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| In [[mathematics]], a '''(right) Leibniz algebra''', named after [[Gottfried Wilhelm Leibniz]], sometimes called a '''Loday algebra''', after {{Link-interwiki|en=Jean-Louis Loday|lang=fr}}, is a module ''L'' over a commutative ring ''R'' with a bilinear product [ _ , _ ] satisfying the '''Leibniz identity'''
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| :<math> [[a,b],c] = [a,[b,c]]+ [[a,c],b]. \, </math>
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| In other words, right multiplication by any element ''c'' is a [[derivation (abstract algebra)|derivation]]. If in addition the bracket is alternating ([''a'', ''a''] = 0) then the Leibniz algebra is a [[Lie algebra]]. Indeed, in this case [''a'', ''b''] = −[''b'', ''a''] and the Leibniz's identity is equivalent to Jacobi's identity ([''a'', [''b'', ''c'']] + [''c'', [''a'', ''b'']] + [''b'', [''c'', ''a'']] = 0). Conversely any Lie algebra is obviously a Leibniz algebra.
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| The tensor module, ''T''(''V'') , of any vector space ''V'' can be turned into a Loday algebra such that
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| :<math> [a_1\otimes \cdots \otimes a_n,x]=a_1\otimes \cdots a_n\otimes x\quad \text{for }a_1,\ldots, a_n,x\in V.</math>
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| This is the free Loday algebra over ''V''.
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| 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
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| of ''L'' is the tensor algebra over the [[Hochschild homology]] of ''A''.
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| A ''[[Zinbiel algebra]]'' is the [[Koszul algebra|Koszul dual]] concept to a Leibniz algebra. It has defining identity:
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| :<math> ( a \circ b ) \circ c = a \circ (b \circ c) + a \circ (c \circ b) . </math>
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| ==References==
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| * {{cite journal | first= Yvette| last=Kosmann-Schwarzbach | title= From Poisson algebras to Gerstenhaber algebras | journal= [[Annales de l'Institut Fourier]] | year=1996 | url= | doi= | volume= 46| issue = 5 | pages= 1243–1274 }}
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| * {{cite journal | first=Jean-Louis | last=Loday | title= {{lang|fr|Une version non commutative des algèbres de Lie: les algèbres de Leibniz}} | journal= Enseign. Math. (2) | year=1993 | url= | doi= | volume= 39| issue = 3–4 | pages= 269–293 }}
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| * {{Cite journal |doi=10.1007/BF01445099 |last=Loday |first=Jean-Louis |lastauthoramp=yes |first2=Pirashvili |last2=Teimuraz |year=1993 |title=Universal enveloping algebras of Leibniz algebras and (co)homology |journal=[[Mathematische Annalen]] |volume=296 |issue= 1|pages=139–158 }}
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| * {{Cite journal | first= A. | last=Bloh | title= On a generalization of the concept of Lie algebra | journal= [[Dokl. Akad. Nauk SSSR]] | year= 1965 | url= | doi= | volume=165 |issue= | pages=471–473 }}
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| * {{Cite journal | first= A. | last=Bloh | title= Cartan-Eilenberg homology theory for a generalized class of Lie algebras | journal= [[Dokl. Akad. Nauk SSSR]] | year= 1967 | url= | doi= | volume= 175 |issue= 8| pages=824–826 }}
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| * {{cite journal | first1=A.S. | last1=Dzhumadil'daev | first2=K.M. | last2=Tulenbaev | title=Nilpotency of Zinbiel algebras | journal=J. Dyn. Control Syst. | volume=11 | number=2 | year=2005 | pages=195–213 }}
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| * {{cite journal | first1=V. | last1=Ginzburg | authorlink=Victor Ginzburg | first2=M. | last2=Kapranov | title=Koszul duality for operads | journal=Duke Math. J. | volume=76 | year=1994 | pages=203–273 | arxiv=0709.1228}}
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| {{DEFAULTSORT:Leibniz Algebra}}
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| [[Category:Lie algebras]]
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| [[Category:Non-associative algebras]]
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