When derivations are exactly homomorphisms?

Question

I would like to ask when a derivation is a Lie homomorphism, specially when the derivation is inner. Thanks for any suggestions. Takjk

Answer 1

A Lie algebra derivation is a linear map $D\colon L\rightarrow L$ satisfying the Leibniz rule, i.e., such that $D([x,y])=[x,D(y)]+[D(x),y]$ for all $x,y\in L$. A linear map $D\colon L\rightarrow L$ is a Lie algebra homomorphism, if $D([x,y])=[D(x),D(y)]$ for all $x,y\in L$. In order for a derivation to be a homomorphism it must satisfy $$D([x,y])=[D(x),D(y)]=[x,D(y)]+[D(x),y]. $$ Let us denote by $H(L)$ the subspace of ${\rm Der}(L)$ consisting of derivations which are homomorphisms. This space can be nontrivial, e.g., for the Heisenberg Lie algebra with basis $(x,y,z)$ and Lie brackets $[x,y]=z$ it consists of all linear mappings $$ D=\begin{pmatrix} \epsilon & 0 & 0 \cr 0 & -\epsilon & 0 \cr \alpha & \beta & 0 \end{pmatrix} $$ with $\epsilon=1$ or $\epsilon=0$. For semisimple Lie algebras every derivation $D$ can be written as $D={\rm ad}(z)$, i.e., it is inner. Then the space $H(L)$ is given by $$ H(L)=\{D={\rm ad}(z) \in End(L)\mid [z,[x,y]]=[[z,x],[z,y]]=[x,[z,y]]+[[z,x],y]\}. $$ For $L$ being, say $\mathfrak{sl}(2)$, this space is trivial.