  We describe the algebraic ingredients of a proof of the conjecture of Frenkel
and Ip that the category of positive representations $\mathcal{P}_\lambda$ of
the quantum group $U_q(\mathfrak{sl}_{n+1})$ is closed under tensor products.
Our results generalize those of Ponsot and Teschner in the rank 1 case of
$U_q(\mathfrak{sl}_2)$. 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
$\mathcal{P}_\lambda \otimes \mathcal{P}_\mu$ into irreducibles.
