ar X iv : h ep - t h / 93 06 05 8 v 1 1 2 Ju n 19 93 Representations of The Coordinate Ring of GL

It is shown that the finite dimensional irreducible representations of the quantum matrix algebra Mq(n) ( the coordinate ring of GLq(n) ) exist only when q is a root of unity ( q = 1 ). The dimensions of these representations can only be one of the following values: p N 2 where N = n(n−1) 2 and k ∈ {0, 1, 2, ...N} For each k the topology of the space of states is (S1)×(N−k) × [0, 1](×(k) (i.e. an N dimensional torus for k = 0 and an N dimensional cube for k = N ).