  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 $\mathsf{U}_q(\mathfrak{g})$ and the Hopf algebra
$\textrm{Pol}(\mathbb{G}_q)$, where $\mathbb{G}_q$ is the Drinfeld-Jimbo
quantization of a compact semisimple simply connected Lie group $\mathbb{G}$
and $\mathfrak{g}$ is a simple complex Lie algebra.
