An analytic characterization of freeness for finitely generated discrete quantum groups

TL;DR

The paper develops a moment-method framework to characterize freeness and freest objects within the category of finitely generated discrete quantum groups. By defining and analyzing moments of the self-adjoint main character with respect to the Haar state, it proves that freest (i.e., most unconstrained) objects minimize these moments, with unitary free quantum groups $\mathbb{F}U(Q)$ uniquely attaining this minimum. It provides a precise monotonicity result: if one quantum group is a quotient of another, its moments are uniformly smaller, and equality of all moments forces isomorphism; this yields explicit moment formulas for $\mathbb{F}U(Q)$ in terms of Catalan numbers. Additionally, the work connects moment minimization to the operator norm, establishing a lower bound $n(\mathbb{F}U(Q))=2\sqrt{2}$ and proving a partial affirmative characterization of freeness in the Kac-type subcategory, where the minimum is realized only by unitary free quantum groups. Overall, the results advance the understanding of freeness vs. freest in quantum groups through a combinatorial-tensor categorical approach, with potential implications for quantum group quotients and representation theory.

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.

Figures (2)

• Figure :
• Figure :

Theorems & Definitions (16)

• Definition 2.1
• Proposition 3.1
• proof
• Corollary 3.2
• Proposition 3.3
• proof
• Theorem 3.4
• proof
• Corollary 3.5
• proof