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:

${\displaystyle (a\circ b)\circ c=a\circ (b\circ c)+a\circ (c\circ b).}$

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

${\displaystyle a\star b=a\circ b+b\circ a}$

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

${\displaystyle (x_{0}\otimes \cdots \otimes x_{p})\circ (x_{p+1}\otimes \cdots \otimes x_{p+q})=x_{0}\sum _{(p,q)}(x_{1},\ldots ,x_{p+q}),}$

where the sum is over all ${\displaystyle (p,q)}$ shuffles.^[1]