Let $\mathfrak{n}$ be a complex nilpotent Lie algebra of dimension $n$, and let $\mathfrak{t}$ be denote a maximal torus of semisimple derivations of $\mathfrak{n}$.
I would be very grateful if you could give me references, bibliography or counterexamples to study the following questions:
- Does there exist a basis $\mathcal{B}= \{e_1,\ldots,e_n\}$ of $\mathfrak{n}$ such that simultaneously diagonalizes $\mathfrak{t}$ and $\operatorname{ad}(X)$ is lower triangular matrix with respect to $\mathcal{B}$; for all $X \in \mathfrak{n}$?
- If $\operatorname{rank}(\mathfrak{n}) \geq 1$ (here, $\operatorname{rank}(\mathfrak{n}) = \operatorname{dim}(\mathfrak{t})$), does $\mathfrak{n}$ admit a semisimple derivation $D$ such that all its eigenvalues are non-negative real numbers?
- If $\operatorname{rank}(\mathfrak{n}) \geq 2$, does $\mathfrak{n}$ admit a semisimple derivation such that all its eigenvalues are positive real numbers?
Concerning the second and third question, It is easy to see that if a complex nilpotent Lie algebra is such that $\operatorname{rank}(\mathfrak{n})\geq 1$, then $\mathfrak{n}$ admits a semisimple derivation with integer eigenvalues.
For the first question, this follows from Lie's Theorem applied to the solvable subalgebra $\mathfrak{t}\ltimes {\rm ad}(\mathfrak{n})$ of $\rm{Der}(\mathfrak{n})$.
For the second question, every nilpotent Lie algebra admits a nonzero outer derivation $D$ with $D^2=0$ by a result of Togo, so with all eigenvalues equal to zero.
For the third question, every Lie algebra admitting a derivation with only positive eigenvalues, is very special, namely is $\mathbb{N}$-graded, and has an invertible derivation. This is a serious restriction for a nilpotent Lie algebra. So I would think there could be a counterexample.