Lie superalgebra

Definition 1.

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

[,]:VVV,vw[v,w], (1)

satisfying the following properties:

  1. 1.

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

  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 [,] is called a Lie superbracket.

Example 1.

A Lie algebraMathworldPlanetmath V can be considered as a Lie superalgebra by setting V=V0 and, therefore, V1={0}.

Example 2.

Any associative superalgebra A has a Lie superalgebra structure where, for any homogeneous elementsPlanetmathPlanetmath a,bA, 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=V0V1.

Let V=V0V1 be a vector superspace. Then any element of V0 is said to be even, and any element of V1 is said to be odd. By the definition of the direct sum, any element v of V can be uniquely written as v=v0+v1, where v0V0 and v1V1.

Definition 3.

A vector vV is homogeneous of degree i if vVi for i=0 or 1.

If vV is homogeneous, then the degree of v is denoted by |v|. In other words, if vVi, then |v|=i by definition.


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