FS-coalgebras and crossed coproducts

In this paper, we introduce FS-coalgebras, which provide solutions of FS-equations and also solution of braid equations considered by Caenepeel, Militaru and Zhu. FS-coalgebras are constructed by using FS-equations and Harrison cocycles. As applications, we prove that every bialgebra H is an FS-bialgebra if and only if there is a two-sided integral alpha in H* such that epsilon(alpha) = 1, and we show that the crossed coproduct H-R introduced by the Harrison cocycle R is an FS-coalgebra when (H, R) is a finite-dimensional quasitriangular Hopf algebra or a Long copaired bialgebra.