Satoshi Goto

We generalize the quantum double construction of subfactors to that from arbitrary flat connections on 4-partite graphs and call it the mixed quantum double construction. If all the four graphs of the original 4 partite graph are connected, it is easy to see that this construction produces Ocneanu's asymptotic inclusion of both subfactors generated by the original flat connection horizontally and vertically. The construction can be applied for example to the non-standard flat connections which appear in the construction of the Goodman-de la HarpeJones subfactors or to those obtained by the composition of flat part of any biunitary connections as in N. Sato's paper [40]. An easy application shows that the asymptotic inclusions of the Goodman-de la Harpe-Jones subfactors are isomorphic to those of the Jones subfactors of type An except for the cases of orbifold type. If two subfactors A subset of B and A subset of C have common A-A bimodule systems, we can construct a flat connection in general. Then by applying our construction to the flat connection, we obtain the asymptotic inclusion of both A subset of B and A subset of C. We also discuss the case when the original 4-partite graph contains disconnected graphs and give some such examples. General phenomena when disconnected graphs appear are explained by using bimodule systems.

By using Ocneanu's result on the classification of all irreducible connections on the Dynkin diagrams, we show that the dual principal graphs as well as the fusion rules of bimodules arising from any Goodman-de la Harpe-Jones subfactors are obtained by a purely combinatorial method. In particular we obtain the dual principal graph and the fusion rule of bimodules arising from the Goodman-de la Harpe-Jones subfactor corresponding to the Dynkin diagram E-8. As an application, we also show some subequivalence among A-D-E paragroups.

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 A(n), D(n), E(6), E(7) and E(8). More precisely, we give a detailed proof of the complete classification of irreducible K-L bi-unitary connections up to gauge choice, where K and L represent the two horizontal graphs which are among the A-D-E Dynkin diagrams. The result also provides a simple proof of the flatness of D(2n), E(6) and E(8) connections as well as an easy computation of the flat part of E(7) as an application. (C) 2009 Elsevier GmbH. All rights reserved.

By generalizing Erlijman's method, we construct a subfactor from a fusion rule algebra with quantum 6j-symbols which produce periodic commuting squares. This construction produces the same subfactor as Ocneanu's asymptotic inclusion for the subfactor which is generated by the original periodic commuting square. This result can be applied to the quantum SU(n)(k) subfactors which is the same as Hecke algebra subfactors of type A of Wenzl for example, which shows that Erlijman's construction gives the same subfactor as the asymptotic inclusion.