Quasitriangular bialgebras are central in the theory of quantum groups Ž . and R-matrices. If two bialgebras or Hopf algebras H and A act on each other, then one can form a bicrossed product H j A, which is defined by w x Ž . co p Majid Mj . Drinfeld double D H is such a bicrossed product H * j H, where H is a finite-dimensional Hopf algebra. The reader is directed to w x w x Ž K or R1 for details of these constructions. Dually, let two bialgebras or . Hopf algebras H and A coact on each other via comodule structure maps and , respectively, then one can form a bicrossed coproduct H j A. w x Ž . Radford R1 showed that the dual Hopf algebra D H * of Drinfeld Ž . Ž . double D H is such a bicrossed coproduct, and that D H * admits a quasitriangular structure if and only if H and H * admit a quasitriangular structure.
In general, we ask when H j A admits a quasitriangular structure, and what forms the universal R-matrices of H j A will take if H j A admits a quasitriangular structure.
In this paper we study the bicrossed coproduct and answer the question above. In Section 1, as preparation we define a weak R-matrix R to be an w invertible element in H m A satisfying two conditions similar to R1, Ž . Ž .x R QT.1 and QT.3 , and then construct a bicrossed coproduct H j A by R. Then we discuss some properties of H j R A.
In Section 2, we discuss the quasitriangular structures of H j A. We show that if H j A is quasitriangular then so are H and A, and H j A can be defined by a weak R-matrix R. Conversely, if both H and A admit a quasitriangular structure, we can construct a quasitriangular structure of H j R A. Then we give the forms of all quasitriangular structures of H j R A.