The variational Poisson cohomology

TL;DR

This paper develops cohomology theories related to Poisson structures and computes their cohomology for certain Hamiltonian operators, providing tools for analyzing integrability of bi-Hamiltonian PDEs.

Contribution

It introduces various cohomology complexes using Lie superalgebra and Poisson vertex algebra frameworks and computes their cohomology for a class of Hamiltonian operators.

Findings

Computed the cohomology of the generalized de Rham complex.

Determined the cohomology of the generalized variational complex for specific Hamiltonian operators.

Established a connection between cohomology complexes and the integrability of bi-Hamiltonian PDEs.

Abstract

It is well known that the validity of the so called Lenard-Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix.