Let $\mathcal{H}_q(d)$ denote the Iwahori-Hecke algebra of type $A$ over a field of characteristic zero. When $q = 1$, this is just the group algebra of the symmetric group on $d$ letters. In this case, it is a bialgebra (Hopf, even). In the more general situation, however, there is no obvious choice of coproduct. However, when $q$ is not a root of unity, the Hecke algebra degenerates into the group algebra of the symmetric group, and so it is in fact a bialgebra in an obscure way.

My question is, is it known when $\mathcal{H}_q(d)$ cannot be given the structure of a bialgebra?