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

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    The dually conjugate Hopf algebras Funp,q(R)Fun_{p,q}(R) and Up,q(R)U_{p,q}(R) associated with the two-parametric (p,q)(p,q)-Alexander-Conway solution (R)(R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebra Up,q(R)U_{p,q}(R) is extracted. The universal T{\cal T}-matrix for Funp,q(R)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 T{\cal T}-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal R{\cal R}-matrix and the FRT matrix generators, L(Β±)L^{(\pm )}, for Up,q(R)U_{p,q}(R) are derived from the T{\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))

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    Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra Up,q(gl(2))U_{p,q}(gl(2)) to the quantum group GLp,q(2)GL_{p,q}(2), we show how the (2j+1)(2j+1)-dimensional representations of GLp,q(2)GL_{p,q}(2) can be obtained by `exponentiating' the well-known (2j+1)(2j+1)-dimensional representations of Up,q(gl(2))U_{p,q}(gl(2)) for jj == 1,3/2,...1,{3/2},... ; jj == 1/2 corresponds to the defining 2-dimensional TT-matrix. The earlier results on the finite-dimensional representations of GLq(2)GL_q(2) and SLq(2)SL_q(2) (or SUq(2)SU_q(2)) are obtained when pp == qq. Representations of UqΛ‰,q(2)U_{\bar{q},q}(2) (q(q ∈\in \C \backslash \R and Uq(2)U_q(2) (q(q ∈\in R\{0})\R \backslash \{0\}) are also considered. The structure of the Clebsch-Gordan matrix for Up,q(gl(2))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 GLp,q(2)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
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