Abstract
In this paper we prove that any harmonic map ϕ from a two-sphere S2 into an arbitrary compact semisimple matrix Lie group G may be reduced to a constant by using the singular dressing actions introduced in (M. J. Bergvelt and M. A. Guest, Action of loop groups on harmonic maps, Trans. Amer. Math. Soc.326 (1991), 861–886); this reduction induces a factorization of ϕ into flag factors S2 → G, and the singular dressing actions are produced from curves of simple factors (rational loops having a minimum number of singularities, whose dressing action can be computed explicitly) for Gℂ. A version of this result for an arbitrary inner symmetric space G/K is established. We also prove generating theorems for the rational loops of the fundamental representations of Sp(n)ℂ and SU(n)ℂ: in both cases the class of generators is slightly larger than the class of simple factors.
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