Let $k$ be a field (algebraically closed for simplicity) and let $A$ be an $n$-dimensional algebra over $k$ (not necessarily commutative or even associative). The group $G=\mbox{Aut}(A)$ is an algebraic group. The elements of the lie algebra $\mathfrak{g}$ of $G$ act (naturally) on $A$ and they act as derivations over $k$. So we have the inclusion $\mathfrak{g}\subseteq \mbox{Der}_k(A)$. My first questions:
- Can this inclusion be strict?
- If so, how can $\mathfrak{g}$ be characterized within $\mbox{Der}_k(A)$? is it an Ideal (in the lie algebra sense)?
- Are there reasonable conditions on $A$ that ensure equality?
Now, for a more specialized setting, in Serre's book "Lie algebras and Lie groups" there is a proof of the theorem that if $\mathfrak{s}$ is a semi-simple real lie algebra with a definite killing form then it is a Lie algebra of some compact real Lie group $G$ (Thm 6.3). The group $G$ is taken to be $\mbox{Aut}(\mathfrak{s})$ and it is stated without explanation that "The Lie algebra of $\mbox{Aut}(\mathfrak{s})$ is the algebra of derivations of $\mathfrak{s}$" (which then equals $\mathfrak{s}$ itself since $\mathfrak{s}$ is semi-simple), so the more specialized question is
- Let $\mathfrak{g}$ be a Lie algebra, under what conditions the lie algebra of $\mbox{Aut}(\mathfrak{g})$ is the whole $\mbox{Der}_k(\mathfrak{g})$?