Victor Ostrik
We classify semisimple module categories over the tensor category of representations of quantum SL(2) extending previous results to the roots of unity and positive characteristic cases.
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We give a 3-universal property for the Karoubi envelope of a 2-category. Using this, we show that the 3-categories of finite semisimple 2-categories (as introduced in arXiv:1812.11933) and of multifusion categories are equivalent.
Compact Semisimple 2-Categories
本文定义了任意域上的紧半单2-范畴，证明其等价于可分模1-范畴的2-范畴，验证了在特定域上的有限性，并将特征零代数闭域上的相关理论推广至该范畴。
Free products of semisimple tesnor categories are constructed with the help of polygonal presentation. The semisimplicity criterion is obtained for the Bisch-Jones' planar algebras as a byproduct.
Tannaka duals of Hopf algebras inside semisimple tensor categories are used to construct orbifold tensor categories, which are shown to include the Tannaka dual of the dual Hopf algebras. The second orbifolds are then canonically isomorphic to the initial tensor categories.
Finite Semisimple Module 2-Categories
该研究针对多融合2-范畴$\boldsymbol{\frak{C}}$上的有限半单模2-范畴，证明其具有典范$\boldsymbol{\frak{C}}$-富集结构，并进一步证明这类模2-范畴等价于$\boldsymbol{\frak{C}}$中刚性代数的模2-范畴。
Skein Categories in Non-semisimple Settings
本文提出了一种非半单设定下的 skein 分类，通过张量理想扩展了现有理论，并揭示了因子化同调与 skein 分类之间的关系。
Constructing non-semisimple modular categories with relative monoidal centers
本文研究非半单模范畴的构造问题，通过明确Müger中心化子的模性条件，推导相对幺半中心及两类量子群表示生成非半单模范畴的条件，拓展了该类范畴的构造路径。
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.
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。
We give a classification of semisimple and separable algebras in a multi-fusion category over an arbitrary field in analogy to Wedderben-Artin theorem in classical algebras. It turns out that, if the multi-fusion category admits a semisimple Drinfeld center, the only obstruction to the separability of a semisimple algebra arises from inseparable field extensions as in classical algebras. Among others, we show that a division algebra is separable if and only if it has a nonvanishing dimension.
Constructing non-semisimple modular categories with local modules
该研究定义非半单模范畴中的刚性Frobenius代数类，证明其局部模范畴仍为模范畴，推广半单情形的已有结论，给出相关示例与关联，并将群代数相关的分类结果推广到任意特征。
Given an action of a finite group G on a fusion category C we give a criterion for the category of G-equivariant objects in C to be group-theoretical, i.e., to be categorically Morita equivalent to a category of group-graded vector spaces. We use this criterion to answer affirmatively the question about existence of non group-theoretical semisimple Hopf algebras asked by P. Etingof, V. Ostrik, and the author in math/0203060. Namely, we show that certain Z/2Z-equivariantizations of fusion categories constructed by D. Tambara and S. Yamagami are equivalent to representation categories of non gro…
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive eleme…
Category theoretic aspects of non-rational conformal field theories are discussed. We consider the case that the category C of chiral sectors is a finite tensor category, i.e. a rigid monoidal category whose class of objects has certain finiteness properties. Besides the simple objects, the indecomposable projective objects of C are of particular interest. The fusion rules of C can be block-diagonalized. A conjectural connection between the block-diagonalization and modular transformations of characters of modules over vertex algebras is exemplified with the case of the (1,p) minimal models.
We show that every essentially small finitely semisimple k-linear additive spherical category in which k=End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as a Weak Hopf Algebra.
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 classify ribbon semisimple monoidal categories with three isomorphism classes of simple objects over the field of complex numbers.
