Multidimensional multiplicative Poisson vertex algebras

TL;DR

This work extends multiplicative Poisson vertex algebras to multidimensional difference algebras with $D$ commuting shifts, establishing a multidimensional master formula that renders the PBW-like computations tractable for Hamiltonian structures on lattices. It proves the equivalence between multidimensional mPVAs and local Hamiltonian difference operators, and provides an explicit classification of 2D scalar Hamiltonian operators up to order $(-2,2)$, revealing a rigid structure: essentially, operators are either of a symmetric normal form or of a restricted $P_i$-type tied to a non-constant function $F$ solving $fF'=F$. The bi-Hamiltonian analysis shows that, in 2D, compatible pairs exist only when the defining functions are proportional, limiting the diversity of multidimensional bi-Hamiltonian systems. These results generalize the 1D theory and illuminate integrability structures for multidimensional differential-difference equations on lattices, with potential extensions to nonlocal and mixed-type PVAs.

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.

Theorems & Definitions (25)

• definition 1: $\lambda$-bracket
• definition 2: Multidimensional Poisson Vertex Algebra
• theorem 1: Master formula
• proof
• proposition 1
• proof
• theorem 2
• proof
• proposition 2
• proof