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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.

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A complete analysis of this is given in the paper by Orellana-Ram. Actually, they consider the action of the affine braid group on $M\otimes V^{\otimes n}$, but you can recover your case by taking $M$ to be the trivial module. I believe that this is is a semisimple representation (assuming you mean $V$ to be finite dimensional), but in any case it is all spelled out in the paper above.

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I'm only on page 10 right now, but so far, it looks like the result that comes closest to what I wanted isn't quite there: they prove in Prop. 3.6 thet their modules $M\otimes V^{\otimes n}$ are what they call calibrated, i.e. semisimple under precisely the abelian subgroup of the affine braid group that comes from the "affine-ness" (the subgroup of $\tilde B_n$ generated by the $X^{\varepsilon_i}$, in the notation of the paper).

I'll look around some more, maybe it's either speled out or it follows from what they're doing. Anyway, thanks for the quick response.

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