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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>d</mtext> </math></EquationSource> </InlineEquation> commuting shifts. After showing the equivalence of this notion to the notion of Hamiltonian difference operator on a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textrm{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>d</mtext> </math></EquationSource> </InlineEquation>-dimensional lattice, we characterize scalar local Hamiltonian difference operators up to the order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((-2,2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and investigate the bi-Hamiltonian pairs they form.</p>

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Multidimensional Multiplicative Poisson Vertex Algebras

  • 期刊论文

  • 发表日期：

    2026年4月4日

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  • Pengfei Yang,
  • Matteo Casati

摘要

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 \(\textrm{d}\) d commuting shifts. After showing the equivalence of this notion to the notion of Hamiltonian difference operator on a \(\textrm{d}\) d -dimensional lattice, we characterize scalar local Hamiltonian difference operators up to the order \((-2,2)\) ( - 2 , 2 ) and investigate the bi-Hamiltonian pairs they form.