# Lie superalgebra

In mathematics, a **Lie superalgebra** is generalisation of a Lie algebra to include a **Z**_{2}-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the *even* elements of the superalgebra correspond to bosons and *odd* elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).

Formally, a **Lie superalgebra** is a (nonassociative) **Z**_{2}-graded algebra, or *superalgebra*, over a field of characteristic 0 (typically **R** or **C**) whose product [·, ·], called the **Lie superbracket** or **supercommutator**, satisfies

and

where *x*, *y*, and *z* are pure in the **Z**_{2}-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 **Z**_{2}-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 disappear, and the superbracket becomes a normal Lie bracket.

One way of thinking about a Lie superalgebra is to consider its even and odd parts, L_{0} and L_{1} separately. Then, L_{0} is a Lie algebra, L_{1} is a linear rep of L_{0}, and there exists a symmetric L_{0}-intertwiner such that for all x,y and z in L_{1},

A ** ^{*} Lie superalgebra** is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the

**Z**

_{2}grading and satisfies [x,y]

^{*}=[y

^{*},x

^{*}] for all x and y in the Lie superalgebra. Its universal enveloping algebra would be an ordinary

^{*}-algebra.

The universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.

## Examples

Given any associative superalgebra *A* one can define the supercommutator on homogeneous elements by

and then extending by linearity to all elements. The algebra *A* together with the supercommutator then becomes a Lie superalgebra.

## Category-theoretic definition

In category theory, a **Lie superalgebra** can be defined as a nonassociative superalgebra whose product satisfies

where σ is the cyclic permutation braiding . In diagrammatic form: