# Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The Quantum Tetrahedron in 3 and 4 Dimensions, we found a description for a Poisson Structure on a Tetrahedron (that it's not smooth), so I was looking for some references about: Poisson structure on non-smooth manifolds with singularities, is there any progress about it?

• I think the objects you are considering (if a classical tetrahedron has to be among them), rather than "singular" smooth manifolds, are manifolds-with-corners in the sense of arxiv.org/abs/0910.3518 – Qfwfq Dec 20 '15 at 16:09
• (continued) They have a notion of tangent bundle with which, I think, you can define a Poisson structure without problems. In the article Joyce says that his theory of manifolds-with-corners is tailored for applications to symplectic geometry. – Qfwfq Dec 20 '15 at 16:16
• you can construct it in the sense of distribution – user21574 Dec 20 '15 at 21:51

If you are thinking of singular spaces that arise in algebraic geometry, then a Poisson structure can be defined as a biderivation on the structure sheaf. This definition straight forwardly generalizes to locally ringed spaces $(X,\mathcal{O})$, where $X$ is a topological space and $\mathcal{O}$ is a ring-valued sheaf on $X$ (the 'structure sheaf'), with $\mathcal{O}(U)$ to be interpreted as the ring of admissible 'smooth functions' on an open subset $U \subset X$. Locally ringed spaces cover a large variety of spaces with some kind of smooth structure but are could be more singular than manifolds. The definition is then as follows:
A Poisson structure on $(X,\mathcal{O})$ is a sheaf morphism $\{-,-\} \colon \mathcal{O} \times \mathcal{O} \to \mathcal{O}$ that is a derivation (satisfies the Leibniz rule) in each argument and also satisfies the Jacobi identity.