Representations of tensor categories coming from quantum linear spaces
Martin Mombelli
对量子线性空间提升的霍普夫代数表示的张量范畴上的精确不可分解模范畴进行了分类。
Exact indecomposable module categories over the tensor category of representations of Hopf algebras that are liftings of quantum linear spaces are classified.
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We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
On the exponent of tensor categories coming from finite groups
我们用群上同调描述了与有限群$G$相关的群论融合范畴$\mathcal C = \mathcal C(G, ω, F, α)$的指数。我们证明了$\C$的指数可以整除$e(ω) \exp G$和$(\exp G)^2$，其中$e(ω)$是3-上循环$ω$的上同调序.特别地，$\exp \C$分裂$(\dim \C)^2$。
This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions which preserve the ascending filtration on tensor algebras, and the second for actions which preserve the descending filtration on …
An Example of Double Cross Coproducts with Non-trivial Left Coaction and Right Coaction in Strictly Braided Tensor Categories
给出了严格辫张量范畴中同时具有非平凡左余作用和非平凡右余作用的双交叉余积的一个例子。
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 …
Composition algebras in symmetric tensor categories
本文定义对称张量范畴中的合成代数，综述相关经典结果与超代数分类，还完成了特征2下对应向量超空间范畴的$Ver_4^+$中单位合成代数的分类。
Lectures on tensor categories
本文综述了张量范畴理论的一些最新进展。题目包括融合环的可实现性，Ocneanu刚性，模范畴，弱霍普夫代数，张量范畴的Morita理论，提升理论，范畴维数，Frobenius- Perron维数，张量范畴的分类。
We extend categorical Morita equivalence to finite tensor categories graded by a finite group $G$. We show that two such categories are graded Morita equivalent if and only if their equivariant Drinfeld centers are equivalent as braided $G$-crossed tensor categories.
Module categories over equivariantized tensor categories
对于满足一定条件的有限张量范畴$\mathcal C$和Hopf单子$T:\mathcal C\to \mathcal C$，我们用左$\mathcal C$ -模范畴和一些额外的数据描述了精确不可分解左$\mathcal C^T$ -模范畴.我们还给出了模范畴的等变化过程的2-范畴解释。
We establish a set of general results to study how the Galois action on modular tensor categories interacts with fusion subcategories. This includes a characterization of fusion subcategories of modular tensor categories which are closed under the Galois action, and a classifcation of modular tensor categories which factor as a product of pointed and transitive categories in terms of pseudoinvertible objects. As an application, we classify modular tensor categories with two Galois orbits of simple objects and a nontrivial grading group.
We present an overview of the notions of exact sequences of Hopf algebras and tensor categories and their connections. We also present some examples illustrating their main features; these include simple fusion categories and a natural question regarding composition series of finite tensor categories.
On odd-dimensional modular tensor categories
本文研究奇数维及低秩MNSD模张量范畴，利用伴随子范畴秩与可逆对象群阶建立秩下界，并据此分类若干奇秩范畴为点状或完美。
Virtually indecomposable tensor categories
设k为任意域。J-P. Serre证明了群的k-表示范畴的Grothendieck环的谱是连通的，并且对于k上的李代数的表示范畴的特征零也是如此。我们说，一个张量范畴C在k上是几乎不可分解的，如果它的Grothendieck环不包含非平凡的中心幂等元。我们证明了以下张量范畴是实际上不可分解的：张量范畴与Chevalley性质;仿射群计划的表示范畴;表示范畴的正式组;表示范畴的仿射超群计划（在特征\ne 2）;表示范畴的正式超群（在特征\ne 2）;对称张量范畴的指数增长性在特征零。特别地，我们得到了Serre定理的另一种证明，并由此推论出任意k上李代数的表示范畴在正特征上也是虚不可分解的（这回答了塞尔的一个问题），（在超情形中使用Deligne定理，和Deligne-Milne的一个定理）导出了任何（超）Tannakian范畴是几乎不可分解的（这回答了塞尔的另一个问题）。
Tensor categories and vacant double groupoids
我们证明了弱霍普夫代数的表示的融合范畴$\Rep(\ku^σ_τ \Tc)$是群论的，这些表示来自于一个空的双广群$\Tc$和一对相容的2-cocyles的$(σ, τ)$.
