Toda frames, harmonic maps and extended Dynkin diagrams

TL;DR

This paper establishes a method to construct genus one surface immersions into certain Lie group quotients using Toda frames, linking harmonic maps, Lie algebra automorphisms, and extended Dynkin diagrams.

Contribution

It provides necessary and sufficient conditions for Toda frame existence and connects harmonic maps into G/T with extended Dynkin diagram involutions.

Findings

Construction of immersions via commuting vector fields

Conditions for Toda frame existence in harmonic maps

Application to G/T spaces from extended Dynkin diagrams

Abstract

We prove that all immersions of a genus one surface into G/T possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional Lie algebra. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and the k-symmetric space structure on G/T is induced from the Coxeter automorphism. We provide necessary and sufficient conditions for the existence of a Toda frame for a harmonic map into G/T and describe those G/T to which the theory applies in terms of involutions of extended Dynkin diagrams.