Representation theory of the Drinfeld doubles of a family of Hopf algebras

We investigate the Drinfeld doubles image of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfeld doubles, from which we deduce properties of their representations, including the Auslander–Reiten quivers of the image. We then determine decompositions of the tensor products of most of the representations described, and in particular give a complete description of the tensor product of two simple modules. This study also leads to explicit examples of Hopf bimodules over the original Hopf algebras.