TGD View on McKay Correspondence, ADE Hierarchy & Inclusions of Hyperfinite Factors

Abstract

There are two mysterious looking correspondences involving ADE groups. McKay correspondence between McKay graphs characterizing tensor products for finite subgroups of SU(2) and Dynkin diagrams of affine ADE groups is the first one. The correspondence between principal diagrams characterizing inclusions of hyper-finite factors of type II₁ (HFFs) with Dynkin diagrams for a subset of ADE groups and Dynkin diagrams for affine ADE groups is the second one. These correspondences are discussed from number theoretic point of view suggested by TGD and based on the interpretation of discrete subgroups of SU(2) as subgroups of the covering group of quaternionic automorphisms SO(3) (analog of Galois group) and generalization of these groups.