Factorization of quasitriangular structures of smash biproduct bialgebras

In this paper, we consider the factorization and reconstruction of quasitriangular structures of smash biproduct bialgebras. Let [Formula: see text] be a smash biproduct bialgebra. Under condition that [Formula: see text] is right conormal, we prove that [Formula: see text] is quasitriangular if and only if there exists a set of normalized elements [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] satisfying a certain series of identities. In this case, the quasitriangular structure of [Formula: see text] is given as [Formula: see text]. Our result generalizes the similar results for Radford’s biproduct Hopf algebras studied by L. Zhao and W. Zhao, for bicrossproduct Hopf algebras studied by Zhao, Wang and Jiao, and for the dual Hopf algebras of double cross product Hopf algebras studied by Jiao.