Hua-Lin Huang, Yu Ye, Qing Zhao
In this paper we investigate pointed Hopf algebras via quiver methods. We classify all possible Hopf structures arising from minimal Hopf quivers, namely basic cycles and the linear chain. This provides full local structure information for general pointed Hopf algebras.
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Quiver Approaches to Quasi-Hopf Algebras
本文给出了拟霍普夫代数的一个可解集，推广了霍普夫代数的可解集理论.作为应用，我们得到了一些一般的结构定理，特别是准Hopf模拟的Cartier定理和Cartier-Gabriel分解定理。
This paper is devoted to the study of the quasitriangularity of Hopf algebras via Hopf quiver approaches. We give a combinatorial description of the Hopf quivers whose path coalgebras give rise to coquasitriangular Hopf algebras. With a help of the quiver setting, we study general coquasitriangular pointed Hopf algebras and obtain a complete classification of the finite-dimensional ones over an algebraically closed field of characteristic 0.
Hopf quivers
本文利用类似于Cayley图的Hopf箭图对路余代数上的分次霍普夫代数结构进行了分类.描述除了涉及余积的规范公式外，还涉及乘积的公式。这使得明确的量子洗牌产品利用自然元素在一个霍普夫双模有一个简单的“几何”的解释，在广义上，而不是系统地工作与右或左coinvariants。
Classification of PM Quiver Hopf Algebras
我们用参数描述了一类非霍普夫代数.这导致了多重Taft代数的分类以及指向的Yetter-Drinfeld模和它们对应的Nichols代数。特别地，当基场$k$是复域，$G$是有限阿贝尔群时，我们对$G$上的多重Taft霍普夫代数，$G$上的多重Taft代数和$^{kG}_{kG} {\cal YD}$上的Nichols代数进行了分类.证明了复半单李代数的量子包络代数是半路霍普夫代数的商.
We classify Hopf actions of Taft algebras T(n) on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful Z_n-action (by directed graph automorphisms) also admits inner faithful actions of a Taft algebra. Several examples for actions of the Sweedler algebra T(2) and for actions of T(3) are presented in detail. We then extend the results on Taft algebra actions on path algebras to actions of the Frobenius-Lusztig kernel u_q(sl_2), and to actions of the Drinfeld double of T(n).
The quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.
Actions of Hopf-Ore Extensions of Group Algebras on Path Algebras of Quivers
本文分类群代数 Hopf-Ore 扩张在箭图路代数上的滤过 Hopf 作用，并将结果应用于 GK 维数一和二的 Noether 素 Hopf 代数等特例。
Hopf quivers and Nichols algebras in positive characteristic
我们应用第一作者和Rosso的一个组合公式，对Hopf代数中的乘积，确定Nichols代数的结构.我们说明了这种技术，明确构造新的例子尼科尔斯代数的正特征。我们进一步描述了相应的Radford双积和这些双积的提升，它们是新的有限维点霍普夫代数.
Let $\k$ be a field and $Q$ a minimal Hopf quiver, i.e., a cyclic quiver or the infinite linear quiver, and let $\rep^{ln}(Q)$ denote the category of locally nilpotent finite dimensional $\k$-representations of $Q.$ The category $\rep^{ln}(Q)$ has natural tensor structures induced from graded Hopf structures on the path coalgebra $\k Q.$ Tensor categories of the form $\rep^{ln} (Q)$ are an interesting class of tame hereditary pointed tensor categories which are not finite. The aim of this paper is to compute the Clebsch-Gordan formulae and Green rings of such tensor categories.
