On the degrees of irreducible representations of Hopf algebras
Martin Lorenz
设H表示特征为0的代数闭域k上的半单霍普夫代数.我们证明了H的任何不可约表示，其特征标属于H^* 的中心的次数必整除H的维数。
Let H denote a semisimple Hopf algebra over an algebraically closed field k of characteristic 0. We show that the degree of any irreducible representation of H whose character belongs to the center of H^* must divide the dimension of H .
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Semisimplicity criteria for irreducible Hopf algebras in positive characteristic
证明了有限维不可约正特征的霍普夫代数$H$是半单的当且仅当它是交换的半单的，当且仅当其基元的限制李代数$P(H)$是环面.这推广了Hochschild关于限制李代数的定理，也推广了Demazure，Gabriel和Sweedler关于群概型的结果，在增加了有限性假设的特殊但本质的情况下.
Coset decomposition for semisimple Hopf algebras
在有限维半单霍普夫代数中引入了双陪集的概念。这是通过考虑对偶霍普夫代数的不可约特征标集上的等价关系来实现的。作为应用，给出了正规Hopf子代数的不可约特征标的限制公式。
Degeneration, rigidity and irreducible components of Hopf algebras
本文讨论退化、形变和刚性的概念，并将它们应用于霍普夫代数簇的几何研究。主要结果是n维刚性霍普夫代数和n=p^2，p素数和n<14的不可约分支的刻画.
We present a unified ring theoretic approach, based on properties of the Casimir element of a symmetric algebra, to a variety of known divisibility results for the degrees of irreducible representations of semisimple Hopf algebras in characteristic 0. All these results are motivated by a classical theorem of Frobenius on the degrees of irreducible complex representations of finite groups.
Masuoka proved (2009) that a finite-dimensional irreducible Hopf algebra $H$ in positive characteristic is semisimple if and only if it is commutative semisimple if and only if the Hopf subalgebra generated by all primitives is semisimple. In this paper, we give another proof of this result by using Hochschild cohomology of coalgebras.
Commutator Hopf subalgebras and irreducible representations
S. Montgomery和S. Witherspoon证明了当有限维的上、下半可解、半单霍普夫代数的维数不能被基域的特征整除时，它们是Froebenius型的.本文证明了有限维半单下可解霍普夫代数在任意特征下都是Froebenius型的.
Let $\left( H,R\right) $ be a finite dimensional semisimple and cosemisimple quasi-triangular Hopf algebra over a field $k$. In this paper, we give the structure of irreducible objects of the Yetter-Drinfeld module category ${} {}_{H}^{H}\mathcal{YD}.$ Let $H_{R}$ be the Majid's transmuted braided group of $\left( H,R\right) ,$ we show that $H_{R}$ is cosemisimple. As a coalgebra, let $H_{R}=D_{1}\oplus\cdots\oplus D_{r}$ be the sum of minimal $H$-adjoint-stable subcoalgebras. For each $i$ $\left( 1\leq i\leq r\right) $, we choose a minimal left coideal $W_{i}$ of $D_{i}$, and we can define th…
For a quasi-triangular Hopf algebra $\left( H,R\right) $, there is a notion of transmuted braided group $H_{R}$ of $H$ introduced by Majid. The transmuted braided group $H_{R}$ is a Hopf algebra in the braided category $_{H}\mathcal{M}$. The $R$-adjoint-stable algebra associated with any simple left $H_{R}$-comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in ${}_{H}^{H} \mathcal{YD}$. In this note, we prove for a semisimple factorizable Hopf algebra $ \left( H,R\right) $ that any simple subcoalgebra of $H_R$ is $H$-stable …
A classical theorem of I. Schur states that the degree of any irreducible complex representation of a finite group $G$ divides the order of $G/\mathscr{Z} G$, where $\mathscr{Z} G$ is the center $G$. This note discusses similar divisibility results for certain classes of Hopf algebras.
