Poisson superalgebra

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

[\cdot, \cdot] : A \times A \to A

.

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

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

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 |

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

| [a, b] | = | a | + | b | - 1

Examples

If $A$ is any associative $ℤ_{2}$ -graded algebra, then, defining a new product $[\cdot, \cdot]$ , called the super-commutator, by $[x, y] := x y - (- 1)^{| x | | y |} y x$ for any pure graded x, y, turns $A$ into a Poisson superalgebra.
The algebra $C^{\infty} (P)$ of smooth functions of a symplectic manifold $(P, Ω)$ is a Poisson Superalgebra if we set $A_{1} = 0$ .

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.

Poisson superalgebra

Examples

See also

References