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 (or mu is an element of R \ {0} in the type A case). It is well known that the category of unitary representations of G(mu) is a braided tensor C*-category. We show that any braided tensor*-functor rho: Rep(G(mu)) -> M to another braided tensor C*-category with irreducible tensor unit is full if vertical bar mu vertical bar not equal 1. In particular, the functor of restriction RepG(mu) -> Rep(K) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley-Lieb category T(+/- d) for d > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.