An analytic characterization of freeness for finitely generated discrete quantum groups

TL;DR

This paper characterizes when finitely generated discrete quantum groups are 'free' by analyzing moments and operator norms of their main characters, identifying unitary free quantum groups as the unique minimizers.

Contribution

It provides an analytic criterion for freeness in finitely generated discrete quantum groups, highlighting the special role of unitary free quantum groups in minimizing moments and operator norms.

Findings

Freeness corresponds to minimal moments of the main character.

Unitary free quantum groups uniquely minimize the moments and operator norm.

The results extend to duals of free quantum groups of Kac type.

Abstract

We prove that a freer quantum group has smaller moments of the self-adjoint main character in the category of finitely generated discrete quantum groups. As a result, the moments are minimized precisely by the unitary free quantum groups $\mathbb{F}U(Q)$. Furthermore, in the spirit of [CC22], we prove that the operator norm of the self-adjoint main character is minimized only by unitary free quantum groups, at least in the subcategory of duals of free quantum groups of Kac type.