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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 Solvable Lie algebra ： ウィキペディア英語版 Solvable Lie algebra In mathematics, a Lie algebra is solvable if its ''derived series'' terminates in the zero subalgebra. The derived Lie algebra is the subalgebra of , denoted :$()$ that consists of all Lie brackets of pairs of elements of . The derived series is the sequence of subalgebras : If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory. Any nilpotent Lie algebra is solvable, ''a fortiori'', but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. A maximal solvable subalgebra is called a ''Borel subalgebra''. The largest solvable ideal of a Lie algebra is called the ''radical''. == Characterizations == Let be a finite-dimensional Lie algebra over a field of characteristic . The following are equivalent. *(i) is solvable. *(ii) , the adjoint representation of , is solvable. *(iii) There is a finite sequence of ideals of : *:$\backslash mathfrak\; =\; \backslash mathfrak\_0\; \backslash supset\; \backslash mathfrak\_1\; \backslash supset\; ...\; \backslash mathfrak\_r\; =\; 0,\; \backslash quad\; \backslash forall\; i\; (\backslash mathfrak\_i)\; \backslash subset\; \backslash mathfrak\_.$ *(iv) is nilpotent.〔 Proposition 1.39.〕 *(v) For -dimensional, there is a finite sequence of subalgebras of : *:$\backslash mathfrak\; =\; \backslash mathfrak\_0\; \backslash supset\; \backslash mathfrak\_1\; \backslash supset\; ...\; \backslash mathfrak\_n\; =\; 0,\; \backslash quad\; \backslash forall\; i\; \backslash operatorname\; \backslash mathfrak\_/\backslash mathfrak\_\; =\; 1,$ :with each an ideal in .〔 Proposition 1.23.〕 A sequence of this type is called an elementary sequence. *(vi) There is a finite sequence of subalgebras of , *:$\backslash mathfrak\; =\; \backslash mathfrak\_0\; \backslash supset\; \backslash mathfrak\_1\; \backslash supset\; ...\; \backslash mathfrak\_r\; =\; 0,$ :such that is an ideal in and is abelian. *(vii) is solvable if and only if its Killing form satisfies for all in and in .〔 Proposition 1.46.〕 This is Cartan's criterion for solvability. 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「Solvable Lie algebra」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.