A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

Let G be a classical compact Lie group and G_\mu the associated compact matrix quantum group deformed by a positive parameter \mu (or a nonzero and real \mu in the type A case). It is well known that the category Rep(G_\mu) of unitary f.d. representations of G_\mu is a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_\mu) to another braided tensor C*-category with irreducible tensor unit is full if |\mu|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.