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

The discrete KP hierarchy and its strict version are both deformations of the commutative algebra $k[\Lambda]$ inside the algebra $\mathrm{Ps}\kern1.1pt\Delta$ of pseudo-difference operators, where $\Lambda$ is the $\mathbb{Z}\times\mathbb{Z}$-matrix corresponding to the shift operator and $k=\mathbb{R}$ or $k=\mathbb{C}$. Under these deformations, the matrix coefficients of the elements of $\mathrm{Ps}\kern1.1pt\Delta$ come from a commutative $k$-algebra $R$. 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 $k$-subalgebras of $R$ that are stable under the basic derivations of $R$ and such that these derivations commute on these $k$-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.