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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 : 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 |- | | | ''p'' | Grassmannian of oriented real ''4''-dimensional subspaces of |- | | | ''p'' | Grassmannian of quaternionic ''1''-dimensional subspaces of |- | | | 10 | Space of symmetric subspaces of isometric to |- | | | 16 | Rosenfeld projective plane over |- | | | 28 | Space of symmetric subspaces of isomorphic to |- | | | 7 | Space of the symmetric subspaces of which are isomorphic to |- | | | 2 | Space of the subalgebras of the octonion algebra which are isomorphic to the quaternion algebra |} 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
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