E. Batista, S. Caenepeel, J. Vercruysse
We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We generalize the fundamental theorem for Hopf modules and some of its applications to Hopf categories.
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We present characterizations of braided co-Frobenius Hopf algebras in the braided tensor category of Yetter-Drinfeld modules over a Hopf algebra extending those already known for co-Frobenius Hopf algebras.
How to disentangle two braided Hopf algebras
给出了如何定义两个辫子霍普夫代数的张量积。
Braided Hopf algebroids and Lie algebroids
本文在辫子幺半范畴中构造辫子Hopf代数胚，引入辫子Lie-Rinehart代数，并证明其右不变向量场与量子主丛上的等变向量场同构，且建立泛包络代数与jet空间的配对。
Structure theorems for bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras and braided Hopf algebras
设H是拟霍普夫代数、弱霍普夫代数或辫状霍普夫代数.设B是H-双余代数，使得存在H-双余代数v的态射：H \rightarrow B.然后我们可以定义一个对象B^{co（H）}，它是H上的一个左-左Yetter-Drinfeld模，具有额外的性质，允许使一个smash积B^{co（H）}# H成为一个H-双同态代数，同构于B。
Braided autoequivalences and the equivariant Brauer group of a quasitriangular Hopf algebra
设$(H, R)$是域$k$上的有限维拟三角霍普夫代数，$_H\mathcal{M}$是$H$的表示范畴.本文研究了$_H\mathcal{M}$上Drinfeld中心$^H_H\mathcal{YD}$可平凡化的辫子自等价.我们建立了这些自等价群与嬗变辫状霍普夫代数$_RH$的量子交换双伽罗瓦对象群之间的群同构。然后我们应用这个同构得到[18]中等变Brauer群$\mathrm{BM}(k, H,R)$的确切序列的范畴解释。为了达到这个目的，我们必须发展Schauenburg在[14，15]中提出的辫子双Galois理论，它把通常的Hopf代数上的Hopf双Galois理论推广到辫子幺半范畴范畴中的辫子Hopf代数上。
Braided Morita invariants of finite-dimensional semisimple and cosemisimple Hopf algebras with braidings are constructed by refining the polynomial invariants introduced by the author. The invariants are computed for the duals of Suzuki's braided Hopf algebras, and as an application of that, the braided Morita equivalence classes over the $8$-dimensional Kac-Paljutkin algebra are determined. This paper also includes the modified results and proofs on determination of the coribbon elements of Suzuki's braided Hopf algebras, that are discussed and given in my previous paper "The coribbon structu…
The Calabi-Yau property of Hopf algebras and braided Hopf algebras
设$H$是有限维半单霍普夫代数，$R$是$H$上Yetter-Drinfeld模范畴中的辫子霍普夫代数.当$R$是Calabi-Yau代数时，给出了$R#H$是Calabi-Yau霍普夫代数的一个充要条件.反之，当$H$是有限群的群代数，Smash积$R#H$是Calabi-Yau代数时，给出了代数$R$是Calabi-Yau代数的一个充要条件.
Braided Hopf algebras over non abelian finite groups
这是一个调查的一般方面的理论编织霍普夫代数代数的重点是一类特殊的编织分次霍普夫代数命名为braidas。兴趣上产生的问题，分类的点霍普夫代数。我们从不同的角度讨论了拉斯蒂格，尼科尔斯和绍恩堡的想法。然后我们集中讨论H上Yetter-Drinfeld范畴中的辫子霍普夫代数，其中H是非交换有限群的群代数.我们提出了一些有限维的例子中所产生的未发表的工作米林斯基和施耐德。
Let $H$ be a Hopf quasigroup with bijective antipode and let $Aut_{HQG}(H)$ be the set of all Hopf quasigroup automorphisms of $H$. We introduce a category ${_{H}\mathcal{YDQ}^{H}}(α,β)$ with $α,β\in Aut_{HQG}(H)$ and construct a braided $T$-category $\mathcal{YDQ}(H)$ having all the categories ${_{H}\mathcal{YDQ}^{H}}(α,β)$ as components.
