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We construct an invariant of closed oriented $3$-manifolds using a finitedimensional, involutory, unimodular and counimodular Hopf algebra $H$. We usethe framework of normal o-graphs introduced by R. Benedetti and C. Petronio, inwhich one can represent a branched ideal triangulation via an oriented virtualknot 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 branchedideal tetrahedron. The invariant takes values in the cyclic quotient$\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$, which is isomorphic to thebase field. In the construction we use only the canonical element and structureconstants of $H$ and we do not use any representations of $H$. This, togetherwith 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 thenumber of group homomorphisms from the fundamental group of the $3$-manifold tothe group.

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