Local and non-local multiplicative Poisson vertex algebras and differential-difference equations

TL;DR

This paper introduces multiplicative Lie conformal and Poisson vertex algebras, exploring their connections to integrable differential-difference equations, $q$-deformed $W$-algebras, and lattice Poisson algebras, advancing the algebraic framework for integrability.

Contribution

It develops the theory of multiplicative Lie conformal and Poisson vertex algebras, including local and non-local cases, and links these to integrability of differential-difference Hamiltonian systems.

Findings

Established relations to $q$-deformed $W$-algebras

Introduced Adler type pseudodifference operators

Applied to integrability of differential-difference equations

Abstract

We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to $q$-deformed $W$-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations.