Let $\mathfrak g=\mathfrak{sl}(2)$.
Let $\zeta$ be a primitive root of unity of even order. Say $\zeta=e^{2\pi i/6}$, for concreteness.
Let $U_q\mathfrak g$ be Lusztig's integral form of the quantized enveloping algebra, specialised at $q=\zeta$. This is an algebra generated by elements $E,F,K$, and by the divided powers $E^{(r)}$, $F^{(r)}$ of $E$ and $F$.
(I work with the convention according to which $[2]_q=q+q^{-1}$)
Let $\mathcal O$ be the category of finite dimensional (type $I$, integrable) representations of $U_q\mathfrak g$.
This category is not semi-simple, but I have the feeling that it should be possible to understand it completely.
> <b>Question:</b> For a given highest weight $\lambda\in \mathbb N$, what are the indecomposable objects in $\mathcal O$ of that given highest weight? (and in particular, how many of them are there?)
For example, for $q=e^{2\pi i/6}$, the answer starts as follows:
$$
\begin{matrix}
\lambda: & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\
\text{# of indec. modules of h. weight $\lambda$}:
& 1 & 1 & 1 & 4 & 4 & 1 & ? & ?
\end{matrix}
$$
> How does the sequence $1,1,1,4,4,1,\ldots$ continue?<br>
(Every third number in that sequence seems to be a $1$ --- what are the other numbers?)
For what it's worth, here are some pictures of the 12 indecomposable modules mentioned above. The $V_n$ are Weyl modules (also called standard modules); the $V_n^*$ are dual Weyl modules; the $L_n$ are simple modules; and the $T_n$ are tilting modules:<br>
$V_0:$ • <br>
$V_1:$ •⇄• (each dot represents a basis vector, the arrows represent $E$ and $F$) <br>
$V_2:$ •⇄•⇄• <br>
$V_3:$ •→•⇄•←• <br>
$V_3^*:$ •←•⇄•→• <br>
$L_3:$ • • (the two dots are conected by $E^{(3)}$ and $F^{(3)}$, not depicted in the picture) <br>
$T_3:$ is an extension of $V_1$ by $V_3$ <br>
$V_4:$ •⇄•→•←•⇄• <br>
$V_4^*:$ •⇄•←•→•⇄• <br>
$L_4:$ •⇄• •⇄• (again there's some $E^{(3)}$ and $F^{(3)}$ not depicted) <br>
$T_4:$ is an extension of $V_0$ by $V_4$ <br>
$V_5:$ •⇄•⇄• •⇄•⇄• <br>