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The Quarterly Journal of Mathematics Advance Access published online on July 11, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/hap022
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

SINGULAR DRESSING ACTIONS ON HARMONIC MAPS

N. Correia {dagger} and R. Pacheco{ddagger}

Departamento de Matemática, Universidade da Beira Interior, Rua Marquês d’Ávila e Bolama 6201-001 Covilhã, Portugal

{ddagger} Corresponding author. E-mail: rpacheco{at}mat.ubi.pt

Received 23 February 2009; revised 27 May 2009
  Abstract

In this paper we prove that any harmonic map {phi} 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 {phi} 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.

{dagger} E-mail: ncorreia@mat.ubi.pt


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