Quantum Metric Structures on Iwahori-Hecke Algebras

TL;DR

This paper explores quantum metric structures on Iwahori-Hecke algebras, establishing conditions under which they form compact quantum metric spaces and analyzing their convergence to group algebras as the deformation parameter varies.

Contribution

It introduces a systematic study of quantum metric structures on Iwahori-Hecke algebras and characterizes when they satisfy Haagerup-type conditions for finite rank right-angled Coxeter systems.

Findings

Algebras satisfy Haagerup-type condition iff the Coxeter diagram's complement has no induced squares.

Iwahori-Hecke algebras inherit compact quantum metric space structures.

Deformed algebras converge to group algebras in quantum Gromov-Hausdorff distance as q approaches 1.

Abstract

Iwahori-Hecke algebras are $q$-deformations of group algebras of Coxeter groups. In this article, we initiate a systematic study of quantum metric structures on Iwahori-Hecke algebras by establishing that, for finite rank right-angled Coxeter systems, the canonical filtrations of the corresponding Iwahori-Hecke algebras satisfy the Haagerup-type condition introduced by Ozawa and Rieffel if and only if the Coxeter diagram's complement contains no induced squares. As a consequence, these algebras naturally inherit compact quantum metric space structures in the sense of Rieffel. Additionally, we investigate continuity phenomena in this framework by demonstrating that, as the deformation parameter $q$ approaches $1$, the deformed Iwahori-Hecke algebras converge to the group algebra of the Coxeter group in Latr\'emoli\`ere's quantum Gromov-Hausdorff propinquity.