# Lie superalgebra

###### Definition 1.

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

 $\begin{split}\displaystyle[\cdot,\cdot]:V\otimes V&\displaystyle\rightarrow V,% \\ \displaystyle v\otimes w&\displaystyle\mapsto[v,w],\end{split}$ (1)

satisfying the following properties:

1. 1.

If $v$ and $w$ are homogeneous vectors, then $[v,w]$ is a homogeneous vector of degree $|v|+|w|\pmod{2}$,

2. 2.

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

3. 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.

 Title Lie superalgebra Canonical name LieSuperalgebra Date of creation 2013-03-22 15:35:44 Last modified on 2013-03-22 15:35:44 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 16 Author bci1 (20947) Entry type Definition Classification msc 81R50 Classification msc 17B60 Classification msc 17B01 Classification msc 81Q60 Synonym Lie super algebra Synonym graded Lie algebra Related topic CartanCalculus Related topic Superalgebra Related topic GradedAlgebra Related topic LieAlgebroids Related topic SuperfieldsSuperspace Related topic SupersymmetryOrSupersymmetries Related topic LieAlgebroids Related topic JordanBanachAndJordanLieAlgebras Related topic LieAlgebra Related topic LieAlgebraCohomology Related topic SuperAlgebra Related topic CartanCalculus Related topic QuantumGravityTheories Related topic Fu Defines vector superspace Defines Lie superbracket Defines supercommutator bracket