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
.
