about derivation for Leibniz algebras

Question

A left Leibniz algebra over the field $k$ is a vector space over $k$ with a bilinear map $[~,~]:L\times L \mapsto L$ satisfying $$[a,[b,c]]=[[a,b],c]+[b,[a,c]].$$ A derivation of $L$ is $\alpha :L \mapsto L$ such that $\alpha ([a,b])=[\alpha(a),b]+[a,\alpha(b)]$ for all $a,b \in L$. I want to know whether the vector space of all derivations of $L$ which is denoted by $Der(L)$ forms a Leibniz algebra or not?

Answer 1

Due to the operation of commutation of linear operators, $Der(L)$ is even a Lie algebra for a Leibniz algebra $L$. So, yes, $Der(L)$ forms a Leibniz algebra. More generally, the derivations $Der(A)=\{D\in End(A)\mid D(a\cdot b)=D(a)\cdot b+a\cdot D(b)\}$ of any algebra $(A,\cdot)$ form a Lie algebra.