A quiver approach to quasi-quantum groups with the Chevalley property
本文通过引入修正广义路余代数，给出具有对偶Chevalley性质的余拟Hopf代数的箭图刻画，并分类了有限表示型的Chevalley性质积分张量范畴。
A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver
V. Ginzburg，并独立地R. Bocklandt和L. Le Bruyn，定义了一个无限维的“项链”李代数，它与任何一个李代数正则相关。根据V.Turaev，P.Etingof和Ginzburg的建议，我们定义了一个上括号，并证明了它定义了一个李双代数结构。给出了一个量子化该李双代数的霍普夫代数，并证明了它是一个满足PBW性质的霍普夫代数.我们提出的表示到空间上的微分算子的表示，它的轨迹表示的李代数的Ginzburg。
For $(Q,W)$ a symmetric quiver with potential satisfying a Künneth-type condition, we construct (positive and negative) extensions of its K-theoretic Hall algebra which are bialgebras. In particular, there are bialgebra extensions of preprojective KHAs of a quiver $Q$ and one can construct their Drinfeld double algebra.
We explore the differential geometry of finite sets where the differential structure is given by a quiver rather than as more usual by a graph. In the finite group case we show that the data for such a differential calculus is described by certain Hopf quiver data as familiar in the context of path algebras. We explore a duality between geometry on the function algebra vs geometry on the group algebra, i.e. on the dual Hopf algebra, illustrated by the noncommutative Riemannian geometry of the group algebra of $S_3$. We show how quiver geometries arise naturally in the context of quantum princi…
We study generalised differential structures $Ω^1,d$ on an algebra $A$, where $A\tens A\to Ω^1$ given by $a\tens b\to a d b$ need not be surjective. The finite set case corresponds to quivers with embedded digraphs, the Hopf algebra left covariant case to pairs $(Λ^1,ω)$ where $Λ^1$ is a right module and $ω$ a right module map, and the Hopf algebra bicovariant case corresponds to morphisms $ω:A^+\to Λ^1$ in the category of right crossed (or Drinfeld-Radford-Yetter) modules over $A$. When $A=U(g)$ the generalised left-covariant differential structures are classified by cocycles $ω\in Z^1(g,Λ^1)…
Hopf algebras with the dual Chevalley property of finite corepresentation type
本文研究了具有有限共表示类型的Hopf代数，证明其与对偶Chevalley性质、coNakayama性质及链接有向图结构之间的等价关系。
We survey some results on counting the rational points of moduli spaces of quiver representations. We then make generalizations to Grassmannians and flags of quiver representations. These results have nice applications to the cluster algebra. Along the way, we use the full Hopf structure of the Hall algebra of a quiver.
Hopf actions of some quantum groups on path algebras
本文研究量子Borel子代数和小量子群在箭图路径代数上的Hopf作用，给出参数化及单位根情形下作用分解的充要条件，并借助张量范畴语言将相应双模范畴等价于明确关系箭图的表示范畴。
Representation theory of the Drinfel'd doubles of a family of Hopf algebras
研究了一类霍普夫代数的Drinfel'd doubles $D(Λ_{n,d})$.我们确定了它们的简单模和不可分解投射模，并得到了这些Drinfel'd double的表示式和关系，由此我们推导出它们的表示式的性质，包括$D(Λ_{n,d})$的Auslander-Reiten箭图.然后，我们确定分解的张量积的大多数表示，特别是给一个完整的描述的张量积的两个简单的模块。本研究还导致显式的例子，霍普夫双模在原来的霍普夫代数。
The necklace Lie coalgebra and renormalization algebras
我们给出了一个自然的单态从项链李余代数，定义为任何一个，Connes和Kreimer的李余代数的树，并延长这一映射从一定的箭理论的Hopf代数Connes和Kreimer的重整化Hopf代数，以及前李版本。这些结果与Turaev在2004年的结果直接类似，通过用箭图上的路径代数替换曲面上的循环代数。我们还因素的态射通过代数的弦图和解释的几何版本。然后，我们解释了所有霍普夫代数如何由预李结构唯一确定，并讨论了霍普夫代数的非交换版本。
Coradically graded Hopf algebras of tame corepresentation type
本文通过链箭图方法，刻画了具有对偶Chevalley性质的有限维Hopf代数的余表示类型，并给出了温和余表示型的充要条件。
We study coquasitriangular pointed Majid algebras via the quiver approaches. The class of Hopf quivers whose path coalgebras admit coquasitriangular Majid algebras is classified. The quiver setting for general coquasitriangular pointed Majid algebras is also provided. Through this, some examples and classification results are obtained.
