Multiplet classification of highest weight modules over quantum universal enveloping algebras: The Uq(sl(3,C)) example

The multiplet classification approach is extended to the quantum universal enveloping algebras Uq(G), where G is any simple Lie algebra. In the case when the deformation parameter q is a root of unity we classify into multiplets and explicitly parametrize all highest weight modules (HWM) over Uq(sl(2,C)) and Uq(sl(3,C)). As a consequence we classify all irreducible HWM over Uq(sl(3,C)) and give the dimensions of some of them. The same is done for Uq(sl(2,C)) reproducing the results obtained earlier by other methods. (author). 28 refs