The Heisenberg double of involutory Hopf algebras and invariants of closed $3$-manifolds

Abstract

We construct an invariant of closed oriented $3$-manifolds using a finite dimensional, involutory, unimodular and counimodular Hopf algebra $H$. We use the framework of normal o-graphs introduced by R. Benedetti and C. Petronio, in which one can represent a branched ideal triangulation via an oriented virtual knot diagram. We assign a copy of a canonical element of the Heisenberg double $\mathcal{H}(H)$ of $H$ to each real crossing, which represents a branched ideal tetrahedron. The invariant takes values in the cyclic quotient $\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$, which is isomorphic to the base field. In the construction we use only the canonical element and structure constants of $H$ and we do not use any representations of $H$. This, together with the finiteness and locality conditions of the moves for normal o-graphs, makes the calculation of our invariant rather simple and easy to understand. When $H$ is the group algebra of a finite group, the invariant counts the number of group homomorphisms from the fundamental group of the $3$-manifold to the group.