Zinbiel algebra

Summary

A Zinbiel algebra is a vector space with a bilinear product $\circ$ such that $(a \circ b) \circ c = a \circ (b \circ c) + a \circ (c \circ b)$. Its symmetrization $a \circ b + b \circ a$ is always associative and commutative, so a Zinbiel product is a way of splitting a commutative product into two halves. The half-shuffle product on tensors and the product $f \circ g = \int_0^x f' g$ of functions are the basic examples. The name is "Leibniz" written backwards: the operad of Zinbiel algebras is Koszul dual to that of Leibniz algebras.