Cartan pair

In the mathematical fields of Lie theory and algebraic topology, the notion of Cartan pair is a technical condition on the relationship between a reductive Lie algebra ${\displaystyle {\mathfrak {g}}}$ and a subalgebra ${\displaystyle {\mathfrak {k}}}$ reductive in ${\displaystyle {\mathfrak {g}}}$.

A reductive pair ${\displaystyle ({\mathfrak {g}},{\mathfrak {k}})}$ is said to be Cartan if the relative Lie algebra cohomology

${\displaystyle H^{*}({\mathfrak {g}},{\mathfrak {k}})}$

is isomorphic to the tensor product of the characteristic subalgebra

${\displaystyle \mathrm {im} {\big (}S({\mathfrak {k}}^{*})\to H^{*}({\mathfrak {g}},{\mathfrak {k}}){\big )}}$

and an exterior subalgebra ${\displaystyle \bigwedge {\hat {P}}}$ of ${\displaystyle H^{*}({\mathfrak {g}})}$, where

• ${\displaystyle {\hat {P}}}$, the Samelson subspace, are those primitive elements in the kernel of the composition ${\displaystyle P{\overset {\tau }{\to }}S({\mathfrak {g}}^{*})\to S({\mathfrak {k}}^{*})}$,
• ${\displaystyle P}$ is the primitive subspace of ${\displaystyle H^{*}({\mathfrak {g}})}$,
• ${\displaystyle \tau }$ is the transgression,
• and the map ${\displaystyle S({\mathfrak {g}}^{*})\to S({\mathfrak {k}}^{*})}$ of symmetric algebras is induced by the restriction map of dual vector spaces ${\displaystyle {\mathfrak {g}}^{*}\to {\mathfrak {k}}^{*}}$.

On the level of Lie groups, if G is a compact, connected Lie group and K a closed connected subgroup, there are natural fiber bundles

${\displaystyle G\to G_{K}\to BK}$,

where ${\displaystyle G_{K}:=(EK\times G)/K\simeq G/K}$ is the homotopy quotient, here homotopy equivalent to the regular quotient, and

${\displaystyle G/K{\overset {\chi }{\to }}BK{\overset {r}{\to }}BG}$.

Then the characteristic algebra is the image of ${\displaystyle \chi ^{*}\colon H^{*}(BK)\to H^{*}(G/K)}$, the transgression ${\displaystyle \tau \colon P\to H^{*}(BG)}$ from the primitive subspace P of ${\displaystyle H^{*}(G)}$ is that arising from the edge maps in the Serre spectral sequence of the universal bundle ${\displaystyle G\to EG\to BG}$, and the subspace ${\displaystyle {\hat {P}}}$ of ${\displaystyle H^{*}(G/K)}$ is the kernel of ${\displaystyle r^{*}\circ \tau }$.