Book Review: Harmonic maps, loop groups, and integrable systems

Abstract

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 τ-functions and the Bruhat decomposition 15 Solutions of the two-dimensional Toda lattice 16 Harmonic maps from C to a Lie group 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 S 2 to CP n 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 T 2 to S 2 26 Epilogue