Skew Hopf algebras, irreducible extensions and the pi-method
延伸至深度2 A| B上的对偶双代数胚S：= \End {}_BA_B和T：=（A_BA_A）^B.在R是B的子代数的情况下，这是很常见的，S和T的两个非退化对将定义代数S的反自同构τ。利用双边深度为2的结构，证明了τ是对极，S是一类称为斜霍普夫代数的霍普夫代数胚.最后一节讨论了τ和非退化对如何通过深度为2的pi-方法推广到模，并且上链复形的某个导出映射是零同伦的。
A Note on Weak Post-Hopf Algebra Structures on the Sweedler Hopf Algebra
本文通过去掉酉约束定义松弛弱 post-Hopf 代数，并完整刻画 Sweedler Hopf 代数上的此类结构，揭示了区别于既有框架的新构型。
Rota-Baxter systems of Hopf algebras and Hopf trusses
本文提出Hopf代数的Rota-Baxter系统概念，探究其与Rota-Baxter Hopf代数的关系、群代数上的对应结构、派生Hopf代数及特征群的局部分解。
It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This r…
Matched pairs of Hopf algebras and Rota-Baxter Hopf algebras
本文研究了权重为-1的罗塔-巴克斯霍普夫代数与霍普夫代数配对之间的关系，并探讨了投影同态对与罗塔-巴克斯霍普夫代数结构的联系。
Depth Two Hopf Subalgebras of Semisimple Hopf algebras
设H是特征为零的代数闭域上的有限维半单霍普夫代数.本文给出了H的一个Hopf子代数是深度为2的子代数的一个简短证明当且仅当它是正规的Hopf子代数。
Connected Hopf Algebras that are Not Hopf Ore Extensions of Enveloping Algebras
本文构造了一族连通Hopf代数，证明其并非原始部分的通用 enveloping 代数的迭代Hopf Ore扩展，并展示了这些代数可表示为 enveloping 代数的迭代交叉积。
$\imath$Hopf algebras associated with self-dual Hopf algebras
受$\imath$Hall代数构造启发，本文引入了对称自对偶Hopf代数相关的$\imath$Hopf代数，基于Green公式类比证明其为有单位的结合代数，并构造了同构于$\mathbb{Z}/4\mathbb{Z}$群代数的4维$\imath$Taft代数。
Let $k$ be an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $9q^2$ over $k$, where $q$ is a prime number. We also prove that odd-dimensional semisimple Hopf algebras over $k$ of dimension less than 600 are of Frobenius type.
The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by A. Van Daele and S. H. Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, mult…
Let $L/F$ be a Galois extension of fields with Galois group isomorphic to the quaternion group of order $ 8 $. We describe all of the Hopf-Galois structures admitted by $ L/F $, and determine which of the Hopf algebras that appear are isomorphic as Hopf algebras. In the case that $ F $ has characteristic zero we also determine which of these Hopf algebras are isomorphic as $ F $-algebras and explicitly compute their Wedderburn-Artin decompositions.
Algebras graded by discrete Doi-Hopf data and the Drinfeld double of a Hopf group-coalgebra
研究了Hopf群余代数的Doi-Hopf数据和Doi-Hopf模。我们引入模块分级的离散Doi-Hopf数据，一个Doi-Hopf数据的Hopf群余代数，我们关联的代数分级的基础离散Doi-Hopf数据，使用粉碎产品类型的建设。Doi-Hopf模范畴同构于该代数上的分次模范畴。这是适用于范畴的Yetter-Drinfeld模在一个霍普夫群余代数，导致建设的Drinfeld双。证明了这个Drinfeld对偶是一个拟三角分次霍普夫代数.
We investigate the structures of Hopf $\ast$-algebra on the Radford algebras over $\mathbb {C}$. All the $*$-structures on $H$ are explicitly given. Moreover, these Hopf $*$-algebra structures are classified up to equivalence.
We investigate the theory of Hopf-Galois extensions for monoidal Hom-Hopf algebras. As the main result of this paper, we prove the Schneider's affineness theorems in the case of monoidal Hom-Hopf algebras in terms of the theory of the total integral and Hom-Hopf Galois extensions. In addition, we obtain the affineness criterion for relative Hom-Hopf module associated with faithfully flat Hom-Hopf Galois extensions.
