Shigeru Yamagami
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.
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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 introduce a variant on the graphical calculus of Cockett and Seely for monoidal functors and illustrate it with a discussion of Tannaka reconstruction, some of which is known and some of which is new. The new portion is: given a separable Frobenius functor F: A --> B from a monoidal category A to a suitably complete or cocomplete braided autonomous category B, the usual formula for Tannaka reconstruction gives a weak bialgebra in B; if, moreover, A is autonomous, this weak bialgebra is in fact a weak Hopf algebra.
What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phung Ho Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstructed bialgebroid H has comodule category equivalent to the category of T-sheaves w.r.t. a monoidal Grothendieck topology on C. We also prove an existence theorem for fiber functors on small additive monoidal categories with bounded fusion and weak kernels. For certain autonomous categories a genera…
Given a horizontal monoid M in a duoidal category F, we examine the relationship between bimonoid structures on M and monoidal structures on the category of right M-modules which lift the vertical monoidal structure of F. We obtain our result using a variant of the Tannaka adjunction. The approach taken utilizes hom-enriched categories rather than categories on which a monoidal category acts ("actegories"). The requirement of enrichment in F itself demands the existence of some internal homs, leading to the consideration of convolution for duoidal categories. Proving that certain hom-functors …
The topic of this paper is a generalization of Tannaka duality to coclosed categories. As an application we prove reconstruction theorems for coalgebras (and bialgebras) in categories of topological vector spaces over a nonarchimedean field K. In particular, our results imply reconstruction and recognition theorems for categories of locally analytic representations of compact $p$-adic groups.
Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful "fibre functors" to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish thi…
Tannaka-Krein reconstruction for fusion 2-categories
本文通过建立截断子高阶范畴与代数范畴的幺半等价，重新证明并推广了Tannaka-Krein重构定理，证明了有限半单2-Hopf代数可由其模融合2-范畴及幺半函子恢复。
Tannaka duality and 1-affineness
本文利用范畴层理论中的仿射性定理，证明 Lurie 关于几何叠的 Tannaka 对偶无需任何驯良性假设，并将结果用于拟凝聚层范畴的张量积和积分变换公式。
We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The $\coend$ of such a functor turns out to be a Hopf algebroid over this ring. Using the result of a previous paper we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.
A classical result of Tannaka duality is the fact that a coalgebra over a field can be reconstructed from its category of finite dimensional representations by using the forgetful functor which sends a representation to its underlying vector space. There is also a corresponding recognition result, which characterizes those categories equipped with a functor to finite dimensional vector spaces which are equivalent to the category of finite dimensional representations of a coalgebra. In this paper we study a generalization of these questions to an arbitrary cosmos, that is, a complete and cocomp…
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.
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.
A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a field extension. If we are working over an arbitrary commutative ring rather than a field, the categories of representations cease to be abelian. We provide a list of sufficient conditions which ensure that an additive tensor category is equivalent to the category of representations of an affine groupoid scheme acting on an affine scheme, or, more generally, to …
We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka-Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite dimensional non-degenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasize…
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。
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.
Notes on semisimple tensor categories of rank two
该研究证明代数闭域上存在无穷多个秩为二的半单张量范畴，丰富了低秩幺半范畴的分类认识。
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 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…
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.
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.
Derived Reconstruction of Semisimple Tensor and Module Categories
本文研究半单张量范畴及其模范畴的有界导出范畴，通过整值特征和稳定化子子群分类幺半与模$t$-结构，证明导出等价可重构原范畴。
On braided simple extensions and braided non-semisimple near-group categories
本文研究了编织简单扩展和非半单近群范畴的性质，证明了非退化编织非半单近群范畴与sRep(W⊕W*)的扩展关系，并揭示了其与皮卡群的联系。
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。
A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a supergroup is reductive if and only if its connected component is reductive. The connected component is reductive if and only if the Lie superalgebra divided by its center is a product of simple Lie algebras of classical type and Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.
Starting from an abelian category A such that every object has only finitely many subobjects we construct a semisimple tensor category T. We show that T interpolates the categories Rep(Aut(p),K) where p runs through certain projective (pro-)objects of A. The main example is A=finite dimensional F_q-vector spaces. Then T can be considered as the category of representations of GL(n,F_q) where n is not a natural number. This work extends a construction of Deligne for symmetric groups.
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.
Hopf algebras are closely related to monoidal categories. More precise, $k$-Hopf algebras can be characterized as those algebras whose category of finite dimensional representations is an autonomous monoidal category such that the forgetful functor to $k$-vectorspaces is a strict monoidal functor. This result is known as the Tannaka reconstruction theorem (for Hopf algebras). Because of the importance of both Hopf algebras in various fields, over the last last few decades, many generalizations have been defined. We will survey these different generalizations from the point of view of the Tanna…
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 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…
Locally finitely presented Grothendieck categories and the pure semisimplicity conjecture
本文研究局部有限呈现的纯半单格罗滕迪克范畴与左纯半单环之间的等价关系，并扩展了Auslander和Ringel-Tachikawa对应到左纯半单环类。
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.
