Phung Ho Hai
A quantum groups of type $A$ is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke symmetry.
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We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax monoidal functors is investigated. We introduce tensor product of representations of bimonoids as a functorial binary operation and show how symmetric lax monoidal functors act on this product. Finally we apply the general theory to classical and quantum representations.
The loop braid group is the motion group of unknotted oriented circles in $\mathbb{R}^3$. In this paper, we study their representations through the approach inspired by two dimensional topological phases of matter. In principle, the motion of loops in $\mathbb{R}^3$ reduces to the motions of points in a two dimensional sliced plane. We realize this physical picture in terms of braided tensor categories and their braid group representations.
We study the construction of premonoidal categories, where the pentagon relation fails, through representations of finite group algebras and their quantum doubles. Both finite group algebras and their quantum doubles have a finite number of irreducible representations. We show that in each case there are at least $2^{n-1}$ inequivalent premonoidal categories of representations, where $n$ is the number of irreducible representations. By construction, for the case of finite group algebras the categories are symmetric whereas for the quantum doubles the categories are braided.
We provide a description of the annular representation category of the free product of two rigid C*-tensor categories.
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.
Quandles with isomorphic representation categories
该研究构造了两类不同构有限拟群的例子，其泛包络群同构，进而二者的加法与张量表示范畴也同构，揭示了拟群与其表示范畴的特殊关系。
The category-valued trace assigns to a bimodule category over a linear monoidal category a linear category. It generalizes Drinfeld centers of monoidal categories and the relative Deligne product of bimodule categories. In this article, we study bimodule categories that are given as categories of bicomodules over a Hopf algebra. Our main result is a representation-theoretic realization of the category-valued trace as a category of generalized Hopf bimodules.
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…
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 describe all fusion subcategories of the representation category of a twisted quantum double of a finite group. In view of the fact that every group-theoretical braided fusion category can be embedded into a representation category of a twisted quantum double of a finite group, this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of representation category of a twisted quantum double of a finite group. We also give a characterization of group-theoretical braided fusion …
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
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.
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…
On rank one 2-representations of web categories
本文研究了SL2、GL2和SO3Web范畴的秩一2-表示分类，借鉴量子群类似结果，通过缩减问题到双线性与三线性形式分类，提出可推广至其他Web范畴的框架。
We develop further the techniques presented in [M. Mombelli. On the tensor product of bimodule categories over Hopf algebras. Preprint arXiv:1111.1610 ] to study bimodule categories over the representation categories of arbitrary finite-dimensional Hopf algebras. We compute the Brauer-Picard group of equivalence classes of exact invertible bimodule categories over the representation categories of a certain large family of pointed non-semisimple Hopf algebras, the so called supergroup algebras. To obtain this result we first give a classification of equivalence classes of exact indecomposable b…
We show that once-extended anomalous 3-dimensional topological quantum field theories valued in the 2-category of k-linear categories are in canonical bijection with modular tensor categories equipped with a square root of the global dimension in each factor.
A strict 2-group is a 2-category with one object in which all morphisms and all 2-morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the present article, we develop the notions of trialgebra and cotrialgebra, generalizations of Hopf algebras with two multiplications and one comultiplication or vice versa, and the notion of Hopf categories, generalizations of monoidal categories with an additional functorial comultiplication. We show that each strict 2-group has a `group algebra' which is a c…
Exact indecomposable module categories over the tensor category of representations of Hopf algebras that are liftings of quantum linear spaces are classified.
This is an expository article invited for the ``Commentary'' section of PNAS in connection with Y.-Z. Huang's article, ``Vertex operator algebras, the Verlinde conjecture, and modular tensor categories,'' appearing in the same issue of PNAS. Huang's solution of the mathematical problem of constructing modular tensor categories from the representation theory of vertex operator algebras is very briefly discussed, along with background material. The hypotheses of the theorems entering into the solution are very general, natural and purely algebraic, and have been verified in a wide range of famil…
Actions of monoidal categories and representations of Cartan type Lie algebras
本文通过交叉同态建立幺半范畴作用，统一构造Lie-Rinehart代数及Leibniz对的弱表示与容许表示，并引入交叉同态上同调理论以研究其形变，同时为广义Witt代数等提供新表示。
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 exhibit a correspondence between subcategories of modules over an algebra and sub-bimodules of the dual of that algebra. We then prove that the semisimplicity of certain such categories is equivalent to the existence of a Peter-Weyl decomposition of the corresponding sub-bimodule. Finally, we use this technique to establish the semisimplicity of certain finite-dimensional representations of the quantum double of sl_2 for generic q.
Higher categories of push-pull spans, II: Matrix factorizations
本文基于前文构建的对称单向$(\infty,3)$-范畴$\mathscr{CRW}$，扩展了Brunner等人关于仿射Rozansky-Witten模型的工作，建立矩阵因子化与同伦2-范畴之间的函子，并计算相应的TFT。
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.
For any finite-dimensional Hopf algebra $H$ we construct a group homomorphism $\biga(H)\to \text{BrPic}(\Rep(H))$, from the group of equivalence classes of $H$-biGalois objects to the group of equivalence classes of invertible exact $\Rep(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case $H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable $\Rep(T_q)$-bimodule categories.
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…
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.
Highest-weight type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows.
We discuss what has been achieved in the past twenty years on the construction and study of a braided finite tensor category structure on a suitable module category for a suitable vertex operator algebra. We identify the main difficult parts in the construction, discuss the methods developed to overcome these difficulties and present some further problems that still need to be solved. We also choose to discuss three among the numerous applications of the construction.
