In mathematics, a Poisson superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded associative unital algebra A = A 0 ⊕ A 1 {\displaystyle A=A_{0}\oplus A_{1}} that is equipped with a second bilinear map,
[ ⋅ , ⋅ ] : A × A → A {\displaystyle [\cdot ,\cdot ]:A\times A\to A} . Let | x | {\displaystyle |x|} denote the parity of a homogeneous element x {\displaystyle x} , then ∀ x , y , z ∈ A {\displaystyle \forall x,y,z\in A} the bracket satisfies:
Graded Antisymmetry: [ x , y ] = − ( − 1 ) | x | | y | [ y , x ] {\displaystyle [x,y]=-(-1)^{|x||y|}[y,x]} . Graded Jacobi Idenitity: [ x , [ y , z ] ] = [ [ x , y ] , z ] + ( − 1 ) | x | | y | [ y , [ x , z ] ] {\displaystyle [x,[y,z]]=[[x,y],z]+(-1)^{|x||y|}[y,[x,z]]} . Graded Leibniz Rule: [ x , y z ] = [ x , y ] z + ( − 1 ) | x | | y | y [ x , z ] {\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:
| [ a , b ] | = | a | + | b | {\displaystyle |[a,b]|=|a|+|b|}
whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:
| [ a , b ] | = | a | + | b | − 1 {\displaystyle |[a,b]|=|a|+|b|-1}
Examples If A {\displaystyle A} is any associative Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra, then, defining a new product [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} , called the super-commutator, by [ x , y ] := x y − ( − 1 ) | x | | y | y x {\displaystyle [x,y]:=xy-(-1)^{|x||y|}yx} for any pure graded x, y, turns A {\displaystyle A} into a Poisson superalgebra. The algebra C ∞ ( P ) {\displaystyle C^{\infty }(P)} of smooth functions of a symplectic manifold ( P , Ω ) {\displaystyle (P,\Omega )} is a Poisson Superalgebra if we set A 1 = 0 {\displaystyle A_{1}=0} .
See also Poisson supermanifold
References Y. Kosmann-Schwarzbach (2001) [1994], "Poisson algebra", Encyclopedia of Mathematics, EMS Press Henneaux, Marc; Teitelboim, Claudio (1992). Quantization of Gauge System. Princeton University Press. ISBN 9780691037691.
