Cartan formula

In mathematics, the Cartan formula can mean:

• one in differential geometry: ${\displaystyle {\mathcal {L}}_{X}=\mathrm {d} \,\iota _{X}+\iota _{X}\mathrm {d} }$, where ${\displaystyle {\mathcal {L}}_{X},\mathrm {d} }$, and ${\displaystyle \iota _{X}}$ are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms. See interior product for the detail. It is also called the Cartan homotopy formula or Cartan magic formula. This formula is named after Élie Cartan.

• one in algebraic topology, which is one of the five axioms of Steenrod algebra.^{[note 1]} It reads:

${\displaystyle {\begin{aligned}Sq^{n}(x\smile y)&=\sum _{i+j=n}(Sq^{i}x)\smile (Sq^{j}y)\quad {\text{or}}\\P^{n}(x\smile y)&=\sum _{i+j=n}(P^{i}x)\smile (P^{j}y)\end{aligned}}}$.

See Steenrod algebra for the detail. The name derives from Henri Cartan, son of Élie.

Footnotes

1. More precisely, these five axioms define 'cohomology operations', which are natural transformations between cohomology functors, which in turn define Steenrod algebra.

See also

• List of things named after Élie Cartan