The variational Poisson cohomology
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Kac_The variational.pdf
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1.1 MB
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Author(s)
De Sole, Alberto
Kac, Victor
Date Issued
March 2013
Journal
Japanese Journal of Mathematics
Publisher
Springer-Verlag
Citation
De Sole, Alberto, and Victor G. Kac. “The Variational Poisson Cohomology.” Japanese Journal of Mathematics 8, no. 1 (March 2013): 1–145.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s11537-013-1124-3
