We introduce a Galois type invariant for the position of a subalgebra inside an algebra, called a paragroup, which has a group-like structure. Paragroups are the natural quantization of (finite) groups. The quantum groups of Drinfeld and Fadeev, as well as quotients of a group by a non-normal subgroup which appear in gauge theory have a paragroup structure.
In paragroups the underlying set of a group is replaced by a graph, the group elements are substituted by strings on the graph and a geometrical connection stands for the composition law. The harmonic analysis is similar to the computation of the partition function in the Andrews-Baxter-Forrester models in quantum statistical mechanics (e.g. harmonic analysis for the paragroup corresponding to the group ℤ2 is done in the Ising model.) The analogue of the Pontryagin van Kampen duality for abelian groups holds in this context, and we can alternatively use as invariant the coupling system, which is similar to the duality coupling between an abelian group and its dual.
We show that for subfactors of finite Jones index, finite depth and scalar centralizer of the Murray-von Neumann factor R the coupling system (or alternatively the paragroup) is a complete conjugacy invariant. In index less than 4 these conditions are always satisfied and the conjugacy classes of subfactors are rigid: there is one for each Coxeter – Dynkin diagram An and D2n and there are two anticonjugate but nonconjugate for each of the E6 and E8 diagrams.
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