- Poisson superalgebra
In

mathematics , a**Poisson superalgebra**is a**Z**_{2}-graded generalization of aPoisson algebra . Specifically, a Poisson superalgebra is an (associative)superalgebra "A" with aLie superbracket :$[cdot,cdot]\; :\; Aotimes\; A\; o\; A$such that ("A", [·,·] ) is aLie superalgebra and the operator:$[x,cdot]\; :\; A\; o\; A$is asuperderivation of "A"::$[x,yz]\; =\; [x,y]\; z\; +\; (-1)^y\; [x,z]\; .,$A supercommutative Poisson algebra is one for which the (associative) product is supercommutative.

This is one possible way of "super"izing the Poisson algebra. This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other is to define an

antibracket algebra instead. This is used in theBRST andBatalin-Vilkovisky formalism.**Examples*** If "A" is any associative

**Z**_{2}graded algebra, then, defining a new product [.,.] by [x,y] =xy-(-1)^{|x||y|}yx for any pure graded x, y turns "A" into a Poisson superalgebra.**ee also***

Poisson supermanifold **References***springer|id=p/p110170|title=Poisson algebra|author=Y. Kosmann-Schwarzbach

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Poisson supermanifold**— In differential geometry a Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (to clarify this: M is not a point set space and so, doesn t really exist, and really, this… … Wikipedia**Poisson bracket**— In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time evolution of a dynamical system in the Hamiltonian formulation. In a more general… … Wikipedia**Poisson algebra**— 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. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central… … Wikipedia**Gerstenhaber algebra**— In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Gerstenhaber (1963) that combines the structures of a supercommutative ring and a… … Wikipedia**Lie algebra representation**— Lie groups … Wikipedia**List of mathematics articles (P)**— NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… … Wikipedia**Batalin-Vilkovisky formalism**— In theoretical physics, Batalin Vilkovisky (BV) formalism was developed as a method for determining the ghost structure for theories, such as gravity and supergravity, whose Hamiltonian formalism has constraints not related to a Lie algebra… … Wikipedia**Schouten-Nijenhuis bracket**— In differential geometry, the Schouten Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different… … Wikipedia**Schouten–Nijenhuis bracket**— In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different… … Wikipedia**Lie algebra**— 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 algebras were introduced to study the concept of infinitesimal transformations. The term… … Wikipedia