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 Jean-Louis Loday (1995). The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.

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

$(x_0\otimes \cdots \otimes x_p)\circ (x_{p+1}\otimes \cdots \otimes x_{p+q})=x_0\sum _{(p,q)}(x_1,\dots ,x_{p+q}),$

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

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. doi:10.1215/s0012-7094-94-07608-4. 
• Loday, Jean-Louis (1995). "Cup-product for Leibniz cohomology and dual Leibniz algebras". Math. Scand. 77 (2): 189–196. http://www.math.uiuc.edu/K-theory/0015/cup_product.pdf. 
• Loday, Jean-Louis (2001). Dialgebras and related operads. Lecture Notes in Mathematics. 1763. Springer Verlag. pp. 7–66. http://www.math.uiuc.edu/K-theory/0333/. 
• Zinbiel, Guillaume W. (2012), "Encyclopedia of types of algebras 2010", in Guo, Li; Bai, Chengming; Loday, Jean-Louis, Operads and universal algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics, 9, pp. 217–298, ISBN 9789814365116, Bibcode: 2011arXiv1101.0267Z, http://www.worldscibooks.com/mathematics/8222.html