Here's something I really feel I should know, but do not:

Let $q$ be some sufficiently nice complex number (just pretend we're working over $\mathbb Q(q)$, for example), and $V$ some simple representation of $U_q(\mathfrak g)$. Then you have the usual representation of the braid group $B_n$ on $V^{\otimes n}$. The question is: is this representation semisimple?

I haven't been able to find a direct reference to the problem, but it really sounds like it should be known so I thought I'd check here.