Cartan subproduct systems
Suvrajit Bhattacharjee, Olof Giselsson, Sergey Neshveyev
TL;DR
Cartan subproduct systems build a bridge between representation theory of quantum groups and operator algebras by forming subproduct systems from highest-weight modules $V_{n\lambda}$ of $G_q$ and analyzing their Cuntz–Pimsner algebras $\mathcal{O}_{\lambda,q}$. The authors introduce a conjectural asymptotic behavior of Clebsch–Gordan coefficients, prove ergodicity of the $G_q$-action under this regime, and identify $\mathcal{O}_{\lambda,q}$ with algebras of quantized functions on homogeneous spaces $G_q^\lambda\backslash G_q$ (or dual), while studying the gauge-invariant Toeplitz algebra as a model for Berezin quantization and the boundary compactifications of discrete quantum spaces. They verify the CG conjecture for $G=SU(N)$ and weights regular or multiples of $\omega_1$, and for $\lambda=\omega_1$ they generalize Arveson’s symmetric subproduct results to the Cartan setting. The work extends Landsman–Rieffel-type convergence results to higher rank and $q\neq1$, providing a framework for continuous-field structures and quantum flag manifold boundaries.
Abstract
Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $λ$, the highest weight $G_q$-modules $V_{nλ}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic behavior of Clebsch-Gordan coefficients we identify the corresponding Cuntz-Pimsner algebras with algebras of quantized functions on homogeneous spaces of $G$. We also show that the gauge-invariant part of the Toeplitz algebra provides a model for convergence of full matrix algebras to quantum flag manifolds, complementing and generalizing results of Landsman and Rieffel for $q=1$ and results of Vaes-Vergnioux in the rank one case for $q\ne1$. We verify our conjecture on Clebsch-Gordan coefficients for $G=SU(n)$ and all weights that are either regular or multiples of the fundamental weight $ω_1$. For $λ=ω_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems.