Cartan pair explained

ak{g} and a subalgebra ak{k} reductive in ak{g} .

A reductive pair

H ^{*(ak{g},ak{k})} is isomorphic to the tensor product of the characteristic subalgebra im ( S( ak{k} ^{*)} \to H ^{*(ak{g},ak{k}))} and an exterior subalgebra w edge \hat P of H ^{*(ak{g})} , where\hat P , the Samelson subspace, are those primitive elements in the kernel of the composition P \overset\tau\to S( ak{g} ^{*)} \to S( ak{k} ^{*)} , P is the primitive subspace of H ^{*(ak{g})} , \tau is the transgression,

• and the map

S( ak{g} ^{*)} \to S( ak{k} ^{*)} of symmetric algebras is induced by the restriction map of dual vector spaces ak{g} ^{*} \to ak{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 x G)/K \simeq G/K is the homotopy quotient, here homotopy equivalent to the regular quotient, andG/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 .

References

• Book: Werner . Greub . Stephen . Halperin . Ray . Vanstone . 10. Subalgebras §4 Cartan Pairs. Cohomology of Principal Bundles and Homogeneous Spaces . Connections, Curvature, and Cohomology . 3 . https://books.google.com/books?id=c724LN914AwC&pg=PA431 . 1976 . Academic Press . 978-0-08-087927-7 . 431–5.