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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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It can be done, with a non trivial amount of work. I have only managed to do this in situations where there is a PBW basis available, though. I'll post here an example of how one can deal with the (simpler!) Weytl algebra later. –  Mariano Suárez-Alvarez Oct 23 '10 at 21:18
    
Scott Morrison has written an excellent quantum groups package in mathematica. Contact him at Scott tqft net and he can tell you more. –  Noah Snyder Oct 24 '10 at 0:58
    
The GAP package Quagroup does computations with quantum groups. See gap-system.org/Packages/quagroup.html –  uunknown Oct 24 '10 at 11:28
    
@Mariano-Suarez-Alvarez, PWB basis = what? –  sleepless in beantown Oct 25 '10 at 22:15
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I just found the link to the QuantumGroups Mathematica package by Scott Morrison mentioned by Noah:

http://katlas.org/wiki/QuantumGroups%60

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This package just seems to deal with quantized enveloping algebras. Is there an equivalent version for quantized coordinate algebras? –  John McCarthy Oct 26 '10 at 17:53
    
@John McCarthy: I don't know. –  mathphysicist Oct 26 '10 at 18:24
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