Torsion-freeness for fusion rings and tensor \(\mathrm C^*\)-categories (Q2418814)

Summary: Torsion-freeness for discrete quantum groups was introduced by \textit{R. Meyer} [Tbil. Math. J. 1, 165--210 (2008; Zbl 1161.18301)] 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 \textit{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 \(\mathrm C^*\)-categories.