Editing Poisson algebra (section)

== Examples ==
Poisson algebras occur in various settings.

===Symplectic manifolds===
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:

:<math>\{F,G\}=dG(X_F) = X_F(G)\,</math>.

This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as

:<math>X_{\{F,G\}}=[X_F,X_G]\,</math>

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

:<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>

Similar considerations apply for [[Poisson manifold]]s, which generalize symplectic manifolds by allowing the symplectic bivector to be rank deficient.

===Lie algebras===
The [[tensor algebra]] of a [[Lie algebra]] has a Poisson algebra structure. A very explicit construction of this is given in the article on [[universal enveloping algebra]]s.

The construction proceeds by first building the [[tensor algebra]] of the underlying vector space of the Lie algebra. The tensor algebra is simply the [[disjoint union]] ([[direct sum]] ⊕) of all tensor products of this vector space. One can then show that the Lie bracket can be consistently lifted to the entire tensor algebra: it obeys both the product rule, and the Jacobi identity of the Poisson bracket, and thus is the Poisson bracket, when lifted. The pair of products {,} and ⊗ then form a Poisson algebra. Observe that ⊗ is neither commutative nor is it anti-commutative: it is merely associative.  

Thus, one has the general statement that the tensor algebra of any Lie algebra is a Poisson algebra.  The universal enveloping algebra is obtained by modding out the Poisson algebra structure.

===Associative algebras===
If ''A'' is an [[associative algebra]], then imposing the commutator [''x'', ''y''] = ''xy'' &minus; ''yx'' turns it into a Poisson algebra (and thus, also a Lie algebra) ''A''<sub>''L''</sub>. Note that the resulting ''A''<sub>''L''</sub> should not be confused with the tensor algebra construction described in the previous section. If one wished, one could also apply that construction as well, but that would give a different Poisson algebra, one that would be much larger.

===Vertex operator algebras===
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.

===Z<sub>2</sub> grading===
Poisson algebras can be given a '''Z'''<sub>2</sub>-[[graded algebra|grading]] in one of two different ways. These two result in the [[Poisson superalgebra]] and the [[Gerstenhaber algebra]]. The difference between the two is in the grading of the product itself. For the Poisson superalgebra, the grading is given by
:<math>|\{a,b\}| = |a|+|b|</math>
whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:
:<math>|\{a,b\}| = |a|+|b| - 1</math>

In both of these expressions <math>|a|=\deg a</math> denotes the grading of the element <math>a</math>; typically, it counts how <math>a</math> can be decomposed into an even or odd product of generating elements. Gerstenhaber algebras conventionally occur in [[BRST quantization]].