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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 isequal to the one defined in \cite{MW} and \cite{CM} using a type of discreteDrinfel'd-Sokolov reduction. We then show that, much as in the continuous case,a family of Hamiltonians defined by fractional powers of difference operatorscommute with respect to both structures, defining the kernel of one of them andcreating an integrable hierarchy in the Liouville sense.

$W_m$-algebras and fractional powers of difference operators