On the Hopf (co)center of a Hopf algebra

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for Uq(g)Uq(g) and the Hopf algebra Pol(Gq)Pol(Gq), where GqGq is the Drinfeld–Jimbo quantization of a compact semisimple simply connected Lie group GG and gg is a simple complex Lie algebra.