Zinbiel algebra

Remarks

Example: let $A = x\Q[x]$ be the polynomials without constant term, with $f \circ g = \int_0^x f'(t)\, g(t)\, dt$. For $f, g \in A$, integration by parts gives $f \circ g + g \circ f = fg$. Then $(a \circ b) \circ c = \int_0^x (a \circ b)' c \, dt = \int_0^x a' b c \, dt$, and $a \circ (b \circ c) + a \circ (c \circ b) = \int_0^x a' (b \circ c + c \circ b)\, dt = \int_0^x a' b c \, dt$. On monomials the product is $x^m \circ x^n = \frac{m}{m + n} x^{m + n}$.

Non-example: $A = \Q$ with the usual product. For $a = b = c = 1$ the left side is $1$ and the right side is $2$.

In any Zinbiel algebra, $a * b = a \circ b + b \circ a$ is commutative and associative. Some authors write the identity with the opposite product, as $a \circ (b \circ c) = (a \circ b) \circ c + (b \circ a) \circ c$.