Representations of The Coordinate Ring of $G{L}_q(n)$}

Abstract

It is shown that the finite dimensional irreducible representations of the quantum matrix algebra $M_q(n)$ ( the coordinate ring of $G{L}_q(n)$ ) 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: $\frac{p^N}{2^k}$ where $N=\frac{n(n-1)}{2}$ 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 $k=N$ ).

20 pages ,report #. 93-020