Relative centers and tensor products of tensor and braided fusion categories
本文部分地利用模范畴的相对中心的概念，研究了辫子融合范畴上模范畴的相对张量积。特别是，我们调查的规范张量范畴结构和编织继承的相对张量积时，模块范畴本身是张量/编织。作为一个基本的例子，我们表明，类别的代表性的纤维产品的有限群可以表示在相对产品，并显示如何倍增编织融合类别所产生的预度量群。此外，我们考虑相对中心的编织融合范畴，并期待在Muger中心的相对张量积。最后，我们用一个深入的例子暗示了有限群共轭类的分类。
We classify finite pointed braided tensor categories admitting a fiber functor in terms of bilinear forms on symmetric Yetter-Drinfeld modules over abelian groups. We describe the groupoid formed by braided equivalences of such categories in terms of certain metric data, generalizing the well-known result of Joyal and Street for fusion categories. We study symmetric centers and ribbon structures of pointed braided tensor categories and examine their Drinfeld centers.
We extend \cite{G} to the nonsemisimple case. We define and study exact factorizations $\B=\A\bullet \C$ of a finite tensor category $\B$ into a product of two tensor subcategories $\A,\C\subset \B$, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories \cite{EG}. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category \cite{G2} by another one. We also provide several examples to illustrate our results.
For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Additionally, we prove that the obstruction to the existence of a trivialization of a categorical group action $T$ on a monoidal category $\mathcal{C}$ is given by an element $O(T)\in H^2(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$. In the case that $O(T)=0$, the set of obstructions form a torsor over $\operatorname{Hom}(G,\operatorname{Aut}_\otimes(\operatorname{Id}_{\mathcal{C}}))$, where $\o…
Duality Theorem and Hom Functor in Braided Tensor Categories
将Blatter-Montgomery对偶定理推广到辫张量范畴。证明了对于任意两个辫子Yetter-Drinfeld模$V$和$W$，$Hom(V,W)$是辫子Yetter-Drinfeld模.
The Green rings of pointed tensor categories of finite type
本文计算了有限型连通点张量范畴的Clebsch-Gordan公式和Green环。
Tannaka Duals in Semisimple Tensor Categories
利用半单张量范畴内霍普夫代数的Tannaka对偶构造了轨道张量范畴，并证明了轨道张量范畴包含对偶霍普夫代数的Tannaka对偶.第二个orbifolds然后规范同构的初始张量范畴。
Module categories over finite pointed tensor categories
研究了有限维拟霍普夫代数的表示范畴上的正合模范畴。作为结果，我们对具有p阶可逆对象的循环群的点张量范畴族上的正合模范畴进行了分类，其中p是素数。
Finite Pre-Tensor Categories that are Morita Equivalent to Finite Tensor Categories
本文研究有限预张量范畴与有限张量范畴的莫里塔等价问题，证明预张量范畴的莫里塔等价当且仅当其Drinfeld中心为有限张量范畴。
Cohomology of Pointed Finite Tensor Categories
本文研究了指有限张量范畴的上同调代数的有限生成性质，证明了所有在交换群上有限的指有限张量范畴具有有限生成的上同调。
Coring categories and Villamayor-Zelinsky sequence for symmetric finite tensor categories
在前面的文章中，我们构造了一个对称有限张量范畴的无穷正合列。它由三种类型的系数，定期重复评估的上同调群。本文解释了序列第二层的中上同调群。引入了核范畴的概念，得到了中上同调群同构于Azumaya拟核范畴群。这个结果是经典的交叉积定理的范畴推广，它将相对Brauer群和第二Galois上同调群关于Galois域扩张联系起来.构造了$vec$上Azumaya拟核范畴的相对群和Azumaya拟核范畴的完全群在对称有限张量范畴上的上极限.我们证明了后两个群是同构的。
Exact sequences of tensor categories with respect to a module category
推广了Bruguières和Natale关于张量范畴的正合序列的定义，并引入了关于模范畴的（有限）张量范畴的正合序列的新概念.我们给出了这个概念的三个定义，并证明了它们的等价性。特别地，张量范畴的Deligne张量积产生我们意义上的正合序列。我们还证明了正合序列的对偶在我们的意义下也是正合序列。这推广了霍普夫代数正合列的相应结论。最后，我们证明了正合列的中间项是半单的，如果其他两项也是半单的。
