Quantum Galois Correspondence for Subfactors

Abstract

Ocneanu has obtained a certain type of quantized Galois correspondence for the Jones subfactors of type An and his arguments are quite general. By making use of them in a more general context, we define a notion of a subequivalent paragroup and establish a bijective correspondence between generalized intermediate subfactors in the sense of Ocneanu and subequivalent paragroups for a given strongly amenable subfactors of type II1 in the sense of Popa, by encoding the subequivalence in terms of a commuting square. For this encoding, we generalize Sato's construction of equivalent subfactors of finite depth from a single commuting square, to strongly amenable subfactors. We also explain a relation between our notion of subequivalent paragroups and sublattices of a Popa system, using open string bimodules.