A braid group action on a $p$-DG homotopy category

On weakly group-theoretical non-degenerate braided fusion categories

We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius-Perron dimension and the Frobenius-Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braid…

The doubles of a braided Hopf algebra

Let A be a Hopf algebra in a braided rigid category B. In the case B admits a coend C, which is a Hopf algebra in B, we defined in 2008 the double D(A) of A, which is a quasitriangular Hopf algebra in B whose category of modules is isomorphic to the center of the category of A-modules as a braided category. Here, quasitriangular means endowed with an R-matrix (our notion of R-matrix for a Hopf algebra in B involves the coend C of B). In general, i.e. when B does not necessarily admit a coend, we construct a quasitriangular Hopf monad d_A on the center Z(B) of B whose category of modules is iso…

$G$-交叉辫子zesting

$G$-crossed braided zesting

本文提出$G$-交叉辫子zesting方法，利用可逆对象的上同调数据从已知$G$-交叉辫子融合范畴构造新范畴，并刻画其与$G$-扩张及Etingof-Nikshych-Ostrik同伦描述的关系。

同伦李代数与一致的微分2辫子

Homotopy Lie algebras and coherent infinitesimal 2-braidings

本文研究了同伦李代数的表示范畴，并通过构造2移位泊松结构得到一致的微分2辫子，同时建立了表示范畴与Chevalley-Eilenberg代数上半自由dg-模之间的对称单向dg-等价。

Solvability of a class of braided fusion categories

We show that a weakly integral braided fusion category C such that every simple object of C has Frobenius-Perron dimension at most 2 is solvable. In addition, we prove that such a fusion category is group-theoretical in the extreme case where the universal grading group of C is trivial.

Transparency condition in the categories of Yetter-Drinfel'd modules over Hopf algebras in braided categories

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…

Crossed actions of matched pairs of groups on tensor categories

We introduce the notion of $(G, Γ)$-crossed action on a tensor category, where $(G, Γ)$ is a matched pair of finite groups. A tensor category is called a $(G, Γ)$-crossed tensor category if it is endowed with a $(G, Γ)$-crossed action. We show that every $(G, Γ)$-crossed tensor category $\mathcal C$ gives rise to a tensor category $\mathcal C^{(G, Γ)}$ that fits into an exact sequence of tensor categories $\operatorname{Rep G} \to \mathcal C^{(G, Γ)} \to \mathcal C$. We also define the notion of a $(G, Γ)$-braiding in a $(G, Γ)$-crossed tensor category, which is connected with certain set-theo…

Some Remarks on Braided Group Reconstruction and Braided Doubles

The cross coproduct braided group $Aut(C) \rcocross B$ is obtained by Tannaka-Krein reconstruction from $C^B\to C$ for a braided group $B$ in braided category $C$. We apply this construction to obtain partial solutions to two problems in braided group theory, namely the tensor problem and the braided double. We obtain $Aut(C)\rcocross Aut(C)\isom Aut(C)\lcross Aut(C)$ and higher braided group `spin chains'. The example of the braided group $B(R)\lcross B(R)\lcross...\lcross B(R)$ is described explicitly by R-matrix relations. We also obtain $Aut(C)\rcocross Aut(C)^*$ as a dual quasitriangular …

The monoidal category of Yetter-Drinfeld modules over a weak braided Hopf algebra

In this paper we introduce the notion of weak operator and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non strict monoidal category. It is also shown that this category is not trivial, that is to say that it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.

On a canonical lift of Artin's representation to loop braid groups

Each pointed topological space has an associated $π$-module, obtained from action of its first homotopy group on its second homotopy group. For the $3$-ball with a trivial link with $n$-components removed from its interior, its $π$-module $\mathcal{M}_n$ is of free type. In this paper we give an injection of the (extended) loop braid group into the group of automorphisms of $\mathcal{M}_n$. We give a topological interpretation of this injection, showing that it is both an extension of Artin's representation for braid groups and of Dahm's homomorphism for (extended) loop braid groups.

Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra

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.

A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

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…

Meromorphic braided category arising in quantum affine algebras

We construct a category of flat vector bundles on an elliptic curve. It arises in the representation theory of quantum affine algebras and carries meromorphic braided structure with singularities on the diagonal of the square of the curve.

An example of a braided category of C*- algebras

A braided category of C*-algebras is constructed. Its objects are C*-algebras endowed with an action of the group R, its morphisms are C*-algebras morphisms intertwining the action of R, the crossed product of its two objects essentially depends on the action of R on considered C*-algebras. Crossed products of any object with the algebra of all continuous vanishing at infinity functions on R and algebras of all continuous functions on one- and twodimensional tori are discussed.

Braid cobordisms, triangulated categories, and flag varieties

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

Virtual braid groups of type B and weak categorification

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

共形块、括号化辫子操作子和 $c=1/2$ Virasoro 顶点算子代数

Conformal blocks, parenthesized braid operad, and $c=1/2$ Virasoro vertex operator algebra

本文通过共形块和路径解析延拓构造顶点算子代数模范畴的伪辫子结构，计算中心荷 $1/2$ 的 Virasoro 顶点算子代数的四点共形块，并证明其模范畴等价于 $\mathbb{Z}_2$ 上的 Tambara-Yamagami 范畴。

