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 $({\Omega }^{•}(X),d_{\mathrm{dR}})$ of differential forms on $X$, a cochain complex with the structure of a dg-algebra.

Tobias: Shouldn’t the last link point to dg Lie algebra instead of dg-algebra? (I just started getting back to differential geometry after a break of several years, so I don’t yet feel confident enough to change anything.)

Every vector field $v\in \Gamma (TX)$ of $X$ induces a derivation on this dg-algebra of degree $-1$

$${\iota }_v:{\Omega }^{•}(X)\to {\Omega }^{•-1}(X)$$\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

$$0\stackrel{{\iota }_v}{\to }[d_{\mathrm{dR}},{\iota }_v]:{\Omega }^{•}(X)\to {\Omega }^{•}(X).$$0 \stackrel{\iota_v}{\to} [d_{dR},\iota_v] : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

One finds that

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

• $[{ℒ}_v,{ℒ}_w]={ℒ}_{[v,w]}$

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

• $[{\iota }_v,{\iota }_w]=0$.

These relations are sometimes called Cartan calculus. The first one is sometimes called Cartan’s magic formula.

In $\mathrm{\infty }$-Lie theory

The relations of Cartan calculus are preciselyy those in an inner derivation Lie 2-algebra.

This allows to generalize Cartan calculus to $\mathrm{\infty }$-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

Original articles by Cartan are

• Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal , Colloque de topologie (espaces fibrés), Bruxelles, 1950, pp. 57–71. Georges Thone, Liège; Masson et Cie., Paris, (1951).

The closely related Cartan model for equivariant cohomology is discussed for instance in

• V. Guillemin, S. Sternberg, Supersymmetry and equivariant de Rham theory, Springer, 1999.

The expression Cartan calculus is also used for noncommutative analogues, see e.g.

• 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.