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 $𝔤$ and a subalgebra $𝔨$ reductive in $𝔤$.

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

$H^{\ast }(𝔤,𝔨)$

is isomorphic to the tensor product of the characteristic subalgebra

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

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

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

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 ${\chi }^{\ast }:H^{\ast }(BK)\to H^{\ast }(G/K)$, the transgression $\tau :P\to H^{\ast }(BG)$ from the primitive subspace P of $H^{\ast }(G)$ 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 $H^{\ast }(G/K)$ is the kernel of $r^{\ast }\circ \tau$.

References

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