P. Bouwknegt, K. Hannabuss, V. Mathai
We define and systematically study nonassociative C*-algebras as C*-algebras internal to a topological tensor category. We also offer a concrete approach to these C*-algebras, as G-invariant, norm closed *-subalgebras of bounded operators on a G-Hilbert space, with deformed composition product. Our central results are those of stabilization and Takai duality for (twisted) crossed products in this context.
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A C*-tensor category with simple unit object is realized by von Neumann algebra bimodules of finite Jones index if and only if it is rigid.
Continuous Tambara-Yamagami tensor categories
该研究将Tambara-Yamagami张量范畴构造从有限阿贝尔群推广到局部紧阿贝尔群，给出其分类，并证明满足对应融合规则的$\mathrm{W}^*$-张量范畴的结合子自动连续。
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.
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。
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.
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。
A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories over a strongly $G$-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of $G$.
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)].
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。
The purpose of this article is to study the existence of Deligne's tensor product of abelian categories by comparing it with the well-known ten- sor product of finitely cocomplete categories. The main result states that the former exists precisely when the latter is an abelian category, and moreover in this case both tensor products coincide. An example of two abelian categories whose Deligne tensor product does not exist is given.
We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf algebra H gives rise to representations of the modular group SL(2,Z) on various morphism spaces. We also explain how every symmetric special Frobenius algebra in a semisimple modular category provides additional structure related to these representations.
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…
A graded tensor category over a group $G$ will be called a crossed product tensor category if every homogeneous component has at least one multiplicatively invertible object. Our main result is a description of the crossed product tensor categories, graded monoidal functors, monoidal natural transformations, and braiding in terms of coherent outer $G$-actions over tensor categories.
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.
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.
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…
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。
We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we…
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.
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.
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 develop a representation theory for $λ$-lattices, arising as standard invariants of subfactors, and for rigid C*-tensor categories, including a definition of their universal C*-algebra. We use this to give a systematic account of approximation and rigidity properties for subfactors and tensor categories, like (weak) amenability, the Haagerup property and property (T). We determine all unitary representations of the Temperley-Lieb-Jones $λ$-lattices and prove that they have the Haagerup property and the complete metric approximation property. We also present the first subfactors with propert…
We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need to require that the finite categories appearing in our construction are semisimple, nor that the finite tensor categories that are assigned to two-dimensional strata are endowed with a pivotal structure. Our prescription can be understood as a state-sum construction. The state-sum variables are assigned to one-dimensional strata and take values in bimodule …
Given a countably generated rigid C^*-tensor category C, we construct a planar algebra P whose category of projections Pro is equivalent to C. From P, we use methods of Guionnet-Jones-Shlyakhtenko-Walker to construct a rigid C^*-tensor category Bim whose objects are bifinite bimodules over an interpolated free group factor, and we show Bim is equivalent to Pro. We use these constructions to show C is equivalent to a category of bifinite bimodules over L(F_infty).
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。
We give a review of some recent developments in the theory of tensor categories. The topics include realizability of fusion rings, Ocneanu rigidity, module categories, weak Hopf algebras, Morita theory for tensor categories, lifting theory, categorical dimensions, Frobenius- Perron dimensions, classification of tensor categories.
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 introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld centre $\mathcal{Z}(\mathcal{A})$, the notion of $\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\mathcal{Z}(\mathcal{A})$. In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group $G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is shown to be equivalent to the 2-category of $G…
We generalize the notion of ends and coends in category theory to the realm of module categories over finite tensor categories. We call this new concept "module (co)end". This tool allows us to give different proofs to several known results in the theory of representations of finite tensor categories. As a new application, we present a description of the relative Serre functor for module categories in terms of a module coend, in a analogous way as a Morita invariant description of the Nakayama functor of abelian categories presented in [J. Fuchs, G. Schaumann and C. Schweigert, Eilenberg-Watts…
The Landau-Ginzburg/Conformal Field Theory correspondence predicts tensor equivalences between categories of matrix factorisations of certain polynomials and categories associated to the $N=2$ supersymmetric conformal field theories. We realise this correspondence for $x^d$ for any $d$, where previous results were limited to odd $d$. Our proof uses the fact that both sides of the correspondence carry the structure of module tensor categories over the category of $\mathbb{Z}_d$-graded vector spaces equipped with a non-degenerate braiding. This allows us to describe the CFT side as generated by …
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.
