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 g and a subalgebra k reductive in g .
A reductive pair ( g , k ) is said to be Cartan if the relative Lie algebra cohomology
H ∗ ( g , k ) is isomorphic to the tensor product of the characteristic subalgebra
i m ( S ( k ∗ ) → H ∗ ( g , k ) ) and an exterior subalgebra ⋀ P ^ of H ∗ ( g ) , where
P ^ , the Samelson subspace, are those primitive elements in the kernel of the composition P → τ S ( g ∗ ) → S ( k ∗ ) , P is the primitive subspace of H ∗ ( g ) , τ is the transgression,and the map S ( g ∗ ) → S ( k ∗ ) of symmetric algebras is induced by the restriction map of dual vector spaces g ∗ → 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 → G K → B K ,
where G K := ( E K × G ) / K ≃ G / K is the homotopy quotient, here homotopy equivalent to the regular quotient, and
G / K → χ B K → r B G .
Then the characteristic algebra is the image of χ ∗ : H ∗ ( B K ) → H ∗ ( G / K ) , the transgression τ : P → H ∗ ( B G ) 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 → E G → B G , and the subspace P ^ of H ∗ ( G / K ) is the kernel of r ∗ ∘ τ .
