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

David Kyed; Sven  Raum; Stefaan Vaes; Matthias Valvekens

doi:10.2140/apde.2017.10.1757

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

David Kyed, Sven Raum, Stefaan Vaes, Matthias Valvekens

Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › peer review

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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.

Originalsprog	Engelsk
Tidsskrift	Analysis & PDE
Vol/bind	10
Udgave nummer	7
Sider (fra-til)	1757-1791
ISSN	2157-5045
DOI	https://doi.org/10.2140/apde.2017.10.1757
Status	Udgivet - 2017

Emneord

math.OA
math.GR
math.QA

Citer dette

Kyed, D., Raum, S., Vaes, S., & Valvekens, M. (2017). L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups. Analysis & PDE, 10(7), 1757-1791. https://doi.org/10.2140/apde.2017.10.1757

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

abstract = "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.",

keywords = "math.OA, math.GR, math.QA",

author = "David Kyed and Sven Raum and Stefaan Vaes and Matthias Valvekens",

note = "Preprint, submitted to {"}Analysis and PDE{"}",

year = "2017",

doi = "10.2140/apde.2017.10.1757",

language = "English",

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Kyed, D, Raum, S, Vaes, S & Valvekens, M 2017, 'L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups', Analysis & PDE, bind 10, nr. 7, s. 1757-1791. https://doi.org/10.2140/apde.2017.10.1757

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 tidsskrift › Tidsskriftartikel › Forskning › peer 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

M3 - Journal article

VL - 10

SP - 1757

EP - 1791

JO - Analysis & PDE

JF - Analysis & PDE

SN - 2157-5045