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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 Quaternion-Kähler symmetric space ： ウィキペディア英語版 Quaternion-Kähler symmetric space In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups. For any compact simple Lie group ''G'', there is a unique ''G''/''H'' obtained as a quotient of ''G'' by a subgroup :$H\; =\; K\; \backslash cdot\; \backslash mathrm(1).\backslash ,$ Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of ''G'', and ''K'' its centralizer in ''G''. These are classified as follows. (\mathrm(p) \times \mathrm(2)) | ''p'' | Grassmannian of complex ''2''-dimensional subspaces of $\backslash mathbb^$ |- | $\backslash mathrm(p+4)\backslash ,$ | $\backslash mathrm(p)\; \backslash cdot\; \backslash mathrm(4)$ | ''p'' | Grassmannian of oriented real ''4''-dimensional subspaces of $\backslash mathbb^$ |- | $\backslash mathrm(p+1)\backslash ,$ | $\backslash mathrm(p)\; \backslash cdot\; \backslash mathrm(1)$ | ''p'' | Grassmannian of quaternionic ''1''-dimensional subspaces of $\backslash mathbb^$ |- | $E\_6\backslash ,$ | $\backslash mathrm(6)\backslash cdot\backslash mathrm(2)$ | 10 | Space of symmetric subspaces of $(\backslash mathbb\; C\backslash otimes\backslash mathbb\; O)P^2$ isometric to $(\backslash mathbb\; C\backslash otimes\; \backslash mathbb\; H)P^2$ |- | $E\_7\backslash ,$ | $\backslash mathrm(12)\backslash cdot\backslash mathrm(1)$ | 16 | Rosenfeld projective plane $(\backslash mathbb\; H\backslash otimes\backslash mathbb\; O)P^2$ over $\backslash mathbb\; H\backslash otimes\backslash mathbb\; O$ |- | $E\_8\backslash ,$ | $E\_7\backslash cdot\backslash mathrm(1)$ | 28 | Space of symmetric subspaces of $(\backslash mathbb\backslash otimes\backslash mathbb\; O)P^2$ isomorphic to $(\backslash mathbb\backslash otimes\backslash mathbb\; O)P^2$ |- | $F\_4\backslash ,$ | $\backslash mathrm(3)\backslash cdot\backslash mathrm(1)$ | 7 | Space of the symmetric subspaces of $\backslash mathbb^2$ which are isomorphic to $\backslash mathbb^2$ |- | $G\_2\backslash ,$ | $\backslash mathrm(4)\backslash ,$ | 2 | Space of the subalgebras of the octonion algebra $\backslash mathbb$ which are isomorphic to the quaternion algebra $\backslash mathbb$ |} The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups. These spaces can be obtained by taking a projectivization of a minimal nilpotent orbit of the respective complex Lie group. The holomorphic contact structure is apparent, because the nilpotent orbits of semisimple Lie groups are equipped with the Kirillov-Kostant holomorphic symplectic form. This argument also explains how one can associate a unique Wolf space to each of the simple complex Lie groups. ==See also== *Quaternionic discrete series representation 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「Quaternion-Kähler symmetric space」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.