Representations of the quantum matrix algebra $M_{q,p}(2)$

TL;DR

This paper investigates the finite-dimensional irreducible representations of the quantum matrix algebra $M_{q,p}(2)$, revealing they exist only when q and p are roots of unity, with the state space topology being a torus or cylinder.

Contribution

It establishes the conditions for the existence of finite-dimensional irreducible representations of $M_{q,p}(2)$ and characterizes their topological structure.

Findings

Finite-dimensional irreducible representations exist only when q and p are roots of unity.

The state space topology is either a torus or a cylinder.

These representations generalize cyclic representations.

Abstract

It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra $ M_{ q,p}(2) $ ( the coordinate ring of $ GL_{q,p}(2) $) exist only when both q and p are roots of unity. In this case th e space of states has either the topology of a torus or a cylinder which may be thought of as generalizations of cyclic representations.