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| In [[mathematics]], a '''Zinbiel algebra''' or '''dual Leibniz algebra''' is a [[module (mathematics)|module]] over a [[commutative ring]] with a [[bilinear product]] satisfying the 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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| Zinbiel algebras were introduced by {{harvs|txt|authorlink=Jean-Louis Loday|first=Jean-Louis|last=Loday|year=1995}}. The name was proposed by J.-M. Lemaire as being "opposite" to Leibniz algebra.<ref name=L0145>{{harvnb|Loday|2001|p=45}}</ref>
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| The symmetrised product
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| :<math> a \star b = a \circ b + b \circ a </math>
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| is associative.
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| A Zinbiel algebra is the [[Koszul dual]] concept to a [[Leibniz algebra]]. The free Zinbiel algebra over ''V'' is the [[tensor algebra]] with product
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| :<math> (x_0 \otimes \cdots \otimes x_p) \circ (x_{p+1} \otimes \cdots \otimes x_{p+q}) =
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| x_0 \sum_{(p,q)} (x_1,\ldots,x_{p+q}) , </math>
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| where the sum is over all (''p'',''q'') [[shuffle product|shuffles]].<ref name=L0145/>
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| ==References==
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| {{reflist}}
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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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| * {{cite journal | first=Jean-Louis | last=Loday | title=Cup-product for Leibniz cohomology and dual Leibniz algebras | journal=Math. Scand. | year=1995|url=http://www.math.uiuc.edu/K-theory/0015/cup_product.pdf | doi= | volume= 77 | issue=2 | pages=189–196 }}
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| *{{cite book | first=Jean-Louis | last=Loday | title=Dialgebras and related operads | publisher=[[Springer Verlag]] | series=Lecture Notes in Maths | year=2001 | url=http://www.math.uiuc.edu/K-theory/0333/ | doi= | volume=1763 | pages=7–66 | ref=harv }}
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| *{{Citation | last1=Zinbiel | first1=Guillaume W. | editor1-last=Guo | editor1-first=Li | editor2-last=Bai | editor2-first=Chengming | editor3-last=Loday | editor3-first=Jean-Louis | title=Operads and universal algebra | url=http://www.worldscibooks.com/mathematics/8222.html | series= Nankai Series in Pure, Applied Mathematics and Theoretical Physics | isbn=9789814365116 | year=2012 | volume=9 | chapter=Encyclopedia of types of algebras 2010 | chapterurl=http://arxiv.org/abs/1101.0267 | pages=217–298}}
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| ==External links==
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| *[http://www-irma.u-strasbg.fr/~loday/Zinbiel.html Biography of Guillaume William Zinbiel]
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| {{DEFAULTSORT:Zinbiel Algebra}}
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| [[Category:Lie algebras]]
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| [[Category:Non-associative algebras]]
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