  We exhibit a correspondence between subcategories of modules over an algebra
and sub-bimodules of the dual of that algebra. We then prove that the
semisimplicity of certain such categories is equivalent to the existence of a
Peter-Weyl decomposition of the corresponding sub-bimodule. Finally, we use
this technique to establish the semisimplicity of certain finite-dimensional
representations of the quantum double of sl_2 for generic q.
