In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Poisson superalgebra is a Z₂-graded

Graded algebra

In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....

 generalization of a Poisson algebra

Poisson algebra

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups...

. Specifically, a Poisson superalgebra is an (associative) superalgebra

Superalgebra

In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....

 A with a Lie superbracket
such that (A, [·,·]) is a Lie superalgebra

Lie superalgebra

In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...

 and the operator
is a superderivation of A:

A supercommutative Poisson algebra is one for which the (associative) product is supercommutative

Supercommutative algebra

In mathematics, a supercommutative algebra is a superalgebra such that for any two homogeneous elements x, y we haveyx = ^In mathematics, a supercommutative algebra is a superalgebra In mathematics, a supercommutative algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two...

.

This is one possible way of "super"izing the Poisson algebra. This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other is to define an antibracket algebra

Antibracket algebra

In mathematics and theoretical physics, a Gerstenhaber algebra is an algebraic structure discovered by Murray Gerstenhaber that combines the structures of a supercommutative ring and a graded Lie superalgebra...

 instead. This is used in the BRST

BRST

BRST may refer to:* BRST formalism and quantization in Yang-Mills theories* Big Red Switch Time , computer jargon for switching your computer off, when all other options for a more elegant shutdown have been exhausted...

 and Batalin-Vilkovisky formalism.

Examples

• If A is any associative Z₂ graded algebra, then, defining a new product [.,.] (which is called the super-commutator) by [x,y]:=xy-(-1)^|x||y|yx for any pure graded x, y turns A into a Poisson superalgebra.