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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 Iwasawa decomposition ： ウィキペディア英語版 Iwasawa decomposition In mathematics, the Iwasawa decomposition KAN of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method. ==Definition== *''G'' is a connected semisimple real Lie group. *$\backslash mathfrak\_0$ is the Lie algebra of ''G'' *$\backslash mathfrak$ is the complexification of $\backslash mathfrak\_0$. *θ is a Cartan involution of $\backslash mathfrak\_0$ *$\backslash mathfrak\_0\; =\; \backslash mathfrak\_0\; \backslash oplus\; \backslash mathfrak\_0$ is the corresponding Cartan decomposition *$\backslash mathfrak\_0$ is a maximal abelian subalgebra of $\backslash mathfrak\_0$ *Σ is the set of restricted roots of $\backslash mathfrak\_0$, corresponding to eigenvalues of $\backslash mathfrak\_0$ acting on $\backslash mathfrak\_0$. *Σ+ is a choice of positive roots of Σ *$\backslash mathfrak\_0$ is a nilpotent Lie algebra given as the sum of the root spaces of Σ+ *''K'', ''A'', ''N'', are the Lie subgroups of ''G'' generated by $\backslash mathfrak\_0,\; \backslash mathfrak\_0$ and $\backslash mathfrak\_0$. Then the Iwasawa decomposition of $\backslash mathfrak\_0$ is :$\backslash mathfrak\_0\; =\; \backslash mathfrak\_0\; \backslash oplus\; \backslash mathfrak\_0\; \backslash oplus\; \backslash mathfrak\_0$ and the Iwasawa decomposition of ''G'' is :$G=KAN$ The dimension of ''A'' (or equivalently of $\backslash mathfrak\_0$) is called the real rank of ''G''. Iwasawa decompositions also hold for some disconnected semisimple groups ''G'', where ''K'' becomes a (disconnected) maximal compact subgroup provided the center of ''G'' is finite. The restricted root space decomposition is :$\backslash mathfrak\_0\; =\; \backslash mathfrak\_0\backslash oplus\backslash mathfrak\_0\backslash oplus\_\backslash mathfrak\_$ where $\backslash mathfrak\_0$ is the centralizer of $\backslash mathfrak\_0$ in $\backslash mathfrak\_0$ and $\backslash mathfrak\_\; =\; \backslash \_0\; \backslash \}$ is the root space. The number $m\_=\; \backslash text\backslash ,\backslash mathfrak\_$ is called the multiplicity of $\backslash lambda$. 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「Iwasawa decomposition」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.