Asymptotic invariants for fusion algebras associated with compact quantum groups
Jacek Krajczok, Adam Skalski
TL;DR
The paper introduces uniform asymptotic invariants for fusion algebras, linking amenability and growth to quantitative measures. It defines and analyzes uniform Følner constants, uniform exponential growth rate, and uniform Kazhdan constants, establishing key inequalities and relations to amenability and growth ω(R). The authors compute these invariants for SU_q(2) and SO_q(3), extend growth-rate results to general q-deformations G_q, and determine the uniform exponential growth rate for the fusion algebra of the free unitary quantum group U_F^+. The findings illuminate how quantum-group fusion rules drive quantitative non-amenability, with explicit formulas and asymptotics (notably r_q for U_F^+), offering benchmarks and open questions for broader fusion-algebra 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 Følner 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.