Cartan calculus

Suppose $M$ is a smooth manifold, and denote by $\mathrm{\Omega }(M)$ the algebra of differential forms on $M$. Then, the Cartan calculus consists of the following three types of linear operators on $\mathrm{\Omega }(M)$:

1. the exterior derivative $d$,

2. the space of Lie derivative operators ${ℒ}_X$, where $X$ is a vector field on $M$, and

3. the space of contraction operators ${\iota }_X$, where $X$ is a vector field on $M$.

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

$d^2$ $=0,$ (1)

$d{ℒ}_X-{ℒ}_{X}d$ $=0,$ (2)

$d{\iota }_X+{\iota }_{X}d$ $={ℒ}_X,$ (3)

${ℒ}_X{ℒ}_Y-{ℒ}_Y{ℒ}_X$ $={ℒ}_{[X,Y]},$ (4)

${ℒ}_X{\iota }_Y-{\iota }_Y{ℒ}_X$ $={\iota }_{[X,Y]},$ (5)

${\iota }_X{\iota }_Y+{\iota }_Y{\iota }_X$ $=0,$ (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 formula or Cartan’s identity

Interpretation as a Lie Superalgebra

Since $\mathrm{\Omega }(M)$ is a graded algebra, there is a natural grading on the space of linear operators on $\mathrm{\Omega }(M)$. Under this grading, the exterior derivative $d$ is degree $1$, the Lie derivative operators ${ℒ}_X$ are degree $0$, and the contraction operators ${\iota }_X$ are degree $-1$.

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

$$AB\pm BA=C,$$ (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 generalization of a Lie bracket.

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

Graded derivations of $\mathrm{\Omega }(M)$

Definition 1.

A degree $k$ linear operator $A$ on $\mathrm{\Omega }(M)$ is a graded derivation if it satisfies the following property for any $p$-form $\omega$ and any differential form $\eta$:

$$A(\omega \wedge \eta )=A(\omega )\wedge \eta +{(-1)}^{kp}\omega \wedge A(\eta ).$$ (9)

All of the Calculus operators are graded derivations of $\mathrm{\Omega }(M)$.

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