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
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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