Clifford theory for tensor categories
群$G$上的分次张量范畴称为强$G$ -分次张量范畴，如果每个齐次分支至少有一个乘可逆对象.我们的主要结果是一个强$G$ -分次张量范畴上的模范畴的描述诱导的张量子范畴与$G$的子群。
On certain integral tensor categories and integral TQFTs
我们构造了某些张量范畴，这些张量范畴被许多简单对象所支配。这些范畴中的对象是代数整数环上的模。我们展示了如何从这些类别中获得定义在代数整数上的TQFT。
Reductions of tensor categories modulo primes
我们学习得很好（即，半单刚性张量范畴模素数的半单约化。一个素数p对于一个半单刚性张量范畴C来说称为好的，如果存在这样的约化（否则称为坏的）。很明显，一个好的素数必须与C的任何简单对象V的Müger平方范数|V| ^2互素。我们表明，使用伊藤-米歇尔定理在有限群理论，群论融合范畴，反之亦然。虽然对于一般的融合类别，逆是假的，但我们得到了许多已知融合类别的好素数和坏素数的结果（例如，用于Verlinde类别）。我们还提出了一些关于好素数和坏素数的问题和解答。
On the tensor product of bimodule categories over Hopf algebras
设H是有限维霍普夫代数。给出了Rep（H）上双模范畴的张量积的一个刻划.当双模范畴是可逆的时，这个描述可以显式地给出。在H是点霍普夫代数的情形下，我们给出了这种描述的一些结果.
We give a short proof for a well-known formula for the rank of a $G$-crossed braided extension of a modular tensor category.
Chromatic maps for finite tensor categories
该研究针对有限张量范畴引入左、右色映射概念，证明其恒存在，并推导出任一球面有限张量范畴都有色映射，为3-流形不变量及(2+1)-TQFT构建提供理论支撑。
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 use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko's mapping class group representations, as discussed in arXiv:2010.14852. This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of…
We show that finitely generated cohomology is invariant under separable equivalences for all algebras. As a result, we obtain a proof of the finite generation of cohomology for finite symmetric tensor categories in characteristic zero, as conjectured by Etingof and Ostrik. Moreover, for such categories we also determine the representation dimension and the Rouquier dimension of the stable category. Finally, we recover a number of results on the cohomology of stably equivalent and singularly equivalent algebras.
Tensor categories
本文给出幺半范畴与张量范畴的基本定义和构造，并引用原始文献，以群论范畴为例说明。
Tensor products of finitely cocomplete and abelian categories
本文的目的是通过比较阿贝尔范畴的Deligne张量积与著名的余完备范畴的Deligne张量积，研究Deligne张量积的存在性.主要结果表明，当后者是一个阿贝尔范畴时，前者恰好存在，而且在这种情况下，两个张量积重合。给出了Deligne张量积不存在的两个阿贝尔范畴的一个例子。
We study exact sequences of finite tensor categories of the form $\Rep G \to \C \to \D$, where $G$ is a finite group. We show that, under suitable assumptions, there exists a group $Γ$ and mutual actions by permutations $\rhd: Γ\times G \to G$ and $\lhd: Γ\times G \to Γ$ that make $(G, Γ)$ into matched pair of groups endowed with a natural crossed action on $\D$ such that $\C$ is equivalent to a certain associated crossed extension $\D^{(G, Γ)}$ of $\D$. Dually, we show that an exact sequence of finite tensor categories $\vect_G \to \C \to \D$ induces an $\Aut(G)$-grading on $\C$ whose neutral…
We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. In particular, we produc…
Crossed product tensor categories
群$G$上的分次张量范畴称为交叉积张量范畴，如果每个齐次分支至少有一个乘可逆对象。我们的主要结果是描述的交叉产品张量范畴，分次monoidal函子，monoidal自然变换，编织在连贯的外部$G$ -行动张量范畴。
We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted to a tensor equivalence under some certain conditions.
