Lie superalgebra
Definition 1.
A Lie superalgebra^{} is a vector superspace equipped with a bilinear map
$$\begin{array}{cc}\hfill [\cdot ,\cdot ]:V\otimes V& \to V,\hfill \\ \hfill v\otimes w& \mapsto [v,w],\hfill \end{array}$$  (1) 
satisfying the following properties:

1.
If $v$ and $w$ are homogeneous^{} vectors, then $[v,w]$ is a homogeneous vector of degree $v+w\phantom{\rule{veryverythickmathspace}{0ex}}(mod2)$,

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

3.
For any homogeneous vectors $u,v,w$, ${(1)}^{uw}[u,[v,w]]+{(1)}^{vu}[v,[w,u]]+{(1)}^{wv}[w,[u,v]]$ = 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
$$[a,b]=ab{(1)}^{ab}ba.$$  (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 welldefined.
Title  Lie superalgebra 
Canonical name  LieSuperalgebra 
Date of creation  20130322 15:35:44 
Last modified on  20130322 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 