Let us consider the quantum group $U_q(\mathfrak{sl}_2)$ (as defined in Kassel's book on quantum groups), for $q$ being a root of unity of order $d$ $\ $ (i.e. $d$ is the smallest integer for which $q^d=1$). If $n$ is the dimension of an irreducible, fin dim rep (over a complex vector space), then it is known that $n$ is bounded by $$ e=\{ \begin{array}{c} d ,& d:odd \\ d/2, & d:even \end{array} $$ As far as i know, there are indecomposable, non-simple modules of dimension higher than $e$. I have made some small search on the structure of such modules, but i have not found anything substantial apart from this article, which however refers to the restricted case. So my questions are:

1. Is there some reference on the structure of indecomposable, non-simple modules of quantum groups at roots of unity?
2. How can the limits of such representations at $q\to 1$, be computed?

I would be interested either on references or on some short -if possible- description of such modules, mainly for the case of $U_q(\mathfrak{sl}_2)$ and more generally for $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a fin dim, simple, complex Lie algebra.

Related:

• Indecomposable modules for the big quantum group
• What are the indecomposable $U_q\mathfrak{sl}(2)$-modules?