Three-manifold invariants associated with restricted quantum groups
type: Item
instance of: scholarly article
title: Three-manifold invariants associated with restricted quantum groups
zbMATH Open document ID: 1257.57012
author: Qi Chen
author: Chih-Chien Yu
author: Yu Zhang
published in: Mathematische Zeitschrift
publication date: 2012-12-19T00:00:00Z
review text: The Witten-Reshetikhin-Turaev invariants of closed 3-manifolds are defined using representation theory of quantum groups at roots of unity. The Hennings invariant of closed manifolds is defined for quantum groups using integrals for the Hopf algebra structures. The main result of the paper under review states that for each root of unity \(\zeta\) the Hennings invariant of a closed 3-manifold \(M\) associated to the restricted quantum \(sl(2)\) at \(\zeta\) is equal, up to sign, to the product of the ``order'' of the first homology group of \(M\) and the Witten-Reshetikhin-Turaev invariant of \(M\) at \(\zeta\). Here the ``order'' must be understood to be \(0\) when the first homology group of \(M\) is infinite. A similar result for the Hennings invariant associated to the small quantum \(sl(2)\) and the \(SO(3)\) Witten-Reshetikhin-Turaev invariants at roots of unity of odd orders has been given by \textit{Q. Chen, S. Kuppum} and \textit{P. Srinivasan} [Math. Proc. Camb. Phil. Soc. 146, No. 1, 151--163 (2009; Zbl 1170.57009)].
reviewed by: Kazuo Habiro
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 81R50
zbMATH DE Number: 6116854
zbMATH Keywords: 3-manifold
zbMATH Keywords: Witten-Reshetikhin-Turaev invariant
zbMATH Keywords: Hennings invariant
zbMATH Keywords: restricted quantum group
MaRDI profile type: Publication
full work available at URL: s00209-011-0969-5
OpenAlex ID: W2097211270
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DOI: 10.1007/S00209-011-0969-5
description: scientific article; zbMATH DE number 6116854