Baxter algebras and Hopf algebras

By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.

     

Homological integral of Hopf algebras

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

     

On the order of the antipode of hopf algebras

In this paper we consider a conjecture on the order of the antipode of semisimple Hopf algebras in the Yetter-Drinfeld category and study a related property of the ordinary Hopf algebras. We show that most known examples of finite-dimensional semisimple Hopf algebras satisfy this property.     

On Finite Quantum Groups at −1

This is a contribution to the classification of finite-dimensional pointed Hopf algebras. We are concerned with the case when the group of group-like elements is Abelian of exponent 2. We attach to such a pointed Hopf algebra a generalized simply-laced Cartan matrix; we conjecture that the Hopf algebra is finite-dimensional if and only if the Cartan matrix is of finite type. We prove the conjecture for the types A_n and A_n⁽¹⁾. We obtain the classification of all possible Hopf algebras with Cartan matrix A_n. We use the lifting method developed by Hans-Jürgen Schneider and the first-named author. Presented by S. MontgomeryMathematics Subject Classifications (2000) Primary: 17B37; secondary: 16W30.This work was partially supported by CONICET, Agencia Córdoba Ciencia – CONICOR, FOMEC and Secyt (UNC).     

Pointed hopf algebras of dimension 32

We give a complete classification of the 32-dimensional pointed Hopf algebras over an algebraically closed field k with chark k ≠ 2. It turns out that there are infinite families of isomorphism classes of pointed Hopf algebras of dimension 32. In [AS1], [BDG] and [Ge] are given families of counterexamples for the tenth Kaplansky conjecture. Up to now, 32 is the lowest dimension where Kaplansky conjecture fails.     

Self-dual weak Hopf algebras

In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.     

Monomial Hopf algebras over fields of positive characteristic

In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n) are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.     

A cancellation conjecture for free associative algebras

We develop a new method to deal with the Cancellation Conjecture of Zariski in different environments. We prove the conjecture for free associative algebras of rank two. We also produce a new proof of the conjecture for polynomial algebras of rank two over fields of zero characteristic.

     

Algebraic isomorphisms of limit algebras

We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.

     

Basic Hopf Algebras of Tame Type

In continuation of the articles (Liu J Algebra 299:841–853, 2006; Huang, J Algebra 321:2650–2669, 2009) we classify all finite-dimensional basic Hopf algebras of tame type over an algebraically closed field of characteristic 0 in this paper. As consequences, we show the following statements: (1) the representation dimension of a tame basic Hopf algebra is exactly 3, (2) for a basic Hopf algebra H, if $\textrm{C}(H)\geq 3$ then it is wild. These conclusions verify a folklore conjecture and one of Rickard’s statements for the class of finite-dimensional basic Hopf algebras.     

On the class equation for Hopf algebras

We give a simple proof of the Kac-Zhu class equation for semisimple Hopf algebras over an algebraically closed field of characteristic 0.

     

The strong no loop conjecture for special biserial algebras

We establish the strong no loop conjecture for some special cases, in particular, for special biserial algebras.

     

Some properties of factorizable Hopf algebras

A direct proof without modular category theory is given of a recent theorem of Etingof and Gelaki (1998) on the dimensions of irreducible representations. Factorizable Hopf algebras are characterized in terms of their Drinfeld double, and their character rings and the group-like elements of their duals are described.

     

Prime-Dimensional Hopf Algebras

Ｐｒｉｍｅ－ＤｉｍｅｎｓｉｏｎａｌＨｏｐｆＡｌｇｅｂｒａｓＣａｉＣｈｕａｎｒｅｎ（蔡传仁）ａｎｄＣｈｅｎＨｕｉｘｉａｎｇ（陈惠香）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＹａｎｇｚｈｏｕＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ，Ｊｉａｎｇｓｕ２２５...     

L'algebre de hopf bitensorielle

For any field k of zero characteristic we give a functor from the category of k-vector spaces into the category of k-Hopf algebras, attaching to any vector space V its bitensorial pointed Hopf algebra A^v. This Hopf algebra is graded, fulfills a universal property, and contains a remarkable subspace P of primitive elements, which as a conjecture may generate the Lie algebra Prim A^v. In case V is finite-dimensional we exhibit a Hopf pairing between A^vand A^v-whose kernel contains the (Hopf) ideal generated by the elements of P of degree ? 2.     

Hopf modules and the double of a quasi-Hopf algebra

We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras. We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case. The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.

     

Pairing and Quantum Double of Multiplier Hopf Algebras

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.     

On the number of types of finite dimensional Hopf algebras

For an odd prime p we construct an infinite class of non-isomorphic Hopf algebras of dimension p ⁴ over an infinite field containing primitive p-th roots of unity, answering in the negative a long standing conjecture of Kaplansky. Oblatum 6-XI-1997 / Published online: 12 November 1998