$L^2$-Betti numbers of rigid $C^*$-tensor categories and discrete quantum groups

Abstract

We compute the $L^2$-Betti numbers of the free $C^{\ast }$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u\left(F\right)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $\mathbb{G}$ are equal to the $L^2$-Betti numbers of the representation category $Rep\left(\mathbb{G}\right)$ and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of $L^2$-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first $L^2$-Betti number in terms of a generating set of a $C^{\ast }$-tensor category.