Let $\frak{g} = \frak{gl}_n(\mathbb{C})$.
What is the Lie algebra of derivations $\text{Der}(\frak{g})$?
Recall a Lie algebra derivation is a linear map $f: \frak{g} \to \frak{g}$ such that $f([x,y]) = [f(x),y] + [x,f(y)]$.
I know that $\frak{sl}_n := h$ is a semi-simple Lie sub-algebra of $\frak{g}$ and we can write $\frak{g} = h \oplus d$ for $\frak{d}$ the Lie sub-algebra of diagonal matrices.
Moreover, for any semi-simple Lie algebra the derivations are all inner (that is, of the form $ad(x))$.
So if $f$ is a derivation on $\frak{g}$ it is easy to see that $f$ preserves $\frak{h}$: because $\frak{h} = [\frak{g}, \frak{g}]$. If it preserved $\frak{d}$ we could write: on $\frak{d}$ a derivation is just a linear map. So is the conclusion is that any derivation is of the form $f = f_1 + ad(h)$ where $f_1$ is any linear map on $\frak{d}$ and $h \in \frak{h}$?
However I don't know if it's true that $f$ preserves the diagonal matrices.
Any assistance?
Best Answer
Since the center $Z(L)$ of $L=\mathfrak{gl}_n(\Bbb C)\cong \mathfrak{sl}_n(\Bbb C)\oplus Z(L)$ is a characteristic ideal of $L$, it is preserved by all derivations of $L$.