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.

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

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