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

Duals of coloured quantum universal enveloping algebras and coloured universal T\cal T-matrices

We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras ${\cal H}^c$ that we recently introduced. We show that if the standard Hopf (super)algebras ${\cal H}_q$ that are the building blocks of ${\cal H}^c$ have Hopf duals ${\cal H}_q^*$, then the latter may be used to construct coloured Hopf duals ${\cal H}^{c*}$, endowed with coloured algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the ${\cal H}_q$'s are quantum universal enveloping algebras of Lie (super)algebras $U_q(g)$, so that the corresponding ${\cal H}_q^*$'s are quantum (super)groups $G_q$. We extend the Fronsdal and Galindo universal ${\cal T}$-matrix formalism to the coloured pairs $(U^c(g), G^c)$ by defining coloured universal ${\cal T}$-matrices. We then show that together with the coloured universal $\cal R$-matrices previously introduced, the latter provide an algebraic formulation of the coloured RTT-relations, proposed by Basu-Mallick. This establishes a link between the coloured extensions of Drinfeld-Jimbo and Faddeev-Reshetikhin-Takhtajan pictures of quantum groups and quantum algebras. Finally, we illustrate the construction of coloured pairs by giving some explicit results for the two-parameter deformations of $\bigl(U(gl(2)), Gl(2)\bigr)$, and $\bigl(U(gl(1/1)), Gl(1/1)\bigr)$.Comment: LaTeX 2.09, 35 pages, no figur