Braided Hopf Algebras
在辫子张量范畴中，我们证明了Maschke定理，并给出了双交叉双积和交叉双积以及双积是双代数的充要条件.在对称张量范畴中，证明了对偶定理，构造了量子偶;在具有普通扭曲的普通向量空间范畴中，得到了代数$R$及其交叉积$R #_σH$的整体维数、弱维数、Jacobson根、Baer根之间的关系.我们还给出了余模分解与余代数分解之间的关系。我们对霍普夫代数进行了分类。在具有dim $L \le 3$的李代数$L$上得到了常值经典杨-巴克斯特方程（CYBE）的所有解.给出了$(L, \hbox {[ ]}, Δ_r, r)$是上边界（或三角）李双代数的充必要条件.
Cyclic homology of Hopf algebras
Connes和Moscovici最近提出了一种适用于霍普夫代数的循环上同调理论。在本文中，我们认为这个对象的同调框架中，在精神的Loday Quillen和Karoubi的工作的循环同调的结合代数。在群代数的情况下，我们解释的分解的经典循环同调的群代数在这个同源。我们还计算了截断的n-代数的两个循环同调。
A Hopf Algebraic Theory of the Quantum Magnusian
本文为量子Magnus展开中的图系数建立了Hopf代数理论，引入混合箭图的收缩Hopf代数，给出连通函数的普适闭式公式，并推导出量子Murua公式及与Tutte和色多项式相关的恒等式。
Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal envelo…
The Hopf algebra $Rep U_q \hat{gl}_\infty$
在极限$N \to \infty$中定义了$U_q \hat{gl}_N$的有限维多项式表示的Grothendieck群上的霍普夫代数结构.得到的霍普夫代数$Rep U_q \hat{gl}_\infty$是它的霍普夫子代数$Rep_a U_q \hat{gl}_\infty$，$a\in\C^\times/q^{2\Z}$的张量积.当$q$为通用时（分别，$q^2$是$l$阶单位元的原根），我们构造了霍普夫代数$Rep_a U_q \hat{gl}_\infty$与prounipotent原代数群$SL_\infty^-$上的正则函数代数之间的同构（分别，$\tilde{GL}_l^-$）。当$q$是一个单位根时，这个同构标识了$Rep_a U_q \hat{gl}_\infty$的Hopf子代数，该子代数由通过关于Frobenius同态的拉回获得的模所张成，该Frobenius同态具有由$\tilde{GL}_l^-$的元素的行列式的系数生成的代数。这给了我们一个基本表象的弗罗贝纽斯拉回的显式公式。此外，我们构造了与无限线性代数相关联的Hall代数的自然作用（分别为，循环的$l$顶点）的$Rep_a U_q \hat{gl}_infty$和描述的张量积的评估表示采取在固定点作为一个模块在这个霍尔代数的跨度。
We find and classify all bialgebras and Hopf algebras or `quantum groups' of dimension $\le 4$ over the field $\Bbb F_2=\{0,1\}$. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras, and among them 25 Hopf algebras with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover s…
Moyal quantization of necklace Lie algebras
我们利用Moyal型公式构造了一个与一个子空间相关联的项链李双代数的霍普夫代数量子化.
