Poisson manifolds are a mathematical setup for classical mechanics with finitely many degrees of freedom.
A Poisson algebra is a commutative unital associative algebra , in this case over the field of real or complex numbers, equipped with a Lie bracket such that, for any , is a derivation of as an associative algebra.
A Poisson manifold is a real smooth manifold equipped with a Poisson structure. A Poisson structure is a Lie algebra bracket on the vector space of smooth functions on which together with the pointwise multiplication of functions makes it a Poisson algebra. As derivations of correspond to smooth tangent vector fields, for each there is a vector given by and called the Hamiltonian vector field corresponding to the function , which is viewed as a classical hamiltonian function.
Alternatively a Poisson structure on a manifold is given by a choice of smooth antisymmetric bivector called a Poisson bivector ; then .
This induces and is equivalently encoded by the structure of a Poisson Lie algebroid.
A morphism of Poisson manifolds is a morphism of smooth manifolds such that, for all , .
Every symplectic manifold carries a natural Poisson structure; however, such Poisson manifolds are very special. It is a basic theorem that Poisson structures on a manifold are equivalent to the smooth foliations of the underlying manifold such that each leaf is a symplectic manifold.
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
Examples of sequences of infinitesimal and local structures
|first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|derivative||Taylor series||germ||smooth function|
|tangent vector||jet||germ of curve||curve|
|Lie algebra||formal group||local Lie group||Lie group|
|Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|