Orderings and *-Orderings on Cocommutative Hopf Algebras

Our aim is to construct new examples of totally ordered and ∗-ordered noncommutative integral domains. We will discuss the following classes of rings: enveloping algebras U(L), group rings G and smash products U(L) G. All of them are examples of Hopf algebras. Characterizations of orderability for enveloping algebras and group rings and of ∗-orderability for enveloping algebras have been found before and will be recalled in the article. Our main results are: for and L finite–dimensional, we characterize the orderability of U(L) G; for , we give a necessary and a sufficient condition for ∗-orderability of G (G orderable, respectively, G residually ‘torsion-free nilpotent’). Moreover, for and L finite-dimensional, we reduce the problem of characterizing the ∗-orderability of U(L) G to the problem of characterizing the ∗-orderability of G. The latter remains open. The research of the first author was supported by the Ministry of Education, Science and Sport of the Republic of Slovenia under grant P1-0222 (Algebraic methods in operator theory). The research of the second and third author was supported by the Natural Sciences and Engineering Research Council of Canada.     

A Hopf Operad of Forests of Binary Trees and Related Finite-Dimensional Algebras

The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaf-labeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests.     

On Hopf Algebras and Their Generalizations

We survey Hopf algebras and their generalizations. In particular, we compare and contrast three well-studied generalizations (quasi-Hopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (Hopf monads and hopfish algebras). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.     

Order Structure on the Algebra of Permutations and of Planar Binary Trees

Let X _n be either the symmetric group on n letters, the set of planar binary n-trees or the set of vertices of the (n – 1)-dimensional cube. In each case there exists a graded associative product on _n0 K[X _n]. We prove that it can be described explicitly by using the weak Bruhat order on S _n, the left-to-right order on planar trees, the lexicographic order in the cube case.     

辫子Hopf代数的积分

本文引进了无限维辫子Hopf代数日的忠实拟对偶H~d和严格拟对偶H~(d′)．证明了每个严格拟对偶H~(d′)是一个H-Hopf模．发现了H~d的极大有理H~d-子模H~(drat)与积分的关系,即：H~(drat)≌∫_(H~d)~l■H．给出了在Yetter-Drinfeld范畴(_B~ByD,C)中的辫子Hopf代数的积分的存在性和唯—性．     

Coloured peak algebras and Hopf algebras

For G a finite abelian group, we study the properties of general equivalence relations on G _n = G ⁿ ⋊ _n , the wreath product of G with the symmetric group _n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of G _n as well as graded connected Hopf subalgebras of ⨁ _{n≥ o} G _n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects. 2000 Mathematics Subject Classification Primary: 16S99; Secondary: 05E05, 05E10, 16S34, 16W30, 20B30, 20E22Bergeron is partially supported by NSERC and CRC, CanadaHohlweg is partially supported by CRC     

Faà di Bruno subalgebras of the Hopf algebra of planar trees from combinatorial Dyson-Schwinger equations

We consider the combinatorial Dyson-Schwinger equation X=B⁺(P(X)) in the non-commutative Connes-Kreimer Hopf algebra of planar rooted trees HNCK, where B⁺ is the operator of grafting on a root, and P a formal series. The unique solution X of this equation generates a graded subalgebra AN,P of HNCK. We describe all the formal series P such that AN,P is a Hopf subalgebra. We obtain in this way a 2-parameters family of Hopf subalgebras of HNCK, organized into three isomorphism classes: a first one, restricted to a polynomial ring in one variable; a second one, restricted to the Hopf subalgebra of ladders, isomorphic to the Hopf algebra of quasi-symmetric functions; a last (infinite) one, which gives a non-commutative version of the Faà di Bruno Hopf algebra. By taking the quotient, the last class gives an infinite set of embeddings of the Faà di Bruno algebra into the Connes-Kreimer Hopf algebra of rooted trees. Moreover, we give an embedding of the free Faà di Bruno Hopf algebra on D variables into a Hopf algebra of decorated rooted trees, together with a non-commutative version of this embedding.     

The Steenrod Algebra and Theories Associated to Hopf Algebras

We show that the homotopy category of products of Z/p-Eilenberg–Mac Lane spaces is an -algebra which algebraically is determined by the Steenrod algebra considered as a Hopf algebra with unstable structure.     

半拟三角弱Hopf代数

作为拟三角弱Hopf代数的推广,我们引入了半拟三角弱Hopf代数的概念.令(H,R,v)是一个半拟三角弱Hopf代数,其中,R是其半拟三角结构.我们指明R保持了拟三角弱Hopf代数中泛R-矩阵的许多基本性质.特别地,讨论了Drinfeld元的性质,证明其是可逆的并且是余作用v的余不变量.另外,证明了半拟三角弱Hopf代数的对极平方是对合的.     

Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

We show that the Connes–Moscovici negative cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a cyclic operad with multiplication is a cocyclic module whose simplicial cohomology is a Batalin–Vilkovisky algebra and whose negative cyclic cohomology is a graded Lie algebra of degree -2. This generalizes the fact that the Hochschild cohomology algebra of a symmetric algebra is a Batalin–Vilkovisky algebra.     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.     

用quiver构造拟三角Hopf代数

利用quiver方法确定了一个广义Taft代数具有拟三角Hopf结构当且仅当它是Sweedler 4维Hopf代数．用不同于文[15]的方法,对任意的正整数n,构造出一类拟三角Hopf代数H(n)．     

Weak Tensor Category and Related Generalized Hopf Algebras

There are at least two kinds of generalization of Hopf algebra, i.e. pre-Hopf algebra and weak Hopf algebra. Correspondingly, we have two kinds of generalized bialgebras, almost bialgebra and weak bialgebra. Let L = （L, ×, I, a, l, r） be a tensor category. By giving up I, l, r and keeping ×, a in L, the first author got so-called pre-tensor category L = （L, ×, a） and used it to characterize almost bialgebra and pre-Hopf algebra in Comm. in Algebra, 32（2）： 397-441 （2004）. Our aim in this paper is to generalize tensor category L = （L, ×, I, a, l, r） by weakening the natural isomorphisms l, r, i.e. exchanging the natural isomorphism ll^-1 = rr^-1 = id into regular natural transformations lll= l, rrr = r with some other conditions and get so-called weak tensor category so as to characterize weak bialgebra and weak Hopf algebra. The relations between these generalized （bialgebras） Hopf algebras and two kinds generalized tensor categories will be described by using of diagrams. Moreover, some related concepts and properties about weak tensor category will be discussed.     

弱Hopf代数上的扭积

在弱Hopf代数上,定义了交叉积概念,并且得到了它的两种特殊形式,冲积和扭积．特别地,给出了扭积为弱Hopf代数的一个充要条件,推广了Hopf代数的相应结论．     

高维的Frobenius型半单Hopf代数

设H是特征为零的代数闭域k上的半单Hopf代数.本文证明了如果dimкH 是小于351的奇数,则H是Frobenius型Hopf代数.     

Twist积和Twist余积Hopf代数

设H,A是两个Hopf代数,构造了twist积A#_σH和twist余积A#~(?)H,证明了文[1]中的double twist和S.Majid构造的Bicrossproduct结构以及通常的smash积都是A#_σH的一种特殊情况;文[2,3]中的twist Hopf代数以及通常的smash余积是A#~(?)H的特殊情况,最后讨论了A#~(?)H上的拟三角结构.     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.