Pavel Etingof, Viktor Ostrik
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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.
Let C be a fusion category which is an extension of a fusion category D by a finite group G. We classify module categories over C in terms of module categories over D and the extension data (c,M,a) of C. We also describe functor categories over C (and in particular the dual categories of C). We use this in order to classify module categories over the Tambara Yamagami fusion categories, and their duals.
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module categories.
We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group $G$ as induced from module categories over fusion subcategories associated with the subgroups of $G$. We define invariant $\C_e$-module categories and extensions of $\C_e$-module categories. The construction of module categories over $\C$ is reduced to determine invariant module categories for subgroups of $G$ and the indecomposable extensions of this modules categories. We associate a $G$-crossed product fusion cat…
In this paper, we re-examine certain integrable modules of Chari-Presslely for an (untwisted) affine Lie algebra $\hat{\g}$ by exploiting basic formal variable techniques. We define and study two categories ${\mathcal{E}}$ and ${\mathcal{C}}$ of $\hat{\g}$-modules using generating functions, where ${\mathcal{E}}$ includes evaluation modules and ${\mathcal{C}}$ unifies highest weight modules, evaluation modules and their tensor product modules, and we classify integrable irreducible $\hat{\g}$-modules in categories ${\mathcal{E}}$ and ${\mathcal{C}}$.
A general procedure is presented which associates to a finite crossed module a premodular category, generalizing the representation categories of a finite group and of its double, and the extent to which the resulting category fails to be modular is explained.
We develop the theory of diagrammatic Temperley-Lieb categories in order to construct examples of spherical fusion categories, which we then use to construct Turaev-Viro skein modules for n-holed disks. This is a thesis submitted for the degree of Bachelor of Science with Honours at the Australian National University.
Semisimplicity of module categories of certain affine vertex operator superalgebras
本文证明了某些仿射顶点算子超代数在共形但非容许水平上的 Kazhdan-Lusztig 范畴是半单的，并由此得到其有限长度广义模范畴具有辫子张量范畴结构。
We introduce Turaev bicategories and Turaev pseudofunctors. On the one hand, they generalize the notions of Turaev categories (and Turaev functors), introduced at the turn of the millennium and originally called "crossed group categories" by Turaev himself, and the notions of bicategories and pseudofunctors, on the other. For bimonads in 2-categories, which we defined in one of our previous papers, we introduce generalized Yetter-Drinfel`d modules in 2-categories. These generalize to the 2-categorical setting the generalized Yetter-Drinfel`d modules (over a field) of Panaite and Staic, and thu…
In this article, we study module categries of simple current extensions of vertex operator algebras. Under certain assumptions, we show that every module for a rational vertex operator algebra be lifted to a twisted module for an extended algebra with an abelian symmetry.
We study versions of the categories of Yetter-Drinfel'd modules over a Hopf algebra $H$ in a braided monoidal category $\C$. Contrarywise to Bespalov's approach, all our structures live in $\C$. This forces $H$ to be transparent or equivalently to lie in Müger's center $\Z_2(\C)$ of $\C$. We prove that versions of the categories of Yetter-Drinfel'd modules in $\C$ are braided monoidally isomorphic to the categories of (left/right) modules over the Drinfel'd double $D(H)\in\C$ for $H$ finite. We obtain that these categories polarize into two disjoint groups of mutually isomorphic braided monoid…
We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra $H$ a Yetter-Drinfeld module braids from the left with $H$-modules. We will generalize classical results to this context and give some application for the categories of Yetter-Drinfeld modules and $H$-modules. In particular we will describe liftings of coradically graded Hopf algebras in the category of Yetter-Drinfeld modules with these techniques.
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…
Module categories for $A_n$ web categories from $\tilde{A}_{n-1}$-buildings
本文为局部有限 $\tilde{A}_{n-1}$ 建筑的顶点向量丛范畴赋予 $A_n$ 编织范畴的模结构，作为 $Rep(SL_n)$ 作用的 $q$-模拟，并证明其等变性，恢复 Jones 的纤维函子。
Let $V=\C^N$ with $N$ odd. We construct a $q$-deformation of $\End_{Sp(N-1)}(V^{\otimes n})$ which contains $\End_{U_q\sl_N}(V^{\otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest the existence of module categories of $Rep(U_q\sl_N)$ which may not come from already known coideal subalgebras of $U_q\sl_N$. We moreover indicate how this can be used to construct module categories of the associated fusion tensor categories as well as subfactors, along the lines of previ…
We generalize the definition of an exact sequence of tensor categories due to Bruguières and Natale, and introduce a new notion of an exact sequence of (finite) tensor categories with respect to a module category. We give three definitions of this notion and show their equivalence. In particular, the Deligne tensor product of tensor categories gives rise to an exact sequence in our sense. We also show that the dual to an exact sequence in our sense is again an exact sequence. This generalizes the corresponding statement for exact sequences of Hopf algebras. Finally, we show that the middle ter…
Let $k$ be an algebraically closed field of characteristic $0$ or $p>2$. Let $\mathcal{G}$ be an affine supergroup scheme over $k$. We classify the indecomposable exact module categories over the tensor category ${\rm sCoh}_{\rm f}(\mathcal{G})$ of (coherent sheaves of) finite dimensional $\mathcal{O}(\mathcal{G})$-supermodules in terms of $(\mathcal{H},Ψ)$-equivariant coherent sheaves on $\mathcal{G}$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\sRep(\mathcal{G})$. When $\mathcal{G}$ is finite, this yields the classification of {\em all} i…
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.
