Essential variational Poisson cohomology

TL;DR

This paper characterizes the structure of variational Poisson cohomology for skewadjoint operators, showing it is isomorphic to a Lie superalgebra of Hamiltonian vector fields and proving the vanishing of essential cohomology classes.

Contribution

It establishes an isomorphism between variational Poisson cohomology and Hamiltonian vector fields for skewadjoint operators and proves the vanishing of essential cohomology classes.

Findings

Cohomology is isomorphic to a Lie superalgebra of Hamiltonian vector fields.

Essential cohomology classes vanish, simplifying the structure.

Results have applications to bi-Hamiltonian structures and deformations.

Abstract

In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology for any quasiconstant coefficient matrix differential operator K of arbitrary order with invertible leading coefficient, provided that the algebra of differential functions is normal and is an algebra over a linearly closed differential field. In the present paper we show that, for K skewadjoint, this cohomology, viewed as a Z-graded Lie superalgebra, is isomorphic to the finite dimensional Lie superalgebra of Hamiltonian vector fields over a Grassman algebra. 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.