Cartan subproduct systems

TL;DR

This paper explores the structure of subproduct systems derived from semisimple compact Lie groups, identifying associated Cuntz-Pimsner algebras with quantized function algebras on homogeneous spaces, and analyzing convergence to quantum flag manifolds.

Contribution

It introduces a conjectural asymptotic behavior of Clebsch-Gordan coefficients and verifies it for certain cases, connecting subproduct systems with quantum homogeneous spaces and flag manifolds.

Findings

Identification of Cuntz-Pimsner algebras with quantized functions on homogeneous spaces

Modeling convergence of matrix algebras to quantum flag manifolds

Verification of the conjecture for SU(n) and specific weights

Abstract

Given a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\lambda$, the highest weight $G_q$-modules $V_{n\lambda}$ 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 $\omega_1$. For $\lambda=\omega_1$, we also provide a detailed description of the Toeplitz and Cuntz-Pimsner algebras, generalizing results of Arveson on symmetric subproduct systems.