The zero curvature form of integrable hierarchies in the \(\mathbb Z \times \mathbb Z\)-matrices (Q2883146)

The integrable hierarchies corresponding to the choosing of a number of commuting directions in the upper triangular \(\mathbb Z \times \mathbb Z\)-matrices are studied. It is shown that an integrable hierarchy consisting of a set of evolution equations for perturbations of the basic directions can be associated to a finite number of commuting directions in the Lie algebra of upper triangular \(\mathbb Z \times \mathbb Z\)-matrices. The central Cauchy theorem and the corresponding geometry in the finite-dimensional situation are described and various decompositions of these matrices are presented. The equations of hierarchy are formulated in the Lax form for the perturbed matrices. It is shown that they possess a minimal realization in which they correspond to commuting flows. The so-called zero curvature form of the integrable hierarchies is obtained and the equivalence of both formulations is proven. The zero curvature equations are formulated in terms of a collection of finite band matrices that can be chosen as the components of a formal differential form with values in the \(\mathbb Z \times \mathbb Z\)-matrices. The linearization of the hierarchy is treated in detail and it is shown that it represents the algebraic substitute for a basis of horizontal sections of the formal connection corresponding to this form. A criterion for the characterization of wave matrices at zero of a certain type is found and the authors remark that the respective matrices are the analogues of the Baker-Akhiezer functions for the KP-hierarchy.