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.
Author(s): Martin A. Guest
Series: London Mathematical Society Student Texts
Edition: Web draft
Publisher: Cambridge University Press
Year: 1997
Language: English
Commentary: Bookmarked copy of the web draft found at: http://www.f.waseda.jp/martin/publications.html
Pages: 153
Part I One-dimensional integrable systems......Page 2
Chapter 1: Lie Groups I. Definitions and examples.......Page 3
Chapter 2: Lie Algebras I. Definitions and examples......Page 6
Chapter 3: Factorizations and homogeneous spaces......Page 11
Chapter 4: Hamilton's equations and Hamiltonian systems......Page 16
Chapter 5: Lax equations......Page 21
Chapter 6: Adler-Kostant-Symes......Page 26
Chapter 7: Adler-Kostant-Symes (continued)......Page 31
Chapter 8: Concluding remarks on one-dimensional Lax equations......Page 36
Part II Two-dimensional integrable systems......Page 44
Chapter 9: Zero-curvature equations......Page 45
Chapter 10: Some solutions of zero-curvature equations......Page 49
Chapter 11: Loop groups and loop algebras......Page 53
Chapter 12: Factorizations and homogeneous spaces I. Decomposition of loop groups, and homogeneous spaces.......Page 56
Chapter 13: The two-dimensional Toda lattice......Page 63
Chapter 14: Tau-functions and the Bruhat decomposition......Page 70
Chapter 15: Solutions of the two-dimensional Toda lattice......Page 77
Chapter 16: Harmonic maps from C to a Lie group......Page 80
Chapter 17: Harmonic maps from C to a Lie group (continued)......Page 86
Chapter 18: Harmonic maps from C to a symmetric space......Page 94
Chapter 19: Harmonic maps from C to a symmetric space (continued)......Page 98
Chapter 20: Application: Harmonic maps from S2 to CP n......Page 102
Chapter 21: Primitive maps......Page 110
Chapter 22: Weierstrass formulae for harmonic maps......Page 118
Part III One-dimensional and two-dimensional integrable systems......Page 127
Chapter 23: From 2 Lax equations to 1 zero-curvature equation......Page 128
Chapter 24: Harmonic maps of finite type I. Harmonic maps of finite type.......Page 132
Chapter 25: Application: Harmonic maps from T 2 to S2......Page 139
Chapter 26: Epilogue......Page 144
