Indecomposable, non-simple, modules of quantum groups at roots of unity

Question

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 positive integer for which $q^d=1$). If $n$ is the dimension of an irreducible, finite-dimensional representation of $U_q(\mathfrak{sl}_2)$ (over a complex vector space), then it is known that $n$ is bounded above by $$ e=\begin{cases} d, & \text{$d$: odd} \\ d/2, & \text{$d$: even.} \end{cases} $$ 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 Chari and Premet - Indecomposable restricted representations of quantum $sl_2$ (pdf abstract MSN), 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. Are there infinite dimensional, indecomposable, non-irreducibles?
3. How can the limits of such representations (either fin or inf dimensional) 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 finite-dimensional, simple, complex Lie algebra.

Related:

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

Answer 1

In the case of the rank 1 simple Lie algebra, your references give a good account of what is known. But in general, it's wise to keep in mind that many of the indecomposable $U_q(\mathfrak{g})$-modules resemble those of the universal enveloping algebra of $\mathfrak{g}$ in prime characteristic (as seen in many papers by Lusztig, Andersen, et al). Verma module analogues give for example a positive answer to your Question 2. Analogues of tilting modules and the like show the variety of examples available, and there are apparently many quotients, etc. So a listing of cases may be impossible in general.

You might however try the papers by Maxim Vybornov.

Answer 2

I apologize for self-citation, however in an old paper of mine, together with R. Giachetti The two-dimensional Euclidean quantum algebra at roots of unity, in the process of describing decomposition of tensor products of irreps at roots of unity we listed some explicit indecomposable modules for $E_q(2)$.

Let me mention that this is a way in which irreps at roots of unity behave like infinite-dimensional ones: in that he tensor products of irreps are not completely reducible any more (and this happens each time the dimension exceeds the roots of unity degree, which then, in a way, plays the role of $\infty$).

I think you may find something of this kind also in papers referring to tensor product decompositions for $SL_q(2)$ at roots of unity.