Affineness on Noetherian graded rings, algebras and Hopf algebras
本文研究Noether分次环及Hopf代数的有限生成性，证明零次部分有限生成即可推出整体有限生成，并应用于分次Hopf代数及其辫子Hopf代数。
By using the language of cogroupoids, we show that Hopf-Galois objects of a twisted Calabi-Yau Hopf algebra with bijective antipode are still twisted Calabi-Yau, and give their Nakayama automorphism explicitly. As applications, cleft Galois objects of twisted Calabi-Yau Hopf algebras and Hopf-Galois objects of the quantum automorphism groups of non-degenerate bilinear forms are proved to be twisted Calabi-Yau.
The Factorization of Braided Hopf Algebras
给出了辫双代数或辫霍普夫代数的双因子分解，并给出了辫霍普夫代数及其因子的积分与半单性之间的关系.
In this paper, the so-called differential graded (DG for short) Poisson Hopf algebra is introduced, which can be considered as a natural extension of Poisson Hopf algebras in the differential graded setting. The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras are discussed.
Galois theory for weak Hopf algebras
我们发展了弱霍普夫代数的霍普夫-伽罗瓦论，并恢复了霍普夫代数的经典结果的类似结果.我们的方法是基于最近推出的伽罗瓦论的核心。研究了弱霍普夫代数是群胚代数或其对偶的情形。得到了强分次环的弱形式和经典Galois扩张.
This paper generalizes the Drinfel'd twist to the multiplier Hopf algebra case. For a multiplier Hopf algebra $A$ with a twist $J$, we construct a new multiplier Hopf algebra $A^{J}$. Furthermore, if $A$ is quasitriangular, then $A^{J}$ is also. Finally, for a counimodular algebraic quantum group $A$, $A^{J}$ is an algebraic quantum group.
From left modules to algebras over an operad: application to combinatorial Hopf algebras
本文的目的有两个方面：在算子上的代数的背景下，研究从S-模到分次向量空间的遗忘函子的行为，并由此理论导出组合霍普夫代数的构造。作为一个副产品，我们得到这些霍普夫代数的自由性和cofreeness结果。设O表示从S-模到分次向量空间的遗忘函子。算子P上的左模被看作是S-模范畴中的P-代数。我们将Patras和Albertenauer在结合情形下得到的结果推广到任意算子P：函子O使P-代数成为P-代数。如果P是一个Hopf操作数，则O将Hopf P-代数发送到Hopf P-代数。若算子P是正则的，则在分次向量空间范畴中得到两种不同的Hopf P-代数结构。本文发展了单位无穷小P-双代数的概念，证明了由Hopf运算所构造的霍普夫代数的自由性和余自由性。最后，我们证明了许多组合霍普夫代数都是由我们的理论衍生出来的，如置换面和结合面上的霍普夫代数.
For a Poisson algebra $A$, by exploring its relation with Lie-Rinehart algebras, we prove a Poincaré-Birkoff-Witt theorem for its universal enveloping algebra $A^e$. Some general properties of the universal enveloping algebras of Poisson Hopf algebras are studied. Given a Poisson Hopf algebra $B$, we give the necessary and sufficient conditions for a Poisson polynomial algebra $B[x; α, δ]_p$ to be a Poisson Hopf algebra. We also prove a structure theorem for $B^e$ when $B$ is a pointed Poisson Hopf algebra. Namely, $B^e$ is isomorphic to $B#_σ\mathcal{H}(B)$, the crossed product of $B$ and $\m…
We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebra. For the projectivity of a projective Hopf algebras we need some finiteness condition in terms of the module of integral. In particular we show the the module of integral has rank one.
Projectivity of Hopf algebras over subalgebras with semilocal central localizations
证明了一类Hopf子代数和右余理想子代数上的霍普夫代数的模结构的一些新结果。
Let $H$ be a pointed Hopf algebra. We show that under some mild assumptions $H$ and its associated graded Hopf algebra $\gr H$ have the same Gelfand-Kirillov dimension. As an application, we prove that the Gelfand-Kirillov dimension of a connected Hopf algebra is either infinity or a positive integer. We also classify connected Hopf algebras of GK-dimension three over an algebraically closed field of characteristic zero.
