Inmathematics, aPoisson algebra is anassociative algebra together with aLie bracket that also satisfiesLeibniz's law; that is, the bracket is also aderivation. Poisson algebras appear naturally inHamiltonian mechanics, and are also central in the study ofquantum groups.Manifolds with a Poisson algebra structure are known asPoisson manifolds, of which thesymplectic manifolds and thePoisson–Lie groups are a special case. The algebra is named in honour ofSiméon Denis Poisson.
A Poisson algebra is avector space over afieldK equipped with twobilinear products, ⋅ and {, }, having the following properties:
The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.
Poisson algebras occur in various settings.
The space of real-valuedsmooth functions over asymplectic manifold forms a Poisson algebra. On a symplectic manifold, every real-valued functionH on the manifold induces a vector fieldXH, theHamiltonian vector field. Then, given any two smooth functionsF andG over the symplectic manifold, the Poisson bracket may be defined as:
This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as
where [,] is theLie derivative. When the symplectic manifold isR2n with the standard symplectic structure, then the Poisson bracket takes on the well-known form
Similar considerations apply forPoisson manifolds, which generalize symplectic manifolds by allowing the symplectic bivector to be rank deficient.
Thetensor algebra of aLie algebra has a Poisson algebra structure. A very explicit construction of this is given in the article onuniversal enveloping algebras.
The construction proceeds by first building thetensor algebra of the underlying vector space of the Lie algebra. The tensor algebra is simply thedisjoint 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.
IfA is anassociative algebra, then imposing the commutator [x,y] =xy −yx turns it into a Poisson algebra (and thus, also a Lie algebra)AL. Note that the resultingAL 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.
For avertex operator algebra (V,Y,ω, 1), the spaceV/C2(V) is a Poisson algebra with {a,b} =a0b anda ⋅b =a−1b. For certain vertex operator algebras, these Poisson algebras are finite-dimensional.
Poisson algebras can be given aZ2-grading in one of two different ways. These two result in thePoisson superalgebra and theGerstenhaber algebra. The difference between the two is in the grading of the product itself. For the Poisson superalgebra, the grading is given by
whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:
In both of these expressions denotes the grading of the element; typically, it counts how can be decomposed into an even or odd product of generating elements. Gerstenhaber algebras conventionally occur inBRST quantization.