The variational Poisson cohomology
Abstract
It is well known that the validity of the so called LenardMagri scheme of integrability of a biHamiltonian 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 (noncanonically) 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2011
 arXiv:
 arXiv:1106.0082
 Bibcode:
 2011arXiv1106.0082D
 Keywords:

 Mathematical Physics;
 Mathematics  Representation Theory;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 37K10 (Primary) 37K30;
 17B80 (Secondary) 37K10 (Primary) 37K30;
 17B80 (Secondary) 37K10 (Primary) 37K30;
 17B80 (Secondary)
 EPrint:
 130 pages, revised version with minor changes following the referee's suggestions