Forked Temperley–Lieb algebras and intermediate subfactors

We consider noncommuting pairs P,Qof intermediate subfactors of an irreducible, finite-index inclusion N ⊂M of II1 factors such that P and Q are supertransitive with Jones index less than 4 over N. We show that up to isomorphism of the standard invariant, there is a unique such pair corresponding to each even value [P : N] = 4cos2 π 2n but none for the odd values [P : N] = 4cos2 π 2n+1 . We also classify the angle values which occur between pairs of intermediate subfactors with small index over their intersection: if [P : N], [Q: N] < 4, then the unique nontrivial angle value is always cos−1 1 [P :N]−1 . © 2007 Elsevier Inc. All rights reserved.