Leibniz Algebra

A Leibniz algebra over a Commutative Ring $R$ is an $R$-Module $M$ equipped with a Bilinear Map $[-,-]:M\otimes M\to M$ that satisfies the Liebniz identity:

• $[[x,y],z]=[x,[y,z]]+[[x,z],y]$

Properties

• If the bracket operation is also Alternating, then the Liebniz algebra is also

a Lie Algebra, and $[a,b]=-[b,a]$.