Английская Википедия:Cartan pair

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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>\mathfrak{g}</math> and a subalgebra <math>\mathfrak{k}</math> reductive in <math>\mathfrak{g}</math>.

A reductive pair <math>(\mathfrak{g},\mathfrak{k})</math> is said to be Cartan if the relative Lie algebra cohomology

<math>H^*(\mathfrak{g},\mathfrak{k})</math>

is isomorphic to the tensor product of the characteristic subalgebra

<math>\mathrm{im}\big(S(\mathfrak{k}^*) \to H^*(\mathfrak{g},\mathfrak{k})\big)</math>

and an exterior subalgebra <math>\bigwedge \hat P</math> of <math>H^*(\mathfrak{g})</math>, where

• <math>\hat P</math>, the Samelson subspace, are those primitive elements in the kernel of the composition <math>P \overset\tau\to S(\mathfrak{g}^*) \to S(\mathfrak{k}^*)</math>,
• <math>P</math> is the primitive subspace of <math>H^*(\mathfrak{g})</math>,
• <math>\tau</math> is the transgression,
• and the map <math>S(\mathfrak{g}^*) \to S(\mathfrak{k}^*)</math> of symmetric algebras is induced by the restriction map of dual vector spaces <math>\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>G \to G_K \to BK</math>,

where <math>G_K := (EK \times G)/K \simeq G/K</math> is the homotopy quotient, here homotopy equivalent to the regular quotient, and

<math>G/K \overset\chi\to BK \overset{r}\to BG</math>.

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

References

• Шаблон:Cite book