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

We extend the notion of dually conjugate Hopf (super)algebras to the colored 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 $Hq*,$ then the latter may be used to construct coloured Hopf duals $Hc*,$ endowed with colored 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 $Hq*$’s are quantum (super)groups $Gq.$ We extend the Fronsdal and Galindo universal 𝒯-matrix formalism to the colored pairs $(Uc(g),Gc)$ by defining colored universal 𝒯-matrices. We then show that together with the colored universal ℛ-matrices previously introduced, the latter provide an algebraic formulation of the colored $RTT$-relations, proposed by Basu-Mallick. This establishes a link between the colored extensions of Drinfeld–Jimbo and Faddeev–Reshetikhin–Takhtajan pictures of quantum groups and quantum algebras. Finally, we illustrate the construction of colored 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)).