In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity:
(a\circb)\circc=a\circ(b\circc)+a\circ(c\circb).
Zinbiel algebras were introduced by . The name was proposed by Jean-Michel Lemaire as being "opposite" to Leibniz algebra.
In any Zinbiel algebra, the symmetrised product
a\starb=a\circb+b\circa
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
(x0 ⊗ … ⊗ xp)\circ(xp+1 ⊗ … ⊗ xp+q)= x0\sum(p,q)(x1,\ldots,xp+q),
where the sum is over all
(p,q)