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 ${\mathcal{L}}_{X}$, 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 $M$:
${d}^{2}$  $=0,$  (1)  
$d{\mathcal{L}}_{X}{\mathcal{L}}_{X}d$  $=0,$  (2)  
$d{\iota}_{X}+{\iota}_{X}d$  $={\mathcal{L}}_{X},$  (3)  
${\mathcal{L}}_{X}{\mathcal{L}}_{Y}{\mathcal{L}}_{Y}{\mathcal{L}}_{X}$  $={\mathcal{L}}_{[X,Y]},$  (4)  
${\mathcal{L}}_{X}{\iota}_{Y}{\iota}_{Y}{\mathcal{L}}_{X}$  $={\iota}_{[X,Y]},$  (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.
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 ${\mathcal{L}}_{X}$ 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 $\mathrm{\Omega}(M)$
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  20130322 15:35:39 
Last modified on  20130322 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 