We introduce a family of braided Hopf algebras that generalizes the rank 1 Hopf algebras introduced by Krop anad Radford and we study its cleft extensions
Irreducible actions and compressible modules
代数$A$上的线性算子的任何有限集产生算子代数$B$和A上的模结构，其自同态环同构于$A$的某些不变元素的子环$A^B$。我们证明了如果$A$是一个临界可压缩左$B$ -模，那么它在$A^B$的分式环上的自内射壳$A$的维数是由$A$的一致维数和生成$B$的线性算子的个数所限定的.这扩展了一个已知的结果不可约的Hopf行动，特别适用于弱Hopf行动。进一步证明了代数A在群作用、群分次和Lie作用下临界可压缩的充要条件。
The Hopf dual $H^\circ$ of any Poisson Hopf algebra $H$ is proved to be a co-Poisson Hopf algebra provided $H$ is noetherian. Without noetherian assumption, it is not true in general. There is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. The Poisson Hopf structures on $A=k[x_1, x_2, \cdots, x_d]$, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, are exactly linear Poisson structures on $A$. The co-Poisson structures on polynomial Hopf algebra $A$ are characterized. Some correspondences between co-P…
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.
We investigate when a skew polynomial extension T = R[x; σ, δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand-Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.
We classify connected Hopf algebras of Gelfand-Kirillov dimension 4 over an algebraic closed field of characteristic zero.
Some lower bounds of GK-dimension of Hopf algebras are given.
Projective modules over Frobenius algebras and Hopf comodule algebras
本文给出了Frobenius代数和某些Hopf Galois扩张的投射模和Grothendieck群K_0和G_0的一些结果。我们的主要技术工具是Higman跟踪Frobenius代数和产品公式的Hattori-Stallings行列的投影在Hopf伽罗瓦扩展。
In this paper we study twisted algebras of multiplier Hopf ($^*$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].
Some applications of Frobenius algebras to Hopf algebras
本文以Frobenius代数理论为基础，给出了一个统一的环论方法，得到了霍普夫代数的一些结果.其中包括S. Zhu等关于不可约表示的次数、所谓的类方程、Grothendieck环的半单轨迹的确定、伴随类的谱以及伴随特征标的一个非零结果.
By using the action of certain Galois groups on complex irreducible characters and conjugacy classes, we define the Galois characters and Galois classes. We will introduce a set of Galois characters, called Galois irreducible characters, and we show that each Galois character is a non-negative linear combination of the Galois irreducible characters. It is shown that whenever the complex characters of the groups of a tower produce a positive self-dual Hopf algebra (PSH), Galois characters of the groups of the tower also produce a PSH. Then we will classify the Galois characters and Galois class…
Hopf algebra deformations of binary polyhedral groups
证明了具有2维自对偶忠实不可约余模的半单霍普夫代数总是作为商为Z_2的阿贝尔扩张得到的。我们证明了这样产生的非平凡霍普夫代数可以看作是二元多面体群的变形，并描述了它的表示范畴。我们还证明了Nichols和Richmond关于有限维余半单霍普夫代数的一个结果的加强。最后，我们给出了某些半单霍普夫代数分类的应用。
A Poincare-Birkhoff-Witt theorem for Hopf algebras with central Hopf algebra coradical
本文证明了在特征为零的域上，具有中心霍普夫代数余根的霍普夫代数有PBW基作为余根上的模。
(Hopf) Algebra Automorphisms of the Hopf algebra ${\check U}^{\geq 0}_{r,s}({\mathfrak sl_{3}})$
本文完全确定了双参数霍普夫代数${\check U}_{r,s}^{\geq 0}({\mathfrak sl_{3}})$的代数自同构群。由此也确定了$\V$的霍普夫代数自同构群.进一步刻画了子代数$U^{+}_{r,s}({\mathfrak sl_{3})}$的所有导子，并计算了它的一次Hochschild上同调群.
