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# Contents

## Idea

What is called Cartan calculus are the structures and relations present in an inner derivation Lie 2-algebra.

The classical examples considers for $X$ a smooth manifold the de Rham complex $(\Omega^\bullet(X), d_{dR})$ of differential forms on $X$, a cochain complex with the structure of a dg-algebra.

(There are of course other differential geometric structures named after Cartan, see also at equivariant de Rham cohomology the section The Cartan model.)

Every vector field $v \in \Gamma(T X)$ of $X$ induces a derivation on this dg-algebra of degree $-1$

$\iota_v : \Omega^\bullet(X) \to \Omega^{\bullet-1}(X)$

given by evaluation of forms on $v$.

As every degree -1-map, this induces a chain homotopy

$0 \stackrel{\iota_v}{\to} [d_{dR},\iota_v] : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.$

One finds that

• $[d_{dR},\iota_v,] = \mathcal{L}_v$ is the Lie derivative on forms along $v$;

• $[\mathcal{L}_v, \mathcal{L}_w] = \mathcal{L}_{[v,w]}$

• $[\mathcal{L}_v, \iota_w] = \iota_{[v,w]}$

• $[\iota_v, \iota_w] = 0$.

These relations are sometimes called Cartan calculus. The first one is sometimes called Cartan’s magic formula or Cartan's homotopy formula.

## In $\infty$-Lie theory

The relations of Cartan calculus are precisely those in an inner derivation Lie 2-algebra.

This allows to generalize Cartan calculus to $\infty$-Lie algebroids, see the section As inner derivations at Weil algebra.

There is also the full automorphism ∞-Lie algebra of any dg-algebra, which subsumes the inner derivation algebras. This is the context in wich the calculus of derived brackets? lives.

## References

Original articles by Cartan are

• Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal , Colloque de topologie (espaces fibrés), Bruxelles, 1950, pp. 57–71. Georges Thone, Liège;

Masson et Cie., Paris, (1951).

For the closely related Cartan model of equivariant de Rham cohomology see the references there.