On the zeroth \(L^2\)-homology of a quantum group
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.46039
author: David Kyed
publication date: 2012-05-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.0656
Mathematics Subject Classification ID: 46L65
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 55N99
Mathematics Subject Classification ID: 18G60
Mathematics Subject Classification ID: 43A07
Mathematics Subject Classification ID: 58B32
zbMATH DE Number: 6032511
zbMATH Keywords: quantum group
zbMATH Keywords: coamenable
zbMATH Keywords: \(L^2\)-Betti number
zbMATH Keywords: \(L^2\)-homology
MaRDI profile type: Publication
arXiv ID: 0906.0656
arXiv classification: math.OA
arXiv classification: math.QA
title: On the zeroth \(L^2\)-homology of a quantum group
published in: Münster Journal of Mathematics
review text: This article slightly improves the main results of the author in [``\(L^2\)-Betti numbers of coamenable quantum groups'', Münster J. Math. 1, No. 1, 143--179 (2008; Zbl 1195.46073)] and [``\(L^2\)-homology for compact quantum groups'', Math. Scand. 103, No. 1, 111--129 (2008; Zbl 1161.46039)] by removing technical assumptions. It is shown that that the zeroth \(L^2\)-homology of a compact quantum group is non-vanishing if and only if the quantum group is coamenable. For a compact quantum group of Kac type, the zeroth \(L^2\)-Betti number is zero if and only if the quantum group is infinite-dimensional. These results are inspired by older vanishing results for the \(L^2\)-homology of discrete amenable groups.
reviewed by: Ralf Meyer
description: scientific article; zbMATH DE number 6032511