Cartan pair

{{short description|Technical condition on the relationship between a reductive Lie algebra and a subalgebra}}

{{hatnote|This notion is not to be confused with a Cartan decomposition.}}

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 $\backslash mathfrak\{g\}$ and a subalgebra $\backslash mathfrak\{k\}$ reductive in $\backslash mathfrak\{g\}$.

A reductive pair $(\backslash mathfrak\{g\},\backslash mathfrak\{k\})$ is said to be Cartan if the relative Lie algebra cohomology

:$H^*(\backslash mathfrak\{g\},\backslash mathfrak\{k\})$

is isomorphic to the tensor product of the characteristic subalgebra

:$\backslash mathrm\{im\}\backslash big(S(\backslash mathfrak\{k\}^*)\; \backslash to\; H^*(\backslash mathfrak\{g\},\backslash mathfrak\{k\})\backslash big)$

and an exterior subalgebra $\backslash bigwedge\; \backslash hat\; P$ of $H^*(\backslash mathfrak\{g\})$, where

• $\backslash hat\; P$, the Samelson subspace, are those primitive elements in the kernel of the composition $P\; \backslash overset\backslash tau\backslash to\; S(\backslash mathfrak\{g\}^*)\; \backslash to\; S(\backslash mathfrak\{k\}^*)$,
• $P$ is the primitive subspace of $H^*(\backslash mathfrak\{g\})$,
• $\backslash tau$ is the transgression,
• and the map $S(\backslash mathfrak\{g\}^*)\; \backslash to\; S(\backslash mathfrak\{k\}^*)$ of symmetric algebras is induced by the restriction map of dual vector spaces $\backslash mathfrak\{g\}^*\; \backslash to\; \backslash mathfrak\{k\}^*$.

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

:$G\; \backslash to\; G\_K\; \backslash to\; BK$,

where

$G\_K\; :=\; (EK\; \backslash times\; G)/K\; \backslash simeq\; G/K$

is the homotopy quotient, here homotopy equivalent to the regular quotient, and

:$G/K\; \backslash overset\backslash chi\backslash to\; BK\; \backslash overset\{r\}\backslash to\; BG$.

Then the characteristic algebra is the image of $\backslash chi^*\backslash colon\; H^*(BK)\; \backslash to\; H^*(G/K)$, the transgression $\backslash tau\backslash colon\; P\; \backslash to\; H^*(BG)$ from the primitive subspace P of $H^*(G)$ is that arising from the edge maps in the Serre spectral sequence of the universal bundle $G\; \backslash to\; EG\; \backslash to\; BG$, and the subspace $\backslash hat\; P$ of $H^*(G/K)$ is the kernel of $r^*\; \backslash circ\; \backslash tau$.