A Geometric Construction of Solutions of the Strict h-Hierarchy

Abstract

Let h be a complex commutative subalgebra of the n×n matrices M_n(ℂ). In the algebra MPsd of matrix pseudodifferential operators in the derivation ∂, we previously considered deformations of h[∂] and of its Lie subalgebra h[∂]_>0 consisting of elements without a constant term. It turned out that the different evolution equations for the generators of these two deformed Lie algebras are compatible sets of Lax equations and determine the corresponding h-hierarchy and its strict version. Here, with each hierarchy, we associate an MPsd-module representing perturbations of a vector related to the trivial solution of each hierarchy. In each module, we describe so-called matrix wave functions, which lead directly to solutions of their Lax equations. We next present a connection between the matrix wave functions of the h-hierarchy and those of its strict version; this connection is used to construct solutions of the latter. The geometric data used to construct the wave functions of the strict h-hierarchy are a plane in the Grassmannian Gr(H), a set of n linearly independent vectors {w_i} in W, and suitable invertible maps δ: S¹ → h, where S₁ is the unit circle in ℂ*. In particular, we show that the action of a corresponding flow group can be lifted from W to the other data and that this lift leaves the constructed solutions of the strict h-hierarchy invariant. For n > 1, it can happen that we have different solutions of the strict h-hierarchy for fixed W and {w_i}. We show that they are related by conjugation with invertible matrix differential operators.