Duality theorem for L-R crossed coproducts

In this paper, the notion of L-R crossed coproduct is introduced as a unified approach for smash coproducts, crossed coproducts and L-R smash coproducts of Hopf algebras.A duality theorem for L-R crossed coproduct is proved.

Constructing the Cocyclic Structures for Crossed Coproduct Coalgebras

In this paper, we construct a cocylindrical object associated to two coalgebras and a cotwisted map. It is shown that there exists an isomorphism between the cocyclic object of the crossed coproduet coalgebra induced from two coalgebras with a cotwisted map and the cocyclic object related to the diagonal of the cocylindrical object.

The Global Dimensions of Crossed Products and Crossed Coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.

On the Structure of the B-cocleft Module Coalgebra

In this paper we develope the notions of crossed coproduct of Hopf algebras and study an equivalent theorem of generalized crossed coproduct of H weakly comodule coalgebras and H module coalgebras. The main result is to prove a structure theorem about B cocleft H module coalgebras.

A Generalized Biproduct Theorem

We develop the Radford's biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H H be two linear maps verifying suitable conditions. We consider in this paper a twisted Hopf crossed coproduct B ×βα H and derive a necessary and sufficient condition for B # ×βα H with a Hopf smash product structure to be a bialgebra which generalizes in [14, Theorem 1.1] and the well-known Radford biproduct theorem [10, Theorem 1] .

Twisted partial coactions of Hopf algebras

In this paper, the notion of a twisted partial Hopf coaction is introduced. The conditions on partial cocycles are established in order to construct partial crossed coproducts. Then the classification of partial crossed coproducts is discussed. Finally, some necessary and sufficient conditions for a class of partial crossed coproducts to be quasitriangular bialgebras are given.