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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
Poisson ring
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(Definition)
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A Poisson ring  is a commutative ring on which a binary operation ![$ [,]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img2.png "$ [,]$") , known as the Poisson bracket is defined. This operation must satisfy the following identities:
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![$ [f,g] = -[g,f]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img3.png "$ [f,g] = -[g,f]$")
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![$ [f + g, h] = [f,h] + [g,h]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img4.png "$ [f + g, h] = [f,h] + [g,h]$")
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![$ [fg,h] = f[g,h] + g[f,h]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img5.png "$ [fg,h] = f[g,h] + g[f,h]$")
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![$ [f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img6.png "$ [f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0$")
If, in addition,  is an algebra over a field, then we call  a Poisson algebra. In this case, we may wish to add the extra requirement
for all scalars  .
Because of properties 2 and 3, for each  , the operation  defined as
![$ ad_g(f) = [f,g]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img13.png "$ ad_g(f) = [f,g]$") is a derivation. If the set
 generates the set of derivations of  , we say that  is non-degenerate.
It can be shown that, if  is non-degenerate and is isomorphic as a commutative ring to the algebra of smooth functions on a manifold  , then  must be a symplectic manifold and ![$ [,]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img20.png "$ [,]$") is the Poisson bracket defined by the symplectic form.
Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics -- the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.
In addition to their use in mechanics, Poisson algebras are also used in the study of Lie groups.
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"Poisson ring" is owned by rspuzio. [ full author list (2) ]
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| Also defines: |
Poisson algebra |
This object's parent.
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Cross-references: Lie groups, singular, functions, Hilbert space, operators, non-commutative algebra, limit, terms, Hamiltonian, geometry, symplectic form, symplectic manifold, manifold, smooth functions, isomorphic, non-degenerate, generates, derivation, properties, scalars, field, algebra, addition, identities, operation, Poisson bracket, binary operation, commutative ring
There are 6 references to this entry.
This is version 6 of Poisson ring, born on 2004-10-24, modified 2006-09-13.
Object id is 6414, canonical name is PoissonAlgebra.
Accessed 2132 times total.
Classification:
| AMS MSC: | 53D05 (Differential geometry :: Symplectic geometry, contact geometry :: Symplectic manifolds, general) |
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Pending Errata and Addenda
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![$\displaystyle [sf,g] = s[f,g]$](http://images.planetmath.org:8080/cache/objects/6414/l2h/img9.png "$\displaystyle [sf,g] = s[f,g]$"){kind=small}