In 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 centr ...

. 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. ...

''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, th ...

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 In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring an ...

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