The doubles of a braided Hopf algebra
设A是辫刚性范畴B中的霍普夫代数.在B允许余端C的情况下，C是B中的一个霍普夫代数，我们在2008年定义了A的双D（A），它是B中的一个拟三角霍普夫代数，其模范畴同构于A-模范畴的中心作为辫子范畴。这里，拟三角意味着被赋予一个R-矩阵（我们对B中的霍普夫代数的R-矩阵的概念涉及B的余端C）。一般地，即当B不一定允许余尾时，我们在B的中心Z（B）上构造一个拟三角Hopf单子d_A，它的模范畴同构于A-模范畴的中心作为辫子范畴.证明了Hopf单子d_A不可用霍普夫代数表示.若B有余边C，则D（A）是Hopf单子d_A与C的向量积.等价地，Hopf单子d_A是霍普夫代数D（A）与霍普夫代数C的交商.
Constructing New Braided $T$-Categories via Weak Monoidal Hom-Hopf Algebras
本文定义并研究了弱monoidal Hom-霍普夫代数，它推广了弱霍普夫代数和monoidal Hom-霍普夫代数.若$H$是一个具有双射对极的弱monoidal Hom-霍普夫代数，且$Aut_{wmHH}(H)$是$H$的所有自同构的集合.然后我们引入了一个带有$α,β\in Aut_{wmHH}(H)$的范畴${_{H}\mathcal{WMHYD}^{H}}(α,β)$，并构造了一个以所有范畴${_{H}\mathcal{WMHYD}^{H}}(α,β)$为分支的辫子$T$ -范畴$\mathcal{WMHYD}(H)$.
Braided Hopf algebras arising from matched pairs of groups
设k是一个域。设（F，G）是一对匹配的群。给出了2-上圈对（σ，τ）的交叉积代数与交叉上积余代数k^G{}^τ#_σ kF合成辫子霍普夫代数的充要条件.我们还讨论了这类辫子霍普夫代数在有限群上的Yetter-Drinfeld模范畴中的对角实现.
Additive Deformations of Braided Hopf Algebras
Wirth意义下的双代数的加法变形，即满足一定相容条件的乘法映射的变形。余代数结构可以推广到辫双代数。霍普夫代数的加法变形定理也可以推广到这种情况。我们认为 *-结构，并证明了一般勋伯格对应在这种情况下。最后我们给出了一些例子。
Transparency condition in the categories of Yetter-Drinfel'd modules over Hopf algebras in braided categories
本文研究了辫子monoidal范畴$\C$中Hopf代数$H$上的Yetter-Drinfel'd模范畴的几个版本.与Bespalov的方法不同，我们所有的结构都存在于$\C$中。这迫使$H$是透明的，或等效地位于$\C$的Müger中心$\Z_2(\C)$。我们证明了$\C$中的Yetter-Drinfel'd模范畴与$H$有限的Drinfel'd double $D(H)\in\C$上的（左/右）模范畴是braided monoidally同构的.我们得到这些范畴可分解为两个互不相交的同构辫幺半群范畴群。我们的结论是，如果$H\in\Z_2(\C)$，那么${}_{D(H)}\C$作为一个子类嵌入到编织中心范畴$\Z_1({}_H\C)$的范畴${}_H\C$的左$H$ -模块在$\C$。对于刚性的、上完备的辫状Hopf代数$\C$和拟三角Hopf代数$H$，使得$H\in\Z_2(\C)$，我们证明了${}_H\C$的整个中心范畴与$\Aut({}_H\C)\rtimes H$上的左模范畴么半群同构-辫状Hopf代数$\Aut({}_H\C)$是${}_H\C$的余端的玻色化.讨论了一类透明霍普夫代数的例子.
