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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 Complex Lie group ： ウィキペディア英語版 Complex Lie group In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way $G\; \backslash times\; G\; \backslash to\; G,\; (x,\; y)\; \backslash mapsto\; x\; y^$ is holomorphic. Basic examples are $\backslash operatorname\_n(\backslash mathbb)$, the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group $\backslash mathbb\; C^$ *). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is an algebraic group. == Examples == *A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way. *A connected compact complex Lie group ''A'' of dimension ''g'' is of the form $\backslash mathbb^g/L$ where ''L'' is a discrete subgroup. Indeed, its Lie algebra $\backslash mathfrak$ can be shown to be abelian and then $\backslash operatorname:\; \backslash mathfrak\; \backslash to\; A$ is a surjective morphism of complex Lie groups, showing ''A'' is of the form described. * $\backslash mathbb\; \backslash to\; \backslash mathbb^$ *, z \mapsto e^z is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since $\backslash mathbb^$ * = \operatorname_1(\mathbb), this is also an example of a representation of a complex Lie group that is not algebraic. * Let ''X'' be a compact complex manifold. Then, as in the real case, $\backslash operatorname(X)$ is a complex Lie group whose Lie algebra is $\backslash Gamma(X,\; TX)$. * Let ''K'' be a connected compact Lie group. Then there exists a unique connected complex Lie group ''G'' such that (i) $\backslash operatorname\; (G)\; =\; \backslash operatorname\; (K)\; \backslash otimes\_$ (ii) ''K'' is a maximal compact subgroup of ''G''. It is called the complexification of ''K''. For example, $\backslash operatorname\_n(\backslash mathbb)$ is the complexification of the unitary group. If ''K'' is acting on a compact kähler manifold ''X'', then the action of ''K'' extends to that of ''G''. 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「Complex Lie group」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.