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Concept
Poisson algebra
Summary
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.
A Poisson algebra is a vector space over a field K equipped with two bilinear products, ⋅ and {, }, having the following properties:
The product ⋅ forms an associative K-algebra.
The product {, }, called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity.
The Poisson bracket acts as a 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}.
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-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 XH, the Hamiltonian vector field. Then, given any two smooth functions F and G 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 the Lie derivative. When the symplectic manifold is R2n with the standard symplectic structure, then the Poisson bracket takes on the well-known form
Similar considerations apply for Poisson manifolds, which generalize symplectic manifolds by allowing the symplectic bivector to be rank deficient.
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.
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Related concepts (9)
Poisson manifold
In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics. A Poisson structure (or Poisson bracket) on a smooth manifold is a functionon the vector space of smooth functions on , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems.
Universal enveloping algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators.