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

Abstract

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $G{L}_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $G{L}_{p,q}(2)$ 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 $G{L}_q(2)$ and $S{L}_q(2)$ (or $S{U}_q(2)$) are obtained when $p$ $=$ $q$. Representations of $U_{\bar{q},q}(2)$ $(q$ $\in$ $\C \backslash \R$ and $U_q(2)$ $(q$ $\in$ $\mathbb{R}\backslash \{0\})$ 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 $G{L}_{p,q}(2)$.

17 pages, LaTeX (latex twice), no figures. Changes consist of more general formula (4.13) for T-matrices, explicit Clebsch-Gordan coefficients, boson realization of group parameters, and typographical corrections

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