Continuous tensor categories from quantum groups I: algebraic aspects

Abstract: 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.