Three-manifold invariants associated with restricted quantum groups
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)].
scientific article; zbMATH DE number 6116854
scientific article