Coend elements of a braided Hopf algebra
本文在辫子刚性幺半范畴中定义并研究了Hopf代数的余端元素，推广了有限维Hopf代数的pivotal和ribbon元素，并基于此为$H$-模范畴的pivotal和ribbon结构提供了新描述。
Relative PBW type theorems for symmetrically braided Hopf algebras
我们证明了在特征零点处所有不可约对称辫霍普夫代数都是PBW型的.从而得到了辫子霍普夫代数作为包含余根的辫子Hopf子代数上的模是PBW型的条件。
Braided autoequivalences and quantum commutative bi-Galois objects
设$(H,R)$是域$k$上的拟三角弱霍普夫代数.证明了$H$上的Yetter-Drinfeld模范畴$^H_H\mathscr{YD}$与范畴$_H\mathscr{M}$中的辫子霍普夫代数${}_RH$上的余模范畴之间存在辫子么半群等价.基于这个等价性，我们证明了辫子霍普夫代数${}_RH$上的每个辫子双Galois对象$A$定义范畴$^H_H\mathscr{YD}$的辫子自等价当且仅当$A$是量子交换的.若$H$在代数闭域上是半单的，即融合情形，则$_H\mathscr{M}$上可平凡化的$^H_H\mathscr{YD}$的每一个辫子自等价都由这样一个量子可换伽罗瓦对象确定. $_H\mathscr{M}$中的量子可换伽罗瓦对象形成了一个群，该群度量了$(H,R)$在霍普夫代数情况下的Brauer群，如[20]中所研究的。
Algebras and Hopf Algebras in Braided Categories
这是一个介绍代数理论的代数和霍普夫代数在编织类别。这样的对象推广了超代数和超Hopf代数，以及色李代数。回顾了辫范畴C的基本事实，研究了这类范畴中Hopf代数的模和余模，回顾了辫-交换或辫-余交换的Hopf代数（辫群）的概念，并给出了辫群aut（C）的重构定理的一个完全图解证明.该理论对拟三角Hopf代数（量子群）理论有重要的意义。它还包括重要的例子，如退化Sklyanin代数和量子平面。
Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories
研究了辫子monoidal范畴C中双代数B上的Hopf（双）模和交叉模.证明了当B是一个具有可逆对极的Hopf代数时，这两个范畴的（辫状）么半群等价。证明了C中Hopf代数上的双代数投影和Hopf双模双代数是同构范畴。作为结果的Radford-Majid准则的辫子Hopf代数是一个交叉产品的推广。本文的结果对辫状Hopf代数上的（双共变）微分演算的构造具有重要意义。
We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.
Bicovariant Differential Calculi and Cross Products on Braided Hopf Algebras
本文研究了辫子范畴中Hopf代数$H$上的Hopf双模和交叉模.它们分别是辫状双协变微分演算及其不变向量场的关键，也是辫状Hopf代数叉积构造的关键。我们证明了Hopf双模和交叉模的概念是等价的。将Radford-Majid准则推广到辫子情形，证明了Hopf代数$H$上的双代数叉积可以用$H$ -交叉模双代数精确刻画.研究了辫子阿贝尔范畴中的（双协变）微分演算理论，并利用一阶双协变微分演算构造了$\NN_0$ -分次双协变微分演算.这些对象被证明是具有普遍双代数性质的编织$\NN_0$ -分级范畴的Hopf代数微分演算。
We prove a coherence theorem for actions of groups on monoidal categories. As an application we prove coherence for arbitrary braided $G$-crossed categories.
Duality Theorem and Hom Functor in Braided Tensor Categories
将Blatter-Montgomery对偶定理推广到辫张量范畴。证明了对于任意两个辫子Yetter-Drinfeld模$V$和$W$，$Hom(V,W)$是辫子Yetter-Drinfeld模.
Hopf cyclic cohomology in braided monoidal categories
我们推广的形式主义的Hopf循环上同调的背景下，辫子范畴。对于辫子monoidal阿贝尔范畴中的霍普夫代数，我们引入了稳定反Yetter-Drinfeld模的概念.在Connes和Moscovici的对合意义下，我们将一个仿上循环对象和一个上循环对象关联到一个具有一个辫子模对的辫子霍普夫代数上.当辫子是对称的时，具有系数的Hopf循环上同调的完整形式可以推广到我们的范畴设置。
Duality Theorems for Infinite Braided Hopf Algebras
设$H$是一个无穷维辫状霍普夫代数，并假设辫状是关于$H$及其拟对偶$H^d$对称的.证明了Blattner-Montgomery对偶定理，即证明了$$ (R # H)# H^{d} \cong R \otimes (H # H^{d}) \hbox {as algebras in braided tensor category} {\cal C}.$$。特别地，我们给出了Yetter-Drinfeld模范畴中无限辫子霍普夫代数的两个对偶定理。
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generali…
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.
