A pr 1 99 8 Hopf algebras and subfactors associated to ver - tex models

IfH is a Hopf algebra whose square of the antipode is the identity, v ∈ L(V )⊗H is a corepresentation, and π : H → L(W ) is a representation, then u = (id ⊗ π)v satisfies the equation (t ⊗ id)u = ((t ⊗ id)u) of the vertex models for subfactors. A universal construction shows that any solution u of this equation arises in this way. A more elaborate construction shows that there exists a “minimal” triple (H, v, π) satisfying (id⊗ π)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 6= id) case.