Classification of Subfactors with the Principal Graph D1n

We show that the number of the conjugacy classes of the AFD type II1 subfactors with the principal graph D1n is n − 2. This gives the last missing number in the complete classfication list of subfactors with index 4 by S. Popa. This also disproves an announcement of A. Ocneanu that such a subfactor is unique for each n. We give two different proofs. One is by an application of an idea of an orbifold model in solvable lattice model theory to Ocneanu′s paragroup theory and the other is by reduction to classification of dihedral group actions. The latter also shows that the AFD type III1 subfactors with the principal graph D1n split as type II1 subfactors tensored with the common AFD type III1 factor. We also discuss a relation between these proofs and a construction of subfactors using Cuntz algebra endomorphisms.