Toda frames, harmonic maps and extended Dynkin diagrams

Emma Carberry and Katharine Turner

Abstract

This paper proves two main theorems. The first is that all cyclic primitive immersions of a genus one surface into \(G/T\) can be constructed by integrating a pair of commuting vector fields on a finite dimensional vector subspace of a Lie algebra. Here \(G\) is any simple real Lie group (not necessarily compact), \(T\) is a Cartan subgroup and \(G/T\) has a \(k\)-symmetric space structure induced from the Coxeter automorphism. If \(G\) is not compact, such a structure may not exist. We characterise the \(G/T\) to which the theory applies in terms of extended Dynkin diagrams, first observing that a Coxeter automorphism preserves the real Lie algebra \(\mathfrak g\) if and only if any corresponding Cartan involution defines a permutation of the extended Dynkin diagram for \(\mathfrak{g}^{\mathbb{C}} =\mathfrak{g}\otimes \mathbb{C}\). The second main result is that every involution of the extended Dynkin diagram for a simple complex Lie algebra \(\mathfrak{g}^{\mathbb{C}}\) is induced by a Cartan involution of a real form of \(\mathfrak{g}^{\mathbb{C}}\).

Keywords: Harmonic Maps, Toda equations.

AMS Subject Classification: Primary 53C43.