Canonical Structure and Symmetries of the Schlesinger Equations

Abstract

The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.