Monomial Hopf Algebras

Categorical Hopf kernels and representations of semisimple Hopf algebras

In the category of semisimple Hopf algebras the Hopf kernels introduced by Andruskiewitsch and Devoto in \cite{AD} coincide with kernels of representation as introduced in \cite{Bker}. Some new results concerning the normality of kernels are also presented. It is proven that the property for Hopf algebras to have all kernels normal Hopf subalgebras is a self dual property.

Monomial Bases of Quantized Enveloping Algebras

We construct a monomial basis of the positive part of the quantized enveloping algebra associated to a finite-dimensional simple Lie algebra. As an application we give a simple proof of the existence and uniqueness of the canonical basis of the positive part.

秩 $2$ Kac-Moody 代数基本表示的单项式实现与 LS 路径

Monomial realizations and LS paths of fundamental representations for rank $2$ Kac-Moody algebras

本文针对秩 $2$ Kac-Moody 代数，显式构造了基本权的 Lakshmibai-Seshadri 路径集合与单项式实现之间的同构，并基于 Weyl 群给出了单项式实现的显式形式。

Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group H^2_L(A) associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg (with a different name) in order to generalize G.I. Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac-Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally the explicit computation of H^2_L(A) for monomial Hopf algebras and for a c…

Vertex operator algebra arising from the minimal series M(3,p) and monomial basis

We study a vertex operator algebra (VOA) V related to the M(3,p) Virasoro minimal series. This VOA reduces in the simplest case p=4 to the level two integrable vacuum module of $\hat{sl}_2$. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of $\hat{sl}_2$. Our main concern is the subspace W generated by this action from the highest weight vector of V. Using the Fourier components of a(z), we present a monomial basis of W and a semi-infinite monomial basis of V. We also give a Gordon type formula for their characters.

单项基与量子仿射代数的规范基

Monomial bases and canonical bases for quantum affine algebras

本文研究了量子仿射代数中单项基与规范基的构造方法，提出了一种基于PBW基的简单算法来计算规范基，并探讨了由此得到的规范基与Lusztig利用有向图几何构造的规范基之间的关系。

Generic base algebras and universal comodule algebras for some finite-dimensional Hopf algebras

After recalling the definitions and the properties of the generic base algebra and of the universal comodule algebra attached to a Hopf algebra by Aljadeff and the second-named author, we determine these algebras for the Taft algebras, the Hopf algebras E(n), and certain monomial Hopf algebras.

Structure of the Malvenuto-Reutenauer Hopf algebra of permutations

We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive elements and its coradical filtration, and show that it decomposes as a crossed product over the Hopf algebra of quasi-symmetric functions. In addition, we describe the structure constants of the multiplication as a certain number of facets of the permutahedron. As a consequence we obtain a new interpretation of the product of monomial quasi-symmetric functions in terms of the facial structure of th…

The universal Hopf algebra associated with a Hopf-Lie-Rinehart algebra

We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure.

Structure theorems for bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras and braided Hopf algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v:H\rightarrow B. Then we can define an object B^{co(H)} which is a left-left Yetter-Drinfeld module over H, having extra properties that allow to make a smash product B^{co(H)}# H which is an H-bicomodule algebra, isomorphic to B.

On Hopf algebras over basic Hopf algebras of dimension 24

We determine finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero, whose Hopf coradical is isomorphic to a non-pointed basic Hopf algebra of dimension $24$ and the infinitesimal braidings are indecomposable objects. In particular, we obtain families of new finite-dimensional Hopf algebras without the dual Chevalley property.

Translation Hopf algebras and Hopf heaps

To every Hopf heap or quantum cotorsor of Grunspan a Hopf algebra of translations is associated. This translation Hopf algebra acts on the Hopf heap making it a Hopf-Galois co-object. Conversely, any Hopf-Galois co-object has the natural structure of a Hopf heap with the translation Hopf algebra isomorphic to the acting Hopf algebra. It is then shown that this assignment establishes an equivalence between categories of Hopf heaps and Hopf-Galois co-objects.

