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 and a subalgebra reductive in .

A reductive pair is said to be Cartan if the relative Lie algebra cohomology

is isomorphic to the tensor product of the characteristic subalgebra

and an exterior subalgebra of , where

  • , the Samelson subspace, are those primitive elements in the kernel of the composition ,
  • is the primitive subspace of ,
  • is the transgression,
  • 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

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  • 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. ISBN 978-0-08-087927-7.