Harmonic Maps, Loop Groups, and Integrable Systems

Martin A. Guest (Tokyo Metropolitan University), J. Bruce

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English

Cambridge University Press
16 June 1997

Differential calculus & equations; Analytic topology

Series: London Mathematical Society Student Texts

This is an accessible introduction to some of the fundamental connections among differential geometry, Lie groups, and integrable Hamiltonian systems.

The text demonstrates how the theory of loop groups can be used to study harmonic maps.

By concentrating on the main ideas and examples, the author leads up to topics of current research.

The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists as well.

By: Martin A. Guest (Tokyo Metropolitan University)
Series edited by: J. Bruce
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 38
Dimensions: Height: 235mm, Width: 158mm, Spine: 20mm
Weight: 465g
ISBN: 9780521580854
ISBN 10: 0521580854
Series: London Mathematical Society Student Texts
Pages: 212
Publication Date: 16 June 1997
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Preface; Acknowledgements; Part I. One-Dimensional Integrable Systems: 1. Lie groups; 2. Lie algebras; 3. Factorizations and homogeneous spaces; 4. Hamilton's equations and Hamiltonian systems; 5. Lax equations; 6. Adler-Kostant-Symes; 7. Adler-Kostant-Symes (continued); 8. Concluding remarks on one-dimensional Lax equations; Part II. Two-Dimensional Integrable Systems: 9. Zero-curvature equations; 10. Some solutions of zero-curvature equations; 11. Loop groups and loop algebras; 12. Factorizations and homogeneous spaces; 13. The two-dimensional Toda lattice; 14. T-functions and the Bruhat decomposition; 15. Solutions of the two-dimensional Toda lattice; 16. Harmonic maps from C to a Lie group G; 17. Harmonic maps from C to a Lie group (continued); 18. Harmonic maps from C to a symmetric space; 19. Harmonic maps from C to a symmetric space (continued); 20. Application: harmonic maps from S2 to CPn; 21. Primitive maps; 22. Weierstrass formulae for harmonic maps; Part III. One-Dimensional and Two-Dimensional Integrable Systems: 23. From 2 Lax equations to 1 zero-curvature equation; 24. Harmonic maps of finite type; 25. Application: harmonic maps from T2 to S2; 26. Epilogue; References; Index.

Reviews for Harmonic Maps, Loop Groups, and Integrable Systems

'... an accessible introduction to some of the fundamental connections beween differental geometry, Lie groups and integrable Hamiltonian systems.' L'Enseignement Mathematique 'It is very rare to find a book that can take a student from the very basics of a subject to the frontiers of active research. The author is to be congratulated on having produced just such a rarity!' Bulletin of the London Mathematics Society 'The book will certainly be appreciated by mathematicians as well as theoretical physics interested in the subject.' European Mathematical Society