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 together with a second product, 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:

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

Here, ${\displaystyle |a|=\deg a}$ is the grading of a (pure) element ${\displaystyle a}$.

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

This is one of two possible ways of "super"izing the Poisson algebra. This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other way is to define an antibracket algebra or Gerstenhaber algebra, used in the BRST and Batalin-Vilkovisky formalism. The difference between these two is in the grading of the Lie bracket. In the Poisson superalgebra, the grading of the bracket is zero:

${\displaystyle |[a,b]|=|a|+|b|}$

whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:

${\displaystyle |[a,b]|=|a|+|b|-1}$

Examples

• If ${\displaystyle A}$ is any associative Z₂ graded algebra, then, defining a new product ${\displaystyle [\cdot ,\cdot ]}$, called the super-commutator, by ${\displaystyle [x,y]:=xy-(-1)^{|x||y|}yx}$ for any pure graded x, y, turns ${\displaystyle A}$ into a Poisson superalgebra.

See also

• Poisson supermanifold

References

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