Adams operations are the natural transformations of the representation ring functor on the category of finite groups, and they are one way to describe the usual lambda-ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, determines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and…
On Higher Representation Theory via Categories of type Charge-Conserving--with--Glue
本文通过电荷守恒类别提出一种新的方法，用于研究更高表示论中的非半单性问题，并应用于braid表示的分类和分析。
Monoidal categories, representation gap and cryptography
针对线性分解攻击对非交换群和幺半群密码学应用的阻碍，本文提出研究仅具有大表示的幺半群，利用格林细胞理论系统分析，并探讨Temperley-Lieb、Brauer和划分范畴等图源幺半范畴的表示下界。
In this note, we describe two analogues of the Verlinde formula for modular categories in a twisted setting. The classical Verlinde formula for a modular category $\mathscr{C}$ describes the fusion coefficients of $\mathscr{C}$ in terms of the corresponding S-matrix $S(\mathscr{C})$. Now let us suppose that we also have an invertible $\mathscr{C}$-module category $\mathscr{M}$ equipped with a $\mathscr{C}$-module trace. This gives rise to a modular autoequivalence $F:\mathscr{C}\xrightarrow{\cong}\mathscr{C}$. In this setting, we can define a crossed S-matrix $S(\mathscr{C},\mathscr{M})$. As o…
Coxeter quiver representations in fusion categories and Gabriel's theorem
本文引入Coxeter箭图表示概念，利用$U_q(\mathfrak{s}\mathfrak{l}_2)$融合范畴推广了经典箭图表示理论，证明了涵盖所有Coxeter--Dynkin图的广义Gabriel定理，并揭示了不可分解表示与正根的双射关系。
We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix--the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.
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…
A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the representation-theoretic properties these algebras possess. Here we will forego technical intricacy as a growing number of researchers study fusion categories disjoint from Lie theory, representation theory, and a laundry list of other obstacles to understanding the mostly combinatorial, geometric, and numerical descriptions of the examples of fusion categories arising from quant…
We study deformation of tube algebra under twisting of graded monoidal categories. When a tensor category $\mathcal{C}$ is graded over a group $Γ$, a torus-valued 3-cocycle on $Γ$ can be used to deform the associator of $\mathcal{C}$. Based on a natural Fell bundle structure of the tube algebra over the action groupoid of the adjoint action of $Γ$, we show that the tube algebra of the twisted category is a 2-cocycle twisting of the original one.
Let $\mathscr{C}$ be a modular tensor category over an algebraically closed field $k$ of characteristic 0. Then there is the ubiquitous notion of the S-matrix $S(\mathscr{C})$ associated with the modular category. The matrix $S(\mathscr{C})$ is a symmetric matrix, its entries are cyclotomic integers and the matrix $(\dim \mathscr{C})^{-\frac{1}{2}}\cdot S(\mathscr{C})$ is a unitary matrix. Here $\dim \mathscr{C}\in k$ denotes the categorical dimension of $\mathscr{C}$ and it is a totally positive cyclotomic integer. Now suppose that we also have a modular autoequivalence $F:\mathscr{C}\to \mat…
We prove Eilenberg-Watts Theorem for 2-categories of the representation categories $\C\x\Mod$ of finite tensor categories $\C$. For a consequence we obtain that any autoequivalence of $\C\x\Mod$ is given by tensoring with a representative of some class in the Brauer-Picard group $\BrPic(\C)$. We introduce bialgebroid categories over $\C$ and a cohomology over a symmetric bialgebroid category. This cohomology turns out to be a generalization of the one we developed in a previous paper and moreover, an analogous Villamayor-Zelinsky sequence exists in this setting. In this context, for a symmetri…
We define annular algebras for rigid $C^{*}$-tensor categories, providing a unified framework for both Ocneanu's tube algebra and Jones' affine annular category of a planar algebra. We study the representation theory of annular algebras, and show that all sufficiently large (full) annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set, hence have equivalent representation theories. Annular algebras admit a universal $C^{*}$-algebra closure analogous to the universal $C^{*}$-algebra for groups. These algebras have interesting cor…
Let G be a classical compact Lie group and G_μthe associated compact matrix quantum group deformed by a positive parameter μ(or a nonzero and real μin the type A case). It is well known that the category Rep(G_μ) of unitary f.d. representations of G_μis a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_μ) to another braided tensor C*-category with irreducible tensor unit is full if |μ|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows…
Morita invariants of quasitriangular coideal subalgebras
本文通过BC和D型辫群的表示，研究准三角余子代数表示范畴的莫里塔不变量，并给出K矩阵的分类结果。
Generalized property F for weakly group-theoretical modular fusion categories
本文证明弱群论模融合范畴具有广义性质F，即其Reshetikhin-Turaev映射类群表示对紧致定向曲面具有有限射影像，推广了早期结果并扩展至完整映射类群。
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of p-groups, the corresponding pure braid group representations factor through a finite p-group, which answers a question asked of the first author by V. Drinfeld.
Indecomposable objects in Khovanov-Sazdanovic's generalizations of Deligne's interpolation categories
该研究针对Khovanov-Sazdanovic提出的Deligne插值范畴的推广形式，通过引入带滤过Krull-Schmidt范畴的相伴分次范畴等工具，完成了不可分解对象的分类并确定了相伴分次Grothendieck环。
We introduce the notion of meromorphic tensor category and illustrate it in several examples. They include representations of quantum affine algebras, chiral algebras of Beilinson and Drinfeld, G-vertex algebras of Borcherds, and representations of GL over a local field. Hopefully the formalism will accomodate various tensor structures arising in relation to the quantized Knizhnik-Zamolodchikov equations and deformed CFT
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