Modular data of non-semisimple modular categories
本文研究非半单模范畴，通过可分解ribbon Hopf代数的表示扩展半单理论，聚焦Cohen-Westreich模数据及其$S$-矩阵对角化混合融合规则，并详细分析小量子群和Nichols Hopf代数实例。
Frobenius-Schur Indicators of Dual Fusion Categories and Semisimple Partially Dualized Quasi-Hopf Algebras
本文研究球面融合范畴的对偶范畴中对象的Frobenius-Schur指标，建立特定对象指标间的关系，并推导出半单Hopf代数及其部分对偶化quasi-Hopf代数的指标公式与指数等式。
Finite-dimensional quantum groups of type Super A and non-semisimple modular categories
本文研究了超A类型有限维量子群的构造及其非半单模范畴的分类，通过辫状德林费尔德双和非半单球形结构，提出了新的链接不变量并区分了某些结。
Semisimple module categories with fusion rules of the compact full flag manifold type
本文分类了A型量子群表示范畴上源自极大环面融合规则的半单左模范畴，揭示其与紧全旗流形上等变泊松结构的联系，并基于形变量子包络代数的BGG范畴给出显式构造。
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…
Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be k-linear for some field k, and every simple object V is strongly simple, meaning that (V) = k. We prove that linearity follows automatically from semisimplicity: Every connected, finite, semisimple, rigid, monoidal category \C is k-linear and finite-dimensional for some field k. Barring inseparable extensions, such a category becomes a multifusion category after passing to an algebraic extension of k. The proof depends on a result in Galois theo…
One of the most fundamental problems in the theory of finite- dimensional Hopf algebras is their classification over an algebraically closed field k of characteristic 0. This problem is extremely difficult, hence people restrict it to certain classes of Hopf algebras, e.g. to semisimple ones. Semisimple Hopf algebras deserve to be considered as "quantum" analogue of finite groups, but even so, the problem remains extremely hard (even in low dimensions) and very little is known. A great boost to the theory of Hopf algebras was given by Drinfeld who invented the fundamental class of (quasi)trian…
We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.
We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category C if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|^2 of any simple object V of C. We show, using the Ito-Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categor…
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…
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.
We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of finite-dimensional pointed Hopf algebras over S_3, and of finite-dimensional Nichols algebras over S_4. The revised version contains an extended introduction with references to recent applications, and a simplified definition of the Weyl groupoid of a semisimple Yetter-Drinfeld module. Key words: Hopf algebras, quantum groups, Weyl groupoid
We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realize…
Let $U_l$ be a quantum group at an $l$th root of unity. Many tilting modules for $U_l$ have been shown to have what we call a balanced semisimple filtration, or a Loewy series whose semisimple layers are symmetric about some middle layer. The existence of such filtrations suggests a remarkably straightforward algorithm for calculating these characters if the irreducible characters are already known. We first show that the results of this algorithm agree with Soergel's character formula for the regular tilting modules. We then show that these balanced semisimple filtrations really do exist for …
Twisted Deligne products of semisimple tensor categories
探讨半单张量范畴$\mathcal C$、$\mathcal D$的扭曲德利涅积分类，给出特定条件下唯一扭曲德利涅积是普通积的结论，借助相关工作实现群论分类及融合范畴精确分解，附录引入范畴$n$-上链概念并回答相关问题。
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.
The 2-Deligne Tensor Product
本文证明了两个紧半单2-范畴的2-Deligne张量积的存在性，并在适当假设下描述了其$Hom$-范畴、连通分支和单对象，同时证明了该张量积仍是紧半单张量2-范畴。
We endow a non-semisimple category of modules of unrolled quantum sl(2) with a Hermitian structure. We also prove that the TQFT constructed in arXiv:1202.3553 using this category is Hermitian. This gives rise to projective representations of the mapping class group in the group of indefinite unitary matrices.
We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.
We develop abstract nonsense for module categories over monoidal categories (this is a straightforward categorification of modules over rings). As applications we show that any semisimple monoidal category with finitely many simple objects is equivalent to the category of representations of a weak Hopf algebra (theorem of T. Hayashi) and classify module categories over the fusion category of $\hat{sl}(2)$ at a positive integer level where we meet once again ADE classification pattern.
