On the braided structures of Radford's biproduct

We obtain the necessary and sufficient conditions for Radford’s biproduct to be a braided Hopf algebra. As an application, a nontrivial example is given.     

Weak Hopf Algebra in Yetter-Drinfeld Categories and Weak Biproducts

The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in the weak Hopf algebra over a field k. For a weak Hopf algebra A in left Yetter-Drinfeld category HHYD. we prove that the weak biproducts of A and H is a weak Hopf algebra.     

Bitwistor and Quasitriangular Structures of Bialgebras

In this article, we first introduce the notion of a bitwistor and discuss conditions under which such bitwistor forms a bialgebra as a generalization of the well-known Radford's biproduct. Then, in order to obtain new quasitriangular bialgebras, we consider a construction called twisted tensor biproduct, which is a special case of bitwistor bialgebra, and give a necessary and sufficient condition for such twisted tensor biproduct to admit quasitriangular structures.     

Double Biproduct Hom-Bialgebra and Related Quasitriangular Structures

Let(H, β) be a Hom-bialgebra such that β~2= id_H.(A, α_A) is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category (_H~H)YD and(B, α_B) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YD_H~H. The authors define the two-sided smash product Hom-algebra(A■H■B, α_A ? β ? α_B) and the two-sided smash coproduct Homcoalgebra(A◇H◇B, α_A ? β ? α_B). Then the necessary and sufficient conditions for(A■H■B, α_A ? β ? α_B) and(A◇H◇B, α_A ? β ? α_B) to be a Hom-bialgebra(called the double biproduct Hom-bialgebra and denoted by(A_◇~■H_◇~■B, α_A ? β ? α_B)) are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra(A◇H, α_A ? β) to be quasitriangular are given.     

Weak crossed biproducts and weak projections

In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we compute explicitly the weak crossed biproduct associated with a groupoid that admits an exact factorization.     

Multiplier Hopf Algebras in Categories and the Biproduct Construction

Let B be a regular multiplier Hopf algebra. Let A be an algebra with a non-degenerate multiplication such that A is a left B-module algebra and a left B-comodule algebra. By the use of the left action and the left coaction of B on A, we determine when a comultiplication on A makes A into a “B-admissible regular multiplier Hopf algebra.” If A is a B-admissible regular multiplier Hopf algebra, we prove that the smash product A # B is again a regular multiplier Hopf algebra. The comultiplication on A # B is a cotwisting (induced by the left coaction of B on A) of the given comultiplications on A and B. When we restrict to the framework of ordinary Hopf algebra theory, we recover Majid’s braided interpretation of Radford’s biproduct. Presented by K. Goodearl.     

On Semisimple Hopf Algebras of Dimension pq r

Let p and q be distinct prime numbers. We prove a result on the existence of nontrivial group-like elements in a certain class of semisimple Hopf algebras of dimension pq ^r . We conclude the classification of semisimple Hopf algebras A of dimension pq ² over an algebraically closed field k of characteristic zero, such that both A and A ^* are of Frobenius type. We also complete the classification of semisimple Hopf algebras of dimension pq ²<100.     

扭Smash双积

设H是双代数 ,A是H 双模代数 ,且为左H 余模余代数 .本文构造一种新的代数—扭Smash双积A×H ,推广了扭Smash积和Smash双积 .我们给出了扭Smash双积A×H作成双代数的充要条件 ,证明了当A ,H都是Hopf代数时 ,A×H 也是Hopf代数 ,利用映射系统刻画了双代数A×H的结构 ,并考虑了它的对偶情况 .