Classical foundations of quantum groups

The concept of classical r matrices is developed from a purely canonical standpoint. The final purpose of this work is to bring about a synthesis between recent developments in the theory of integrable systems and the general theory of quantization as a deformation of classical mechanics. The concept of quantization algebra is here dominant; in integrable systems this is the set of dynamical variables that appear in the Lax pair. The nature of this algebra, a solvable Lie algebra in such models as the Sine-Gordon and Toda field theories but semisimple in the case of spin systems, provides a useful scheme for the classification of integrable models. A completely different classification is obtained by the nature of the r matrix employed; there are three kinds: rational, trigonometric, and elliptic. All cases are studied in detail, with numerous examples. Some of the problems connected with quantization are discussed.This paper is dedicated to my friend Asim Barut.