Representations of the coordinate ring of GL q (3)

It is shown that finite-dimensional irreducible representations of the quantum matrix algebraM_q(3) (the coordinate ring of GL_q(3)) exist only whenq is a root of unity (q^p = 1). The dimensions of these representations can only be one of the following values:p³,p³/2,p³/4, orp³/8. The topology of the space of states ranges between two extremes, from a three-dimensional torusS¹ ×S¹ ×S¹ (which may be thought of as a generalization of the cyclic representation) to a three-dimensional cube [0, 1] × [0, 1] × [0, 1].