$W_m$-algebras and fractional powers of difference operators

In this paper we describe a Poisson pencil associated to the lattice $W_m$-algebras defined in \cite{IM}, and we prove that the Poisson pencil is equal to the one defined in \cite{MW} and \cite{CM} using a type of discrete Drinfel'd-Sokolov reduction. We then show that, much as in the continuous case, a family of Hamiltonians defined by fractional powers of difference operators commute with respect to both structures, defining the kernel of one of them and creating an integrable hierarchy in the Liouville sense.