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:

Zinbiel algebras were introduced by Jean-Louis Loday (1995). The name was proposed by J.-M. Lemaire as being "opposite" to Leibniz 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 (p,q) shuffles.