The modular automorphism group of a Poisson manifold

Abstract

The modular flow of Poisson manifold is a 1-parameter group of automorphisms determined by the choice of a smooth density on the manifold. When the density is changed, the generator of the group changes by a hamiltonian vector field, so one has a 1-parameter group of “outer automorphisms” intrinsically attached to any Poisson manifold. The group is trivial if and only if the manifold admits a measure which is invariant under all hamiltonian flows. The notion of modular flow in Poisson geometry is a classical limit of the notion of modular automorphism group in the theory of von Neumann algebras. In addition, the modular flow of a Poisson manifold is related to modular cohomology classes for associated Lie algebroids and symplectic groupoids. These objects have recently turned out to be important in Poincaré duality theory for Lie algebroids.