Tao Yang
For an infinite dimensional Hopf quasigroup, if the faithful integral exists, then it is unique up to scalar. Base on the faithful integrals, we construct the integral dual of a class of infinite dimensional Hopf quasigroups, and show that the integral dual has a similar structure to Hopf coquasigroup, which is a regular multiplier Hopf coquasigroup with a faithful integral.
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In this paper we introduce the notion of factorization in the Hopf quasigroup setting and we prove that, if $A$ and $H$ are Hopf quasigroups such that their antipodes are isomorphisms, a Hopf quasigroup $X$ admits a factorization as $X=AH$ iff $X$ is isomorphic to a double cross product $A\bowtie H$ as Hopf quasigroups.
The matched pair theory (of groups) is studied for a class of quasigroups; namely, the $m$-inverse property loops. The theory is upgraded to the Hopf level, and the "$m$-invertible Hopf quasigroups" are introduced.
Distributive laws and Hopf quasigroups
本文提出 Hopf 拟群间的 $a$-monoidal 分配律，证明其可诱导使 $A\otimes H$ 成为 Hopf 拟群的圈积，并统一解释多种叉积与碎积构造。
In this paper, we generalizing the main result in Liu[10] to weak Hopf coquasigroups case. We first define and study group-cograded weak Hopf quasigroups, which generalize both group-cograded Hopf quasigroups and weak Hopf group-coalgebras. Then we introduce the notion of p-Yetter-Drinfeld weak quasimodule over group-cograded weak Hopf quasigroups H. If the antipode of H is bijective, we show that the category YDWQ(H) of Yetter-Drinfeld weak quasimodules over H is a crossed category, and the subcategory YD(H) of Yetter-Drinfeld modules is a braided crossed category.
Weak Hopf quasigroups and matched pairs of quasigroupoids
本文提出拟广群体的精确分解与匹配对概念，证明匹配对可构造双重交叉积，并揭示其与弱 Hopf 拟群的关联。
Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler.
Let $H$ be a crossed group-cograded Hopf quasigroup. We first introduce the notion of $p$-Yetter-Drinfeld quasimodule over $H$. If the antipode of $H$ is bijective, we show that the category $\mathscr Y\mathscr D\mathscr Q(H)$ of Yetter-Drinfeld quasimodules over $H$ is a crossed category, and the subcategory $\mathscr Y\mathscr D(H)$ of Yetter-Drinfeld modules is a braided crossed category.
Firstly, we introduce a class of new algebraic systems which generalize Hopf quasigroups and Hopf $π-$algebras called $Q$-graded Hopf quasigroups, and research some properties of them. Secondly, we define the representations of $Q$-graded Hopf quasigroups, i.e $Q$-graded Hopf quasimodules, research the construction method and fundamental theorem of them. Thirdly, we research the smash products of $Q$-graded Hopf quasigroups.
Let $A$ be a multiplier Hopf coquasigroup. If the faithful integrals exist, then they are unique up to scalar. Furthermore, if $A$ is of discrete type, then its integral duality $\widehat{A}$ is a Hopf quasigroup, and the biduality $\widehat{\widehat{A}}$ is isomorphic to the original $A$ as multiplier Hopf coquasigroups. This biduality theorem also holds for a class of Hopf quasigroups with faithful integrals.
Hopf quasigroups and the algebraic 7-sphere
本文引入了Hopf拟群和Hopf余拟群$H$的概念，推广了逆性质拟群$G$的经典概念，分别表示为拟群代数$k G$和代数拟群$k[G]$ .证明了Hopf代数的一些基本结果，如对极$S:H\to H$的反（余）乘性，如果$H$是交换的或余交换的，则$S^2=\id$是反乘的，以及交叉（余）积的理论.我们还介绍了一个Moufang的Hopf（co）拟群的概念，并表明，坐标代数$k[S^{2^n-1}]$的可并行领域是代数拟群（交换的Hopf coquasigroups在我们的制定）和Moufang。我们利用描述的复合代数，如八元数通过上链$F$在\cite{Ma99}介绍。我们构造了一个非交换非平凡Moufang的Hopf余拟群的例子$k[S^7]\rtimes\Z_2^3$.本文利用Hopf余拟群方法研究了$k[S^7]$上的微分几何，并给出了$S^7$是可并行的一个简短的代数证明.着眼于组合的左，右不变向量场的$k[S^7]$，我们提供了一个新的描述的结构常数的李代数$g_2$的结构常数$F$的八元数。在最后一节中，我们给出了$q$ -变形量子群$\C_q[S^3]$的一个新的描述，它通常被看作是一个Moufang Hopf余拟群（因为它实际上是一个Hopf代数），但现在是根据通过Cayley-Dickson过程建立的$F$。
In this paper we introduce the notion of weak non-asssociative Doi-Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also we prove that there exists a categorical equivalence that admits as particular instances the ones constructed in the literature for Hopf algebras, weak Hopf algebras, Hopf quasigroups, and weak Hopf quasigroups.
In this paper we introduce the theory of multiplication alteration by two-cocycles for nonassociative structures like nonassociative bimonoids with left (right) division. Also we explore the connections between Yetter-Drinfeld modules for Hopf quasigroups, projections of Hopf quasigroups, skew pairings, and quasitriangular structures, obtaining the nonassociative version of the main results proved by Doi and Takeuchi for Hopf algebras.