We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to …
This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions which preserve the ascending filtration on tensor algebras, and the second for actions which preserve the descending filtration on …
Representations of matched pairs of groupoids and applications to weak Hopf algebras
我们引入了匹配群胚对的集合论表示范畴。这是一个幺半范畴范畴，它被赋予一个对应于匹配对中群胚公共基上的箭点范畴的幺半群函子（遗忘函子）。我们研究monoidal函子之间的两个这样的类别的代表保持健忘函子。我们表明，这样一个monoidal函子的中心化是一个新的匹配对，我们明确构造的表示范畴。我们引入了匹配群胚对的{ \em double}和匹配态射对的{ \em广义double}的概念。我们证明了遗忘函子的中心化子是对偶匹配对的表示范畴，而单位函子（中心）的中心化子是对偶匹配对的表示范畴。我们使用这些结构来分类的辫子的类别表示的匹配对。这样的辫子是参数化的某些群理论的结构，我们称之为{ \em匹配对旋转}。最后，我们将结果表示为一对匹配群胚所对应的弱霍普夫代数。一对匹配的旋转使相应的弱霍普夫代数具有拟三角结构.匹配对的弱霍普夫代数的Drinfeld二重是与二重匹配对相关联的弱霍普夫代数。
We define and study the category $\RepQ$ of representations of a quiver in $\VFun$ - the category of vector spaces "over $\Fun$". $\RepQ$ is an $\Fun$-linear category possessing kernels, co-kernels, and direct sums. Moreover, $\RepQ$ satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra $\HQ$ of $\RepQ$, which behaves in some ways like the specialization at $q=1$ of the Hall algebra of $\on{Rep}(\Q, \mathbf{F}_q)$. We prove the existence of a Hopf algebra homomorphism of $ ρ': \U(\n_+) \rightarrow \HQ$, from the enveloping algebra of t…
Hopf algebras with the dual Chevalley property of discrete corepresentation type
本文研究了具有离散共表示类型的双Chevalley性质的Hopf代数的分类与构造，提出了相关Hopf代数的结构特征和非半单构造方法。
Old and New Minimalism: a Hopf algebra comparison
本文通过Hopf代数比较极简主义的旧新表述，证明新表述中内外部Merge源自同一运算，结构更优，并对比了外化模型与平面嵌入方案。
Some Properties and Examples of Triangular Pointed Hopf Algebras
霍普夫代数理论中的一个基本问题是由Drinfeld引入的有限维（极小）三角霍普夫代数（A，R）的分类和构造。直到最近Etingof和作者完全解决了这个问题的半单A的代数闭域的特点0和p>>dim（A）（任何p，如果一个假设，A也cosemiomSimple）。在本文中，我们采取了第一步解决这个问题的有限维点的霍普夫代数上的代数闭域k的特征0。首先证明了任意三角尖霍普夫代数A的对极的四次方是恒等式。我们这样做，通过专注于极小三角尖的霍普夫代数（A，R）（每个三角形的霍普夫代数包含一个极小三角形的子霍普夫代数）和证明，群的群样元素的A（必须是交换）的群代数承认一个极小的三角形结构，因此，A的结构的一个biproduct。我们还将我们关于对极阶的结果推广到任何有限维拟三角霍普夫代数A，其Drinfeld元u在A的任何不可约表示中充当标量（例如，当A^* 是指向的时）。其次，我们给出了一种构造有限维点霍普夫代数的方法，并对所有的极小三角结构进行了分类。最后证明了由群元和斜本原元生成的极小三角Hopf代数同构于用本文方法构造的极小三角霍普夫代数.
The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finite representation type. To this end, we first investigate the modules o…
We investigate the quasi-Hopf twist of the quantum toroidal algebra of $\mathfrak{gl}_1$ as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter $p$ is identified as the elliptic modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on $\mathbb{R}^4 \times T^2$. The same $R$ matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum…
Recently, Li and Yamazaki proposed a new class of infinite-dimensional algebras, quiver Yangian, which generalizes the affine Yangian $\mathfrak{gl}_{1}$. The characteristic feature of the algebra is the action on BPS states for non-compact toric Calabi-Yau threefolds, which are in one-to-one correspondence with the crystal melting models. These algebras can be bootstrapped from the action on the crystals and have various truncations. In this paper, we propose a $q$-deformed version of the quiver Yangian, referred to as the quiver quantum toroidal algebra (QQTA). We examine some of the consist…
We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of Varagnolo-Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized K-theoretic Hall algebra associated to the quiver by Schiffmann-Vasserot. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in Neguţ-Sala-Schiffmann to give a generators-and-relations description of the Hall algebra of any curv…
Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY) and shifted QY, were introduced, and they act on BPS states for non-compact toric Calabi-Yau threefolds. In particular, shifted QY acts on general subcrystals of the original BPS crystal. A trigonometric deformation called quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure which is useful in deriving representations. In this paper, we define the shifted QQTA and study a class of their representations. We define 1d a…