PBW bases for a class of braided Hopf algebras
证明了P（R）的辫子子空间V生成的辫子霍普夫代数R的Poincare-Birkhoff-Witt型基的存在性，如果V上的辫子满足三角条件.我们将所得结果应用于具有交换余根的点霍普夫代数和低维单模U_q（sl_2）的Nichols代数。
The Hopf automorphism group and the quantum Brauer group in braided monoidal categories
为了对域$k$上的Hopf代数$H$的量子Brauer群给出一个更精确的估计，我们构造了一个包含某个辫子幺半群范畴中的Hopf代数的量子Brauer群的正合序列.设$B$是$H$上的Yetter-Drinfel'd模范畴$\C=_H ^H\YD$中的一个Hopf代数.考虑$\C$中$B$的量子Brauer群$\BQ(\C; B)$，它同构于Radford双积Hopf代数$B\rtimes H$的通常量子Brauer群$\BQ(k; B\rtimes H)$.我们证明了在一定的双曲性条件下，$\C$中的辫子存在着$B$的Hopf自同构群对前者的内在作用.我们证明了子群$\BM(\C; B)$ -在$\C$中的$B$上的模代数的Brauer群-在此作用下对于一族Radford双积Hopf代数是不变的。我们研究了$\BM(k; B\rtimes H)$的类似不变性。我们将最近的结果应用到后一个群上，生成了$B\rtimes H$的量子Brauer群的一个新子群.特别地，我们得到了一些已知的Hopf代数的量子Brauer群的新信息。
A decomposition Theorem for pointed braided Hopf algebras
本文推广了带点辫子Hopf代数的分解定理，证明在底层普通Hopf代数余半单时，其张量积分解与Yetter-Drinfeld范畴的余模结构相容。
Braided $\mathbb{Z}_q$-extensions of pointed fusion categories
我们分类编织$\mathbb{Z}_q$ -扩展的点融合类别，其中$q$是一个素数。作为应用，我们对Frobenius-Perron维数为$q^3$的模范畴进行了分类.
Quantization of Poisson Hopf algebras
本文提出一种在有理线性对称单向范畴中对泊松霍夫代数进行量子化的办法，该方法兼容张量积，并可生成辫状霍夫代数。核心思想源于群的神经是称 simplicial 集合，而霍夫代数的神经则为辫状，泊松霍夫代数的神经则为无限小辫状。
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 $\o…
Weak Hopf monoids in braided monoidal categories
我们发展了辫状幺半范畴范畴中的弱双序理论，并证明了它们在一定意义上是量子范畴。弱Hopf幺半群被证明是量子群胚。每个可分的Frobenius幺半群R导致一个弱Hopf幺半群R \otimes R。
The notion of a coherent unit action on algebraic operads was first introduced by Loday for binary quadratic nonsymmetric operads and generalized by Holtkamp, to ensure that the free objects of the operads carry a Hopf algebra structure. There was also a classification of such operads in the binary quadratic nonsymmetric case. We generalize the notion of coherent unit action to braided operads and show that free objects of braided operads with such an action carries a braided Hopf algebra structure. Under the conditions of binary, quadratic and nonsymmetric, we give a characterization and clas…
Braided Hopf Algebras and Differential Calculus
证明了量子群上的双协变微分代数可以理解为辫子Hopf代数与量子群的量子偶之间的叉积的投影.由此产生的超Hopf代数可以通过将外导数扩展到张量积来再现。
Braided Hopf algebras obtained from coquasitriangular Hopf algebras
设$(H, σ)$是余拟三角霍普夫代数，不一定是有限维的.根据Doi和Takeuchi的方法，在有限维拟三角霍普夫代数的情况下，我们定义了$H$的有限对偶$H^0$的子霍普夫代数$H_σ$.利用广义量子二重结构和带投影的霍普夫代数理论，将$H_σ^{\rm cop}$上的Yetter-Drinfeld模范畴中的辫子霍普夫代数结构与$H$联系起来.针对$H={\rm SL}_q(N)$，我们得到了在$U_q^{\rm ext}({\rm sl}_N)^{\rm cop}$上的左Yetter-Drinfeld模范畴内赋予${\rm SL}_q(N)$辫状霍普夫代数结构的显式公式.
