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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cartan pair ： ウィキペディア英語版 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 $\backslash mathfrak$ and a subalgebra $\backslash mathfrak$ reductive in $\backslash mathfrak$. A reductive pair $(\backslash mathfrak,\backslash mathfrak)$ is said to be Cartan if the relative Lie algebra cohomology :$H^$ *(\mathfrak,\mathfrak) is isomorphic to the tensor product of the characteristic subalgebra :$\backslash mathrm\backslash big(S(\backslash mathfrak^$ *) \to H^ *(\mathfrak,\mathfrak)\big) and an exterior subalgebra $\backslash bigwedge\; \backslash hat\; P$ of $H^$ *(\mathfrak), where *$\backslash hat\; P$, the ''Samelson subspace'', are those primitive elements in the kernel of the composition $P\; \backslash overset\backslash tau\backslash to\; S(\backslash mathfrak^$ *) \to S(\mathfrak^ *), *$P$ is the primitive subspace of $H^$ *(\mathfrak), *$\backslash tau$ is the transgression, *and the map $S(\backslash mathfrak^$ *) \to S(\mathfrak^ *) of symmetric algebras is induced by the restriction map of dual vector spaces $\backslash mathfrak^$ * \to \mathfrak^ *. 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 :$G/K\; \backslash overset\backslash chi\backslash to\; BK\; \backslash overset\backslash to\; BG$. Then the characteristic algebra is the image of $\backslash chi^$ *\colon H^ *(BK) \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 $r^$ * \circ \tau. ==References== * Werner Greub, Stephen Halperin, and Ray Vanstone ''Connections, Curvature, and Cohomology'' Volume III, Academic Press (1976). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cartan pair」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.