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| {{Orphan|date=March 2013}}
| | Wilber Berryhill is what his wife loves to call him and he totally loves this name. Invoicing is what I do. To play lacross is something he would never give up. Her family members lives in Ohio but her husband wants them to transfer.<br><br>my site; [http://Www.weddingwall.Com.au/groups/easy-advice-for-successful-personal-development-today/ clairvoyants] |
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| In [[mathematics]], a '''Poisson ring''' is a [[commutative ring]] on which an [[anticommutativity|anticommutative]] and [[distributivity|distributive]] [[binary operation]] <math>[\cdot,\cdot]</math> satisfying the [[Jacobi identity]] and the [[product rule]] is defined. Such an operation is then known as the [[Poisson bracket]] of the Poisson ring.
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| 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 algebra]]s as well. This observation is important in studying the [[classical limit]] of [[quantum mechanics]]—the [[non-commutative algebra]] of [[Operator (mathematics)|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.
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| ==Definition==
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| The Poisson bracket must satisfy the identities
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| * <math>[f,g] = -[g,f]</math> (skew symmetry)
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| * <math>[f + g, h] = [f,h] + [g,h] </math> (distributivity)
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| * <math>[fg,h] = f[g,h] + [f,h]g</math> ([[derivation (abstract algebra)|derivation]])
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| * <math>[f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0</math> ([[Jacobi identity]])
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| for all <math>f,g,h</math> in the ring.
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| A [[Poisson algebra]] is a Poisson ring that is also an [[algebra over a field]]. In this case, add the extra requirement
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| :<math>[sf,g] = s[f,g]</math>
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| for all scalars ''s''.
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| For each ''g'' in a Poisson ring ''A'', the operation <math>ad_g</math> defined as <math>ad_g(f) = [f,g]</math> is a [[derivation (abstract algebra)|derivation]]. If the set <math>\{ ad_g | g \in A \}</math> generates the set of derivations of ''A'', then ''A'' is said to be '''non-degenerate'''.
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| If a non-degenerate Poisson ring is [[ring isomorphism|isomorphic as a commutative ring]] to the [[algebra of smooth functions]] on a manifold ''M'', then ''M'' must be a [[symplectic manifold]] and <math>[\cdot,\cdot]</math> is the Poisson bracket defined by the [[symplectic form]].
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| ==References==
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| * {{planetmath reference|id=6422|title=If the algebra of functions on a manifold is a Poisson ring then the manifold is symplectic}}
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| {{PlanetMath attribution|id=6414|title=Poisson Ring}}
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| {{DEFAULTSORT:Poisson Ring}}
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| [[Category:Ring theory]]
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| [[Category:Symplectic geometry]]
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Wilber Berryhill is what his wife loves to call him and he totally loves this name. Invoicing is what I do. To play lacross is something he would never give up. Her family members lives in Ohio but her husband wants them to transfer.
my site; clairvoyants