  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.
