I'm trying to learn how to do algebraic manipulations in Mathematica but not finding the help very helpful. I'm going to ask about a specific quantum group example related to a previous question of mine. For $SU_q(N)$, how would I use Mathematica to show that
$$
S(u^1_2)u^3_1 = q^{1}u^3_1S(u^1_2).
$$
I am, of course, assuming that such a thing can be done in Mathematica. If I am wrong in this assumption, could someone please direct me a package that can do this calculation? Gap, Magma?
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I just found the link to the QuantumGroups Mathematica package by Scott Morrison mentioned by Noah: 

