Gromov-Hausdorff Convergence of Spectral Truncations for Quantum Groups

TL;DR

This paper establishes the Gromov-Hausdorff convergence of spectral truncations for a broad class of quantum groups, extending previous results from tori to include groups like SU(N) and SO(N).

Contribution

It introduces a new framework for spectral truncations on quantum groups and proves their convergence, broadening the scope of quantum metric geometry.

Findings

First Gromov-Hausdorff convergence result for spectral truncations on quantum groups.

Applicable to compact and discrete quantum groups with polynomial growth.

Extends convergence results from tori to groups like SU(N) and SO(N).

Abstract

We study the quantum Gromov-Hausdorff convergence of spectral truncations for compact quantum groups. Using a proper length function, we define a Dirac operator and the associated spectral truncations. This work extends the previous convergence results for tori (Leimbach-van Suijlekom) to a broad class of quantum groups, and provides the first Gromov-Hausdorff convergence result for spectral truncations on quantum groups, encompassing both compact and discrete quantum groups. Our results are applicable to $SU(N)$,$SO(N)$ and discrete quantum groups with strong polynomial growth.