Continuous tensor categories from quantum groups I: algebraic aspects

We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations P_λ of the quantum group Uq(slₙ₊₁) is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of Uq(sl₂). In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of P_λ⊗ P_μ into irreducibles.