On the representation of differentiations by Hamiltonians, operating from an algebra of local observable spin systems

Let $\overline{\mathfrak A}$ and $\mathfrak A$ be algebras of local and quasilocal observable spin systems corresponding to the group $Z^r$, $D:\mathfrak A\to\overline{\mathfrak A}$ be a differentiation invariant with respect to displacements. The question of representation of $D$ in the form of formal Hamiltonian $H=\sum_{k\in Z^r}T_kX$ formed by the displacements of an element $X\in\overline{\mathfrak A}$ is considered. It is shown that such a representation exists if the condition $\overline{\mathfrak A}$ holds, where $[H,a]\in\overline{\mathfrak A}$; $a\in\mathfrak A$ means an element obtained from the elements $[T_kX,a]$ by some $r$-multiple process of summation.