This paper is devoted to constructing quasitriangular bialgebras (or Hopf algebras). The tool we use is a new coproduct, we call it the unified coproduct, in the construction of which a Hopf algebra and an algebra are connected by three algebra maps: two coactions and a generalized cocycle. Both the crossed coproduct of a bialgebra coacting on a coalgebra and the bicrossed coproduct of two bialgebras are special cases of the unified coproduct. Then we establish a bijective correspondence between the set of all quasitriangular structures on the arbitrary unified coproduct and a certain set of datum related to the components of the unified coproduct. As the main application, we derive the necessary and sufficient conditions for the crossed coproduct (or bicrossed coproduct) to be a quasitriangular Hopf algebra.

In this work, the notion of an L-R crossed product is introduced as a unified approach for L-R smash product and crossed product. Then the duality theorem for L-R crossed product is given. As the applications of the main result, some classical results in some materials can be obtained.

In this work, the notion of a twisted partial Hom-Hopf action is introduced, and the conditions on partial cocycles are established in order to construct partial Hom-crossed products. Also, the equivalence of partial Hom-crossed products is discussed. Finally, we shall describe the coquasitriangular structures on partial Hom-crossed products

In this paper, we use elementary linear algebra methods to explore possible Hopf algebra structures within the generalized quaternion algebra. The sufficient and necessary conditions that make the generalized quaternion algebra a Hopf algebra are given. It is proven that not all of the generalized quaternion algebras have Hopf algebraic structures. When the generalized quaternion algebras have Hopf algebraic structures, we describe all the Hopf algebra structures. Finally, we shall prove that all the Hopf algebra structures on the generalized quaternion algebras are isomorphic to Sweedler Hopf algebra, which is consistent with the classification of 4-dimensional Hopf algebras.