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.

55 pages, 38 figures

References in corpus (9)

• Total positivity, Grassmannians, and networks
• Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of representations of U_q(sl(2,R))
• Liouville bootstrap via harmonic analysis on a noncompact quantum group
• Modular Double of Quantum Group
• Positive Representations of Split Real Simply-laced Quantum Groups
• On the relation between the modular double of U_q(sl(2,R)) and the quantum Teichmueller theory
• A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties
• On tensor product decomposition of positive representations of ${\mathcal{U}}_{q\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$
• Cluster Realization of $U_q(\mathfrak{g})$ and Factorization of the Universal $R$-Matrix

Cited by in corpus (9)

• Positive Representations of Split Real Simply-laced Quantum Groups
• Superintegrable Systems on Moduli Spaces of Flat Connections
• On $b$-Whittaker functions
• Semiclassical geometry of integrable systems
• A cluster realization of $U_q(\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}})$ from quantum character varieties
• Heisenberg double and Drinfeld double of the quantum superplane
• Parabolic Positive Representations of ${\mathcal{U}}_q({\mathfrak{g}}_{\mathbb{R}})$
• Characteristic equation for symplectic groupoid and cluster algebras
• Quantum traces for ${\mathrm{SL}}_n(\mathbb{C})$: The case $n=3$