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 $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ 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.

v4: Minor changes, final version, to appear in Analysis and PDE. v3: We corrected a mistake in Section 3.2 without impact on the main results of the paper. v2: No major changes, but several remarks and a few references were added