Three-manifold invariants associated with restricted quantum groups

Abstract

We show a simple relation between Witten–Reshetikhin–Turaev SU(2) invariant and the Hennings invariant associated with the restricted quantum \({{\mathfrak{sl}_{2}}}\) . These invariants are defined in very different methods: the former uses the representation theory of quantum \({{\mathfrak{sl}_{2}}}\) 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.