Continuous tensor categories from quantum groups I: algebraic aspects

2017/08/27 by Schrader, Gus, Shapiro, Alexander
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1708.08107

Abstract

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 U_q(\mathfraksl_n+1) is closed under tensor products. Our results generalize those of Ponsot and Teschner in the rank 1 case of U_q(\mathfraksl₂). 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.

Related

• On tensor product decomposition of positive representations of U_qq(\mathfraksl(2,ℝ))
• Representations of algebraic quantum groups and reconstruction theorems for tensor categories
• On tensor products of positive representations of split real quantum Borel subalgebra U_qq(b_R)
• Quantum G₂ categories have property (T)
• A theory of induction and classification of tensor C*-categories