PlanetPhysics/Poisson Ring

A Poisson ring $A$ is a [[../PAdicMeasure/|commutative ring]] on which a binary [[../Cod/|operation]] $[,]$, known as the Poisson bracket is defined. This operation must satisfy the following [[../Identity2/|identities]]:

1. $[f,g]=-[g,f]$
2. $[f+g,h]=[f,h]+[g,h]$
3. $[fg,h]=f[g,h]+g[f,h]$
4. $[f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0$

If, in addition, $A$ is an algebra over a [[../CosmologicalConstant/|field]], then we call $A$ a Poisson algebra . In this case, we may wish to add the extra requirement $[sf,g]=s[f,g]$ for all [[../Vectors/|scalars]] $s$.

Because of properties 2 and 3, for each $g\in A$, the operation $a{d}_g$ defined as $a{d}_g(f)=[f,g]$ is a derivation. If the set $\{a{d}_g|g\in A\}$ generates the set of derivations of $A$, we say that $A$ is non-degenerate .

It can be shown that, if $A$ is non-degenerate and is isomorphic as a commutative ring to the algebra of smooth [[../Bijective/|functions]] on a [[../NoncommutativeGeometry4/|manifold]] $M$, then $M$ must be a symplectic manifold and $[,]$ is the Poisson bracket defined by the symplectic form.

Many important operations and results of symplectic geometry and [[../Hamiltonian2/|Hamiltonian]] [[../Mechanics/|mechanics]] may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of [[../QuantumParadox/|quantum mechanics]] --- the [[../MoritaInvariant/|non-commutative algebra]] of [[../QuantumOperatorAlgebra4/|operators]] on a [[../NormInducedByInnerProduct/|Hilbert space]] has the Poisson algebra of functions on a symplectic manifold as a singular limit and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

In addition to their use in mechanics, Poisson algebras are also used in the study of [[../BilinearMap/|Lie groups]].