Cylinder Coalgebras and Cylinder Coproducts for Quasitriangular Hopf Algebras

This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras,and points out that there exists an anti-coaigebra isomorphism (H,■)■(H,■),where (H,■) is the cylinder coproduct,and (H,■) is the braided coproduct given by Kass.For any finite dimensional Hopf algebra H,the Drinfel'd double (D(H),■D(H)) is proved to be the cylinder coproduct.Let (H,H,R) be copaired Hopf algebras.If R∈Z(H■ H) with inverse R~(-1) and skew inverse ■,then the twisted coalgebra (H■)~(R~(-1)) is constructed via twice twists,whose comultiplication is exactly the cylinder coproduct.Moreover,for any generalized Long dimodule,some solutions for Yang-Baxter equations,four braid pairs and Long equations are constructed via cylinder twists.