Finite-dimensional Hopf algebras over the smallest non-pointed basic Hopf algebra

We classify finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero whose Hopf coradcial is isomorphic to the smallest non-pointed basic Hopf algebra, under the assumption that the diagrams are strictly graded. In particular, we obtain some new Nichols algebras of non-diagonal type and new finite-dimensional Hopf algebras without the dual Chevalley property.

Quadratic Algebras arising from Hopf operads generated by a single element

The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.

Classification of Quiver Hopf Algebras and Pointed Hopf Algebras of Type One

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.

Hopf-Galois algebras and their Poisson structures

As is known to all, Hopf-Galois objects have a significant research value for analyzing tensor categories of comodules and classification questions of pointed Hopf algebras, and are natural generalizations of Hopf algebras with a Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce the conception of Poisson Hopf-Galois algebras, and establish the relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we study Poisson Hopf-Galois structures on Poisson polynomi…

Doi-Hopf Modules over Weak Hopf Algebras

The theory of Doi-Hopf modules is generalized to Weak Hopf algebras.

Quasi-Hopf $*$-Algebras

We introduce quasi-Hopf $*$-algebras i.e. quasi-Hopf algebras equipped with a conjugation (star) operation. The definition of quasi-Hopf $*$-algebras proposed ensures that the class of quasi-Hopf $*$-algebras is closed under twisting and additionally, that any Hopf $*$-algebra becomes a quasi-Hopf $*$-algebra via twisting. The basic properties of these algebras are developed. The relationship between the antipode and star structure is investigated. Quasi-triangular quasi-Hopf $*$-algebras are introduced and studied.

The Calabi-Yau property of Hopf algebras and braided Hopf algebras

Let $H$ be a finite dimensional semisimple Hopf algebra and $R$ a braided Hopf algebra in the category of Yetter-Drinfeld modules over $H$. When $R$ is a Calabi-Yau algebra, a necessary and sufficient condition for $R#H$ to be a Calabi-Yau Hopf algebra is given. Conversely, when $H$ is the group algebra of a finite group and the smash product $R#H$ is a Calabi-Yau algebra, we give a necessary and sufficient condition for the algebra $R$ to be a Calabi-Yau algebra.

Cofree Hopf algebras on Hopf bimodule algebras

We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.

On Hopf algebras over the unique $12$-dimensional Hopf algebra without the dual Chevalley property

Let $\mathds{k}$ be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over $\mathds{k}$ whose Hopf coradical is isomorphic to the unique $12$-dimensional Hopf algebra $\mathcal{C}$ without the dual Chevalley property, such that the diagrams are strictly graded and the corresponding infinitesimal braidings are indecomposable objects in ${}_{\mathcal{C}}^{\mathcal{C}}\mathcal{YD}$. In particular, we obtain new Nichols algebras of dimension $18$ and $36$ and two families of new Hopf algebras of dimension $216$.

Some Hopf Algebras related to $\mathfrak{sl}_2$

We define a series of Artin-Schelter Gorenstein Hopf algebras $H_β$ with injective dimensions 3. Radford's Hopf algebra and Gelaki's Hopf algebra are homomorphic images of $H_β$. We determine its Grothendieck ring $G_0(H_β)$. Meanwhile we can obtain Grothendieck rings of Gelaki's Hopf algebras and Radford's Hopf algebras $U_{(N,ν,ω)}$ in \cite{R}, and non-isomorphic Hopf algebras with isomorphic Grothendieck rings.

Examples of pointed color Hopf algebras

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras.

A quasi-Hopf algebra freeness theorem

We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite-dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

On braided co-Frobenius Hopf algebras

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.

Semisimple Hopf algebras of dimension 60

We determine the isomorphism classes of semisimple Hopf algebras of dimension 60 which are simple as Hopf algebras.

Morita invariance of equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids

We investigate the equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids and derive their Morita invariance. For this, we use the tools developed by McCarthy for $k$-linear categories and subsequently by Kaygun and Khalkhali for a certain class of categories, called as $H$-categories, for a Hopf algebra $H$.

