与子组的$G$。我们定义了不变的$\C_e$ -模范畴和$\C_e$ -模范畴的扩张.通过构造$\C$上的模范畴来确定$G$上子群的不变模范畴以及这些模范畴的不可分解扩张.我们将一个$G$ -交叉积融合范畴与每个$G$ -不变$\C_e$ -模范畴相关联，并给出了一个分级融合范畴是群论融合范畴的判据.给出了不可分解模范畴可扩张的充要条件。
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.
On the Dualizability of Fusion 2-Categories
本文在任意域上构造了可分模2-范畴的相对2-Deligne张量积，证明了可分紧半单张量2-范畴在Morita 4-范畴中完全可对偶，并推广到特征零域，给出场论解释。
We describe equivalence classes of exact indecomposable module categories over a finite graded tensor category. When applied to a pointed fusion category, our results coincide with the ones obtained in [S. Natale, On the equivalence of module categories over a group-theoretical fusion category, SIGMA, Symmetry Integrability Geom. Methods Appl. 13 Paper 042, 9 p. (2017)].
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
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.
On bicrossed product of fusion categories and exact factorizations
本文推广群的匹配对概念至融合环与融合范畴，定义其双交叉积并构造精确分解，证明融合环精确分解可表为双交叉积，刻画融合范畴精确分解的伴随子范畴与泛分次群，还给出新例子的融合规则及结合约束。
Group-theoretical fusion categories are defined by data concerning finite groups and their cohomology: A finite group $G$ endowed with a three-cocycle $ω$, and a subgroup $H\subset G$ endowed with a two-cochain whose coboundary is the restriction of $ω$. The objects of the category are $G$-graded vector spaces with suitably twisted $H$-actions; the associativity of tensor products is controlled by $ω$. Simple objects are parametrized in terms of projective representations of finite groups, namely of the stabilizers in $H$ of right $H$-cosets in $G$, with respect to two-cocycles defined by the …
Faithful simple objects, orders and gradings of fusion categories
我们建立了融合范畴中简单对象的阶与泛分次群的结构之间的关系。我们考虑了具有忠实单对象的融合范畴，并证明了它的泛分次群必是循环的。反之，我们证明了具有循环泛分次群的辫子幂零融合范畴总是有忠实的单对象。研究了融合范畴的泛分次问题。作为一个应用，我们在这个类中的模块范畴进行分类，并描述了辫子Tambara-Yamagami融合范畴的模块化。
On Z/2Z-extensions of pointed fusion categories
给出了0-分支为点融合范畴的Z/2 Z-分次融合范畴的一个分类。一些具体的例子被认为是。
Frobenius-Perron Dimensions of Conjugacy Classes and an Ito-Michler-Type Result in Modular Fusion Categories
本文研究有限群共轭类大小的算术条件在融合范畴中的类比，并为模融合范畴建立了Ito-Michler型结果。
Morita equivalence methods in classification of fusion categories
我们描述了一种方法来分类的融合范畴的森田等价。这通常是通过分析Drinfeld中心的融合范畴，并找到其中的Tannakian子范畴。
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.
We give a nontrivial lower bound for global dimension of a spherical fusion category.
Generalized Tambara-Yamagami categories
融合规则通过允许多值乘法来推广群。组是简单电流指数为1的融合规则。我们分类幂零（在Gelaki和Nikshych的意义上）简单的当前指数2的融合规则，并表征相关的融合类别。
Nilpotent fusion categories
本文将有限幂零群的概念范畴地推广到融合范畴。为此，我们首先分析了一个融合范畴C的泛分次的平凡分量，然后引入了C的上中心序列。对于具有交换Grothendieck环的融合范畴（例如，编织融合分类），我们还引入了下中心序列。我们研究幂零融合范畴的算术和结构性质，并将我们的理论应用到模范畴和半单霍普夫代数。特别地，我们证明了在模的情况下，两个中心列在M的意义下是彼此的中心化子。凶手。
Frobenius-Schur indicators for a class of fusion categories
本文给出了群论拟霍普夫代数的一个明确的描述，直到规范等价。我们使用这种描述来计算Frobenius-Schur指标群理论融合类别。
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.
Pivotal fusion categories of rank 3 (with an Appendix written jointly with Dmitri Nikshych)
我们分类的秩3，承认一个枢纽结构的代数闭域的特征为零的所有融合类别。在附录（与D.Nikshych合著）中，我们也给出了近群范畴的Grothendieck环的一些限制。
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 …
Ng and Schauenburg generalized higher Frobenius-Schur indicators to pivotal fusion categories and showed that these indicators may be computed utilizing the modular data of the Drinfel'd center of the given category. We consider two classes of fusion categories generated by a single non-invertible simple object: near groups, those fusion categories with one non-invertible simple object, and Haagerup-Izumi categories, those with one non-invertible simple object for every invertible object. Examples of both types arise as representations of finite or quantum groups or as Jones standard invariant…
In this paper, we study the structure of a generalized near-group fusion category and classified it when it is slightly degenerate.
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.
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