Cartan pair

This notion is not to be confused with a Cartan decomposition.

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

$\mathrm{i}\mathrm{m}(S({\mathfrak{𝔨}}^{*})\to H^{*}(\mathfrak{𝔤},\mathfrak{𝔨}))$

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 $P\stackrel{\tau }{\to }S({\mathfrak{𝔤}}^{*})\to S({\mathfrak{𝔨}}^{*})$,
• $P$ 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

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

Then the characteristic algebra is the image of ${\chi }^{*}:H^{*}(BK)\to H^{*}(G/K)$, the transgression $\tau :P\to H^{*}(BG)$ 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 $r^{*}\circ \tau$.

References

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