Wm -algebras and fractional powers of difference operators

Abstract

Abstract In this paper we define a Poisson pencil associated to a lattice Wm -algebras defined in a recent paper by Izosimov and Marí Beffa (2023 Int. Math. Res. Not. 2023 17021–59). We then prove that this Poisson pencil is equal to the one defined in 2013 by Marí Beffa and Wang (2013 Nonlinearity 26 2515) and the author 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.