# nLab Cartan calculus

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# 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 $v$;

• $[d_{dR},\mathcal{L}_v]=0$;

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

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

• $[\iota_v, \iota_w] = 0$;

• $[d_{dR},\iota_v]=0$.

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.

Cartan calculus on diffeological spaces requires a nontrivial condition, which is explored and developed in

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