Poisson superalgebra

In mathematics, a Poisson superalgebra is a $\mathbb {Z} _{2}$ -graded associative unital algebra $A=A_{0}\oplus A_{1}$ that is equipped with a second bilinear map,

[\cdot ,\cdot ]:A\times A\to A

.

Let $|x|$ denote the parity of a homogeneous element $x$ , then $\forall x,y,z\in A$ the bracket satisfies:

Graded Antisymmetry: $[x,y]=-(-1)^{|x||y|}[y,x]$ .
Graded Jacobi Idenitity: $[x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]]$ .
Graded Leibniz Rule: $[x,yz]=[x,y]z+(-1)^{|x||y|}y[x,z]$ .

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:

|[a,b]|=|a|+|b|

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

|[a,b]|=|a|+|b|-1

Examples

If $A$ is any associative $\mathbb {Z} _{2}$ -graded algebra, then, defining a new product $[\cdot ,\cdot ]$ , called the super-commutator, by $[x,y]:=xy-(-1)^{|x||y|}yx$ for any pure graded x, y, turns $A$ into a Poisson superalgebra.
The algebra $C^{\infty }(P)$ of smooth functions of a symplectic manifold $(P,\Omega )$ is a Poisson Superalgebra if we set $A_{1}=0$ .