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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Generators for rational loop groups
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by Neil Donaldson, Daniel Fox and Oliver Goertsches PDF
Trans. Amer. Math. Soc. 363 (2011), 3531-3552 Request permission


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
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  • Daniel Fox
  • Affiliation: Mathematics Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom
  • Email:
  • Oliver Goertsches
  • Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
  • Email:
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 363 (2011), 3531-3552
  • MSC (2000): Primary 22E67, 37K25, 53C35
  • DOI:
  • MathSciNet review: 2775817