Abstract.Let (A, α) be a finite-dimensional Hom-Hopf algebra.In this paper we mainly construct the Drinfel'd double D(A) = (A op ⊲⊳ A * , α ⊗ (α −1 ) * ) in the setting of Hom-Hopf algebras by two ways, one of which generalizes Majid's bicrossproduct for Hopf algebras (see [7]) and another one is to introduce the notion of dual pairs of of Hom-Hopf algebras. Then we study the relation between the Drinfel'd double D(A) and Heisenberg double H(A) = A#A * , generalizing the main result in [5]. Especially, the examples given in the paper are not obtained from the usual Hopf algebras.

Abstract. The aim of this paper is to generalize the theory of Hopf-Ore extension on Hopf algebras to Hopf group coalgebras. First the concept of Hopf-Ore extension of Hopf group coalgebra is introduced. Then we will give the necessary and sufficient condition for the Ore extensions to become a Hopf group coalgebra, and certain isomorphism between Ore extensions of Hopf group coalgebras are discussed.

Let (H,?) be a Hom-Hopf algebra and (A,?) be a Hom-algebra. In this paper we will construct the Hom-crossed product (A#?H???), and prove that the extension A ? A#?H is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make (A#?H???) into a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on (H,?).

We mainly construct a bicrossproduct for a finite-dimensional monoidal Hom-Hopf algebra (H, α), generalizing Majid's bicrossproduct. Naturally, the Hom-type bicrossproduct leads to the Drinfel'd double (H op H * , α⊗(α −1) *) with a quasitriangular structure R satisfying the quantum Hom-Yang-Baxter equations.