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Cartan calculus
Contents
Context
$\infty$ -Lie theory
∞-Lie theory (higher geometry )

Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Cohomology
Homotopy
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 $v$ ;

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

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

$[\iota_v, \iota_w] = 0$ .

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
Named after Élie Cartan .

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

See also

The expression Cartan calculus is also used for noncommutative geometry -analogues such as for quantum groups , see

P. Schupp, Cartan calculus: differential geometry for quantum groups , Quantum groups and their applications in physics (Varenna, 1994), 507–524, Proc. Internat. School Phys. Enrico Fermi, 127, IOS, Amsterdam, 1996.
Last revised on October 30, 2020 at 04:22:11.
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