Torsion-freeness for fusion rings and tensor C*-categories

TL;DR

This paper extends the concept of torsion-freeness from discrete quantum groups to fusion rings and tensor C*-categories, establishing new criteria and properties relevant for quantum group theory and noncommutative geometry.

Contribution

It introduces torsion-freeness for fusion rings and shows its implications for discrete quantum groups, including preservation under certain products and applications to free unitary quantum groups.

Findings

Discrete quantum groups are torsion-free if their fusion rings are torsion-free.

Duals of free unitary quantum groups are strongly torsion-free.

Torsion-freeness is preserved under Cartesian and free products.

Abstract

Torsion-freeness for discrete quantum groups was introduced by R. Meyer in order to formulate a version of the Baum-Connes conjecture for discrete quantum groups. In this note, we introduce torsion-freeness for abstract fusion rings. We show that a discrete quantum group is torsion-free if its associated fusion ring is torsion-free. In the latter case, we say that the discrete quantum group is strongly torsion-free. As applications, we show that the discrete quantum group duals of the free unitary quantum groups are strongly torsion-free, and that torsion-freeness of discrete quantum groups is preserved under Cartesian and free products. We also discuss torsion-freeness in the more general setting of abstract rigid tensor C*-categories