For a finite tensor category $\mathcal C$ and a Hopf monad $T:\mathcal C\to \mathcal C$ satisfying certain conditions we describe exact indecomposable left $\mathcal C^T$-module categories in terms of left $\mathcal C$-module categories and some extra data. We also give a 2-categorical interpretation of the process of equivariantization of module categories.
We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of a finite dimensional pointed Hopf algebras over an abelian group.
We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson …
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.
This is the eighth part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part VIII), we construct the braided tensor category structure, using the previously developed results.
The deformation cohomology of a tensor category controls deformations of its monoidal structure. Here we describe the deformation cohomology of tensor categories generated by one object (the so-called Schur-Weyl categories). Using this description we compute the deformation cohomology of free symmetric tensor categories generated by one object with an algebra of endomorphism free of zero-divisors. We compare the answers with the exterior invariants of the general linear Lie algebra.
We introduce Nakayama functors for coalgebras and investigate their basic properties. These functors are expressed by certain (co)ends as in the finite case discussed by Fuchs, Schaumann, and Schweigert. This observation allows us to define Nakayama functors for Frobenius tensor categories in an intrinsic way. As applications, we establish the categorical Radford $S^4$-formula for Frobenius tensor categories and obtain some related results. These are generalizations of works of Etingof, Nikshych, and Ostrik on finite tensor categories and some known facts on co-Frobenius Hopf algebras.
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…
Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasiz…
We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\cal B$ (consisting of invertible central $\cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categorie…
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。
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。
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.
This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining operator superalgebras and vertex tensor categories in the N=1 case.
In this paper, we compute the Clebsch-Gordan formulae and the Green rings of connected pointed tensor categories of finite type.
We introduce Manifold tensor categories, which make precise the notion of a tensor category with a manifold of simple objects. A basic example is the category of vector spaces graded by a Lie group. Unlike classic tensor category theory, our setup keeps track of the smooth (and topological) structure of the manifold of simple objects. We set down the necessary definitions for Manifold tensor categories and a generalisation we term Orbifold tensor categories. We also construct a number of examples, most notably two families of examples we call Interpolated Tambara-Yamagami categories and Interp…
In this paper we study the relative tensor product of module categories over braided fusion categories using, in part, the notion of the relative center of a module category. In particular we investigate the canonical tensor category structure and braiding inherited by the relative tensor product when module categories are themselves tensor/braided. As a basic example we show that the category of representations of the fibre product of finite groups may be expressed in terms of the relative product and show how to multiply braided fusion categories arising from pre-metric groups. Also, we cons…
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.
This is the second part in a series of papers presenting a theory of tensor products for module categories for a vertex operator algebra. In Part I (hep-th/9309076), the notions of $P(z)$- and $Q(z)$-tensor product of two modules for a vertex operator algebra were introduced and under a certain hypothesis, two constructions of a $Q(z)$-tensor product were given, using certain results stated without proof. In Part II, the proofs of those results are supplied.
We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.
We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these categories.
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 compute the group of braided tensor autoequivalences and the Brauer-Picard group of the representation category of the small quantum group $\mathfrak{u}_q(\mathfrak{g})$, where $q$ is a root of unity.
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.
Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k-representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over k. We say that a tensor category C over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. We prove that the following tensor categories are virtually indecomposable: Tensor categories with the Chevalley property; representation categories of affine group schemes; representation categories of formal groups; rep…
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…
Lectures on Symmetric Tensor Categories
本研究是两场对称张量范畴讲座讲义的扩展版，综述了该理论当前研究现状的若干方面，还介绍了现有文献未涵盖的相关应用。
Blatter-Montgomery duality theorem is generalized into braided tensor categories. It is shown that $Hom(V,W)$ is a braided Yetter-Drinfeld module for any two braided Yetter-Drinfeld modules $V$ and $W$.
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