In mathematics, a Poisson superalgebra is a -graded associative unital algebra that is equipped with a second bilinear map, . Let denote the parity of a homogeneous element , then the bracket satisfies: Graded Antisymmetry: . Graded Jacobi Idenitity: . Graded Leibniz Rule: .
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:
whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:
Examples If is any associative -graded algebra, then, defining a new product , called the super-commutator, by for any pure graded x, y, turns into a Poisson superalgebra. The algebra of smooth functions of a symplectic manifold is a Poisson Superalgebra if we set .
See also Poisson supermanifold
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