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 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.

Question: For a given highest weight $\lambda\in \mathbb N$, how many indecomposable objects are there in $\mathcal O$ of that given highest weight?

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?