From Wikipedia, the free encyclopedia
In
mathematics, a Zinbiel algebra or dual Leibniz algebra is a
module over a
commutative ring with a
bilinear product satisfying the defining identity:
![{\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d917c3730c67620c31f8c6d7c2fe5fa7a8a676)
Zinbiel algebras were introduced by
Jean-Louis Loday (
1995). The name was proposed by Jean-Michel Lemaire as being "opposite" to
Leibniz algebra.
[1]
In any Zinbiel algebra, the symmetrised product
![{\displaystyle a\star b=a\circ b+b\circ a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd98eab54866a0877ff351cd8f4385114e399c4b)
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
![{\displaystyle (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}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cfdaf997f8145c0c8116c3f87a6e73a5c16c5396)
where the sum is over all
shuffles.
[1]
References
- Dzhumadil'daev, A.S.; Tulenbaev, K.M. (2005). "Nilpotency of Zinbiel algebras". J. Dyn. Control Syst. 11 (2): 195–213.
-
Ginzburg, Victor;
Kapranov, Mikhail (1994). "Koszul duality for operads".
Duke Mathematical Journal. 76: 203–273.
arXiv:
0709.1228.
doi:
10.1215/s0012-7094-94-07608-4.
MR
1301191.
-
Loday, Jean-Louis (1995).
"Cup-product for Leibniz cohomology and dual Leibniz algebras" (PDF). Math. Scand. 77 (2): 189–196.
-
Loday, Jean-Louis (2001).
Dialgebras and related operads. Lecture Notes in Mathematics. Vol. 1763.
Springer Verlag. pp. 7–66.
- Zinbiel, Guillaume W. (2012), "Encyclopedia of types of algebras 2010", in Guo, Li; Bai, Chengming; Loday, Jean-Louis (eds.),
Operads and universal algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, vol. 9, pp. 217–298,
arXiv:
1101.0267,
Bibcode:
2011arXiv1101.0267Z,
ISBN
9789814365116