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Transactions of the American Mathematical Society

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Generators for rational loop groups

Authors: Neil Donaldson, Daniel Fox and Oliver Goertsches
Journal: Trans. Amer. Math. Soc. 363 (2011), 3531-3552
MSC (2000): Primary 22E67, 37K25, 53C35
Published electronically: February 14, 2011
MathSciNet review: 2775817
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Abstract: Uhlenbeck proved that a set of simple elements generates the group of rational loops in $\mathrm {GL}(n,\mathbb {C})$ that satisfy the $\mathrm {U}(n)$-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality and enables us to write a natural set of simple elements. Using these simple elements we prove generator theorems for the fundamental representations of the remaining neo-classical groups and most of their symmetric spaces. We also obtain explicit dressing and permutability formulae.

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Additional Information

Neil Donaldson
Affiliation: Department of Mathematics, University of California, Irvine, Irvine, California 92697

Daniel Fox
Affiliation: Mathematics Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom

Oliver Goertsches
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Keywords: Loop group, integrable system, submanifold geometry, generator theorem, simple element, neo-classical
Received by editor(s): January 23, 2009
Received by editor(s) in revised form: March 31, 2009, and April 13, 2009
Published electronically: February 14, 2011
Additional Notes: The third author was supported by the Max-Planck-Institut für Mathematik in Bonn and a DAAD-postdoctoral scholarship. He would like to thank the University of California, Irvine, and especially Chuu-Lian Terng, for their hospitality.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.