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# 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. the exterior derivative $d$,

2. the space of Lie derivative operators $\mathcal{L}_{X}$, 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 $M$:

$\displaystyle d^{2}$ | $\displaystyle=0,$ | (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 $\Omega(M)$.

## Mathematics Subject Classification

81R15*no label found*17B70

*no label found*81R50

*no label found*53A45

*no label found*81Q60

*no label found*58A15

*no label found*14F40

*no label found*13N15

*no label found*

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