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][1] [abstract](https://www-ems-ph-org.ezproxy.tcu.edu/journals/show_abstract.php?issn=0034-5318&vol=30&iss=2&rank=11) [MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=1265478)), 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:**

 - https://mathoverflow.net/q/307436/85967
 - https://mathoverflow.net/q/239893/85967


 


  [1]: https://www.jstage.jst.go.jp/article/kyotoms1969/30/2/30_2_335/_pdf