A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the dual of a pointed semisimple category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs (G, w), where G is a finite group and w is a 3-cocycle on G. A gr…
A non-semisimple Witt class
本文提出了一种非半单Witt类的辫状有限张量范畴，展示了其与半单范畴的区别。
Let $k$ be a field, and $H$ a Hopf algebra with bijective antipode. If $H$ is commutative, noetherian, semisimple and cosemisimple, then the category ${}_{H}{\mathcal {YD}}^H$ of Yetter-Drinfeld modules is semisimple. We also prove a similar statement for the category of Long dimodules, without the assumption that $H$ is commutative.
Non compact (2+1)-TQFTs from non-semisimple spherical categories
该研究提出曲面容许skein模与色范畴概念，构造了有限维非紧(2+1)-TQFT，证明其可延拓为真正TQFT的充要条件，并验证球面张量范畴均为色范畴，拓展了(2+1)-TQFT的构造范围。
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.
We point out that results of Shimizu on internal characters imply a useful non-semisimple variant of the categorical Verlinde formula for factorisable finite tensor categories. When combined with results on pseudo-trace functions by Miyamoto and Arike-Nagatomo, one can make a precise conjecture for a non-semisimple modular Verlinde formula which relates modular properties of pseudo-trace functions and the product in the Grothendieck ring of the corresponding vertex operator algebra.
For each finite semisimple tensor category, we associate a quantum group (face algebra) whose comodule category is equivalent to the original one, in a simple natural manner. To do this, we also give a generalization of the Tannaka-Krein duality, which assigns a face algebra for each tensor category equipped with an embedding into a certain kind of bimodule category.
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
Skein (3+1)-TQFTs from Non-Semisimple Ribbon Categories
本文基于绳结理论，定义了非半单有限单模绳结张量范畴的(3+1)维拓扑量子场论，提出'色非退化'条件，并推广了Crane-Yetter-Kauffman和Lyubashenko等不变量。
In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category. The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories. After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about represent…
We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence we classify exact module categories over some families of pointed tensor categories with cyclic group of invertible objets of order p, where p is a prime number.
Polynomial families of quantum semisimple coadjoint orbits via deformed quantum enveloping algebras
本研究利用形变量子包络代数构造Drinfeld-Jimbo形变表示范畴上的半单左模范畴多项式族，建立与中间Levi子代数诱导模范畴的比较定理，相关内容在Lusztig整性框架下讨论。
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.
Positivstellensatz for C*-tensor categories
本文在刚性半单C*-张量范畴中探讨半预C*-代数，并利用环表示技术将Ozawa的群性质(T)判据推广至该范畴。
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。
Let λbe a partition of a positive integer n. Let C be a symmetric rigid tensor category over a field k of characteristic 0 or char(k)>n, and let V be an object of C. In our main result (Theorem 4.3) we introduce a finite set of integers F(λ) and prove that if the Schur functor \mathbb{S}_λ V of V is semisimple and the dimension of V is not in F(λ), then V is semisimple. Moreover, we prove that for each d in F(λ) there exist a symmetric rigid tensor category C over k and a non-semisimple object V in C of dimension d such that \mathbb{S}_λ V is semisimple (which shows that our result is the best…
We classify module categories over the category of representations of quantum $SL(2)$ in a case when $q$ is not a root of unity. In a case when $q$ is a root of unity we classify module categories over the semisimple subquotient of the same category.
We continue our study of the McKay Correspondence for grading preserving actions of semisimple Hopf algebras H on (noncommutative) Artin-Schelter regular algebras A. Here, we establish correspondences between module categories over A^H, over A\#H, and over \End_{A^H} A. We also study homological properties of (endomorphism rings of) maximal Cohen-Macaulay modules over A^H.
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 …
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 use a version of Haboush's theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic $p$ to characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover, any isomorphism between such structures can be reduced modulo $p$. This fills a gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer-Pic…
In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular category, the twists are roots of unity dividing the algebraic integer D^{5/2}, where D is the global dimension of the category (the sum of squares of dimensions of simple objects). Both results generalize Vafa's theorem, saying that in a modular category twists are roots of unity, and square of the braiding has finite order. We also discuss the notion of the qua…
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their associativity and commutativity morphisms.
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