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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 Linear complex structure ： ウィキペディア英語版 Linear complex structure In mathematics, a complex structure on a real vector space ''V'' is an automorphism of ''V'' that squares to the minus identity, ''−I''. Such a structure on ''V'' allows one to define multiplication by complex scalars in a canonical fashion so as to regard ''V'' as a complex vector space. Every complex vector space can be equipped with a compatible complex structure, however, there is in general no canonical such structure. Complex structures have applications in representation theory as well as in complex geometry where they play an essential role in the definition of almost complex manifolds, by contrast to complex manifolds. The term "complex structure" often refers to this structure on manifolds; when it refers instead to a structure on vector spaces, it may be called a "''linear'' complex structure". ==Definition and properties== A complex structure on a real vector space ''V'' is a real linear transformation :$J\; :V\; \backslash rightarrow\; V$ such that :$J^2\; =\; -\backslash rm\_V.$ Here means composed with itself and is the identity map on . That is, the effect of applying twice is the same as multiplication by . This is reminiscent of multiplication by the . A complex structure allows one to endow with the structure of a complex vector space. Complex scalar multiplication can be defined by :$(x\; +\; iy)v\; =\; xv\; +\; yJ(v)$ for all real numbers and all vectors in . One can check that this does, in fact, give the structure of a complex vector space which we denote . Going in the other direction, if one starts with a complex vector space then one can define a complex structure on the underlying real space by defining for all . More formally, a linear complex structure on a real vector space is an algebra representation of the complex numbers , thought of as an associative algebra over the real numbers. This algebra is realized concretely as :$\backslash mathbf\; =\; \backslash mathbf()/(x^2+1),$ which corresponds to . Then a representation of is a real vector space , together with an action of on (a map ). Concretely, this is just an action of , as this generates the algebra, and the operator representing (the image of in ) is exactly . If has complex dimension then must have real dimension . That is, a finite-dimensional space admits a complex structure only if it is even-dimensional. It is not hard to see that every even-dimensional vector space admits a complex structure. One can define on pairs of basis vectors by and and then extend by linearity to all of . If is a basis for the complex vector space then is a basis for the underlying real space . A real linear transformation is a ''complex'' linear transformation of the corresponding complex space if and only if commutes with , i.e. if and only if :$AJ\; =\; JA.$ Likewise, a real subspace of is a complex subspace of if and only if preserves , i.e. if and only if :$JU\; =\; U.$ 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「Linear complex structure」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.