University of Oulu

Biswarup Das, Matthew Daws, Pekka Salmi, Admissibility Conjecture and Kazhdan's Property (T) for quantum groups, Journal of Functional Analysis, Volume 276, Issue 11, 2019, Pages 3484-3510, ISSN 0022-1236, https://doi.org/10.1016/j.jfa.2018.09.001

Admissibility Conjecture and Kazhdan’s Property (T) for quantum groups

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Author: Das, Biswarup1,2; Daws, Matthew3; Salmi, Pekka1
Organizations: 1Department of Mathematical Sciences, University of Oulu, Finland
2Instytut Matematyczny, Uniwersytet Wrocławski, Poland
3Jeremiah Horrocks Institute, University of Central Lancashire, UK
Format: article
Version: accepted version
Access: embargoed
Persistent link: http://urn.fi/urn:nbn:fi-fe202003107787
Language: English
Published: Elsevier, 2019
Publish Date: 2020-11-06
Description:

Abstract

We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdan’s Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and Sołtan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are:

(i) All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible.

(ii) A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum.

(iii) A very short proof of the fact that quantum groups with Property (T) are unimodular.

(iv) A generalisation of a quantum version of a theorem of Bekka–Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations.

(v) A generalisation of a quantum version of Kerr–Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations.

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Series: Journal of functional analysis
ISSN: 0022-1236
ISSN-E: 1096-0783
ISSN-L: 0022-1236
Volume: 276
Issue: 11
Pages: 3484 - 3510
DOI: 10.1016/j.jfa.2018.09.001
OADOI: https://oadoi.org/10.1016/j.jfa.2018.09.001
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Subjects:
Copyright information: © 2018 © 2018 Elsevier Inc. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
  https://creativecommons.org/licenses/by-nc-nd/4.0/