Mixed Quantum Double Construction of Subfactors

Abstract

We generalize the quantum double construction of subfactors to that from arbitrary flat connections on 4-partite graphs and call it the \textit{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 Harpe-Jones 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 $A_n$ except for the cases of orbifold type. If two subfactors $A\subset B$ and $A \subset C$ 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 $A\subset B$ and $A \subset C$. 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.