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

DOI:
https://doi.org/10.1090/S0002-9947-2011-05120-2

Published electronically:
February 14, 2011

MathSciNet review:
2775817

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Uhlenbeck proved that a set of simple elements generates the group of rational loops in that satisfy the -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

Email:
ndonalds@math.uci.edu

**Daniel Fox**

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

Email:
foxd@maths.ox.ac.uk

**Oliver Goertsches**

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

Email:
ogoertsc@math.uni-koeln.de

DOI:
https://doi.org/10.1090/S0002-9947-2011-05120-2

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.