Poisson manifold

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.

A Poisson structure (or Poisson bracket) on a smooth manifold ${\displaystyle M}$ is a function$${\displaystyle \{\cdot ,\cdot \}:{\mathcal {C}}^{\infty }(M)\times {\mathcal {C}}^{\infty }(M)\to {\mathcal {C}}^{\infty }(M)}$$on the vector space ${\displaystyle {C^{\infty }}(M)}$ of smooth functions on ${\displaystyle M}$, making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977^[1] and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.^[2]

A Poisson structure on a manifold ${\displaystyle M}$ gives a way of deforming the product of functions on ${\displaystyle M}$ to a new product that is typically not commutative. This process is known as deformation quantization, since classical mechanics can be based on Poisson structures, while quantum mechanics involves non-commutative rings.