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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 |
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Vol. 10 (2017), No. 7, 1757–1791
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Abstract |
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We compute the -Betti numbers of the free -tensor categories, which are the representation categories of the universal unitary quantum groups . We show that the -Betti numbers of the dual of a compact quantum group are equal to the -Betti numbers of the representation category and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of -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 -Betti number in terms of a generating set of a -tensor category. |
Keywords
$L^2$-Betti numbers, rigid $C^*$-tensor categories,
discrete quantum groups, subfactors, compact quantum groups
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Mathematical Subject Classification 2010
Primary: 46L37
Secondary: 16E40, 18D10, 20G42
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Milestones
Received: 9 February 2017
Revised: 3 May 2017
Accepted: 11 June 2017
Published: 1 August 2017
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Authors |
| David Kyed | |
| Department of Mathematics and
Computer Science University of Southern Denmark DK-5230 Odense Denmark |
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| Sven Raum | |
| Sciences de Base Section de Mathématiques École polytechnique Fédérale de Lausanne CH-1015 Lausanne Switzerland |
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| Stefaan Vaes | |
| Department of Mathematics KU Leuven 3001 Leuven Belgium |
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| Matthias Valvekens | |
| Department of Mathematics KU Leuven 3001 Leuven Belgium |
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