Hopf Algebras and Subfactors Associated to Vertex Models

IfHis a Hopf algebra whose square of the antipode is the identity,v∈L(V)⊗His a corepresentation, andπ: H→L(W) is a representation, thenu=(id⊗π) vsatisfies the equation (t⊗id) u−1=((t⊗id) u)−1of the vertex models for subfactors. A universal construction shows that any solutionuof 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. Ifuis unitary we construct aC*-norm onHand we find a new description of the standard invariant of the subfactor associated tou. We discuss also the “twisted” (i.e.,S2≠id) case.