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。
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)].
It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.
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.
We classify the module categories over the double (possibly twisted) of a finite group.
We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the small quantum groups $u_q(\mathfrak{sl}_2)$.
Classifying Module Categories for Generalized Temperley-Lieb-Jones *-2-Categories
本文提出通过加权双向公平和平衡图对广义Temperley-Lieb-Jones *-2-范畴的单位ary模块进行分类。
We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. Gómez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and [R. Wisbauer, On Galois comodules (2004), to appear in Comm. Algebra.]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.
Monoidal categories graded by crossed modules and 3-dimensional HQFTs
本文引入由交叉模分级的幺半范畴与融合范畴，并利用球面χ-融合范畴通过状态和方法构造以同伦2型Bχ为目标的三维同伦量子场论。
This is the second part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In thi…
Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. As a consequence a generalization of the Radford-Majid criterion for a braided Hopf algebra to be a cross product is obtained. The results of this paper turn out to be fundamental for the construction of (bicovariant) differential calculi on braided Hopf alg…
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.
This is the second part of the paper. Results of the first part about crossed modules are applied here to study of quantum groups in braided categories. Correct cross product in the class of quantum braided groups is built. Criterion when quantum braided group is cross product is otained.
We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to equivariant Morita equivalence. Specifically, any indecomposable exact module categories is equivalent to the category of finite-dimensional modules over a left comodule algebra. This is an alternative approach to the results of Etingof and Ostrik. For this, we study the stabilizer introduced by Yan and Zhu and show that it coincides with the internal Hom. We a…
This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans - crossed modules of monoids relative to this class of spans - simplicial monoids of so-called Moore length 1 relative to this class of spans. The most important examples of monoids that are covered are small categories (treated as monoids in categories of spans) and bimonoids in symmetric monoidal categories (regarded as monoids in categories of comonoids). In this…
Fully exact and fully dualizable module categories
本文提出完美模范畴作为有限张量2-范畴的模型，研究了完全精确模范畴的性质及其在不同霍普夫代数中的应用。
We introduce a tensor category O_+ (resp. O_{-}) of certain modules of gl_{\infty} with non-negative (resp. non-positive) integral central charges with the usual tensor product. We also introduce a tensor category O_f consisting of certain modules over GL(N) for all N. We show that the tensor categories O_+, O_{-} and O_f are semisimple abelian and all equivalent to each other. We give a formula to decompose a tensor product of two modules in each of these categories. We also introduce a tensor category O^w of certain modules over W_{1 +\infty} with non-negative integral central charges. We sh…
We consider traces on module categories over pivotal fusion categories which are compatible with the module structure. It is shown that such module traces characterise the Morita classes of special haploid symmetric Frobenius algebras. Moreover, they are unique up to a scale factor and they equip the dual category with a pivotal structure. This implies that for each pivotal structure on a fusion category over the complex numbers there exists a conjugate pivotal structure defined by the canonical module trace.
We prove in full generality that the generalized quantum affine Schur-Weyl duality functor, introduced by Kang-Kashiwara-Kim, gives an equivalence between the category of finite-dimensional modules over a quiver Hecke algebra and a certain full subcategory of finite-dimensional modules over a quantum affine algebra which is a generalization of the Hernandez-Leclerc's category $\mathcal{C}_Q$. This was previously proved in untwisted $ADE$ types by Fujita using the geometry of quiver varieties, which is not applicable in general. Our proof is purely algebraic, and so can be extended uniformly to…
Given a finite dimensional Hopf algebra H and an exact indecomposable module category M over Rep(H), we explicitly compute the adjoint algebra A_M as an object in the category of Yetter-Drinfeld modules over H, and the space of class functions CF(M) associated to M, as introduced by K. Shimizu [ Further results on the structure of (Co)ends in fintite tensor categories, preprint arXiv:1801.02493]. We use our construction to describe these algebras when H is a group algebra and a dual group algebra. This result allows us to compute the adjoint algebra for certain group-theoretical fusion categor…
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.
