Cartan pair
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 [math]\displaystyle{ \mathfrak{g} }[/math] and a subalgebra [math]\displaystyle{ \mathfrak{k} }[/math] reductive in [math]\displaystyle{ \mathfrak{g} }[/math].
A reductive pair [math]\displaystyle{ (\mathfrak{g},\mathfrak{k}) }[/math] is said to be Cartan if the relative Lie algebra cohomology
- [math]\displaystyle{ H^*(\mathfrak{g},\mathfrak{k}) }[/math]
is isomorphic to the tensor product of the characteristic subalgebra
- [math]\displaystyle{ \mathrm{im}\big(S(\mathfrak{k}^*) \to H^*(\mathfrak{g},\mathfrak{k})\big) }[/math]
and an exterior subalgebra [math]\displaystyle{ \bigwedge \hat P }[/math] of [math]\displaystyle{ H^*(\mathfrak{g}) }[/math], where
- [math]\displaystyle{ \hat P }[/math], the Samelson subspace, are those primitive elements in the kernel of the composition [math]\displaystyle{ P \overset\tau\to S(\mathfrak{g}^*) \to S(\mathfrak{k}^*) }[/math],
- [math]\displaystyle{ P }[/math] is the primitive subspace of [math]\displaystyle{ H^*(\mathfrak{g}) }[/math],
- [math]\displaystyle{ \tau }[/math] is the transgression,
- and the map [math]\displaystyle{ S(\mathfrak{g}^*) \to S(\mathfrak{k}^*) }[/math] of symmetric algebras is induced by the restriction map of dual vector spaces [math]\displaystyle{ \mathfrak{g}^* \to \mathfrak{k}^* }[/math].
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
- [math]\displaystyle{ G \to G_K \to BK }[/math],
where [math]\displaystyle{ G_K := (EK \times G)/K \simeq G/K }[/math] is the homotopy quotient, here homotopy equivalent to the regular quotient, and
- [math]\displaystyle{ G/K \overset\chi\to BK \overset{r}\to BG }[/math].
Then the characteristic algebra is the image of [math]\displaystyle{ \chi^*\colon H^*(BK) \to H^*(G/K) }[/math], the transgression [math]\displaystyle{ \tau\colon P \to H^*(BG) }[/math] from the primitive subspace P of [math]\displaystyle{ H^*(G) }[/math] is that arising from the edge maps in the Serre spectral sequence of the universal bundle [math]\displaystyle{ G \to EG \to BG }[/math], and the subspace [math]\displaystyle{ \hat P }[/math] of [math]\displaystyle{ H^*(G/K) }[/math] is the kernel of [math]\displaystyle{ r^* \circ \tau }[/math].
References
- Greub, Werner; Halperin, Stephen; Vanstone, Ray (1976). "10. Subalgebras §4 Cartan Pairs". Cohomology of Principal Bundles and Homogeneous Spaces. Connections, Curvature, and Cohomology. 3. Academic Press. pp. 431–5. ISBN 978-0-08-087927-7. https://books.google.com/books?id=c724LN914AwC&pg=PA431.
 |  |
