Hecke algebras and the Schlichting completion for discrete quantum groups

TL;DR

This paper extends classical concepts of Hecke algebras and Schlichting completions to the setting of discrete quantum groups, analyzing their algebraic and analytic properties with new quantum constructions.

Contribution

It introduces Hecke algebras for discrete quantum groups with commensurated subgroups and adapts Schlichting completion to quantum groups, providing new tools for their analysis.

Findings

Hecke algebras for quantum groups exhibit modular properties.

Schlichting completion can be adapted to quantum groups, yielding locally compact quantum groups.

Detailed analysis of quantum HNN extension examples.

Abstract

We introduce Hecke algebras associated to discrete quantum groups with commensurated quantum subgroups. We study their modular properties and the associated Hecke operators. In order to investigate their analytic properties we adapt the construction of the Schlichting completion to the quantum setting, thus obtaining locally compact quantum groups with compact open quantum subgroups. We study in detail a class of examples arising from quantum HNN extensions.