Suppose $M$ is a smooth manifold, and denote by $\Omega(M)$ the algebra of differential forms on $M$ . 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 $\lie_X$ , 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 $M$ :

		(1)
		(2)
		(3)
		(4)
		(5)
		(6)

The identity () 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 $\lie_X$ are degree $0$ , and the contraction operators $\iota_X$ are degree $-1$ .

The identities ()-() may each be written in the form \begin{equation} AB \pm BA = C, \end{equation}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 \begin{equation}\label{supercom} [A,B] = C, \end{equation}where the bracket in () 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 $\Omega(M)$

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