Matrix Quantum Groups of Type

To a Hecke symmetry R there associate the matrix bialgebra ER and the matrix Hopf algebra HR, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and compute the integral on the function ring of the quantum group, i.e, on HR. Further, we show that the fusion coefficients of simple representations depend only on the rank of the symmetry. In the general case, we show that the quantum semi-group is " Zariski " dense in the quantum group. This enables us to study the semi-simplicity of the associated quantum group in some case.