Poisson superalgebra

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

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

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

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

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:

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

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

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

• If ${\displaystyle A}$ is any associative ${\displaystyle \mathbb {Z} _{2}}$-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.
• The algebra ${\displaystyle C^{\infty }(P)}$ of smooth functions of a symplectic manifold ${\displaystyle (P,\Omega )}$ is a Poisson Superalgebra if we set ${\displaystyle A_{1}=0}$.