The Toda Lattice in the Complex Domain

Abstract

This chapter discusses the Toda lattice in the complex domain. The Toda lattice is an integrable Hamiltonian system. Its many special properties can be explained by various analytical, algebraic, and geometric constructions. Because of its rich and rigid structure, it is a paradigm for integrability: most features of integrable systems are likely to be revealed in the clearest possible way by the Toda lattice. The chapter discusses in a simple setting, some implications of the so-called Kovalevskaya-Painlevé method. This is a test for integrability, first used by S. Kovalevskaya in her pioneering work on the motion of a rigid body. All the integrable systems that have, to date, been subjected to a Kovalevskaya analysis, define flows on Lie algebras—possibly infinite-dimensional. Their solutions are provided by triangular factorization in the associated Lie group. The basic information about the lowest balance can be obtained by calculation, with no appeal to factorization or Schubert cells.