Dear Mr Pop,
I don't know exactly what quantum matrices and iterated Ore rings are,
but from what you write I understand that, in any case, calculations with
formal expressions are involved. I don't know of any already existing
package to work with this. If the formal expressions that you work with
involve several different but commuting variables, then MAPLE looks like
the right system to use. If you just have one variable or if you work in
a non-commutative context, then it might be better to write some programs
in GAP where your objects are represented by GAP "records", with operations
specified in an "operations record". For example, in the GAP share package
"weyl" we have also realized Hecke algeras associated with finite Coxeter
groups in this way, and this works quite well. (Hecke algebras are certain
non-commutative algebras over polynomials rings, with a basis labelled by
the elements of a finite Coxeter group, say the symmetric group.) -- This
is not much, but maybe it helps you a bit further.

Sincerely yours, Meinolf Geck
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Meinolf Geck                    UFR de Math'ematiques et UMR 9994 du CNRS
geck@math.jussieu.fr              Universit'e Paris 7, 2 Place Jussieu,    
http://www.mathp7.jussieu.fr/~geck                 F--75251 Paris, France
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