The variational Poisson cohomology

It is well known that the validity of the so called LenardMagri 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. 1. Dipartimento di Matematica, Università di Roma “La Sapienza”, 00185 Roma, Italy [email protected] . 2. Department of Mathematics, M.I.T., Cambridge, MA 02139, USA [email protected] . A.D.S. was partially supported by PRIN and AST grants. V.K. was partially supported by an NSF grant, and an ERC advanced grant. Parts of this work were done while A.D.S. was visiting the Department of Mathematics of M.I.T., while V.K. was visiting the newly created Center for Mathematics and Theoretical Physics in Rome, and A.D.S and V.K. were visiting the MSC and the Department of Mathematics of Tsinghua University in Beijing. 1 2 ALBERTO DE SOLE1 AND VICTOR G. KAC2