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 $\mathrm{\Omega }(M)$.