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 $\mathfrak$ and a subalgebra $\mathfrak$ reductive in $\mathfrak$.

A reductive pair $(\mathfrak,\mathfrak)$ is said to be Cartan if the relative Lie algebra cohomology
:$H^*(\mathfrak,\mathfrak)$
is isomorphic to the tensor product of the characteristic subalgebra
:$\mathrm\big(S(\mathfrak^*) \to H^*(\mathfrak,\mathfrak)\big)$
and an exterior subalgebra $\bigwedge \hat P$ of $H^*(\mathfrak)$, where
*$\hat P$, the ''Samelson subspace'', are those primitive elements in the kernel of the composition $P \overset\tau\to S(\mathfrak^*) \to S(\mathfrak^*)$,
*$P$ is the primitive subspace of $H^*(\mathfrak)$,
*$\tau$ is the transgression,
*and the map $S(\mathfrak^*) \to S(\mathfrak^*)$ of symmetric algebras is induced by the restriction map of dual vector spaces $\mathfrak^* \to \mathfrak^*$.

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
:$G/K \overset\chi\to BK \overset\to BG$.
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 $r^* \circ \tau$.

References

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Cohomology theories
Homological algebra
Lie algebras