L2-Betti numbers of rigid C∗-tensor categories and discrete quantum groups

Kyed, David; Raum, Sven; Vaes, Stefaan; Valvekens, Matthias

doi:10.2140/apde.2017.10.1757

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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

DOI: 10.2140/apde.2017.10.1757

Abstract

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 $G$ 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 $C^{*}$ -tensor category.

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Keywords

$L^2$-Betti numbers, rigid $C^*$-tensor categories, discrete quantum groups, subfactors, compact quantum groups

Mathematical Subject Classification 2010

Primary: 46L37

Secondary: 16E40, 18D10, 20G42

Milestones

Received: 9 February 2017

Revised: 3 May 2017

Accepted: 11 June 2017

Published: 1 August 2017

Authors

David Kyed
Department of Mathematics and Computer Science University of Southern Denmark DK-5230 Odense Denmark

Sven Raum
Sciences de Base Section de Mathématiques École polytechnique Fédérale de Lausanne CH-1015 Lausanne Switzerland

Stefaan Vaes
Department of Mathematics KU Leuven 3001 Leuven Belgium

Matthias Valvekens
Department of Mathematics KU Leuven 3001 Leuven Belgium

Vol. 10, No. 7, 2017