Representations of The Coordinate Ring of $GL_{q}(3)$

TL;DR

This paper classifies finite-dimensional irreducible representations of the quantum matrix algebra $ M_q(3) $ at roots of unity, revealing specific dimension constraints and topological structures of the state space.

Contribution

It provides a complete classification of irreducible representations of $ M_q(3) $ at roots of unity, including their dimensions and topological characteristics.

Findings

Representations exist only when q is a root of unity.

Possible dimensions are $ p^3 $, $ p^3/2 $, $ p^3/4 $, or $ p^3/8 $.

The topology of the state space varies from a 3-torus to a cube.

Abstract

It is shown that the finite dimensional ireducible representations of the quantum matrix algebra $ M_q(3) $ ( the coordinate ring of $ GL_q(3) $ ) 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^3 , { p^3 \over 2 } , { p^3 \over 4 } $ or $ { p^3 \over 8 } $ . The topology of the space of states ranges between two extremes , from a 3-dimensional torus $ S^1 \times S^1 \times S^1 $ ( which may be thought of as a generalization of the cyclic representation ) to a 3-dimensional cube $ [ 0 , 1 ]\times [ 0 , 1 ]\times [ 0 , 1 ] $ .