Essential Variational Poisson Cohomology

Abstract

In our recent paper “The variational Poisson cohomology” (2011) we computed the dimension of the variational Poisson cohomology H[subscript K](V) for any quasiconstant coefficient ℓ × ℓ matrix differential operator K of order N with invertible leading coefficient, provided that V is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the Z -graded Lie superalgebra H[subscript K](V) is isomorphic to the finite dimensional Lie superalgebra [˜ over H](Nℓ,S) . We also prove that the subalgebra of “essential” variational Poisson cohomology, consisting of classes vanishing on the Casimirs of K, is zero. This vanishing result has applications to the theory of bi-Hamiltonian structures and their deformations. At the end of the paper we consider also the translation invariant case.