The infinitesimal Hopf algebra and the poset of planar forests
引入了平面树的无穷小霍普夫代数，推广了非交换Connes-Kreimer霍普夫代数的构造.定义了一个非退化对和一个对偶基，并给出了平面森林顶点序对的组合解释。此外，余积和配对也可以借助平面森林集合上的偏序来描述，使其同构于Tamari偏序集。作为推论，对偶基可以用莫比乌斯反演来计算。
Let $H$ be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra $A$, homogeneously, inner-faithfully, preserving the grading on $A$, and so that $A$ is an $H$-module algebra. When the fixed subring $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$. We show that each of the semisimple Hopf algebras $H_{2n^2}$ of Pansera, and $\mathcal{A}_{4m}$ and $\mathcal{B}_{4m}$ of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.
In 1999, Y. Kashina introduced the exponent of a Hopf algebra. In this paper, we prove that the exponent of a finite dimensional non-cosemisimple Hopf algebra with Chevalley property in characteristic 0 is infinite, and the exponent of a finite dimensional non-cosemisimple pointed Hopf algebra in positive characteristic is finite.
Correspondence theorems for infinite Hopf-Galois extensions
该研究将 Hopf-Galois 理论扩展至无限域扩张，定义子扩张，并为可分（含无限）情形建立 Galois 对应，同时改进有限可分情形的已有结论。
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。
Failure of flatness over finite-dimensional Hopf subalgebras
本文证明任意有限维非半单 Hopf 代数作为子代数时存在非平坦的包含代数，并指出一类无限维 Hopf 代数保证所有 Hopf 代数在其子代数上忠实平坦。
Character groups of Hopf algebras as infinite-dimensional Lie groups
本文从无限维李理论的角度研究了霍普夫代数的特征群。对于分次连通的霍普夫代数我们构造了特征标群上的无穷维李群结构，其特征标群的值在局部凸代数中.这种结构使特征标群成为Milnor意义下正则的Baker-Campbell-Hausdorff-李群。此外，我们表明，某些子群的Hopf理想成为封闭的李群的特征群。如果霍普夫代数不是分次的，则它的特征标群一般不是李群.然而，我们证明了对于任何霍普夫代数，其值在弱完备代数中的特征标群是Hofmann和Morris意义下的亲李群.
Classification of connected Hopf algebras of dimension $p^3$ I
设$p$是素数，$k$是特征为$p$的代数闭域.本文给出了维数为$p^3$的连通霍普夫代数的分类，但当霍普夫代数的本原空间是二维阿贝尔空间时除外.每个同构类都由具有关系和霍普夫代数结构的生成元$x, y, z$表示。设$μ$是$(p^2+p-1)$ -次单位根的乘法群.当本原空间为一维且$p$为奇数时，存在同构类的无限族，该同构类自然地被$A_{k}^1/μ$参数化。
Homological Integral of Hopf Algebras
对一大类无穷维霍普夫代数引入了左、右同调积分.利用同调积分证明了无限维霍普夫代数的Maschke定理的一个版本。Maschke定理的推广和同调积分是研究一维Gelfand-Kirillov Noether正则霍普夫代数的关键。
Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras
本文研究了某些著名Hopf代数家族的无穷小R矩阵的分类，基于双代数的准三角结构定义并给出了具体结果。
Hopf algebras under finiteness conditions
本文简要地综述了无限维霍普夫代数的一些最新研究进展，这些代数要么是Noether维数，要么是有限Gelfand-Kirillov维数。列出了一些未决问题。
An algebraic framework for the Drinfeld double based on infinite groupoids
本文基于无限群胚构造 Drinfeld 双重，证明其保持弱乘子 Hopf 代数及代数量子群胚性质，并建立模对应与拟三角结构。
We describe all finite-dimensional pointed Hopf algebras whose infinitesimal braiding is a fixed Yetter-Drinfeld module decomposed as the sum of two simple objects: a point and the one of transpositions of the symmetric group in three letters. We give a presentation by generators and relations of the corresponding Nichols algebra and show that Andruskiewitsch-Schneider Conjecture holds for this kind of pointed Hopf algebras.
This is a brief survey on the current state of play on the programs to classify infinite dimensional Hopf algebras of Gel'fand-Kirillov dimension one or two. We list a number of open questions and suggest directions for future work.
Pointed and copointed Hopf algebras as cocycle deformations
我们证明了在Andruskiewitsch和Schneider的分类方案中具有相同图的所有有限维点霍普夫代数都是彼此的余圈变形。这是这样做的第一个合适的表征这样的霍普夫代数，它允许应用的结果增冈约森田竹内等价和Schauenburg约霍普夫伽罗瓦扩展。变形上循环和变形的“无穷小”部分决定了变形乘法，并且可以用Hochschild上同调来明确描述。应用程序，和结果，共点霍普夫代数也被认为是。
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.
Homological Integrals for Weak Hopf Algebras
本文引入了无限维弱Hopf代数的同调积分概念，并借此证明了若干结构定理，包括Artin-Schelter性质与Van den Bergh条件的等价性、对极映射的自动可逆性，以及关于仿射中心上有限弱Hopf代数的分解定理。
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