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

We extend the notion of dually conjugate Hopf (super)algebras to the coloured Hopf (super)algebras Hc that we recently introduced. We show that if the standard Hopf (super)algebras Hq that are the building blocks of Hc have Hopf duals H∗ q , then the latter may be used to construct coloured Hopf duals Hc∗, endowed with coloured algebra and antipode maps, but with a standard coalgebraic structure. Next, we review the case where the Hq’s are quantum universal enveloping algebras of Lie (super)algebras Uq(g), so that the corresponding H∗ q ’s are quantum (super)groups Gq. We extend the Fronsdal and Galindo universal T -matrix formalism to the coloured pairs (U c(g), Gc) by defining coloured universal T -matrices. We then show that together with the coloured universal R-matrices previously introduced, the latter provide an algebraic formulation of the coloured RTT -relations, proposed by BasuMallick. 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 (U(gl(2)), Gl(2)), and (U(gl(1/1)), Gl(1/1)). PACS: 02.10.Tq, 02.20.Sv, 03.65.Fd, 11.30.Na Running title: Duals of coloured quantum algebras To be published in J. Math. Phys. ∗Directeur de recherches FNRS; Electronic mail: [email protected] 1