Lie superalgebra

Definition 1.

A Lie superalgebra is a vector superspace equipped with a bilinear map

(1)

satisfying the following properties:

1. If $v$ and $w$ are homogeneous vectors, then $[v,w]$ is a homogeneous vector of degree $|v|+|w|(mod2)$,

2. For any homogeneous vectors $v,w$, $[v,w]={(-1)}^{|v||w|+1}[w,v]$,

3. For any homogeneous vectors $u,v,w$, ${(-1)}^{|u||w|}[u,[v,w]]+{(-1)}^{|v||u|}[v,[w,u]]+{(-1)}^{|w||v|}[w,[u,v]]$ = 0.

The map $[\cdot ,\cdot ]$ is called a Lie superbracket.

Example 1.

A Lie algebra $V$ can be considered as a Lie superalgebra by setting $V=V_0$ and, therefore, $V_1=\{0\}$.

Example 2.

Any associative superalgebra $A$ has a Lie superalgebra structure where, for any homogeneous elements $a,b\in A$, the Lie superbracket is defined by the equation

$$[a,b]=ab-{(-1)}^{|a||b|}ba.$$(2)

The Lie superbracket (2) is called the supercommutator bracket on $A$.

Example 3.

The space of graded derivations of a supercommutative superalgebra, equipped with the supercommutator bracket, is a Lie superalgebra.

Definition 2.

A vector superspace is a vector space $V$ equipped with a decomposition $V=V_0\oplus V_1$.

Let $V=V_0\oplus V_1$ be a vector superspace. Then any element of $V_0$ is said to be even, and any element of $V_1$ is said to be odd. By the definition of the direct sum, any element $v$ of $V$ can be uniquely written as $v=v_0+v_1$, where $v_0\in V_0$ and $v_1\in V_1$.

Definition 3.

A vector $v\in V$ is homogeneous of degree $i$ if $v\in V_i$ for $i=0$ or $1$.

If $v\in V$ is homogeneous, then the degree of $v$ is denoted by $|v|$. In other words, if $v\in V_i$, then $|v|=i$ by definition.

Remark.

The vector $0$ is homogeneous of both degree $0$ and $1$, and thus $|0|$ is not well-defined.