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2017 $L^2$-Betti numbers of rigid $C^*$-tensor categories and discrete quantum groups
David Kyed, Sven Raum, Stefaan Vaes, Matthias Valvekens
Anal. PDE 10(7): 1757-1791 (2017). DOI: 10.2140/apde.2017.10.1757


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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David Kyed. Sven Raum. Stefaan Vaes. Matthias Valvekens. "$L^2$-Betti numbers of rigid $C^*$-tensor categories and discrete quantum groups." Anal. PDE 10 (7) 1757 - 1791, 2017.


Received: 9 February 2017; Revised: 3 May 2017; Accepted: 11 June 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 1378.46056
MathSciNet: MR3683927
Digital Object Identifier: 10.2140/apde.2017.10.1757

Primary: 46L37
Secondary: 16E40 , 18D10 , 20G42

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

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.10 • No. 7 • 2017
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