  Let $k$ be a field, and $H$ a Hopf algebra with bijective antipode. If $H$ is
commutative, noetherian, semisimple and cosemisimple, then the category
${}_{H}{\mathcal {YD}}^H$ of Yetter-Drinfeld modules is semisimple. We also
prove a similar statement for the category of Long dimodules, without the
assumption that $H$ is commutative.
