Comonoidal W*-Morita equivalence for von Neumann bialgebras

Commer, Kenny De

doi:10.4171/jncg/86

Cited by 5 publications

(16 citation statements)

References 31 publications

(73 reference statements)

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“…The following theorem was proven in [11], Proposition 2.1 and Theorem 0.7. N )) the von Neumann algebra which is generated by elements of the form xy * , where x, y ∈ N .…”

Section: Reflecting a Compact Woronowicz Algebra Across A Galois Co-omentioning

confidence: 84%

“…Now it can be shown that the quantum group SU q (1, 1) contains only two group-like unitaries. By Proposition 3.5 of [11], this implies that the associated [ ( N, N )]-projective corepresentations still form a (countably) infinite family (which in fact will be parametrized by N, as obtained from {0 + , 0 − } ∪ N 0 by identifying 0 + with 0 − ). This family of ergodic actions is discussed from an algebraic viewpoint in [10].…”

Section: Remarksmentioning

confidence: 98%

“…In this section, we will make the connection with the theory of Galois objects from [11]. We first introduce the notion of the dual of a compact Woronowicz algebra.…”

Section: Galois Objects For Discrete Woronowicz Algebrasmentioning

confidence: 99%

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On projective representations for compact quantum groups

Commer

2011

Journal of Functional Analysis

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We study actions of compact quantum groups on type I -factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz's results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).It is well known that for compact groups, one can easily extend the main theorems of the Peter-Weyl theory to cover also projective representations. In this article, we will see that if one tries to do the same for Woronowicz's compact quantum groups, one confronts at least one surprising novelty: not all irreducible projective representations of a compact quantum group have to be finite-dimensional. On the other hand, one will still be able to decompose any projective rep-

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“…The following theorem was proven in [11], Proposition 2.1 and Theorem 0.7. N )) the von Neumann algebra which is generated by elements of the form xy * , where x, y ∈ N .…”

Section: Reflecting a Compact Woronowicz Algebra Across A Galois Co-omentioning

confidence: 84%

Section: Remarksmentioning

confidence: 98%

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On projective representations for compact quantum groups

Commer

2011

Journal of Functional Analysis

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“…. It is known that x G Ω and p G are monoidally co-Morita equivalent in the sense of [12], [15]. The corresponding co-linking quantum groupoid takes the form…”

Section: Twisted Crossed Productsmentioning

confidence: 99%

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

Commer¹,

Martos²,

Nest³

2021

Preprint

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We study the theory of projective representations for a compact quantum group G, i.e. actions of G on BpHq 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 KpHq, 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 p 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.

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“…This paper reports on preliminary work related to the quantization of non-compact semi-simple Lie groups. The main idea behind such a quantization is based on the reflection technique developed in [5] and [11] (see also [7] and [6] for concrete, small-dimensional examples relevant to the topic of this paper). Briefly, this technique works as follows.…”

Section: Introductionmentioning

confidence: 99%

Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

Commer¹

2013

SIGMA

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Abstract. Let g be a compact simple Lie algebra. We modify the quantized enveloping * -algebra associated to g by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping * -algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.

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Comonoidal W*-Morita equivalence for von Neumann bialgebras

Cited by 5 publications

References 31 publications

On projective representations for compact quantum groups

On projective representations for compact quantum groups

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

Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure

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