Question about $\mathfrak{g}$-module homomorphism

Question

I am reading some notes on Lie algebras and they say something which I don't quite understand. Let $M$ and $N$ be two $\mathfrak{g}$-modules. The space of $\mathfrak{g}$-module homomorphism is denoted Hom$_{\mathfrak{g}}(M,N)$. When $M=N$ we write End$_{\mathfrak{g}}(M)$ for this space. Then they say that Hom$_{\mathfrak{g}}(M,N)$ is a subspace of Hom$_{\mathbb{C}}(M,N)$ and End$_{\mathfrak{g}}(M)$ is a subspace of End$_{\mathbb{C}}(M)$. Can someone explain why this is true?

Answer 1

The elements of $\operatorname{Hom}_{\mathfrak g}(M,N)$ are the linear maps $\varphi$ from $M$ into $N$ such that$$(\forall X,Y\in M):\varphi\bigl([X,Y]\bigr)=\bigl[\varphi(X),\varphi(Y)\bigr].$$In particular, the elements of $\operatorname{Hom}_{\mathfrak g}(M,N)$ are linear maps, and therefore they are elements of $\operatorname{Hom}_{\mathbb C}(M,N)$.