Asymptotic invariants for fusion algebras associated with compact quantum groups

TL;DR

This paper introduces new asymptotic invariants for fusion algebras linked to compact quantum groups, generalizing classical concepts and analyzing their properties, including growth rates and amenability, with explicit computations for various quantum groups.

Contribution

It defines uniform Følner and Kazhdan constants for fusion algebras, relates them to amenability and growth, and computes these invariants for key quantum groups and deformations.

Findings

Computed invariants for fusion algebras of quantum SU_q(2) and SO_q(3).

Determined exponential growth rates for q-deformations of compact Lie groups.

Established relationships between invariants, amenability, and growth in quantum group contexts.

Abstract

We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous concepts studied for classical discrete groups. Specifically we introduce uniform F\o lner constants and the uniform Kazhdan constant for a regular representation of a fusion algebra, and establish a relationship between these, amenability, and the exponential growth rate considered earlier by Banica and Vergnioux. Further we compute the invariants for fusion algebras associated with % discrete duals of quantum $SU_q(2)$ and $SO_q(3)$ and determine the uniform exponential growth rate for the fusion algebras of all $q$-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups.