On the First Cohomology of Infinitesimal Poisson Algebras

Abstract

For the so-called infinitesimal Poisson algebras encoding first-order jets of Poisson submanifolds, we provide a description of their first cohomology in terms of intrinsic cohomologies of the underlying Poisson submanifold. We establish a natural mapping from their first cohomology to the first Poisson cohomology of the corresponding Poisson submanifold. Moreover, we formulate necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras. Finally, we consider the special cases of symplectic leaves and, more generally, Poisson submanifolds with partially split first-order jets. In particular, we derive cohomological necessary conditions for the partially split property.