Three-manifold invariants associated with restricted quantum groups

We show a simple relation between Witten–Reshetikhin–Turaev SU(2) invariant and the Hennings invariant associated with the restricted quantum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak{sl}_{2}}}$$\end{document} . These invariants are defined in very different methods: the former uses the representation theory of quantum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathfrak{sl}_{2}}}$$\end{document} while the latter uses the integral of the Hopf algebra. But they turn out to be the same for most rational homology 3-spheres up to a sign. This relation can be used to prove the integrality of the former invariant.