Quantum double construction for subfactors arising from periodic commuting squares

Abstract

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.