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Poisson ring (Definition)

A Poisson ring $A$ is a commutative ring on which a binary operation $[,]$, known as the Poisson bracket is defined. This operation must satisfy the following 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 field, then we call $A$ a Poisson algebra. In this case, we may wish to add the extra requirement

$\displaystyle [sf,g] = s[f,g]$
for all scalars $s$.

Because of properties 2 and 3, for each $g \in A$, the operation $ad_g$ defined as $ad_g(f) = [f,g]$ is a derivation. If the set $\{ ad_g \vert g \in A \}$ 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 functions on a 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 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 corresponding properties of the Poisson algebra.

In addition to their use in mechanics, Poisson algebras are also used in the study of Lie groups.



"Poisson ring" is owned by rspuzio. [ full author list (2) ]

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See Also: Hilbert space, deformation quantization

Also defines:  Poisson algebra

Cross-references: Lie groups, Hilbert space, operators, non-commutative algebra, quantum mechanics, mechanics, Hamiltonian, manifold, functions, scalars, field, identities, operation, commutative ring
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This is version 3 of Poisson ring, born on 2009-03-09, modified 2009-06-22.
Object id is 580, canonical name is PoissonRing.
Accessed 660 times total.

Classification:
Physics Classification: 45.20.-d (Formalisms in classical mechanics)
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