On some classification of finite-dimensional Hopf algebras over the Hopf algebra $H_{b:1}^*$ of Kashina

Let $H$ be the dual of $16$-dimensional nontrivial semisimple Hopf algebra $H_{b:1}$ in the classification work of Kashina \cite{K00}. We completely determine all finite-dimensional Nichols algebras satisfying $\mathcal{B}(N)\cong \bigotimes_{i\in I}\mathcal{B}(N_i)$, where $N=\bigoplus_{i\in I}N_i$, each $N_i$ is a simple object in $_H^H\mathcal{YD}$. Under this assumption, we classify all those Hopf algebras of finite-dimensional growth from the semisimple Hopf algebra $H$ via the relevant Nichols algebras $\mathcal B(N)$.

Radford's S^4 formula for co-Frobenius Hopf algebras

This note extends Radford's formula for the fourth power of the antipode of a finite dimensional Hopf algebra to co-Frobenius Hopf algebras and studies equivalent conditions to a Hopf algebra being involutory for finite dimensional and co-Frobenius Hopf algebras.

Quasitriangular (G-cograded) multiplier Hopf algebras

We put the known results on the antipode of a usual quasitriangular Hopf algebra into the framework of multiplier Hopf algebras. We illustrate with examples which can not be obtained using classical Hopf Algebras. The focus of the present paper lies on the class of the so-called group cograded multiplier Hopf algebras. By doing so, we bring the results of quasitriangular Hopf group-coalgebras (as introduced by Turaev) to the more general framework of multiplier Hopf algebras.

秩一尖点Hopf代数上的Hopf q-括号结构

Hopf q-braces structures on rank one pointed Hopf algebras

本文完整分类了秩一尖点Hopf代数上的Hopf q-括号结构，计算其基座并识别Hopf斜括号，进而给出同构判定条件及所有弱编织算子。

Basic quasi-Hopf algebras over cyclic groups

Let $m$ a positive integer, not divisible by 2,3,5,7. We generalize the classification of basic quasi-Hopf algebras over cyclic groups of prime order given in \cite{EG3} to the case of cyclic groups of order $m$. To this end, we introduce a family of non-semisimple radically graded quasi-Hopf algebras $A(H,s)$, constructed as subalgebras of Hopf algebras twisted by a quasi-Hopf twist, which are not twist equivalent to Hopf algebras. Any basic quasi-Hopf algebra over a cyclic group of order $m$ is either semisimple, or is twist equivalent to a Hopf algebra or a quasi-Hopf algebra of type $A(H,s…

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild co…

Families of Hopf algebras of trees and pre-Lie algebras

Using methods from math-ph/9907010, we study families of Hopf algebra structures on coloured trees.

关于Hopf代数的双交叉模

On bicrossed modules of Hopf algebras

本文利用Hopf代数胚定义非交换非余交换的Hopf 2-代数，证明带Peiffer恒等式的双交叉积Hopf代数可产生此类结构，并给出Sweedler Hopf代数的应用实例。

Involutory quasi-Hopf algebras

We introduce and investigate the basic properties of an involutory (dual) quasi-Hopf algebra. We also study the representations of an involutory quasi-Hopf algebra and prove that an involutory dual quasi-Hopf algebra with non-zero integral is cosemisimple.

Calabi-Yau pointed Hopf algebras of finite Cartan type

We study the Calabi-Yau property of pointed Hopf algebra $U(\mc{D},\lmd)$ of finite Cartan type. It turns out that this class of pointed Hopf algebras constructed by N. Andruskiewitsch and H.-J. Schneider contains many Calabi-Yau Hopf algebras. To give concrete examples of new Calabi-Yau Hopf algebras, we classify the Calabi-Yau pointed Hopf algebras $U(\mc{D},\lmd)$ of dimension less than 5.

A Poincare-Birkhoff-Witt theorem for Hopf algebras with central Hopf algebra coradical

We show that over fields of characteristic zero a Hopf algebra with central Hopf algebra coradical has a PBW basis as a module over the coradical.

Word Hopf algebras

Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generalizations such as the word Hopf algebra and the double word Hopf algebra (and many more).

