Poisson algebra - Citizendia

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics and Theoretical physics, quantum groups are certain Noncommutative algebras that first appeared in the theory of Quantum integrable systems 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the The algebra is named in honour of Siméon-Denis Poisson. Siméon-Denis Poisson (21 June 1781 &ndash 25 April 1840 was a French Mathematician, Geometer, and Physicist.

1 Definition
2 Examples
3 See also
4 References

Definition

A Poisson algebra is a vector space over a field K equipped with two bilinear products, $\cdot$ and { , }, having the following properties:

The product $\cdot$ forms an associative K-algebra. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

The product { , }, called the Poisson bracket, forms a Lie algebra, and so it is anti-symmetric, and obeys the Jacobi identity. In Mathematics and Classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie In Mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation

The Poisson bracket acts as a derivation of the associative product $\cdot$ , so that for any three elements x, y and z in the algebra, one has {x, yz} = {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.

Examples

Poisson algebras occur in various settings.

Symplectic manifolds

The space of real-valued smooth functions over a symplectic manifold forms a Poisson algebra. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the On a symplectic manifold, every real-valued function $H$ on the manifold induces a vector field $X H$ , the Hamiltonian vector field. In Mathematics and Physics, a Hamiltonian vector field on a Symplectic manifold is a Vector field, defined for any energy function Then, given any two smooth functions $F$ and $G$ over the symplectic manifold, the Poisson bracket {,} may be defined as:

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

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

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

where [,] is the Lie derivative. In Mathematics, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of one Vector field along the When the symplectic manifold is $\mathbb R^{2n}$ with the standard symplectic structure, then the Poisson bracket takes on the well-known form

$\{F,G\}=\sum_{i=1}^n \frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}.$

Similar considerations apply for Poisson manifolds, which generalize symplectic manifolds by allowing the symplectic bivector to be vanishing on some (or trivially, all) of the manifold. In Mathematics, a Poisson manifold is a Differential manifold M such that the algebra C &infin( M) of Smooth functions

Associative algebras

If A is a noncommutative associative algebra, then the commutator [x,y]≡xy−yx turns it into a Poisson algebra. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

Vertex operator algebras

For a vertex operator algebra $(V, Y,ω,1)$ , the space $V / C 2 (V)$ is a Poisson algebra with ${a, b} = a 0 b$ and $a \cdot b =a_{-1}b$ . In Mathematics, a vertex operator algebra (VOA is an algebraic structure that plays an important role in Conformal field theory and related areas of physics For certain vertex operator algebras, these Poisson algebras are finite dimensional.

References

Y. In Mathematics, a Poisson superalgebra is a Z 2- graded generalization of a Poisson algebra. In mathematics and Theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an Algebraic structure Kosmann-Schwarzbach (2001), “Poisson algebra”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

The Encyclopaedia of Mathematics is a large reference work in Mathematics.

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