p-Adicizable Discrete Variants of Classical Lie Groups & Coset Spaces in TGD Framework

Abstract

p-Adization of quantum TGD is one of the long term projects of TGD. The notion of finite measurement resolution reducing to number theoretic existence in p-adic sense is the fundamental notion. p-Adic geometries replace discrete points of discretization with p-adic analogs of monads of Leibniz making possible to construct differential calculus and formulate p-adic variants of field equations allowing to construct p-adic cognitive representations for real space-time surfaces. This leads to a construction for the hierarchy of p-adic variants of imbedding space inducing in turn the construction of p-adic variants of space-time surfaces. Number theoretical existence reduces to conditions demanding that all ordinary (hyperbolic) phases assignable to (hyperbolic) angles are expressible in terms of roots of unity (roots of e). The construction reduces to the construction of p-adicizable discrete subgroups of classical Lie groups. The construction starts from SU(2) and U(1) and proceeds iteratively. Remarkably, the finite discrete p-adicizable subgroups of SU(2) correspond to those appearing in the hierarchy of inclusions of hyperfinite factors and include the groups assignable to Platonic solids. One can see them as cognitively especially simple finite p-adicizable objects providing p-adic approximation of sphere. The Platonic solids have analogs also for larger classical Lie groups.