Representations of The Coordinate Ring of $ GL_{q}(n) $}

TL;DR

This paper investigates the finite dimensional irreducible representations of the quantum matrix algebra associated with $GL_q(n)$, revealing their existence only at roots of unity and characterizing their dimensions and topologies.

Contribution

It establishes the conditions for the existence of irreducible representations of $ M_q(n) $ at roots of unity and describes their dimensions and topological structures.

Findings

Representations exist only when q is a root of unity.

Dimensions are of the form p^N / 2^k, with specific N and k.

Topology of state spaces varies from tori to cubes depending on k.

Abstract

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 $ ).