On weakly group-theoretical non-degenerate braided fusion categories
证明了弱群论非退化辫状融合范畴的Witt类属于由非退化尖辫状融合范畴和Ising辫状范畴生成的子群.这特别适用于可解的非退化编织融合范畴。我们还利用辫子融合范畴的Frobenius-Perron维数的因子分解和简单对象的Frobenius-Perron维数给出了辫子融合范畴是弱群论的或可解的充分条件.作为应用，我们证明了Frobenius-Perron维数为小于1800的自然数或小于小于33075的奇数自然数的非退化辫子融合范畴是弱群论的.
Structure theorems for braided Hopf algebras
本文在辫子 Hopf 代数背景下发展了 Poincaré-Birkhoff-Witt 和 Cartier-Milnor-Moore 定理的版本，并利用辫子算子在辫子幺半范畴中引入了 Lie 代数的新类似物。
We show that a class of braided Hopf algebras, which includes the braided $SU_q(2)$ is obtained by twisting. We show further examples and demonstrate that twisting of bicovariant differential calculi gives braided bicovariant differential calculi.
We describe Lagrangian algebras in twisted Drinfeld centres for finite groups. Using the full centre construction, we establish a 1-1 correspondence between Lagrangian algebras and module categories over pointed fusion categories.
On braided near-group categories
我们证明了$\mathbb{C}$上的任意一个不可逆简单对象的融合范畴都是球面的。此外，我们对所有配备编织的类别进行分类。
The octonions form an Azumaya algebra in certain braided linear Gr-categories
通过将八元数看作是某些张量范畴中的一个结合代数，或者更确切地说是一个被2-上链扭曲的群代数，我们证明了八元数在某些适当的辫状线性Gr范畴中构成一个Azumaya代数.
Weak braiding for algebras in braided monoidal categories
本文提出弱辫子张量范畴的定义，证明辫子G交叉范畴、辫子张量范畴中交换代数的模范畴均属此类，且合理假设下经扭曲函子得到的弱辫子范畴为群交叉辫子化范畴。
Integrals of Braided Hopf Algebras
引入了无限辫子霍普夫代数$H$的忠实拟对偶$H^d$和严格拟对偶$H^{d'}$，证明了每个严格拟对偶$H^{d'}$都是$H$ -Hopf模.找到了积分与$H^{d}$的极大有理$H^{d}$ -子模$H^{d rat}$之间的联系，即证明了$H^{d rat}\cong \int ^l_{H^d} \otimes H$.给出了辫子霍普夫代数在Yetter-Drinfeld范畴$(^B_B{\cal YD},C)$中积分的存在唯一性.
The monoidal category of Yetter-Drinfeld modules over a weak braided Hopf algebra
本文在严格幺半范畴范畴中引入了弱算子的概念和具有可逆对极的弱辫子霍普夫代数上的Yetter-Drinfeld模理论。我们证明了此类对象的类构成一个非严格的幺半范畴。它也表明，这个范畴是不是平凡的，也就是说，它允许对象产生的伴随作用（coaction）与弱辫霍普夫代数。
We introduce the notions of sovereign, spherical and balanced quasi-Hopf algebra. We investigate the connections between these, as well as their connections with the class of pivotal, involutory and ribbon quasi-Hopf algebras, respectively. Examples of balanced and ribbon quasi-Hopf algebras are obtained from a sort of double construction which associates to a braided category (resp. rigid braided) a balanced (resp. ribbon) one.
