The variational Poisson cohomology

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.

130 pages, revised version with minor changes following the referee's suggestions

References in corpus (1)

• Calculus structure on the Lie conformal algebra complex and the variational complex

Cited by in corpus (26)

• Non-local Poisson structures and applications to the theory of integrable systems
• Bihamiltonian Cohomologies and Integrable Hierarchies I: A Special Case
• Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras
• Essential variational Poisson cohomology
• An operadic approach to vertex algebra and Poisson vertex algebra cohomology
• On deformations of multidimensional Poisson brackets of hydrodynamic type
• Some algebraic properties of differential operators
• Poisson cohomology of scalar multidimensional Dubrovin-Novikov brackets
• Rational matrix pseudodifferential operators
• Classification of linearly compact simple Nambu-Poisson algebras
• On classical finite and affine W-algebras
• Dirac operators and the Very Strange Formula for Lie superalgebras
• Chiral Homology of elliptic curves and the Zhu algebra
• The calculus of multivectors on noncommutative jet spaces
• Three computational approaches to weakly nonlocal Poisson brackets
• Computation of cohomology of Lie conformal and Poisson vertex algebras
• A new approach to the Lenard-Magri scheme of integrability
• Singular degree of a rational matrix pseudodifferential operator
• Hamiltonian structures for integrable nonabelian difference equations
• Chiral vs classical operad
• Double Multiplicative Poisson Vertex Algebras
• On Miura maps for W-superalgebras
• MasterPVA and WAlg: Mathematica packages for Poisson vertex algebras and classical affine $\mathcal{W}$-algebras
• The right-hand side of the Jacobi identity: to be naught or not to be?
• A geometric construction of integrable Hamiltonian hierarchies associated with the classical affine W-algebras
• Classical and variational Poisson cohomology