Deformations of (Bi) tensor Categories
我们定义了代数和双代数范畴的上同调和形式变形理论。我们提出了一些方法来寻找与Lusztig的工作相关的量子群的范畴的非平凡变形。
Free Products of Semisimple Tensor Categories
利用多边形表示构造了半单tesnor范畴的自由积。作为副产品，得到了Bisch-Jones平面代数的半单性准则.
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。
Actions of Tensor Categories and Cylinder Braids
范畴行动的辫子张量范畴的定义和示出的范畴结构的讨论有关的组在圆柱体中的辫子是正确的框架。构造了立体环面中链环的Kauffman多项式。
Central Exact Sequences of Tensor Categories, Equivariantization and Applications
定义了张量范畴在张量群概型作用下的等变分，给出了张量范畴的正合列在有限群或有限群概型作用下等变分的充要条件.引入张量范畴的中心正合列的概念，并利用它给出了融合范畴等变分的一些已知刻画的另一种表述，并将这些刻画推广到有限张量范畴在有限群概型作用下的等变分.特别地，我们得到了有限阿贝尔群作用下等变分的一个简单刻画。作为一个应用，我们证明了：如果$\C$是一个融合范畴，且$F: \C \to \D$是Frobenius-Perron指标$p$的一个支配张量函子，则$F$是一个等变分化，如果$p=2$，或如果$\C$是弱整数，且$p$是$\FPdim \C$的最小素因子.
Descent and forms of tensor categories
本文发展了任意域上张量范畴的下降理论和形式。我们描述的一般计划，使用代数和同伦语言的分类形式，并给出明确的分类形式的例子。我们还讨论了弱融合环的范畴化问题，并对最简单的此类环族，确定哪些是可范畴化的。
In this paper we follow the constructions of Turaev's book [Tu] closely, but with small modifications, to construct of a modular functor, in the sense of Kevin Walker, from any modular tensor category. We further show that this modular functor has duality and if the modular tensor category category is Hermitian or unitary, then the resulting modular functor is also Hermitian or unitary respectively.
Given a tensor functor between tensor categories $\mathcal{C}$ and $\mathcal{D}$, we give criteria that, under certain assumptions, the Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of the other one. We also give an affirmative answer to Natale's question asking if the class of Frobenius tensor categories is closed under exact sequences.
Categorified trace for module tensor categories over braided tensor categories
给定一个辫子枢轴范畴$\mathcal C$和一个枢轴模张量范畴$\mathcal M$，我们定义一个函子$\mathrm{Tr}_{\mathcal C}:\mathcal M \to \mathcal C$，称为关联范畴化迹。根据Bezrukavnikov，Finkelberg和Ostrik的结果，函子$\mathrm{Tr}_{\mathcal C}$配备了自然同构$τ_{x,y}:\mathrm{Tr}_{\mathcal C}(x \otimes y) \to \mathrm{Tr}_{\mathcal C}(y \otimes x)$，我们称之为traciator。这种情况使其本身成为“圆柱上的弦”的数学演算，其中曳迹符对应于将弦缠绕在圆柱的背面。我们表明，$\mathrm{Tr}_{\mathcal C}$实际上有一个更丰富的图形演算中的管被允许分支和编织。给定代数对象$A$和$B$，证明了$\mathrm{Tr}_{\mathcal C}(A)$和$\mathrm{Tr}_{\mathcal C}(A \otimes B)$也是代数对象.而且，只要满足某些温和的假设，只要$A$和$B$是半单的，$\mathrm{Tr}_{\mathcal C}(A)$和$\mathrm{Tr}_{\mathcal C}(A \otimes B)$就是半单的。
Constructing Modular Tensor Categories with Hopf Algebras
本文通过研究Hopf代数的中心扩张、商代数可因子化条件及ribbon元构造方法，得到两类非半单和一族半单的ribbon可因子化Hopf代数，证明其表示范畴为素模张量范畴且无法通过常规张量积构造，为模张量范畴的构建提供了新途径。
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.
