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

Abstract

We investigate the Drinfeld doubles D ( Λ n , d ) 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 D ( Λ n , d ) . 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.