On the Range of the Index of Subfactors

A simple numerical argument is given that the minimal (Jones) index of a subfactor N ⊂ > M is strongly restricted if for L ⊂ N with the same index, the subfactor L ⊂ M contains a sector with index from the Jones series 4 cos2 π/m. E.g., N ⊂ M might be the Jones extension of, or isomorphic with, L ⊂ N. As a corollary extending results of Longo, the range of the index of braided endomorphisms is completely computed up to the value Ind = 6. An algebraic version of the argument is outlined and is expected to generalize to braided endomorphisms the square of which contains a sector with index from the Hecke or Birman-Wenzl-Murakami series. This would allow to push the determination of the range of the index beyond 6.