L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups

David Kyed, Sven Raum, Stefaan Vaes, Matthias Valvekens

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Resumé

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.
OriginalsprogEngelsk
TidsskriftAnalysis & PDE
Vol/bind10
Udgave nummer7
Sider (fra-til)1757-1791
ISSN2157-5045
DOI
StatusUdgivet - 2017

Emneord

  • math.OA
  • math.GR
  • math.QA

Citer dette

Kyed, David ; Raum, Sven ; Vaes, Stefaan ; Valvekens, Matthias. / L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups. I: Analysis & PDE. 2017 ; Bind 10, Nr. 7. s. 1757-1791.
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L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups. / Kyed, David; Raum, Sven ; Vaes, Stefaan; Valvekens, Matthias.

I: Analysis & PDE, Bind 10, Nr. 7, 2017, s. 1757-1791.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups

AU - Kyed, David

AU - Raum, Sven

AU - Vaes, Stefaan

AU - Valvekens, Matthias

N1 - Preprint, submitted to "Analysis and PDE"

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - math.OA

KW - math.GR

KW - math.QA

U2 - 10.2140/apde.2017.10.1757

DO - 10.2140/apde.2017.10.1757

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SP - 1757

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