A Poisson superalgebra A is a Z2-graded algebra with two products, . and [,] (which both share the same grading) such that . turns A into an associative algebra, [,] turns A into a Lie superalgebra and the superLeibniz law stating that for any pure element x, [x,.] is a derivation/antiderivation.
- [x,yz]=[x,y]z+(-1)xyy[x,z]
for any pure elements x,y,z of A.
If in addition, . is supercommutative, we say A is a supercommutative Poisson superalgebra.
This is one possible way of "super"izing the Poisson algebra. This gives the classical dynamics of fermion fields (formally only! We don't observe classical anticommuting fermion fields as a correspondence principle!) 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. (Not that I'm saying only physicists study such algebras)
Examples
- If A is any associative Z2 graded algebra, then, defining a new product [.,.] by [x,y]=xy-(-1)xyyx for any pure graded x, y turns A into a Poisson superalgebra.
See also Poisson algebra, Poisson supermanifold, antibracket algebra, Lie superalgebra, associative algebra, supercommutative algebra.