Zinbiel algebra

In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:

: $(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b).$

Zinbiel algebras were introduced by {{harvs|txt|authorlink=Jean-Louis Loday|first=Jean-Louis|last=Loday|year=1995}}. The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.{{harvnb|Loday|2001|p=45}}

In any Zinbiel algebra, the symmetrised product

: $a \star b = a \circ b + b \circ a$

is associative.

A Zinbiel algebra is the Koszul dual concept to a Leibniz algebra. The free Zinbiel algebra over V is the tensor algebra with product

: $(x_0 \otimes \cdots \otimes x_p) \circ (x_{p+1} \otimes \cdots \otimes x_{p+q}) =
x_0 \sum_{(p,q)} (x_1,\ldots,x_{p+q}),$

where the sum is over all $(p,q)$ shuffles.

References

{{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 }}
{{cite journal | first1=Victor | last1=Ginzburg | author-link1=Victor Ginzburg | first2=Mikhail | last2=Kapranov |author-link2=Mikhail Kapranov| title=Koszul duality for operads | journal=Duke Mathematical Journal | volume=76 | year=1994 | pages=203–273 | arxiv=0709.1228 | doi=10.1215/s0012-7094-94-07608-4|mr=1301191}}
{{cite journal | first=Jean-Louis | last=Loday | author-link=Jean-Louis 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 | volume= 77 | issue=2 | pages=189–196 }}
{{cite book | first=Jean-Louis | last=Loday | author-link=Jean-Louis Loday| title=Dialgebras and related operads | publisher=Springer Verlag | series=Lecture Notes in Mathematics | year=2001 | url=http://www.math.uiuc.edu/K-theory/0333/ | volume=1763 | pages=7–66 }}
{{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 | arxiv=1101.0267 | pages=217–298| bibcode=2011arXiv1101.0267Z }}

Category:Lie algebras

Category:Non-associative algebras

Category:Algebra of random variables