Why Leibniz algebras are called on this way?

Question

I´m interested to know the reason to call the Leibniz algebras on this way.

I know that previously they were known as $D$-algebras

Why Loday chose this name?

Did Leibniz study something related?

Thanks!

Answer 1

It is called after Gottfried Wilhelm Leibniz for the derivation rule, also called Leibniz rule. In the Lie case it is just the Jacobi identity, namely that the adjoint operators are derivations of the Lie algebra: $$[x,[y,z]]=[[x,y],z]+[y,[x,z]]$$ Loday chose the word Leibniz algebra for a nonassociative algebra, where we only have the Jacobi identity but not the skew-symmetry. The dual operad he called Zinbiel, which is Leibniz read backwards.

Answer 2

Leibniz algebras are defined as vector spaces $L$ endowed with a bilinear application $[,]\colon L\times L\to L$ such that the identity $$[x,[y,z]]=[[x,y],z]+[y,[x,z]]$$ holds for all $x,y,z\in L$. This can be equivalently rephrased as asking that for all $x$, the morphism $$\delta_x\colon L\to L\colon y\mapsto [x,y]$$ satisfies the identity $$\delta_x([y,z])=[\delta_x(y),z]+[y,\delta_x(z)],$$ and this is often called the Leibniz identity, hence the name. It means that $\delta_x$ is a derivation of $(L,[,])$, and in fact the identity above also implies that $\delta:L\to Der(L)$ is a morphism of Leibniz algebras.