Let $g$ be a simple Lie algebra and $U_q(g)$ the corresponding quantum group. What are the relation between integrable representations and highest weight representations of $U_q(g)$? Are all highest weight representations of $U_q(g)$ integrable? Thank you very much.
Edit: here an integrable module is a weight module such that the actions of $e_i$ and $f_i$ are locally nilpotent (i.e. for any vector $v$ in the module, there exists a positive integer $k$, possibly dependent on $v$, such that ${\displaystyle e_{i}^{k}.v=f_{i}^{k}.v=0}$ for all $i$).
A weight module is a module with a basis of weight vectors. A weight vector is a nonzero vector $v$ such that $k_{\lambda} . v = d_{\lambda} v$ for all $\lambda$, where $d_{\lambda}$ are complex numbers for all weights $\lambda$ such that $$ {\displaystyle d_{0}=1}, \\ {\displaystyle d_{\lambda }d_{\mu }=d_{\lambda +\mu }}, $$ for all weights $\lambda, \mu$.