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?
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$\begingroup$ 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 $\endgroup$– QfwfqDec 20, 2015 at 16:09
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$\begingroup$ (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. $\endgroup$– QfwfqDec 20, 2015 at 16:16
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$\begingroup$ you can construct it in the sense of distribution $\endgroup$– user21574Dec 20, 2015 at 21:51
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2 Answers
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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.
Again in the context of algebraic geometry, the above definition appears as Definition 1 in Gualtieri & Pym (PLMS 2012, arXiv), which gives Polishchuk (JMathSci 1997) as the historical reference.
answered Dec 20, 2015 at 15:37
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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).
