∞-Lie theory (higher geometry)
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
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 $-1$
given by contraction of forms with $v$.
As every degree -1-map, this induces a chain homotopy
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.
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.
noncommutative differential calculus (in the sense of Tsygan, Tamarkin and Nest)
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.
See also
The expression Cartan calculus is also used for noncommutative geometry-analogues such as for quantum groups, see
Last revised on January 3, 2024 at 13:33:18. See the history of this page for a list of all contributions to it.