In mathematics, a Poisson superalgebra is a <math>\mathbb{Z}_2</math>-graded associative unital algebra <math>A = A_0 \oplus A_1</math> that is equipped with a second bilinear map,
:<math>[\cdot,\cdot] : A\times A\to A</math>.
Let <math>|x|</math> denote the parity of a homogeneous element <math>x</math>, then <math> \forall x,y,z \in A</math> the bracket satisfies:
- Graded Antisymmetry: <math> [x,y] = - (-1)^{|x||y|} [y,x] </math>.
- Graded Jacobi Idenitity: <math> [x,[y,z]] = [[x,y],z] + (-1)^{|x||y|}[y,[x,z]] </math>.
- Graded Leibniz Rule: <math> [x,yz] = [x,y]z + (-1)^{|x||y|} y[x,z]</math>.
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:
:<math>|[a,b]| = |a|+|b|</math>
whereas in the Gerstenhaber algebra, the bracket decreases the grading by one:
:<math>|[a,b]| = |a|+|b| - 1</math>
Examples
- If <math>A</math> is any associative <math>\mathbb{Z}_2</math>-graded algebra, then, defining a new product <math>[\cdot,\cdot]</math>, called the super-commutator, by <math>[x,y]:=xy-(-1)^{|x||y|}yx</math> for any pure graded x, y, turns <math>A</math> into a Poisson superalgebra.
- The algebra <math>C^\infty(P)</math> of smooth functions of a symplectic manifold <math>(P,\Omega)</math> is a Poisson Superalgebra if we set <math>A_1 = 0</math>.
See also
- Poisson supermanifold