Computing in quantum groups

There is the package QuaGroup by de Graaf for both GAP and Magma: see QuaGroup. I've used it in both systems and found it to be extremely helpful. Since there's a GAP package, you also have the option of using it inside Sage.

You will potentially want to be careful about exactly which PBW basis you and the computer are working with, of course.

One very effective tool for automated computation in the positive or negative half of a quantum group is the shuffle algebra realization; see Leclerc's paper (and the references therein) for some of the mathematical details. I don't know of a specific implementation of it off the top of my head, though I'm sure one exists (maybe someone else knows of one) and it isn't too hard to implement yourself. However, since multiplication is based on shuffling words, computation time gets quite long if you consider products with lots of elements. (For instance, the not-at-all-optimized implementation I wrote for myself starts taking minutes with monomials of 7 or so elements.)