Poisson algebra

Examples

Poisson algebras occur in various settings.

Symplectic manifolds

The space of real-valued smooth functions 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_H, the Hamiltonian vector field. Then, given any two smooth functions F and G over the symplectic manifold, the Poisson bracket may be defined as:

${\displaystyle \{F,G\}=dG(X_F)=X_F(G)}$.

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

${\displaystyle X_{\{F,G\}}=[X_F,X_G]}$

where [,] is the Lie derivative. When the symplectic manifold is R²ⁿ with the standard symplectic structure, then the Poisson bracket takes on the well-known form

${\displaystyle \{F,G\}=\sum _{i=1}^n\frac{\mathrm{\partial }F}{\mathrm{\partial }{q}_i}\frac{\mathrm{\partial }G}{\mathrm{\partial }{p}_i}-\frac{\mathrm{\partial }F}{\mathrm{\partial }{p}_i}\frac{\mathrm{\partial }G}{\mathrm{\partial }{q}_i}.}$

Similar considerations apply for Poisson manifolds, 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 algebras.

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 − yx turns it into a Poisson algebra (and thus, also a Lie algebra) A_L. Note that the resulting A_L 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/C₂(V) is a Poisson algebra with {a, b} = a₀b and a ⋅ b = a₋₁b. For certain vertex operator algebras, these Poisson algebras are finite-dimensional.

Z2 grading

Poisson algebras can be given a Z₂-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

${\displaystyle |\{a,b\}|=|a|+|b|}$

whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:

${\displaystyle |\{a,b\}|=|a|+|b|-1}$

In both of these expressions ${\displaystyle |a|=\mathrm{deg}a}$ denotes the grading of the element ${\displaystyle a}$; typically, it counts how ${\displaystyle a}$ can be decomposed into an even or odd product of generating elements. Gerstenhaber algebras conventionally occur in BRST quantization.