Introduction

Abstract

The three chapters that follow are conceived as independent surveys dealing with various group-theoretical constructions of finite-dimensional integrable systems. The major part of Chapter 1 is devoted to the so-called Calogero-Sutherland systems, which historically are among the first examples of this kind. A somewhat wider range of applications is provided by the Kostant-Adler scheme and its generalization known as the r-matrix construction; these are discussed in Chapter 2 with an emphasis on concrete examples. A natural class of Lie algebras where the r-matrix construction gives interesting results includes semi-simple Lie algebras and loop algebras (or affine Lie algebras); the latter appear already in Chapter 1 in connection with periodic Toda lattices and are exploited more profoundly in Chapter 2. The r-matrix approach applied to loop algebras gives a natural passage to algebraic-geometric methods; explanation of these links is another major theme of Chapter 2. (We also recommend it to the reader to consult the survey “Integrable systems. 1” by B. Dubrovin, I. Krichever and S. Novikov, EMS vol. 4, Springer-Verlag 1990.) Finally, Chapter 3 is concerned entirely with the quantization problem for a particular, though very interesting, family of integrable systems, the nonperiodic Toda lattices.KeywordsIntegrable SystemToda LatticeLoop AlgebraQuantization ProblemBibliographical NoteThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.