Editing Poisson superalgebra

{{Short description|Z2-graded generalization of a Poisson algebra}}
{{one source|date=July 2023}}
In [[mathematics]], a '''Poisson superalgebra''' is a '''Z'''<sub>2</sub>-[[graded algebra|graded]] generalization of a [[Poisson algebra]]. Specifically, a Poisson superalgebra is an (associative) [[superalgebra]] ''A'' together with a second product, a [[Lie superbracket]]
:<math>[\cdot,\cdot] : A\otimes A\to A</math>
such that (''A'', [·,·]) is a [[Lie superalgebra]] and the operator
:<math>[x,\cdot] : A\to A</math>
is a [[superderivation]] of ''A'':
:<math>[x,yz] = [x,y]z + (-1)^{|x||y|}y[x,z].</math>

Here, <math>|a|=\deg a</math> is the grading of a (pure) element <math>a</math>.

A supercommutative Poisson algebra is one for which the (associative) product is [[supercommutative algebra|supercommutative]].

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 quantization|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 algebra|associative]] '''Z'''<sub>2</sub> 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.

==See also==
*[[Poisson supermanifold]]

==References==
*{{springer|id=p/p110170|title=Poisson algebra|author=[[Yvette Kosmann-Schwarzbach|Y. Kosmann-Schwarzbach]]}}

[[Category:Super linear algebra]]
[[Category:Symplectic geometry]]