Zinbiel algebra

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Zinbiel algebra (also called a dual Leibniz algebra) is a module over a commutative ring with a bilinear product that satisfies a specific defining rule. It was introduced by Jean-Louis Loday in 1995, and the name, chosen by Jean-Michel Lemaire, means the opposite of a Leibniz algebra. In a Zinbiel algebra, the symmetrized product x*y + y*x is associative. Zinbiel algebras are the Koszul duals of Leibniz algebras. The free Zinbiel algebra on a space V is the tensor algebra equipped with a product defined by summing over all p,q-shuffles (all ways to interleave p and q factors).