Forked Temperley–Lieb algebras and intermediate subfactors

Abstract

We consider noncommuting pairs P, Q of intermediate subfactors of an irreducible, finite-index inclusion N ⊂ M of II 1 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 ] = 4 cos 2 π 2 n but none for the odd values [ P : N ] = 4 cos 2 π 2 n + 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 .