Can anyone point me to a classification/construction of the irreducibles for $U_q(\mathfrak{sl}_n)$, or the associated small quantum groups, when the parameter $q$ is a root of unity and $n>2$? Neither Jantzen or Lusztig's quantum group books seem to help.

Edit: perhaps the best way to clarify what I mean when I refer to the `small quantum group' is to give the definition: take the $\mathbb{C}$-algebra (other fields will do) generated by (for $n=3$) $E_1, E_2, F_1, F_2, K_1, K_2$ subject to the following relations.

$$ E_1^2 E_2 - [2] E_1E_2E_1 + E_2 E_1^2 =0 $$
$$ E_2^2 E_1 - [2] E_2E_1E_2 + E_1 E_2^2 =0 $$
and the same relations on the $F$s, where $[2]$ is the quantum integer $q+q^{-1}$
$$ [E_i, F_j] = \delta_{ij} (K_i-K_i^{-1})/(q-q^{-1}) $$
$$ K_i E_j K_i^{-1} = q^{a_{ij}} E_j $$
$$ K_i F_j K_i^{-1} = q^{-a_{ij}} F_j $$
where $[a_{ij}]$ is the Cartan matrix for $\mathfrak{sl}_3$.
$$E_i^N = F_i^N = 0, K_i^N=1$$
and also define $E_{1+2} = qE_2E_1 - E_1E_2$ and $F_{1+2}$ similarly and impose $E_{1+2}^N=F_{1+2}=0$. I'm convinced this last is necessary for the algebra to be finite-dimensional, though I have seen papers omitting it --- the small quantum group above should be a finite dimensional Hopf algebra with dimension $N^8$, with a PBW basis described by Lusztig. $q$ is a primitive $N$th root of unity in the field used.

J. Amer. Math. Soc.. $\endgroup$ – Jim Humphreys Mar 2 '11 at 13:24`$N \geq 3$`

in this case, though Andersen and others have studied the extra complications when`$N=2$`

(or`$N=3$`

for type`$G_2$`

). Even though`$\mathfrak{sl}_n$`

is fairly well-behaved, it's extremely hard in practice to work out explicit results about the (finite dimensional) simple modules for quantum groups even in terms of characters and dimensions when`$n>4$'. The modular analogue also needs`

$N=p$` to be very large compared with the Coxeter number`$n$`

. $\endgroup$ – Jim Humphreys Mar 2 '11 at 15:45