We give a degree 8 separable extension having two non-isomorphic Hopf-Galois structures with isomorphic underlying Hopf algebras.
We studied both the double cross product and the bicrossproduct constructions for the Hom-Hopf algebras of general $(α,β)$-type. This allows us to consider the universal enveloping Hom-Hopf algebras of Hom-Lie algebras, which are of $(α,{\rm Id})$-type. We show that the universal enveloping Hom-Hopf algebras of a matched pair of Hom-Lie algebras form a matched pair of Hom-Hopf algebras. We observe also that, the semi-dualization of a double cross product Hom-Hopf algebra is a bicrossproduct Hom-Hopf algebra. In particular, we apply this result to the universal enveloping Hom-Hopf algebras of a…
Normal Hopf subalgebras of semisimple Hopf algebras
本文引入了半单霍普夫代数表示的核的概念。在某些特殊情况下，证明了与群表示的核类似的性质。特别地，半单霍普夫代数H的每个正规Hopf子代数都是H的一个表示的核.
Subcoalgebras and endomorphisms of free Hopf algebras
对于域上的矩阵余代数$C$，确定了$C$上的自由霍普夫代数，$C$上具有一个对射的自由霍普夫代数，以及对于某个固定的$d$，对射满足$C$上$S^{2d}={\rm id}$的自由霍普夫代数的所有小子余代数.利用这些信息，我们找到了这些自由霍普夫代数的自同态，并确定了霍普夫代数、具有双射对极的霍普夫代数和具有整除2d阶对极的霍普夫代数范畴的中心.
Rota-Baxter operators on cocommutative Hopf algebras and Hopf braces
本文研究了余交换Hopf代数上Rota-Baxter算子与Hopf brace及Yang-Baxter方程的关系，重点将其嵌入Rota-Baxter Hopf代数，并建立与双射1-cocycle及对称Hopf brace的联系。
We construct a class of non-commutative, non-cocommutative, semisimple Hopf algebras of dimension $2n^2$ and present conditions to define an inner faithful action of these Hopf algebras on quantum polynomial algebras, providing, in this way, more examples of semisimple Hopf actions which do not factor through group actions. Also, under certain condition, we classify the inner faithful Hopf actions of the Kac-Paljutkin Hopf algebra of dimension $8$, $H_8$, on the quantum plane.
Fundamental theorem of Poisson $(A,H)$-Hopf module for weak Hopf algebras
该研究将Poisson Hopf模基本定理推广至弱Hopf代数情形，同时推导了Poisson Hopf模范畴的相对投射性，拓展了相关代数结构理论的适用范围。
Partial Actions of a Hopf algebra on its base field and the corresponding partial smash product algebra
本研究引入λ-Hopf代数概念并证明所有Hopf代数均为λ-Hopf代数，提出Hopf代数在基域上部分作用的计算方法，还给出若干Hopf代数族的全部此类部分作用。
Post-Hopf group algebras, Hopf group braces and Rota-Baxter operators on Hopf group algebras
该研究提出Hopf群brace、后Hopf群代数、Rota-Baxter Hopf群代数三类推广概念，并证明余交换条件下前两者可互推、第三者可导出Hopf群brace，厘清了三类代数结构的关联。
Partial (co)actions of Hopf algebras and partial Hopf-Galois theory
引入了霍普夫代数$H$在代数上的部分（余）作用。为此，我们首先介绍的概念，宽松的核心，推广Wisbauer的概念弱核心。我们还有松懈环的双重概念。给出了部分$H$ -余代数的几个对偶结果，发展了部分$H$ -余代数的伽罗瓦论.
Local Quasitriangular Hopf Algebras
我们发现了一类新的霍普夫代数--局部拟三角霍普夫代数，它推广了拟三角霍普夫代数.利用这些霍普夫代数，我们系统地得到了杨-巴克斯特方程的解.局部拟三角霍普夫代数上具有有限圈的模范畴是一个辫张量范畴。
Cosemisimple Hopf algebras with antipode of arbitrary finite order
证明了存在任意有限偶数阶的余半单霍普夫代数.我们还讨论了这类霍普夫代数的Schur指标。
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