Cylinder Coalgebras and Cylinder Coproducts for Quasitriangular Hopf Algebras

Abstract:

      This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras, and points out that there exists an anti-coalgebra isomorphism $(H, \overline{\Delta})\cong (H, \tilde{\Delta})$, where $(H, \overline{\Delta})$ is the cylinder coproduct, and $(H, \tilde{\Delta})$ is the braided coproduct given by Kass. For any finite dimensional Hopf algebra $H$, the Drinfel'd double $(D(H), \overline{\Delta}_{D(H)})$ is proved to be the cylinder coproduct. Let $(H, H, R)$ be copaired Hopf algebras. If $R\in Z(H\otimes H)$ with inverse $R^{-1}$ and skew inverse $\Re$, then the twisted coalgebra $(H^{\Re})^{R^{-1}}$ 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.