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 of , 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 , .
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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 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 of , 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 , . (en)
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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 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 of , where
* , the Samelson subspace, are those primitive elements in the kernel of the composition ,
* is the primitive subspace of ,
* is the transgression,
* and the map of symmetric algebras is induced by the restriction map of dual vector spaces . 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 , the transgression from the primitive subspace P of is that arising from the edge maps in the Serre spectral sequence of the universal bundle , and the subspace of is the kernel of . (en)
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