On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams

We give an exposition of Ocneanu's theory of double triangle algebras for subfactors and its application to the classification of irreducible bi-unitary connections on the Dynkin diagrams AnAn, DnDn, E6E6, E7E7 and E8E8. More precisely, we give a detailed proof of the complete classification of irreducible K–LK–L bi-unitary connections up to gauge choice, where K and L   represent the two horizontal graphs which are among the A–D–EA–D–E Dynkin diagrams. The result also provides a simple proof of the flatness of D2nD2n, E6E6 and E8E8 connections as well as an easy computation of the flat part of E7E7 as an application.