A Number Of Studies Related To Smash Coproduct

In this paper, firstly, we study the braided structure of the so-called twisted biproduct Hopf algebra B H. A necessary and sufficient condition for the twisted biproduct Hopf algebra B H to be braided is obtained. Secondly, we study the quasitriangular structure over the double crossproduct Hopf algebra by a similar method and find a necessary and sufficient condition for the double crossproduct Hopf algebra to be quasitriangular. Finally, a necessary and sufficient condition for the twisted Smash coproduct comodule categories to be braided monoidal categories is given.