# Cartan calculus

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

1. 1.

the exterior derivative $d$,

2. 2.

the space of Lie derivative operators $\mathcal{L}_{X}$, where $X$ is a vector field on $M$, and

3. 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 $M$:

 $\displaystyle d^{2}$ $\displaystyle=0,$ (1) $\displaystyle d\mathcal{L}_{X}-\mathcal{L}_{X}d$ $\displaystyle=0,$ (2) $\displaystyle d\iota_{X}+\iota_{X}d$ $\displaystyle=\mathcal{L}_{X},$ (3) $\displaystyle\mathcal{L}_{X}\mathcal{L}_{Y}-\mathcal{L}_{Y}\mathcal{L}_{X}$ $\displaystyle=\mathcal{L}_{[X,Y]},$ (4) $\displaystyle\mathcal{L}_{X}\iota_{Y}-\iota_{Y}\mathcal{L}_{X}$ $\displaystyle=\iota_{[X,Y]},$ (5) $\displaystyle\iota_{X}\iota_{Y}+\iota_{Y}\iota_{X}$ $\displaystyle=0,$ (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 $\Omega(M)$ is a graded algebra, there is a natural grading on the space of linear operators on $\Omega(M)$. Under this grading, the exterior derivative $d$ is degree $1$, the Lie derivative operators $\mathcal{L}_{X}$ 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 $\Omega(M)$

###### Definition 1.

A degree $k$ linear operator $A$ on $\Omega(M)$ 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 $\Omega(M)$.

 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