Finite dimensional representations of the quantum group $GL_{p,q}(2)$ using the exponential map from $U_{p,q}(gl(2))$

TL;DR

This paper demonstrates how finite-dimensional representations of the quantum group $GL_{p,q}(2)$ can be constructed via exponential mapping from the quantum algebra $U_{p,q}(gl(2))$, extending known results and analyzing representation structures.

Contribution

It introduces a method to derive finite-dimensional representations of $GL_{p,q}(2)$ using the exponential map from $U_{p,q}(gl(2))$, generalizing previous quantum group representations.

Findings

Finite-dimensional representations obtained via exponential map.

Extension of known $q$-deformed group representations to $(p,q)$-deformations.

Analysis of Clebsch-Gordan coefficients for representation reduction.

Abstract

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},... $; $j$ $=$ 1/2 corresponds to the defining 2-dimensional $T$-matrix. The earlier results on the finite-dimensional representations of $GL_q(2)$ and $SL_q(2)$ (or $SU_q(2)$) are obtained when $p$ $=$ $q$. Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\R \backslash \{0\})$ are also considered. The structure of the Clebsch-Gordan matrix for $U_{p,q}(gl(2))$ is studied. The same Clebsch-Gordan coefficients are applicable in the reduction of the direct product representations of the quantum group $GL_{p,q}(2)$.