Hopf algebras and subfactors associated to vertex models

Abstract

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.