Poisson ring

In mathematics, a Poisson ring "A" is a commutative ring on which a binary operation $[,]$, known as the Poisson bracket, is defined.

Many important operations and results of symplectic geometry and Hamiltonian 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 quantum mechanics -- the non-commutative algebra of operators on a 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 correstponding properties of the Poisson algebra.

Definition

This operation must satisfy the following identities:

* $[f,g]\; =\; -\; [g,f]$ (skew symmetry)
* $[f\; +\; g,\; h]\; =\; [f,h]\; +\; [g,h]$ (distributivity)
* $[fg,h]\; =\; f\; [g,h]\; +\; [f,h]\; g$ (derivation)
* $[f,\; [g,h]\; +\; [g,\; [h,f]\; +\; [h,\; [f,g]\; =\; 0$ (Jacobi identity)

for all $f,g,h$ in the ring. If, in addition, "A" is an algebra over a field, then "A" is a Poisson algebra. In this case, add the extra requirement

:$[sf,g]\; =\; s\; [f,g]$

for all scalars "s". For each $g\; in\; A$, the operation $ad\_g$ defined as $ad\_g(f)\; =\; [f,g]$ is a derivation. If the set $\{\; ad\_g\; |\; g\; in\; A\; \}$ generates the set of derivations of "A", then "A" is said to be non-degenerate. It can be shown that, if "A" is non-degenerate and is isomorphic as a commutative ring to the algebra of smooth functions on a manifold "M", then "M" must be a symplectic manifold and $[,]$ is the Poisson bracket defined by the symplectic form.

References

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