We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also non-semisimple examples such as categories of representations of the small quantum group at good roots of unity. Via the cobordism hypothesis, we obtain new invertible 4-dimensional framed topological field theories, which we regard as a non-semisimple framed version of the Crane-Yetter-Kauffman invariants, after Freed--Teleman and Walker's construction in the s…
Crossed actions of matched pairs of groups on tensor categories
在张量范畴上引入了$(G, Γ)$ -交叉作用的概念，其中$(G, Γ)$是有限群的匹配对.一个张量范畴被称为$(G, Γ)$ -交叉张量范畴，如果它被赋予一个$(G, Γ)$ -交叉作用.我们证明了每个$(G, Γ)$交叉的张量范畴$\mathcal C$产生一个适合于张量范畴$\operatorname{Rep G} \to \mathcal C^{(G, Γ)} \to \mathcal C$的精确序列的张量范畴$\mathcal C^{(G, Γ)}$。我们还定义了一个$(G, Γ)$ -编织的概念，在一个$(G, Γ)$ -交叉张量范畴，这是连接到一定的QYBE集理论的解决方案。这推广了Turaev提出的$G$ -交叉辫张量范畴的概念。证明了如果$\mathcal C$是一个具有$(G, Γ)$ -辫子的$(G, Γ)$ -交叉张量范畴，则张量范畴$\mathcal C^{(G, Γ)}$是一个标准形式的辫子张量范畴.
In this short note we investigate the process of constructing auto-equivalences of modular tensor categories using invertible objects. We derive conditions on the invertible object for the resulting auto-equivalence to be either monoidal, braided, or pivotal. We also discuss the composition of these auto-equivalences constructed from invertible objects. To demonstrate the practicality of this construction, we construct auto-equivalences of several real-world examples of modular tensor categories.
Non-cyclotomic fusion categories
Etingof、Nikshych和Ostrik在arXiv：math.QA/0203060中提出了一个问题，即每个融合类别是否都可以在一个分圆域上完全定义。我们发现，这是不是这样的：特别是一个融合类别来自Haagerup子因子arXiv：math.OA/9803044和一个来自新构建的扩展Haagerup子因子arXiv：0909.4099不能完全定义在一个分圆域。另一方面，我们证明了Haagerup子因子的偶部分的二重化在分圆域上是完全定义的。我们确定这些融合类别的定义的最小字段，计算的伽罗瓦群，并确定其伽罗瓦共轭。
Making non-trivially associated tensor categories from left coset representatives
本文首先给出群上子群左作用的陪集表示集的一个代数结构。由此我们构造了一个非平凡关联的张量范畴。还给出了一个双重构造，这使得可以构造一个非平凡关联的辫状张量范畴。在这个范畴中，我们明确地重建了一个辫子霍普夫代数，其表示构成了范畴本身。
On the unitarity and modularity of ribbon tensor categories associated with affine Lie algebras
本文通过数值计算和两个猜想，确定了与仿射李代数相关的带状张量范畴在哪些 $q$ 值下为（伪）酉和模的，并与文献结果完全一致，补充了缺失情况。
Center construction for group-crossed tensor categories
本文定义了Natale意义下群交叉张量范畴的交叉中心，证明其构成辫子张量范畴，并推广了分次中心与等变中心构造。
From Hopf algebras to tensor categories
本文是关于球面霍普夫代数的一个综述。给出了判定霍普夫代数是球面代数的准则，并收集了实例。我们讨论了倾斜模作为一种手段，以获得一个合适的霍普夫代数的表示范畴的非退化商的融合子范畴。
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