Ocneanu has introduced an algebraic notion of a pargroup for classification of subfactors. A paragroup is sometimes called a "quantized Galois group". Many people have studied a correspondence between subgroups of an operator algebraic Galois group and intermediate algebras, as an analogue of the classical Galois correspondence. We have studied a "quantization" of this operator algebraic Galois correspondence this year.
We consider a pair N\subset M of operator algebras, called II_1 factors, in subfactor theory. An algebraic system, called a paragroup, arising from this pair is regarded as a quantization of a notion of a (finite) group. Thus, we expect that we can get a quantized Galois correspondence by finding a correspondence between intermediate algebras of N\subset M and "subparagroups". However, if we call a paragroup corresponding to an intermediate algebra a "subparagroup", then a "subparagroup" of a "subparagroup" would not be a "subparagroup" of the original paragroup any more, which causes a trouble. In order to avoid this trouble, we use notions of an equivalent paragroup, introduced by N. Sato based on Ocneanu's idea, and a fusion rule subalgebra with quantum 6j-symbols. We then define a subequivalent paragroup. Combining this with Ocneanu's notion of generalized intermediate subfactor, we have established a Galois type correspondence between subequivalent paragroups and generalized intermediate subfactors. This is a generalization of Ocneanu's work on the Jones subfactors of type A.
If the original subfactor N\subset M is strongly amenable in the sense of Popa, we can encode equivalence/subequivalence in a single commuting square. For this encoding, we use a generalization of Sato's characterization of equivalent subfactors with finite depth in terms of a commuting square. In this generalization, we have to replace the compactness argument with something else. We have also clarified a relation between sublattices of a Popa system and our subequivalent paragroups.