The dual (p,q)(p,q)-Alexander-Conway Hopf algebras and the associated universal T{\cal T}-matrix

The dually conjugate Hopf algebras $Fun_{p,q}(R)$ and $U_{p,q}(R)$ associated with the two-parametric $(p,q)$-Alexander-Conway solution $(R)$ of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebra $U_{p,q}(R)$ is extracted. The universal ${\cal T}$-matrix for $Fun_{p,q}(R)$ is derived. While expressing an arbitrary group element of the quantum group characterized by the noncommuting parameters in a representation independent way, the ${\cal T}$-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal ${\cal R}$-matrix and the FRT matrix generators, $L^{(\pm )}$, for $U_{p,q}(R)$ are derived from the ${\cal T}$-matrix.Comment: LaTeX, 15 pages, to appear in Z. Phys. C: Particles and Field

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

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 $GL_{p,q}(2)$.Comment: 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 correction