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 is a functionon the vector space of smooth functions on , 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 gives a way of deforming the product of functions on 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.
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