rt.representation theory - Are there interesting monoidal structures on representations of quantum affine algebras?

The fusion product for affine Lie algebras is closely related to the existence of "evaluation homomorphisms" from the loop algebra to the finite-dimensional semisimple Lie algebra g, which split the natural inclusion of g as the subalgebra of constant loops. In the quantum case there is no evaluation map from the quantum affine algebra to the finite-type quantum algebra outside of type A (this is proved - at least for Yangians - in Drinfeld's original paper I'm pretty sure).

You see consequences of this in lots of places: e.g. for representations of g, the evaluation homomorphisms mean any irreducible representation for g can be lifted to an irreducible representation of the affine Lie algebra Lg. On the other hand, irreducible representations of the associated quantum groups do not (necessarily) lift to representations of quantum affine algebras, and so one asks about "minimal affinizations" -- irreducible finite dimensional representations of the quantum affine algebra which have the given irreducible as a constituent when restricted to the finite-type quantum group.

That said, the "ordinary" tensor product for finite dimensional representations of quantum affine algebras is pretty interesting -- it's not braided any more for example.