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 $$\mathfrak{g}$$ 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 $$r^* \circ \tau$$.