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.