Poisson superalgebra
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]} .