In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:
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
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
where the sum is over all shuffles.[1]