Geometric constructions in algebra of densities

The algebra of densities $\Den(M)$ is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra $\Den(M)$ is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra $\Den(M)$.It allows {a natural definition of}bracket operations on vector densities of various weights on a (super)manifold $M$,similar to how the classical Fr\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from ``vector fields'' (derivations) on $\Den(M)$ to ``multivector fields''. This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.~Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.