nLab Cartan calculus



\infty-Lie theory

∞-Lie theory (higher geometry)


Smooth structure

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Lie theory

∞-Lie groupoids

∞-Lie algebroids

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\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 XX a smooth manifold the de Rham complex (Ω (X),d dR)(\Omega^\bullet(X), d_{dR}) of differential forms on XX, 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 vΓ(TX)v \in \Gamma(T X) of XX induces a derivation on this dg-algebra of degree 1-1

ι v:Ω (X)Ω 1(X) \iota_v \,\colon\, \Omega^\bullet(X) \to \Omega^{\bullet-1}(X)

given by contraction of forms with vv.

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

0ι v[d dR,ι v]:Ω (X)Ω (X). 0 \stackrel{\iota_v}{\to} [d_{dR},\iota_v] \,\colon\, \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

One finds that:

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

  • [d dR, v]=0[d_{dR},\mathcal{L}_v]=0;

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

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

  • [ι v,ι w]=0[\iota_v, \iota_w] = 0;

  • [d dR,ι v]=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.

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 January 3, 2024 at 13:33:18. See the history of this page for a list of all contributions to it.