The algebraic structure of lax equations for infinite matrices

In this paper we discuss the algebraic structure of the tower of differential difference equations that one can associate with any commutative subalgebra of $M_k(\mathbb{C})$. These equations can be formulated conveniently in so-called Lax equations for infinite upper- resp. lowertriangular matrices and they are shown in a purely algebraic way to be equivalent with zero curvature equations for a collection of finite band matrices. The uppertriangular and lowertriangular systems corresponding to the same algebra are shown to be compatible. Finally the linearizations of the aforementioned systems are treated, which form the basis of the construction of solutions of these hierarchies. As such this work is an extension of that of Ueno and Takasaki and furnishes a complete algebraic context for it