Vol. 10, No. 7, 2017

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
Other MSP Journals
$L^2$-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.

$L^2$-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