Cartan calculus

Suppose M is a smooth manifold, and denote by Ω(M) the algebra of differential formsMathworldPlanetmath on M. Then, the Cartan calculus consists of the following three types of linear operators on Ω(M):

  1. 1.
  2. 2.

    the space of Lie derivativeMathworldPlanetmathPlanetmath operators X, where X is a vector field on M, and

  3. 3.

    the space of contraction operators ιX, where X is a vector field on M.

The above operators satisfy the following identities for any vector fields X and Y on M:

d2 =0, (1)
dX-Xd =0, (2)
dιX+ιXd =X, (3)
XY-YX =[X,Y], (4)
XιY-ιYX =ι[X,Y], (5)
ιXιY+ιYιX =0, (6)

where the brackets on the right hand side denote the Lie bracket of vector fields.

The identity (3) is known as Cartan’s magic formulaMathworldPlanetmathPlanetmath or Cartan’s identity

Interpretation as a Lie Superalgebra

Since Ω(M) is a graded algebra, there is a natural grading on the space of linear operators on Ω(M). Under this grading, the exterior derivative d is degree 1, the Lie derivative operators X are degree 0, and the contraction operators ιX are degree -1.

The identities (1)-(6) may each be written in the form

AB±BA=C, (7)

where a plus sign is used if A and B are both of odd degree, and a minus sign is used otherwise. Equations of this form are called supercommutation relations and are usually written in the form

[A,B]=C, (8)

where the bracket in (8) is a Lie superbracket. A Lie superbracket is a generalizationPlanetmathPlanetmath of a Lie bracket.

Since the Cartan Calculus operators are closed under the Lie superbracket, the vector spaceMathworldPlanetmath spanned by the Cartan Calculus operators has the structureMathworldPlanetmath of a Lie superalgebra.

Graded derivations of Ω(M)

Definition 1.

A degree k linear operator A on Ω(M) is a graded derivation if it satisfies the following property for any p-form ω and any differential form η:

A(ωη)=A(ω)η+(-1)kpωA(η). (9)

All of the Calculus operators are graded derivations of Ω(M).

Title Cartan calculus
Canonical name CartanCalculus
Date of creation 2013-03-22 15:35:39
Last modified on 2013-03-22 15:35:39
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 14
Author bci1 (20947)
Entry type Definition
Classification msc 81R15
Classification msc 17B70
Classification msc 81R50
Classification msc 53A45
Classification msc 81Q60
Classification msc 58A15
Classification msc 14F40
Classification msc 13N15
Synonym Lie superalgebra
Related topic LieSuperalgebra3
Related topic LieDerivative
Related topic DifferentialForms
Defines anticommutator bracket
Defines Cartan’s magic formula
Defines supercommutation relation
Defines graded derivation