Toda frames, harmonic maps and extended Dynkin diagrams

We consider a natural subclass of harmonic maps from a surface into G/T, namely cyclic primitive maps. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and both are chosen so that there is a Coxeter automorphism on G(C)/T-C which restricts to give a k-symmetric space structure on G/T. When G is compact, any Coxeter automorphism restricts to the real form. It was shown in [3] that cyclic primitive immersions into compact G/T correspond to solutions of the affine Toda field equations and all those of a genus one surface can be constructed by integrating a pair of commuting vector fields on a finite dimensional vector subspace of a Lie algebra. We generalise these results, removing the assumption that G is compact. The first major obstacle is that a Coxeter automorphism may not restrict to a non-compact real form. We characterise, in terms of extended Dynkin diagrams, those simple real Lie groups G and Cartan subgroups T such that G/T has a k-symmetric space structure induced from a Coxeter automorphism. A Coxeter automorphism preserves the real Lie algebra g if and only if any corresponding Cartan involution defines a permutation of the extended Dynkin diagram for g(C) = g circle times C; we show that every involution of the extended Dynkin diagram for a simple complex Lie algebra g(C) is induced by a Cartan involution of a real form of sf. (C) 2017 Published by Elsevier B.V.

Published in:
Differential Geometry And Its Applications, 52, 142-157
Amsterdam, Elsevier

 Record created 2017-07-10, last modified 2018-12-03

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