The quantized enveloping algebra $U_q$ is constructed as a quotient of the generalized quantum double $ U^{\leq 0}_q \cmdbicross_τ U^{\geq 0}_q $ associated to a natural skew pairing $ τ: U^{\leq 0}_q \otimes U^{\geq 0}_q \to k $. This double is generalized by $\cmdcalD = (\cmdfrakB (V) \cmddotrtimes F) \cmdbicross_τ (\cmdfrakB (W) \cmddotrtimes G)$, where $F$, $G$ are abelian groups, $ V \in {}^F_F \cmdcalYD $, $ W \in {}^G_G \cmdcalYD $ are Yetter-Drinfeld modules and $ \cmdfrakB (V) $, $ \cmdfrakB (W) $ are their Nichols algebras. We prove some results on Hopf ideals of $ \cmdcalD $, includ…
Protomodularity of cocommutative Hopf monoids in duoidal categories and quasitriangular Hopf algebras
本文将余交换Hopf代数范畴的原模范性扩展到拟三角情形，通过修改对极概念等方法，研究双幺半范畴中Hopf幺半群相关性质，证明了一些分解和引理，得到多个范畴的原模范性及相关结果，并扩展了双交叉积。
In this work we investigate partial actions of a Hopf algebra H on nonunital algebras and the associated partial smash products. We show that our partial actions correspond to nonunital algebras in the category of partial representations of H. The central problem of existence of a globalization for a partial action is studied in detail, and we provide sufficient conditions for the existence (and uniqueness) of a minimal globalization for associative algebras in general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we define Morita equivalence for partial Hopf actions, and w…
The coset $G$-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization $Q$ to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that $Q$ has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of $Q$ and its tensor pow…
A method of construction of finite-dimensional triangular semisimple Hopf algebras
本文的目的是给出一种构造有限维半单三角霍普夫代数的新方法，其中包括非平凡的极小代数（即非群代数）.证明了这类霍普夫代数是相当丰富的.它还发现了这样的霍普夫代数与有限群上的双射1-上循环和Drinfeld定义的量子杨-巴克斯特方程的集合论解的意想不到的联系。
We construct a minimalistic presentation of Drinfeld super Yangians in the case of special linear superalgebra associated with an arbitrary Dynkin diagram. This gives us a possibility to introduce Hopf superalgebra structure on Drinfeld super Yangians. Using complete Weyl group we classify Drinfeld super Yangians endowed with mentioned Hopf superalgebra structures. Also it is constructed an isomorphism between completions of Drinfeld super Yangians and quantum loop superalgebras.
Quiver Heisenberg algebras: a cubic analogue of preprojective algebras
本文研究有向图海森堡代数作为预项目代数的中心扩张，探讨其在正特征下的表示理论，揭示其与路径代数的表示理论关系，并提供精确的Auslander-Reiten序列描述。
Hopf images of coactions and effective symmetry of quantum principal bundles
本文研究了量子主纤维 bundle 的有效对称性，通过Hopf图像的约简方法，实现了对量子主纤维 bundle 的分类和刚性结果的确定。
In this paper, we introduce twisted and folded AR-quivers of type $A_{2n+1}$, $D_{n+1}$, $E_6$ and $D_4$ associated to (triply) twisted Coxeter elements. Using the quivers of type $A_{2n+1}$ and $D_{n+1}$, we describe the denominator formulas and Dorey's rule for quantum affine algebras $U'_q(B^{(1)}_{n+1})$ and $U'_q(C^{(1)}_{n})$, which are important information of representation theory of quantum affine algebras. More precisely, we can read the denominator formulas for $U'_q(B^{(1)}_{n+1})$ (resp. $U'_q(C^{(1)}_{n})$) using certain statistics on any folded AR-quiver of type $A_{2n+1}$ (resp…
The quantum $ε$lectrical Hopf algebra and categorification of Fock space
本文引入电KLR代数，证明其范畴化类型A电Lie代数的量子版本，并利用其循环商范畴化更高层Fock空间。
On Spineless Cacti, Deligne's Conjecture and Connes--Kreimer's Hopf Algebra
利用无刺仙人掌的小圆盘操作数的细胞模型，我们给出了三个先验不同代数结构的最小公共拓扑操作形式：（1）Deligne猜想关于Hochschild复形的解，（2）Connes和Kreimer的霍普夫代数，（3）Chas和Sullivan的弦拓扑.
