  If H is a Hopf algebra whose square of the antipode is the identity, $v\in\l
(V)\otimes H$ is a corepresentation, and $\pi :H\to\l (W)$ is a representation,
then $u=(id\otimes\pi)v$ satisfies the equation $(t\otimes id)u^{-1}=((t\otimes
id)u)^{-1}$ of the vertex models for subfactors. A universal construction shows
that any solution $u$ of this equatio n arises in this way. A more elaborate
construction shows that there exists a ``minimal'' triple $(H,v,\pi)$
satisfying $(id\otimes\pi)v=u$. This paper is devoted to the study of this
latter construction of Hopf algebras. If $u$ is unitary we construct a
$\c^*$-norm on $H$ and we find a new description of the standard invariant of
the subfactor associated to $u$. We discuss also the ``twisted'' (i.e. $S^2\neq
id$) case.
