| Cylinder Coalgebras and Cylinder Coproducts for Quasitriangular Hopf Algebras |
| Received:March 04, 2005 |
| Key Words:
quasitriangular Hopf algebras cylinder coalgebras cylinder coproducts braided coproducts.
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| Fund Project:the National Natural Science Foundation of China (10571153), and Postdoctoral Science Foundation of China (2005037713) |
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| 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. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2006.04.001 |
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