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

We compute the L 2 -Betti numbers of the free C * -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(double-struck) are equal to the L 2 -Betti numbers of the representation category Rep(G(double-struck)) 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 * -tensor category.

Publication year:2017

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BOF-publication weight:6

CSS-citation score:2

Authors:International

Authors from:Higher Education

Accessibility:Open