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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.[1]

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.[1]

References

Loday 2001, p. 45

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. 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, 9, pp. 217–298, arXiv:1101.0267, Bibcode:2011arXiv1101.0267Z, ISBN 9789814365116