(Hopf) Algebra Automorphisms of the Hopf algebra ${\check U}^{\geq 0}_{r,s}({\mathfrak sl_{3}})$

In this paper, we completely determine the group of algebra automorphisms for the two-parameter Hopf algebra ${\check U}_{r,s}^{\geq 0}({\mathfrak sl_{3}})$. As a result, the group of Hopf algebra automorphisms is determined for $\V$ as well. We further characterize all the derivations of the subalgebra $U^{+}_{r,s}({\mathfrak sl_{3})}$, and calculate its first degree Hochschild cohomology group.

On bases of quantized enveloping algebras

We give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW bases and canonical bases are also discussed.

Hom-Lie-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…

Cyclic Cohomology of Hopf algebras and Hopf Algebroids

We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.

多重ζ值霍普夫代数的同构性

Isomorphism between Hopf algebras for multiple zeta values

本文证明了多重ζ值的准洗牌代数与洗牌代数通过准对称函数具有霍普夫代数同构性，并与已知的霍夫曼等人同构性进行了比较。

Lie-Hopf algebras and their Hopf cyclic cohomology

The correspondence between Lie algebras, Lie groups, and algebraic groups, on one side and commutative Hopf algebras on the other side are known for a long time by works of Hochschild-Mostow and others. We extend this correspondence by associating a noncommutative noncocommutative Hopf algebra to any matched pair of Lie algebras, Lie groups, and affine algebraic groups. We canonically associate a modular pair in involution to any of these Hopf algebras. More precisely, to any locally finite representation of a matched pair object as above we associate a SAYD module to the corresponding Hopf al…

余交换Hopf代数和Hopf brace上的Rota-Baxter算子

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。

A class of semisimple Hopf algebras acting on quantum polynomial algebras

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.

Hopf-Frobenius Algebras and a Simpler Drinfeld Double

The zx-calculus and related theories are based on so-called interacting Frobenius algebras, where a pair of dagger-special commutative Frobenius algebras jointly form a pair of Hopf algebras. In this setting we introduce a generalisation of this structure, Hopf-Frobenius algebras, starting from a single Hopf algebra which is not necessarily commutative or cocommutative. We provide a few necessary and sufficient conditions for a Hopf algebra to be a Hopf-Frobenius algebra, and show that every Hopf algebra in the category of finite dimensional vector spaces is a Hopf-Frobenius algebra. In additi…

弱Hopf代数的Poisson $(A,H)$-Hopf模基本定理

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。

Hopf代数在其基域上的部分作用及对应的部分冲积代数

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。

Hopf-cyclic cohomology of bicrossed product Hopf algebras

In this dissertation we study the coefficients spaces (SAYD modules) of Hopf-cyclic cohomology theory over a certain family of bicrossed product Hopf algebras, and we compute the Hopf-cyclic cohomology of such Hopf algebras with coefficients. We associate a Hopf algebra, what we call a Lie-Hopf algebra, to any matched pair of Lie groups, Lie algebras and affine algebraic groups via the semi-dualization procedure of Majid. We then identify the SAYD modules over Lie-Hopf algebras with the representations and corepresentations of the total Lie group, Lie algebra or the affine algebraic group of t…

Explicit Realization of Hopf Cyclic Cohomology Classes of Bicrossed Product Hopf Algebras

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomology classes of our Hopf algebra $\cH$ and then realize these classes in terms of explicit representative cocycles in the cyclic cohomology of the convolution algebra $\cA$.

Partial (co)actions of Hopf algebras and partial Hopf-Galois theory

We introduce partial (co)actions of a Hopf algebra $H$ on an algebra. To this end, we introduce first the notion of lax coring, generalizing Wisbauer's notion of weak coring. We also have the dual notion of lax ring. Several duality results are given, and we develop Galois theory for partial $H$-comodule algebras.

Local Quasitriangular Hopf Algebras

We find a new class of Hopf algebras, local quasitriangular Hopf algebras, which generalize quasitriangular Hopf algebras. Using these Hopf algebras, we obtain solutions of the Yang-Baxter equation in a systematic way. The category of modules with finite cycles over a local quasitriangular Hopf algebra is a braided tensor category.