Context
Let $\mathfrak g$ be a Lie algebra over a field $k$.
We call an element $x \in \mathfrak g$ ad-semisimple if $ad_x \in \mathfrak gl(\mathfrak g)$ is a semisimple endomorphism.
Similarly, we call an element $x \in \mathfrak g$ ad-nilpotent if $ad_x \in \mathfrak gl(\mathfrak g)$ is a nilpotent endomorphism.
In the following let $\mathfrak g$ be a complex semisimple Lie algebra and $\rho: \mathfrak g \rightarrow \mathfrak gl(V)$ a finite-dimensional $\mathfrak g$-representation. One can show that if $x \in \mathfrak g$ is ad-semisimple then $\rho (x)$ is semisimple (equivalently diagonalizable since we consider $\mathbb C$-vector spaces). Similarly, $x \in \mathfrak g$ ad-nilpotent implies that $\rho (x)$ is nilpotent. This follows from the fact that given the unique absolute Jordan decomposition (also known as the additive Jordan-Chevalley decomposition) $x=s+n$ (which exists in any complex semisimple Lie algebra) the Jordan decomposition of $\rho (x)$ is given by $\rho(x)=\rho(s)+\rho(n)$.
Questions
- Do these two implications ($x$ ad-semisimple/ad-nilpotent implies $\rho (x)$ semisimple/nilpotent) fail when $\mathfrak g$ is not a complex semisimple Lie algebra?
- If so, what are counterexamples?
Best Answer
Hint 1: If $L$ is any non-perfect, finite dimensional Lie algebra (over a field $k$), the natural map $L \rightarrow L/[L,L]$ is a surjection from $L$ to a finite dimensional abelian Lie algebra.
Hint 2: a) The diagonal matrices inside $\mathfrak{gl}_n(k)$ form an $n$-dimensional abelian Lie algebra, all of whose nonzero elements are not nilpotent as matrices.
b) The matrices of the form $\pmatrix{0 &* &*&*&\dots&*\\ 0&0&0&0&...&0\\\vdots&&&\ddots &&\vdots}$ inside $\mathfrak{gl}_n(k)$ form an $(n-1)$-dimensional abelian Lie algebra, all of whose nonzero elements are not semisimple as matrices.
Can you combine the hints to find, for any (finite-dim.) non-perfect Lie algebra (over an arbitrary field $k$), a representation $\rho$ such that some $ad$-nilpotent element $x \in L$ is sent to a non-nilpotent matrix $\rho(x)$, and/or some $ad$-semisimple element is sent to a non-semisimple matrix?