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| In [[mathematics]], a '''Poisson algebra''' is an [[associative algebra]] together with a [[Lie algebra|Lie bracket]] that also satisfies [[product rule|Leibniz' law]]; that is, the bracket is also a [[derivation (abstract algebra)|derivation]]. Poisson algebras appear naturally in [[Hamiltonian mechanics]], and are also central in the study of [[quantum group]]s. [[Manifold]]s with a Poisson algebra structure are known as [[Poisson manifold]]s, of which the [[symplectic manifold]]s and the [[Poisson-Lie group]]s are a special case. The algebra is named in honour of [[Siméon Denis Poisson]].
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| ==Definition==
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| A Poisson algebra is a [[vector space]] over a [[field (mathematics)|field]] ''K'' equipped with two [[bilinear map|bilinear]] products, ⋅ and {, }, having the following properties:
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| * The product ⋅ forms an [[associative algebra|associative ''K''-algebra]].
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| * The product {, }, called the [[Poisson bracket]], forms a [[Lie algebra]], and so it is anti-symmetric, and obeys the [[Jacobi identity]].
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| * The Poisson bracket acts as a [[Derivation (abstract algebra)|derivation]] of the associative product ⋅, so that for any three elements ''x'', ''y'' and ''z'' in the algebra, one has {''x'', ''y'' ⋅ ''z''} = {''x'', ''y''} ⋅ ''z'' + ''y'' ⋅ {''x'', ''z''}.
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| The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.
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| == Examples ==
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| Poisson algebras occur in various settings.
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| ===Symplectic manifolds===
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| The space of real-valued [[smooth function]]s over a [[symplectic manifold]] forms a Poisson algebra. On a symplectic manifold, every real-valued function ''H'' on the manifold induces a vector field ''X<sub>H</sub>'', the [[Hamiltonian vector field]]. Then, given any two smooth functions ''F'' and ''G'' over the symplectic manifold, the Poisson bracket may be defined as:
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| :<math>\{F,G\}=dG(X_F) = X_F(G)\,</math>.
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| This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as
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| :<math>X_{\{F,G\}}=[X_F,X_G]\,</math>
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| where [,] is the [[Lie derivative]]. When the symplectic manifold is '''R'''<sup>2''n''</sup> with the standard symplectic structure, then the Poisson bracket takes on the well-known form
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| :<math>\{F,G\}=\sum_{i=1}^n \frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}.</math>
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| Similar considerations apply for [[Poisson manifold]]s, which generalize symplectic manifolds by allowing the symplectic bivector to be vanishing on some (or trivially, all) of the manifold.
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| ===Associative algebras===
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| If ''A'' is an [[associative algebra]], then the commutator [''x'',''y'']≡''xy''−''yx'' turns it into a Poisson algebra.
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| ===Vertex operator algebras===
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| For a [[vertex operator algebra]] ''(V,Y, ω, 1)'', the space ''V/C<sub>2</sub>(V)'' is a Poisson algebra with ''{a, b}'' = ''a<sub>0</sub>b'' and ''a'' ⋅ ''b'' = ''a<sub>−1</sub>b''. For certain vertex operator algebras, these Poisson algebras are finite dimensional.
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| ==See also==
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| *[[Poisson superalgebra]]
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| *[[Antibracket algebra]]
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| *[[Moyal bracket]]
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| ==References==
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| *{{springer|id=p/p110170|title=Poisson algebra|author=Y. Kosmann-Schwarzbach}}
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| *{{cite book|first = K. H.|last = Bhaskara|first2 = K.|last2 = Viswanath|title = Poisson algebras and Poisson manifolds|location = |publisher = Longman|year = 1988|isbn = 0-582-01989-3}}
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| [[Category:Algebras]]
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| [[Category:Symplectic geometry]]
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