Question

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.