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In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after , is a module ''L'' over a commutative ring ''R'' with a bilinear product (_ , _ ) satisfying the Leibniz identity : In other words, right multiplication by any element ''c'' is a derivation. If in addition the bracket is alternating (() = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case () = −() and the Leibniz's identity is equivalent to Jacobi's identity (. \, In other words, right multiplication by any element ''c'' is a derivation. If in addition the bracket is alternating (() = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case () = −() and the Leibniz's identity is equivalent to Jacobi's identity (] = 0). Conversely any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras, and the investigation of which theorems and properties of Lie algebras are still valid for Leibniz algebras is a recurrent theme in the literature. For instance, it has been shown that Engel's theorem still holds for Leibniz algebras and that a weaker version of Levi-Malcev theorem also holds. The tensor module, ''T''(''V'') , of any vector space ''V'' can be turned into a Loday algebra such that : This is the free Loday algebra over ''V''. Leibniz algebras were discovered by A. Bloh in 1965 who called them D-algebras. They attracted interest after Jean-Louis Loday noticed that the classical Chevalley–Eilenberg boundary map in the exterior module of a Lie algebra can be lifted to the tensor module which yields a new chain complex. In fact this complex is well-defined for any Leibniz algebra. The homology ''HL''(''L'') of this chain complex is known as Leibniz homology. If ''L'' is the Lie algebra of (infinite) matrices over an associative ''R''-algebra A then Leibniz homology of ''L'' is the tensor algebra over the Hochschild homology of ''A''. A ''Zinbiel algebra'' is the Koszul dual concept to a Leibniz algebra. It has defining identity: : ==Notes== 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「Leibniz algebra」の詳細全文を読む スポンサード リンク
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