Poisson superalgebra

Short description: Z2-graded generalization of a Poisson algebra

In mathematics, a Poisson superalgebra is a ${\mathbb{\mathbb{Z}}}_2$-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 $\mathrm{\forall }x,y,z\in A$ 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: $[x,yz]=[x,y]z+(-1{)}^{|x||y|}y[x,z]$.

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:

$|[a,b]|=|a|+|b|$

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

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

Examples

• If $A$ is any associative ${\mathbb{\mathbb{Z}}}_2$-graded algebra, then, defining a new product $[\cdot ,\cdot ]$, called the super-commutator, by $[x,y]:=xy-(-1{)}^{|x||y|}yx$ for any pure graded x, y, turns $A$ into a Poisson superalgebra.
• The algebra $C^{\mathrm{\infty }}(P)$ of smooth functions of a symplectic manifold $(P,\mathrm{\Omega })$ is a Poisson Superalgebra if we set $A_1=0$.

See also

• Poisson supermanifold

References

• Hazewinkel, Michiel, ed. (2001), "Poisson algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p110170 
• Henneaux, Marc; Teitelboim, Claudio (1992). Quantization of Gauge System. Princeton University Press. ISBN 9780691037691.