A Lie theoretic Galois theory for the spectral curves of an integrable system. I
The authors give a classification of the spectral curves for a completely integrable system. They study an integrable system presented in the Lax form \(dA(t,s)/dt=[A(t,s),B(t,s)]\), where \(A(t,s)\), \(B(t,s)\) are elements of a simple complex Lie algebra \(g\) and functions of time \(t\) and rational functions of a point \(s\) of an algebraic curve. For each representation \(\rho\) of \(g\) the spectral curves are defined as irreducible components of a variety presented by an equation \(\text{det}(\rho A(t,s)-z)=0\) which are independent of time. By means of the Galois theory for the spectral curves the authors indicate these curves for generic initial conditions, of the Lax equation by the conjugacy classes of parabolic subgroups of the Weyl group for the appropriate Lie algebra. These results are illustrated by detailed example of the generalized periodic Toda lattice.
scientific article; zbMATH DE number 95697
scientific article