Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map

Publikation: Working paper › Preprint › Forskning

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Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map. / Commer, Kenny De ; Martos Prieto, Ruben; Nest, Ryszard.

arxiv.org, 2021.

Publikation: Working paper › Preprint › Forskning

Harvard

Commer, KD, Martos Prieto, R & Nest, R 2021 'Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map' arxiv.org.

APA

Commer, K. D., Martos Prieto, R., & Nest, R. (2021). Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map. arxiv.org.

Vancouver

Commer KD, Martos Prieto R, Nest R. Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map. arxiv.org. 2021.

Author

Commer, Kenny De ; Martos Prieto, Ruben ; Nest, Ryszard. / Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map. arxiv.org, 2021.

Bibtex

@techreport{5f38c78484a048998b0e1001e25c8626,
title = "Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map",
abstract = "We study the theory of projective representations for a compact quantum group G, i.e. actions of G on B(H) for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators K(H), if and only if the associated 2-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of G^ in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C*-algebras. We define deformed crossed products with respect to Ω, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups.",
keywords = "Faculty of Science, assembly map, Baum-Connes conjecture, cleftness, 2-cocycle, compact objects, crossed products, Galois co-objects, projective representations, quantum groups, regularity, torsion, triangulated categories, twisting",
author = "Commer, {Kenny De} and {Martos Prieto}, Ruben and Ryszard Nest",
year = "2021",
language = "English",
publisher = "arxiv.org",
type = "WorkingPaper",
institution = "arxiv.org",

}

RIS

TY - UNPB

T1 - Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map

AU - Commer, Kenny De

AU - Martos Prieto, Ruben

AU - Nest, Ryszard

PY - 2021

Y1 - 2021

N2 - We study the theory of projective representations for a compact quantum group G, i.e. actions of G on B(H) for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators K(H), if and only if the associated 2-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of G^ in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C*-algebras. We define deformed crossed products with respect to Ω, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups.

AB - We study the theory of projective representations for a compact quantum group G, i.e. actions of G on B(H) for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators K(H), if and only if the associated 2-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of G^ in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C*-algebras. We define deformed crossed products with respect to Ω, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups.

KW - Faculty of Science

KW - assembly map

KW - Baum-Connes conjecture

KW - cleftness

KW - 2-cocycle

KW - compact objects

KW - crossed products

KW - Galois co-objects

KW - projective representations

KW - quantum groups

KW - regularity

KW - torsion

KW - triangulated categories

KW - twisting

M3 - Preprint

BT - Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map

PB - arxiv.org

ER -

ID: 311872028