On the Classification of Finite-Dimensional Triangular Hopf Algebras
C上有限维拟三角霍普夫代数的分类和显式构造是霍普夫代数理论中的一个基本问题。这些霍普夫代数构成了一类非常重要的霍普夫代数，由德林费尔德引入（例如，量子双霍普夫代数）。然而，这个有趣的问题原来是非常困难的，它仍然是广泛开放的。人们可以希望，解决三角形情况下的问题将有助于理解一般情况。有限维三角霍普夫代数至今还没有完整的分类。然而，这个问题解决了半简单的情况下，由Etingof和作者，在最小三角指出的情况下，由作者，更一般的三角霍普夫代数与Chevalley财产的Andruskiewitsch，Etingof和作者。本文的目的是报告所有这些进展，并详细解释这一理论所涉及的数学和思想。在半单的情况下，分类依赖于德利涅的定理Tannakian范畴和Movshev的理论在本质上。我们详细地解释了Movshev的理论，并请读者参阅math.QA/0007147以获得对第一个方面的详细讨论。我们还讨论了拟三角半单Hopf代数中类群元的存在性以及余三角半单霍普夫代数的表示论。最后，我们列出了相关的开放问题的文件。
Classification of Finite-Dimensional Triangular Hopf Algebras With The Chevalley Property
回想一下，三角形霍普夫代数A被称为具有Chevalley性质，如果任意两个简单A-模的张量积是半单的，或者等价地，如果A的根是一个Hopf理想。研究这类三角霍普夫代数有两个原因：第一，它包含了所有已知的有限维三角霍普夫代数的例子;第二，它可以在某种意义上被“完全理解”。也就是说，它是在我们以前的工作与Andruskiewitsch表明，任何有限维三角霍普夫代数的Chevalley性质是通过扭曲的有限维三角霍普夫代数与R-矩阵的秩至多为2，反过来，是通过“修改”的群代数有限超群。这提供了一个分类这样的霍普夫代数。本文的目标是使这种分类更有效和明确，即参数化同构类的有限维三角霍普夫代数的Chevalley性质的群论对象，类似于我们以前的工作中所做的半单的情况下。这是在定理2.2中实现的，其中这些类与某些七元组的数据放在双射中。在半单情形下，七元组简化为四元组，我们恢复了前一篇文章的结果。在最小三角尖的情况下，我们恢复了分类定理，从以前的文件的第二作者。
Hopf algebras and subfactors associated to vertex models
如果H是一个Hopf代数，其对极的平方是单位元，$v\inł(V)\otimes H$是余表示，$π:H\toł(W)$是表示，则$u=(id\otimesπ)v$满足子因子顶点模型的方程$(t\otimes id)u^{-1}=((t\otimes id)u)^{-1}$.一个普遍的构造表明，这个方程的任何解$u$都是这样产生的。一个更精细的构造表明，存在一个满足$(id\otimesπ)v=u$的"最小"三元$(H,v,π)$。本文致力于研究霍普夫代数的后一种构造。如果$u$是酉的，我们构造一个$\c^*$-范数$H$，我们发现一个新的描述的标准不变量的子因子相关的$u$。我们还讨论了"扭曲"（即$S^2\neq id$）的情况。
CoHA of Cyclic Quivers and an Integral Form of Affine Yangians
本文研究了循环有向图的CoHA，揭示其与有理Yangian的积分形式的关系，并探讨了相关代数结构与猜想。
Construction of a $C^*$-algebraic quantum groupoid from a weak multiplier Hopf algebra
本文从弱乘子Hopf代数出发，通过引入准不变性条件，利用左积分构造Hilbert空间上的算子代数，证明可构建可分型C*-代数量子广群。
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