What does it mean for a lie subalgebra to be "stable" under inner derivations?

Question

I was wondering if I could get some clarification about what exactly this definition (found in my course notes) means.

First of all, the author defines the set of "inner derivations" of a lie algebra $\frak{g}$ to be any derivation $D[x,y]=[Dx,y]+[x,Dy]$ that can be written as $\frak{ad}$$(x)$ for some $x\in\frak{g}$. There is a theorem saying that the set of inner derivations of $\frak{g}$ forms an ideal of the set of derivations of $\frak{g}$.

Then, an ideal of $\frak{g}$ is then re-defined as "a subalgebra of $\frak{g}$ that is stable under inner derivations of $\frak{g}$".

Similarly, a characteristic ideal is defined as a subalgebra of $\frak{g}$ that is stable under all derivations of $\frak{g}$.

What exactly does "stable" mean in this context?

Answer 1

"Stable" just means "closed": in other words, if $D$ is any (inner) derivation of $\mathfrak{g}$ and $x$ is in your subalgebra, then $Dx$ is required to be in the subalgebra as well.

Answer 2

You could easily recover the meaning of "stable" from the "re-definition" of what ideal is.

So, let $\mathfrak a\subseteq \mathfrak g$ be ideal. That means that for all $X\in\mathfrak g$, $[X,\mathfrak a]\subseteq \mathfrak a$, or using adjoint representation, $(\operatorname{ad} X)(\mathfrak a)\subseteq \mathfrak a$. Rewrite this as $(\forall X\in\mathfrak g)(\forall Y\in\mathfrak a) (\operatorname{ad}X)Y \in\mathfrak a$, and finally, $(\operatorname{Inn}\mathfrak g)Y\subseteq \mathfrak a,$ for all $Y\in\mathfrak a$, where $\operatorname{Inn}\mathfrak g$ stands for inner derivations of $\mathfrak g$.

In ordinary English, stable under inner derivations means that if you act with inner derivation on $\mathfrak a$, you won't get out of $\mathfrak a$.

Answer 3

The adjoint representation of a Lie algebra $L$ can also viewed as an $L$-module. Then the ideals are the submodules of the $L$-module $L$, i.e., stable under $ad$. So we have $ad(L)(I)\subseteq I$, which is just the definition of an ideal. Sometimes the words is also "invariant subspace".