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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.
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Yuki Arano, Kenny De Commer, Torsion-freeness for fusion rings and tensor C*-categories. J. Noncommut. Geom. 13 (2019), no. 1, pp. 35–58