Multidimensional multiplicative Poisson vertex algebras

TL;DR

This paper introduces multidimensional multiplicative Poisson vertex algebras, generalizing existing structures to difference algebras with multiple shifts, and explores their relation to Hamiltonian difference operators and bi-Hamiltonian pairs.

Contribution

It defines the multidimensional multiplicative Poisson vertex algebra and characterizes scalar local Hamiltonian difference operators up to order (-2,2).

Findings

Established the equivalence to Hamiltonian difference operators.

Characterized scalar local Hamiltonian difference operators.

Investigated bi-Hamiltonian pairs in this context.

Abstract

In this paper we introduce the notion of multidimensional multiplicative Poisson vertex algebra, the generalization of the notion of multiplicative Poisson vertex algebra to a difference algebra endowed with D commuting shifts. After showing the equivalence of this notion to the notion of Hamiltonian difference operator on a D-dimensional lattice, we characterize scalar local Hamiltonian difference operators up to the order (-2,2) and investigate the bi-Hamiltonian pairs they form.