Minimal realizations and scaling invariance of the discrete KP hierarchy and its strict version

Theoretical and Mathematical Physics. 2022. Vol. 213. No. 1. P. 1348–1361.

Helminck G. F., Poberezhny V. A., Polenkova S. V.

The discrete KP hierarchy and its strict version are both deformations of the commutative algebra k[Λ]k[Λ] inside the algebra PsΔPsΔ of pseudo-difference operators, where ΛΛ is the Z×ZZ×Z-matrix corresponding to the shift operator and k=Rk=R or k=Ck=C. Under these deformations, the matrix coefficients of the elements of PsΔPsΔ come from a commutative kk-algebra RR. We discuss both deformations from a wider perspective and consider them in a presetting instead of a setting. In this more general setup, we present a number of kk-subalgebras of RR that are stable under the basic derivations of RR and such that these derivations commute on these kk-subalgebras. This is used to introduce the minimal realizations of both deformations. We relate these realizations to solutions in different settings and use them to show that both hierarchies possess invariant scaling transformations.

Keywords: Lax equations, псевдоразностные операторы, уравнения Лакса, pseudo-difference operators, minimal realizations, scaling transformations, минимальные реализации, масштабные преобразования