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.

Question: 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?
(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:
$V_0:$ •
$V_1:$ •⇄•             (each dot represents a basis vector, the arrows represent $E$ and $F$)
$V_2:$ •⇄•⇄•
$V_3:$ •→•⇄•←•
$V_3^*:$ •←•⇄•→•
$L_3:$ •             •    (the two dots are conected by $E^{(3)}$ and $F^{(3)}$, not depicted in the picture)
$T_3:$ is an extension of $V_1$ by $V_3$
$V_4:$ •⇄•→•←•⇄•
$V_4^*:$ •⇄•←•→•⇄•
$L_4:$ •⇄•         •⇄•    (again there's some $E^{(3)}$ and $F^{(3)}$ not depicted)
$T_4:$ is an extension of $V_0$ by $V_4$
$V_5:$ •⇄•⇄•   •⇄•⇄•

  • 2
    $\begingroup$ Hi, André. This recent paper treats in detail the indecomposable modules over Temperley-Lieb algebras, which are related to $U_q \mathfrak{sl}(2)$. I don't know if it's directly relevant, but I hope that I might be useful. $\endgroup$ – Igor Khavkine May 27 '16 at 16:49

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