Infinite dimensional symmetric spaces and Lax equations compatible with the infinite Toda chain

In this paper we present a natural embedding of the infinite Toda chain in a set of Lax equations in the algebra LTLT consisting of Z×ZZ×Z-matrices that possess only a finite number of nonzero diagonals above the main central diagonal. This hierarchy of Lax equations describes the evolution of deformations of a set of commuting anti-symmetric matrices and corresponds to splitting this algebra into its anti-symmetric part and the subalgebra of matrices in LTLT that have no component above the main diagonal. We show that the projections of these deformations satisfy a set of zero curvature relations, which demonstrates the compatibility of the system. Further we introduce a suitable LTLT-module in which we can distinguish elements, the so-called wave matrices, that will lead you to solutions of the hierarchy. We conclude by showing how wave matrices of the infinite Toda chain hierarchy can be constructed starting from an infinite dimensional symmetric space.