Classification of Paragroup Actions on Subfactors

Abstract

We define “a crossed product by a paragroup action on a subfactor” as a certain commuting square of type ${\mathrm{II}}_1$ factors and give their complete classification in a strongly amenable case (in the sense of S. Popa) in terms of a new combinatorial object which generalizes Ocneanu's paragroup.

As applications, we show that the subfactor $N\subset M$ of Goodman–de la Harpe–Jones with index $3+\sqrt{3}$ is not conjugate to its dual $M\subset M_1$ by showing the fusion algebras of $N$-$N$ bimodules and $M$-$M$ bimodules are different, although the principal graph and the dual principal graph are the same. We also make an analogue of the coset construction in RCFT for subfactors in our settings.