Lie superalgebra

Lie superalgebra

In mathematics, a Lie superalgebra is a kind of generalisation of a Lie algebra. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions.

A Lie superalgebra is a Z₂-graded algebra over a field of characteristic 0 (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies

where x, y, and z are pure in the Z₂-grading. Here |x| denotes the degree of x (either 0 or 1).

Lie superalgebras are a natural generalization of normal Lie algebras to include a Z₂-grading. Indeed, the above conditions on the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. The last condition is sometimes called the super Jacobi identity.

Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the funny signs disapper, and the superbracket becomes a normal Lie bracket.