Poisson structure on manifolds with corners

Abstract

Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf $\OO_{X}$ on a manifold with corners $X$, we can interpret $\OO_{X}(U)$ as the ring of admissible smooth functions where $U$ is an open subset on $X$, in this way, a poisson structure on $(X, \OO_{X})$ is a sheaf morphism $$ \{-, -\}: \OO_{X} \times \OO_{X} \longrightarrow \OO_{X} $$ which satisfies the Leibniz rule an also the Jacobi Identity.