Universal Semisimple Reconstruction and Obstruction Theory
针对有限长度阿贝尔范畴建立半单障碍理论，构造典范障碍子对象与半单化反射子，其迭代可还原经典根基滤过，揭示经典根基理论是范畴框架的模论实例。
I review recent developments of "non-semisimple" modular tensor categories in the sense of Lyubashenko and the categorical Verlinde formula for such categories (this is a proceedings article for Meeting for Study of Number theory, Hopf algebras and related topics held at University of Toyama, Japan, 12-15 February 2017).
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…
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。
We introduce the Frobenius-Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius-Schur theorem, including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism how the `twisted' theory arises from the ordinary case. As a demonstration, we give a twisted Frobenius-Schur theorem for semisimple quasi-Hopf algebras. We also give several applications to the quantum SL_2.
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…
Fibre functors and reconstruction of Hopf algebras
本文提出Tannaka-Krein型重构定理，证明当张量范畴具有相容的monoidal模范畴结构时，可通过相对coend重构辫子Hopf代数，并应用于对数Kazhdan-Lusztig猜想。
For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties - expressible entirely within the language of tensor categories - which are necessary and suf…
Tannaka duality and its extensions by Lurie, Schäppi et al. reveal that many schemes as well as algebraic stacks may be identified with their tensor categories of quasi-coherent sheaves. In this thesis we study constructions of cocomplete tensor categories (resp. cocontinuous tensor functors) which usually correspond to constructions of schemes (resp. their morphisms) in the case of quasi-coherent sheaves. This means to globalize the usual local-global algebraic geometry. For this we first have to develop basic commutative algebra in an arbitrary cocomplete tensor category. We then discuss ten…
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.
Rigidity of non-negligible objects of moderate growth in braided categories
本文研究了辫状范畴中非可忽略对象的刚性性质，并探讨了其在半单范畴和对数共形场论中的应用。
Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors
本文研究带彩色区域的广义Temperley-Lieb分类，确定其半单性并分解张量积的简单对象，利用双变量Chebyshev多项式和彩色Jones-Wenzl投影子证明了一个猜想。
Motivated by algebraic quantum field theory, we study presheaves of symmetric tensor categories defined over the base of a space, intended as a spacetime. Any section of a presheaf (that is, any "superselection sector", in the applications that we have in mind) defines a holonomy representation whose triviality is measured by Cheeger-Chern-Simons characteristic classes, and a non-abelian unitary cocycle defining a Lie group gerbe. We show that, given an embedding in a presheaf of full subcategories of the one of Hilbert spaces, the section category of a presheaf is a Tannaka-type dual of a loc…
We focus on the problem of producing new modular tensor categories from Hopf algebras. To do this, we first give a general method to construct factorizable Hopf algebras. Then we apply the method to construct two families of ribbon factorizable Hopf algebras which are not quantum groups or Drinfel'd double. One family are point Hopf algebras which give non-semisimple modular tensor categories, while the other family are C^*-Hopf algebras which give new unitary modular tensor categories. Lastly, we determine the fusion rings of these unitary modular tensor categories.
For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that t…
Growth problems in diagram categories
研究图示/插值范畴中生成对象张量幂的分解数增长速率的渐进行为，推导出半单情况下的渐近公式。
We show that certain tensor product multiplicities in semisimple braided sovereign tensor categories must be even. The quantity governing this behavior is the Frobenius-Schur indicator. The result applies in particular to the representation categories of large classes of groups, Lie algebras, Hopf algebras and vertex algebras.
We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A=F-Vect, where F is a field. An object X in A with two-sided dual X^ gives rise to a Frobenius alg…
A fundamental theorem of P. Deligne (2002) states that a pre-Tannakian category over an algebraically closed field of characteristic zero admits a fiber functor to the category of supervector spaces (i.e., is the representation category of an affine proalgebraic supergroup) if and only if it has moderate growth (i.e., the lengths of tensor powers of an object grow at most exponentially). In this paper we prove a characteristic p version of this theorem. Namely we show that a pre-Tannakian category over an algebraically closed field of characteristic p>0 admits a fiber functor into the Verlinde…
We extend the calculus of relations to embed a regular category A into a family of pseudo-abelian tensor categories T(A,d) depending on a degree function d. Under the condition that all objects of A have only finitely many subobjects, our main results are as follows: 1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N is semisimple provided that A is exact and Mal'cev. Thereby, we produce many new semisimple, hence abelian, tensor categories. 2. Using lattice theory, we give a simple numerical criterion for the vanishing of N. 3. We determine all degree functions for w…
Multiplicities and dimensions in enveloping tensor categories
本文针对由正则 Mal'cev 范畴构造的半单张量范畴，利用基本对象的子商分解计算单对象的张量积重数、范畴维数，并确定其 Grothendieck 环。
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