Non-commutative phase and the unitarization of GL_{p,q}(2)

TL;DR

This paper investigates how imposing hermitian conjugate relations on the two-parameter quantum group GL_{p,q}(2) introduces a non-commutative phase, leading to the unitarization and new representation of the quantum group U_{p,q}(2).

Contribution

It demonstrates the role of a non-commutative phase in unitarizing the two-parameter quantum group and constructs its representation via operator actions.

Findings

Introduction of a non-commutative phase for unitarization

Construction of the U_{p,q}(2) representation

Achievement of a real pq parameter in the quantum group

Abstract

In this paper, imposing hermitian conjugate relations on the two-parameter deformed quantum group GL_{p,q}(2) is studied. This results in a non-commutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group U_{p,q}(2) with pq real via a non-commutative phase, the representation of the algebra is built by means of the action of the operators constituting the U_{p,q}(2) matrix on states.