mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Poisson superalgebra is a Z₂- graded generalization of a

Poisson algebra 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 i ...

. Specifically, a Poisson superalgebra is an (associative)

superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. Th ...

''A'' with a Lie superbracket :

: A\otimes A\to A

such that (''A'', �,· is a

Lie superalgebra In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2 grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the ...

and the operator :

: A\to A

is a superderivation of ''A'': :

,yz =,y + (-1)^y,z \,

A supercommutative Poisson algebra is one for which the (associative) product is

supercommutative In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have :yx = (-1)^xy , where , ''x'', denotes the grade of the element and is 0 or 1 ( ...

. 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 the BRST and Batalin-Vilkovisky formalism.

Examples

* If ''A'' is any associative Z₂ graded algebra, then, defining a new product ,.(which is called the super-commutator) by ,y=xy-(-1)^{, x, , y,}yx for any pure graded x, y turns ''A'' into a Poisson superalgebra.

References

*{{springer, id=p/p110170, title=Poisson algebra, author= Y. Kosmann-Schwarzbach Super linear algebra Symplectic geometry

Examples

See also

References