What we can say about dimension of derivation as a function of dim L?

Question

I study in derivation Lie algebra. Now I have some questions A: What we can say about dimension of derivation as a function of dim L? B: For finitely generated Lie algebra L, what we can say about its derivation? Finitely generated? Finite dimension? Please introduce the articles you are interested in And guide me on this topic Thank you

Answer 1

Question $A$: Let $\dim (L)=n$. Then $n\le \dim Der(L)\le n^2$. For semisimple Lie algebras $L$, among others, we have $L\cong ad(L)=Der(L)$, so that the lower bound is attained, and for abelian Lie algebras we have $Der(L)=End(L)$, which is of dimension $n^2$, so that the upper bound is attained. For more involved estimates see for example Togo's article Dimensions of the Derivation Algebras of Lie Algebras.

Question $B$: A finitely generated Lie algebra may be infinite and may have an infinite-dimensional derivation algebra. Take the free Lie algebra on $2$ generators.