  It is shown that the finite dimensional irreducible representations of the
quantum matrix algebra $ M_q(n) $ ( the coordinate ring of $ GL_q(n) $ ) exist
only when q is a root of unity ( $ q^p = 1 $ ). The dimensions of these
representations can only be one of the following values: $ {p^N \over 2^k } $
where $ N = {n(n-1)\over 2 } $ and $ k \in \{ 0, 1, 2, . . . N \} $ For each $
k $ the topology of the space of states is $ (S^1)^{\times(N-k)} \times [ 0 , 1
] ^{(\times (k)} $ (i.e. an $ N $ dimensional torus for $ k=0 $ and an $ N $
dimensional cube for $ k = N $ ).
