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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.

Named after Élie Cartan.

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