Harmonic Maps, Loop Groups, and Integrable Systems
Harmonic Maps, Loop Groups, and Integrable Systems
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.
- Deals with the intersection of three important areas: harmonic maps, loop groups, and integrable systems
- Guides the reader from elementary topics to current research
- Accessible: emphasises main ideas and examples
Reviews & endorsements
"This is an accessible and very interesting text..." Monashefte fur Mathematik
"This is an accessible and very interesting text..." Monashefte fur Mathematik
"...a very well written, easily accessible introduction to how loop group techniques are used in the description of harmonic maps from Riemann surfaces to compact Lie groups and compant symmetric spaces...The book presents in a unifying way a very nice introduction to a new part of harmonice map theory, is easily accessible, fun to read and has a modest price. It is an ideal text for a beginning graduate student and any newcomer to the field." Bulletin of the American Mathematical Society
Product details
January 1997Paperback
9780521589321
212 pages
229 × 153 × 13 mm
0.29kg
Available
Table of Contents
- 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.
