Classify indecomposable modules for Drinfeld doubles of rank-one pointed Hopf algebras

Determine a complete classification of all finite-dimensional indecomposable modules over the Drinfeld double D(H_𝒟) of a finite-dimensional pointed rank-one Hopf algebra H_𝒟 over an algebraically closed field of characteristic zero, under the assumption that the group of group-like elements G(H_𝒟) is abelian.

Background

Drinfeld doubles of pointed rank-one Hopf algebras play a central role in the representation theory of quasi-triangular Hopf algebras. Prior work by Krop and Radford classified all simple and projective indecomposable modules for D(H_𝒟) when G(H_𝒟) is abelian, but did not provide a complete description of all indecomposable modules.

The paper highlights that, despite existing partial results, the full classification problem for finite-dimensional indecomposable D(H_𝒟)-modules had not been settled at the time of writing and motivates addressing this gap under the standard setting of characteristic zero and abelian group of group-like elements.