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

Abstract

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 $\bigl(U(gl(1/1)), Gl(1/1)\bigr)$.