, Volume 8, Issue 1, pp 1-145
Date: 20 Mar 2013

The variational Poisson cohomology

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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 superalgebra 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.

Communicated by: Yasuyuki Kawahigashi
Parts of this work were done while the first author was visiting the Department of Mathematics of MIT, and while the second author was visiting the newly created Center for Mathematics and Theoretical Physics in Rome, and the paper was completed while both authors were visiting the MSC and the Department of Mathematics of Tsinghua University in Beijing.
The first author was partially supported by PRIN and AST grants.
The second author was partially supported by an NSF grant, and an ERC advanced grant.
To the memory of Boris Kupershmidt (11/27/1946–12/12/2010)