nLab Cartan calculus

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 $-1$

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

given by contraction of forms with $v$.

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

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

One finds that:

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

• $[d_{dR},\mathcal{L}_v]=0$;

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

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

• $[\iota_v, \iota_w] = 0$;

• $[d_{dR},d_{dR}]=0$.

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.

Related concepts

• horizontal differential form

• basic differential form

• equivariant de Rham cohomology

• noncommutative differential calculus (in the sense of Tsygan, Tamarkin and Nest)

References

Named after Élie Cartan.

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

• Christian Blohmann, Elastic diffeological spaces, arXiv:2301.02583.

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

See also

• Planet Math, Cartan calculus

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.