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In Lie theory and representation theory, the Levi decomposition, conjectured by Killing and Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form : where ''z'' is in the nilradical (Levi–Malcev theorem). ==Application== To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple. 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「Levi decomposition」の詳細全文を読む スポンサード リンク
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