The existence of a canonical lifting of even Poisson Structures to the Algebra of Densities

TL;DR

This paper constructs a canonical way to lift even Poisson structures from a manifold to its algebra of densities, classifies all such extensions, and highlights the existence of a natural lift unique to even structures.

Contribution

It introduces a canonical lifting method for even Poisson structures to the algebra of densities and classifies all possible extensions, a novel approach in this area.

Findings

Existence of a natural lift for even Poisson structures.

Classification of all extensions of Poisson structures to densities.

Resemblance to Frolicher-Nijenhuis bracket construction.

Abstract

In this note we construct a canonical lifting of arbitrary Poisson structures on a manifold to its algbera of densities. Using this construction we proceed to classify all extensions of a fixed structure on the original manifold to its algebra of densities. The question is analogous to the problem studied by H.M.Khudaverdian and Th.Voronov for odd Poisson structures and differential operators. Although the questions are similar the results are distinctly marked, namely in the case of even Poisson structures there always exists a lift which is naturally defined. The proof of this result bears a remarkable resemblance to the construction of the Frolicher-Nijenhuis bracket.