This is pretty specific, but there are some experts around.

So, in Chari & Pressley, it's explained that in the standard *-structure, every irreducible, finite-dimensional representation of a quantum group (at a generic parameter) is unitary. Is it written somewhere what the "right" unitary structure on a tensor product of these representations is?

I ask because if one categorifies such representations, one gets a unitary structure essentially for free, so it would extremely useful if someone had already written down one I could match up with.