Vol. 10, No. 7, 2017

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

David Kyed, Sven Raum, Stefaan Vaes and Matthias Valvekens

Vol. 10 (2017), No. 7, 1757–1791

We compute the L2-Betti numbers of the free C-tensor categories, which are the representation categories of the universal unitary quantum groups Au(F). We show that the L2-Betti numbers of the dual of a compact quantum group G are equal to the L2-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 L2-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 L2-Betti number in terms of a generating set of a C-tensor category.

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$L^2$-Betti numbers, rigid $C^*$-tensor categories, discrete quantum groups, subfactors, compact quantum groups
Mathematical Subject Classification 2010
Primary: 46L37
Secondary: 16E40, 18D10, 20G42
Received: 9 February 2017
Revised: 3 May 2017
Accepted: 11 June 2017
Published: 1 August 2017
David Kyed
Department of Mathematics and Computer Science
University of Southern Denmark
DK-5230 Odense
Sven Raum
Sciences de Base
Section de Mathématiques
École polytechnique Fédérale de Lausanne
CH-1015 Lausanne
Stefaan Vaes
Department of Mathematics
KU Leuven
3001 Leuven
Matthias Valvekens
Department of Mathematics
KU Leuven
3001 Leuven