We revisit the notion of interacting Frobenius Hopf algebras for ZX-calculus in quantum computing, with focus on allowing the algebras to be noncommutative and coalgebras to be noncocommutative. We introduce the notion of *-structures in ZX-calculus at this algebraic level and construct examples based on the quantum group u_q(sl_2) at a root of unity. We provide an abstract formulation of the Hadamard gate at this level and clarify its relationship to Hopf algebra self-duality. We then solve the problem of extending the notion of interacting Hopf algebras and ZX-calculus to take place in a bra…
On Müger's centralizer for a certain class of braided fusion categories
我们提供了一个一般公式的Mueger的中心化的任何融合子范畴的辫子融合范畴包含一个tannakian子范畴。这需要一个描述的Mueger的中心化的所有融合子范畴的一组理论编织融合范畴。
We study infinitesimal gauge transformations of an equivariant noncommutative principal bundle as a braided Lie algebra of derivations. For this, we analyse general $K$-braided Hopf and Lie algebras, for $K$ a (quasi)triangular Hopf algebra of symmetries, and study their representations as braided derivations. We then study Drinfeld twist deformations of braided Hopf algebras and of Lie algebras of infinitesimal gauge transformations. We give examples coming from deformations of abelian and Jordanian type. In particular we explicitly describe the braided Lie algebra of gauge transformations of…
Braided Zestings of Verlinde Modular Categories and Their Modular Data
本文针对单位根处量子群相关的模范畴，分类并构造了所有辫状zesting数据，还基于对应李代数的根系推导了新模范畴模数据的闭式公式。
Integrals for braided Hopf algebras
设H是刚性辫子monoidal范畴中幂等元可裂的Hopf代数.我们证明了存在的积分（在）H的特点是普遍的性质，采用有关的结果Hopf模，并表明，他们的共同目标（源）对象IntH是可逆的。得到了对极四次方的Radford公式的全辫形式。研究了积分与叉积和嬗变的关系。结果适用于拓扑Hopf代数，例如，一个环面与一个洞，这并不具有加法结构。
This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra …
We give a short proof for a well-known formula for the rank of a $G$-crossed braided extension of a modular tensor category.
Braided cofree Hopf algebras and quantum multi-brace algebras
给出了辫余自由余代数上霍普夫代数结构的系统构造.相关的底层结构是辫代数和辫余代数。我们提供了这些代数和余代数与量子群有关的一些有趣的例子。我们引入量子多支撑代数作为自然框架，它是辫子代数和$\textbf{B}_\infty$ -代数的推广。这门新学科使人们能够用统一的方法来表示一些重要的代数结构。特别有趣的例子是量子拟洗牌代数。
Relative centers and tensor products of tensor and braided fusion categories
本文部分地利用模范畴的相对中心的概念，研究了辫子融合范畴上模范畴的相对张量积。特别是，我们调查的规范张量范畴结构和编织继承的相对张量积时，模块范畴本身是张量/编织。作为一个基本的例子，我们表明，类别的代表性的纤维产品的有限群可以表示在相对产品，并显示如何倍增编织融合类别所产生的预度量群。此外，我们考虑相对中心的编织融合范畴，并期待在Muger中心的相对张量积。最后，我们用一个深入的例子暗示了有限群共轭类的分类。
We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.
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.
Braided symmetries of $SU_{q,ϕ}(2)$ and Podleś Spheres
本文引入辫子量子 Hopf 代数 $\cU_{q, ϕ}$，证明量子球为其上的辫子 Hopf 模，并建立相关辫子代数结构以显式刻画 Podleś 球的对称性。
Yetter-Drinfeld categories for quasi-Hopf algebras
证明了拟霍普夫代数$H$上的Yetter-Drinfeld模的所有可能范畴都是同构的.我们还证明了有限维左Yetter-Drinfeld模范畴$\yd^{\rm fd}$是刚性的，并明确地计算了$\yd^{\rm fd}$中的正则同构.最后，证明了在\cite{bn,bpv}中引入的辫状Hopf代数$H_0$的某些子代数同构于辫状霍普夫代数，如果$H$是有限维三角拟霍普夫代数.
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