TITLE: Poisson structures on non-smooth manifolds with singularities
QUESTION [8 upvotes]: 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?

REPLY [2 votes]: More generally, since Poisson brackets are defined on associative algebras (in fact also NC ones), the key step is usually to find the "right" associative algebra of functions (or rather seaf of algebras, as Igor Khavkine says, in some settings) to describe your space. It is in this way, for example, that orbifold singularities can also be included in Poisson geometry setting (e.g. http://arxiv.org/abs/0807.0027).