Vol. 10, No. 7, 2017

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

Volume 11
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261

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
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
$L^2$-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
Abstract

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.

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