This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category of $\A^\bullet$, which corresponded to the category of coherent sheaves on a complex manifold. In this paper we enlarge this category to include objects which correspond to quasi-coherent sheaves. We then apply this framework to proving an equivalence of categories between derived categories on the noncommutative complex torus and on a holomorphic gerbe on t…
Let $k$ be an algebraically closed field of characteristic $p\ge 0$. Let $G$ be an affine group scheme over $k$. We classify the indecomposable exact module categories over the rigid tensor category $\text{Coh}_f(G)$ of coherent sheaves of finite dimensional $k-$vector spaces on $G$, in terms of $(H,ψ)-$equivariant coherent sheaves on $G$. We deduce from it the classification of indecomposable {\em geometrical} module categories over $\Rep(G)$. When $G$ is finite, this yields the classification of {\em all} indecomposable exact module categories over the finite tensor category $\Rep(G)$. In pa…
We define a notion of tensor product of bimodule categories and prove that with this product the 2-category of C-bimodule categories for fixed tensor C is a monoidal 2-category in the sense of Kapranov and Voevodsky. We then provide a monoidal-structure preserving 2-equivalence between the 2-category of C-bimodule categories and Z(C)-module categories (module categories over the center). For finite group G we show that de-equivariantization is equivalent to tensor product over category Rep(G) of finite dimensional representations. We derive Rep(G)-module fusion rules and determine the group of…
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 …
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 study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
There are various generalizations of bialgebras to their ''many object'' versions, such as quantum categories, bialgebroids and weak bialgebras. These can also be thought of as quantum analogues of small categories. In this paper we study modules over these structures, which are quantum analogues of profunctors (also called distributors) between small categories.
Grothendieck-Verdier duality in categories of bimodules and weak module functors
该研究探讨幺半范畴中Grothendieck-Verdier对偶结构下关联两个张量积的分配子成为同构的条件，借助模范畴视角与内Hom、内上Hom的弱模函子结构展开分析，并以结合代数的双模范畴为例进行具体阐释。
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…
In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a (strictly object-adapted) cellular category. We use this result to specify a subset of cellular basis elements, which generates all morphisms in this category. In a different direction we generalize the earlier results to the case where G is replaced by the infinitesimal thickenings G_rT of a maximal torus T in G by the Frobenius subgroup schemes G_r. Here …
This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal st…
Given a weight-one element $u$ of a vertex operator algebra $V$, we construct an automorphism of the category of generalized $g$-twisted modules for automorphisms $g$ of $V$ fixing $u$. We apply this construction to the case that $V$ is an affine vertex operator algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules.
Reconstruction of module categories in the infinite and non-rigid settings
本研究扩展了模范畴重构理论，通过拟有限余代数对象和松弛模单子，实现无限与非刚性情形下模范畴的重构，并给出具体形式、相关定理及范畴证明。
The balanced tensor product M (x)_A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M x N. The balanced tensor product M [x]_C N of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M x N. We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.
In this paper, we define the KW cell system on a graph $Γ$, depending on parameters $N\in \mathbb{N}$, $q$ a root of unity, and $ω$ an $N$-th root of unity. This is a polynomial system of equations depending on $Γ$ and the parameters. Using the graph planar algebra embedding theorem, we prove that when $q = e^{2πi \frac{1}{2(N+k)}}$, solutions to the KW cell system on $Γ$ classify module categories over $\overline{\mathrm{Rep}(U_q(sl_N))^ω}$ whose action graph for the object $Λ_1$ is $Γ$. The KW cell system is a generalisation of the Etingof-Ostrik and the De Commer-Yamashita classifying data …
The categorical formulation of the Eilenberg-Watts calculus relates, for any pair of finite categories M and N, the finite categories Fun^{le}(N,M) and Fun^{re}(N,M) of linear left or right exact functors and the Deligne product \bar N \boxtimes M by adjoint equivalences. We establish a variant of this calculus for the case that the finite categories M and N are module categories over a finite tensor category. This provides in particular canonical and explicitly computable equivalences between categories of left or right exact module functors (or, more generally, balanced functors) and certain…
Nondegenerate module categories
本文将Shimizu（2019）的辫子有限张量范畴非退化等价理论部分推广到辫子模范畴，证明其“非退化”与“可因子化”等价，还在Hopf情形下建立拟三角余模代数的可因子化性与表示范畴非退化的等价关系。
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。
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.
Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to discretely strongly graded modules a contragredient module, obtained as a gradewise dual. In this paper, we show that this gradewise dual endows the module category with the structure of a ribbon Grothendieck-Verdier category. This duality structure is more general than that of a rigid monoidal category; in contrast to rigidity, it naturally accommodates the fac…
The effectiveness of the aplication of constructions in $G$-graded $k$-categories to the computation of the fundamental group of a finite dimensional $k$-algebra, alongside with open problems still left untouched by those methods and new problems arisen from the introduction of the concept of fundamental group of a $k$-linear category, motivated the investigation of $H$-module categories, i.e., actions of a Hopf algebra $H$ on a $k$-linear category. The $G$-graded case corresponds then to actions of the Hopf algebra $k^G$ on a $k$-linear category. In this work we take a step further and introd…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。