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# Lie superalgebra

###### Definition 1.

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

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

satisfying the following properties:

1. If $v$ and $w$ are homogeneous vectors, then $[v,w]$ is a homogeneous vector of degree $|v|+|w|\;\;(\mathop{{\rm mod}}2)$,

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.

## Mathematics Subject Classification

81R50*no label found*17B60

*no label found*17B01

*no label found*81Q60

*no label found*

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