The present dissertation consists of four interconnected projects. In the first, we introduce and study what we call factorizable module algebras. These are $U_q(\mathfrak{g})$-module algebras $A$ which factor, potentially ...

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 ...