Crossed modules and quantum groups in braided categories

Let $A$ be a Hopf algebra in a braided category $\cal C$. Crossed modules over $A$ are introduced and studied as objects with both module and comodule structures satisfying a compatibility condition. The category $\DY{\cal C}^A_A$ of crossed modules is braided and is a concrete realization of a known general construction of a double or center of a monoidal category. For a quantum braided group $(A,\overline A,{\cal R})$ the corresponding braided category of modules ${\cal C}_{\cO{A,\overline A}}$ is identified with a full subcategory in $\DY{\cal C}_A^A$. The connection with cross products is …

The braided monoidal structures on the category of vector spaces graded by the Klein group

Let $k$ be a field, $k^*=k\setminus\{0\}$ and $C_2$ the cyclic group of order 2. In this note we compute all the braided monoidal structures on the category of $k$-vector spaces graded by the Klein group $C_2\times C_2$. Actually, for the monoidal structures we will compute the explicit form of the 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$, while for the braided monoidal structures we will compute the explicit form of the abelian 3-cocycles on $C_2\times C_2$ with coefficients in $k^*$. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided…

Crossed modules and quantum groups in braided categories I

Let $H$ be a Hopf algebra in braided category $\cal C$. Crossed modules over $H$ are objects with both module and comodule structures satisfying some comatibility condition. Category ${\cal C}^H_H$ of crossed modules is braided and is concrete realization of general categorical construction. For quantum braided group $(H,{\cal R})$ corresponding braided category ${\cal C}^{\cal R}_H$ of modules is identifyed with full subcategory in ${\cal C}_H^H$. Connection with crossproducts is discussed. Correct cross product in the class of quantum braided groups is built. Radford's--Majid's theorem gives…

Classification of charge-conserving loop braid representations

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that …

Braided tensor categories and extensions of vertex operator algebras

Let $V$ be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of $V$ and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of $V$-modules are equivalent.

Representations of the homotopy surface category of a simply connected space

We introduce the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.

Braided Near-group Categories

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.

Hom-Tensor Categories and the Hom-Yang-Baxter Equation

We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obta…

Diagrams for primitive cycles in spaces of pure braids and string links

The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of B…

辫子幺半 2-范畴中的局部模

Local Modules in Braided Monoidal 2-Categories

本文在辫子幺半 2-范畴中细化了局部模概念，证明了局部模 2-范畴具有辫子幺半结构与对偶，并在辫子融合 2-范畴中确立其为辫子多融合 2-范畴，最后探讨了 Lagrangian 代数及其与拓扑物相边界分类的联系。

辫子群在之字形代数的$A_\infty$-范畴上的作用

A braid group action on an $A_\infty$-category for zigzag algebras

本研究构造之字形代数的微分分次增强，证明其极小模型$A_\infty$-结构非形式，建立该框架下的辫子群作用并给出辛几何诠释，关联$p$-微分分次结构与Koszul对偶。

Classification of spin-chain braid representations

A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ charge-conserving representation (or spin-chain representation) is a strict monoidal functor $F$ from $\mathsf{C}$ to the category $\mathrm{Match}^N$ of rank-$N$ charge-conserving matrices that is natural in the sense that $F(1)=1$}. In this work we construct all spin-chain braid representations, and classify up to suitable notions of isomorphism.

Braid group action on the module category of quantum affine algebras

Let $\mathfrak{g}_0$ be a simple Lie algebra of type ADE and let $U'_q(\mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(\mathfrak{g}_0)$ on the quantum Grothendieck ring $K_t(\mathfrak{g})$ of Hernandez-Leclerc's category $C_{\mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors $\{\mathscr{S}_i\}_{i\in \mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism bet…

Symmetries of a rigid braided category

We identify natural symmetries of each rigid higher braided category. Specifically, we construct a functorial action by the continuous group $Ω\mathsf{O}(n)$ on each $\mathcal{E}_{n-1}$-monoidal $(g,d)$-category $\mathcal{R}$ in which each object is dualizable (for $n\geq 2$, $d \geq 0$, $d \leq g \leq \infty$). This action determines a canonical action by the continuous group $Ω\mathbb{R}\mathbb{P}^{n-1}$ on the moduli space of objects of each such $\mathcal{R}$. In cases where the parameters $n$, $d$, and $g$ are small, we compare these continuous symmetries to known symmetries, which manife…

通过B型之字形代数证明2-辫子群的忠实性

Faithfulness of the 2-Braid Group via Zigzag Algebra in Type B

本文通过证明B型之字形代数上的双模范畴是B型Soergel双模范畴的商范畴，为Rouquier关于B型2-辫子群忠实性的猜想提供了另一种证明。

Braided and coboundary monoidal categories

In this expository paper, we discuss and compare the notions of braided and coboundary monoidal categories. Coboundary monoidal categories are analogues of braided monoidal categories in which the role of the braid group is replaced by the cactus group. We focus on the categories of representations of quantum groups and crystals and explain how while the former is a braided monoidal category, this structure does not pass to the crystal limit. However, the categories of representations of quantum groups of finite type also possess the structure of a coboundary category which does behave well in…

纯辫群在范畴O模上的作用

Pure braid group actions on category O modules

本文证明可对称化Kac-Moody代数量子包络代数的量子Weyl群算子能在范畴O模上产生纯辫群典范作用，并证明该作用描述了有理Casimir联络的单值性，同时推广至抛物范畴O。

Homotopy Quantum Field Theories and the Homotopy Cobordism Category in Dimension 1+1

We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord}(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in parti…

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $\int_S\mathcal A$ of a surface $S$, determined by the choice of a braided tensor category $\mathcal A$, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a {\em braided module category} for $\mathcal A$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em quantum moment maps}. We then show that the quan…