Non Inner Automorphism of Lie Algebras

Question

I have seen some examples of inner automorphisms of Lie algebras. Can anyone please give me an example of an automorphism of Lie algebras that is not inner (with proof). Note - An automorphism is said to be inner if it is of the form $exp(adx)$ for $adx$ nilpotent where $adx(y)$=$[x,y]$. Thanks for any help.

Answer 1

The simplest example I know is ${\mathfrak s}{\mathfrak l}_n({\mathbb C})$ for $n\geq 3$, which has the non-inner automorphism $A\mapsto -A^t$. This is not inner because is doesn't preserve conjugacy classes of matrices (look at the eigenvalues) which however any inner automorphism $\text{exp}(\text{ad}(X))=\text{conj}(e^X)$ does.

For a general semisimple complex Lie-algebra you get a non-inner automorphism for any non-trivial symmetry of the Dynkin diagram. In the example above you are using the horizontal symmetry of $A_{n-1}$ (note that for $n=2$ this is trivial, and $A\mapsto -A^t$ is indeed inner there).

Answer 2

The Lie algebra ${\mathfrak s}{\mathfrak l}_n({\mathbb C})$ has an automorphism given by $A\mapsto -A^t$. It is not inner for $n>2$, but inner for $n=2$, in which case it is given by $$ A\mapsto -A^t=X^{-1}AX, \quad \text{with} \quad X=\begin{pmatrix} 0 & 1 \cr -1 & 0 \end{pmatrix}. $$ This answers the question in the comment.