Poisson superalgebra

In mathematics, a Poisson superalgebra is a Z₂-graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra A with a Lie superbracket

${\displaystyle [\cdot ,\cdot ]:A\otimes A\to A}$

such that (A, [·,·]) is a Lie superalgebra and the operator

${\displaystyle [x,\cdot ]:A\to A}$

is a superderivation of A:

${\displaystyle [x,yz]=[x,y]z+(-1)^{|x||y|}y[x,z].\,}$

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 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 [x,y]:=xy-(-1)^|x||y|yx for any pure graded x, y turns A into a Poisson superalgebra.

See also

• Poisson supermanifold

References

• Y. Kosmann-Schwarzbach (2001) [1994], "Poisson algebra", Encyclopedia of Mathematics, EMS Press