## 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.

: Primary 53C43.

This paper is available as a pdf (452kB) file.

 Thursday, February 13, 2014