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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 Quaternionic projective space ： ウィキペディア英語版 Quaternionic projective space In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension ''n'' is usually denoted by :$\backslash mathbb^n$ and is a closed manifold of (real) dimension 4''n''. It is a homogeneous space for a Lie group action, in more than one way. ==In coordinates== Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written :$()$ where the $q\_i$ are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion ''c''; that is, we identify all the :$()$. In the language of group actions, $\backslash mathbb^n$ is the orbit space of $\backslash mathbb^\backslash setminus\backslash$ by the action of $\backslash mathbb^$, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside $\backslash mathbb^$ one may also regard $\backslash mathbb^$ as the orbit space of $\backslash mathbb^$ by the action of $\backslash text(1)$, the group of unit quaternions.〔Gregory L. Naber, ''Topology, geometry, and gauge fields: foundations'' (1997), p. 50.〕 The sphere $\backslash mathbb^$ then becomes a principal Sp(1)-bundle over $\backslash mathbb^n$: :$\backslash mathrm(1)\; \backslash to\; \backslash mathbb^\; \backslash to\; \backslash mathbb^n.$ There is also a construction of $\backslash mathbb^$ by means of two-dimensional complex subspaces of $\backslash mathbb^$, meaning that $\backslash mathbb^$ lies inside a complex Grassmannian. 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「Quaternionic projective 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.