Minimal realizations of the KP hierarchy, its strict version and their reductions

Abstract

We discuss here realizations of the KP hierarchy, its strict version and their reductions, the n n -KdV hierarchy and the strict n n -KdV hierarchy, that have a minimal dependence between the different coefficients of a solution of them, the so-called minimal realizations. Thereto we discuss all these deformations from a wider perspective and consider them in a presetting instead of a setting. In this more general set-up we will present a number of C \mathbb {C} -subalgebras of R R that are stable under the basic C \mathbb {C} -linear derivations of R R and such that these derivations commute on these C \mathbb {C} -subalgebras. This we apply at the introduction of the minimal realizations of all deformations, we show how these realizations relate to solutions in different settings and use them to show that all hierarchies under consideration possess invariant scaling transformations.