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
, where 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
χ
τ :
{\displaystyle \tau \colon P\to H^{*}(BG)}
from the primitive subspace P of
is that arising from the edge maps in the Serre spectral sequence of the universal bundle
{\displaystyle G\to EG\to BG}
, and the subspace
∘ τ