∞-Lie theory

# 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 $\left({\Omega }^{•}\left(X\right),{d}_{\mathrm{dR}}\right)$ 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 $v\in \Gamma \left(TX\right)$ of $X$ induces a derivation on this dg-algebra of degree $-1$

${\iota }_{v}:{\Omega }^{•}\left(X\right)\to {\Omega }^{•-1}\left(X\right)$\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 }\left[{d}_{\mathrm{dR}},{\iota }_{v}\right]:{\Omega }^{•}\left(X\right)\to {\Omega }^{•}\left(X\right)\phantom{\rule{thinmathspace}{0ex}}.$0 \stackrel{\iota_v}{\to} [d_{dR},\iota_v] : \Omega^\bullet(X) \to \Omega^\bullet(X) \,.

One finds that

• $\left[{d}_{\mathrm{dR}},{\iota }_{v},\right]={ℒ}_{v}$ is the Lie derivative on forms along $v$;

• $\left[{ℒ}_{v},{ℒ}_{w}\right]={ℒ}_{\left[v,w\right]}$

• $\left[{ℒ}_{v},{\iota }_{w}\right]={\iota }_{\left[v,w\right]}$

• $\left[{\iota }_{v},{\iota }_{w}\right]=0$.

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

## In $\infty$-Lie theory

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

This allows to generalize Cartan calculus to $\infty$-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.
Revised on October 1, 2011 19:16:50 by Tobias Fritz (147.83.178.20)