Scaling Invariance of the k[S]-Hierarchy and Its Strict Version

Let LT_N(R) denote the algebra of N × N-matrices with coefficients from the commutative k-algebra R, k = R or C, that possess only a finite number of nonzero diagonals above the central diagonal. In a previous paper we discussedintegrable deformations inside LT_N(R) of various commutative subalgebras of LT_N(k) that contain Sⁿ, where S is the N × N-matrix corresponding to the shift operator. Here we focus on two deformations of k[S], called the k[S]-hierarchy and its strict version and we discuss the scaling invariance that they possess. To do so, it is necessary to discuss both deformations from a wider perspective and consider them in a presetting instead of the usual setting. In this more general set-up we will present two k-subalgebras of R that are stable under the basic derivations of R and such that these derivations commute on these k-subalgebras. This we apply at the introduction of the minimal realizations of both deformations, we show how these realizations relate to solutions in different settings and use them to show that both hierarchies possess invariant scaling transformations.