Algebraic Geometrical Methods in Hamiltonian Mechanics: Discussion

A few years ago the `hidden symmetries' of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group [Note: See the image of page 391 for this formatted text] SL(2, C)^hat{} in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group [Note: See the image of page 391 for this formatted text] (SL(2, C) × SL(2, C))^hat{} in V × V^*. The affine analogue of the classical invariant theory will provide an explicit description of these equations in bilinear form. Another application of the representation theory will allow us to pass to nonlinear matrix equations, which for [Note: See the image of page 391 for this formatted text] SL(2, C)^hat{} yield the familiar form of Korteweg-de Vries and sine-Gordon equations.