Cartan pair

Last updated July 14, 2021

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 ${\mathfrak {g}}$ and a subalgebra ${\mathfrak {k}}$ reductive in ${\mathfrak {g}}$ .

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

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

is isomorphic to the tensor product of the characteristic subalgebra

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

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

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

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

G\to G_{K}\to BK

,

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

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

.

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

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References

Greub, Werner; Halperin, Stephen; Vanstone, Ray (1976). "10. Subalgebras §4 Cartan Pairs". Cohomology of Principal Bundles and Homogeneous Spaces. Connections, Curvature, and Cohomology. 3. Academic Press. pp. 431–5. ISBN 978-0-08-087927-7.

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