Abstract
Abstract 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 α in H ∗ {H^{\ast}} such that ε ( α ) = 1 {\varepsilon(\alpha)=1} , and we show that the crossed coproduct H R {H^{R}} introduced by the Harrison cocycle R is an FS-coalgebra when ( H , R ) {(H,R)} is a finite-dimensional quasitriangular Hopf algebra or a Long copaired bialgebra.
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