and the map of symmetric algebras is induced by the restriction map of dual vector spaces .
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
,
where
is the homotopy quotient, here homotopy equivalent to the regular quotient, and
.
Then the characteristic algebra is the image of , the transgression from the primitive subspace P of is that arising from the edge maps in the Serre spectral sequence of the universal bundle, and the subspace of is the kernel of .
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. ISBN978-0-08-087927-7.