## Generators for rational loop groups

HTML articles powered by AMS MathViewer

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

## 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.## References

- Martina Brück, Xi Du, Joonsang Park, and Chuu-Lian Terng,
*The submanifold geometries associated to Grassmannian systems*, Mem. Amer. Math. Soc.**155**(2002), no. 735, viii+95. MR**1875645**, DOI 10.1090/memo/0735 - F. E. Burstall,
*Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems*, Integrable systems, geometry, and topology, AMS/IP Stud. Adv. Math., vol. 36, Amer. Math. Soc., Providence, RI, 2006, pp. 1–82. MR**2222512**, DOI 10.1090/amsip/036/01 - F. E. Burstall and M. A. Guest,
*Harmonic two-spheres in compact symmetric spaces, revisited*, Math. Ann.**309**(1997), no. 4, 541–572. MR**1483823**, DOI 10.1007/s002080050127 - F. E. Burstall and F. Pedit,
*Harmonic maps via Adler-Kostant-Symes theory*, Harmonic maps and integrable systems, Aspects Math., E23, Friedr. Vieweg, Braunschweig, 1994, pp. 221–272. MR**1264189**, DOI 10.1007/978-3-663-14092-4_{1}1 - Bo Dai and Chuu-Lian Terng,
*Bäcklund transformations, Ward solitons, and unitons*, J. Differential Geom.**75**(2007), no. 1, 57–108. MR**2282725** - N.M. Donaldson and C.-L. Terng,
*Conformally flat submanifolds in spheres and integrable systems*, 2007, Eprint: arXiv:math/0803.2754v2, 2008. - N.M. Donaldson,
*Symmetric $r$-spaces: Submanifold geometry and Transformation theory*, Ph.D. thesis, University of Bath, 2006. - Dirk Ferus and Franz Pedit,
*Isometric immersions of space forms and soliton theory*, Math. Ann.**305**(1996), no. 2, 329–342. MR**1391218**, DOI 10.1007/BF01444224 - Reese Harvey and H. Blaine Lawson Jr.,
*Calibrated geometries*, Acta Math.**148**(1982), 47–157. MR**666108**, DOI 10.1007/BF02392726 - Andrew Pressley and Graeme Segal,
*Loop groups*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR**900587** - Chuu-Lian Terng,
*Soliton equations and differential geometry*, J. Differential Geom.**45**(1997), no. 2, 407–445. MR**1449979** - Chuu-Lian Terng and Karen Uhlenbeck,
*Bäcklund transformations and loop group actions*, Comm. Pure Appl. Math.**53**(2000), no. 1, 1–75. MR**1715533**, DOI 10.1002/(SICI)1097-0312(200001)53:1<1::AID-CPA1>3.3.CO;2-L - Chuu-Lian Terng and Erxiao Wang,
*Transformations of flat Lagrangian immersions and Egoroff nets*, Asian J. Math.**12**(2008), no. 1, 99–119. MR**2415015**, DOI 10.4310/AJM.2008.v12.n1.a8 - Karen Uhlenbeck,
*Harmonic maps into Lie groups: classical solutions of the chiral model*, J. Differential Geom.**30**(1989), no. 1, 1–50. MR**1001271**

## 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
- 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: https://doi.org/10.1090/S0002-9947-2011-05120-2
- MathSciNet review: 2775817