$W_m$-algebras and fractional powers of difference operators

TL;DR

This paper establishes a Poisson pencil structure for lattice $W_m$-algebras, demonstrating that fractional powers of difference operators generate commuting Hamiltonians and form an integrable hierarchy.

Contribution

It introduces a Poisson pencil for lattice $W_m$-algebras and proves its equivalence to known structures using discrete Drinfel'd-Sokolov reduction, extending continuous integrable systems to the discrete setting.

Findings

Poisson pencil for lattice $W_m$-algebras is described.

Fractional powers of difference operators produce commuting Hamiltonians.

An integrable hierarchy in the Liouville sense is constructed.

Abstract

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.