Multi-Sided Braid Type Subfactors, II

We show that the multi-sided inclusion R of braid-type subfactors of the hyperfinite II1 factor R, introduced in Multi-sided braid type subfactors [E3], contains a sequence of intermediate subfactors: R\otimes l \subset R\otimes l-1 \subset ... That is, every t-sided subfactor is an intermediate subfactor for the inclusion R\otimes l \subset R, for 2 \leq t \leq l. Moreover, we also show that if t > m then R\otimes t \subset R\otimes m is conjugate to R\otimes t-m+1 \subset R. Thus, if the braid representation considered is associated to one of the classical Lie algebras then the asymptotic inclusions for the Jones-Wenzl subfactors are intermediate subfactors.