Let $\frak{g}$ be a finite-dimensional complex nilpotent Lie algebra. Given $\xi\in\frak{g}$, what is known about the intersection of $im(ad\_{\xi})$ (the image of $ad\_{\xi}:\frak{g}\rightarrow\frak{g}$) and $z(\frak{g})$ (the centre of $\frak{g}$)? In particular, under what conditions is it true that the intersection is non-trivial?


Alternatively, under what conditions will $im(ad\_{\xi})$ contain a 1-dimensional ideal?


Thanks



Nilpotent Lie Algebras
Whenever $ad\_{\xi}$ is an endomorphism of $\mathfrak{g}$ whose corresponding partition $\pi: 1^{s\_{1}}2^{s\_{2}} \cdots \;$ of $\dim \mathfrak{g}$ is such that $s\_{1} =0$, then we have $im\; ad\_{\xi} \cap Z(\mathfrak{g}) \neq \{0\}$. 


As $\mathfrak{g}$ is nilpotent, $ad\_{\xi}$ acts as a nilpotent endomomorphism of $\mathfrak{g}$, for every $\xi \in \mathfrak{g}$, and we can associate to $ad\_{\xi}$ a partition $\pi(\xi)$ of $n=\dim \mathfrak{g}$ as follows: the Jordan form of $ad\_{\xi}$ is 


$$J=\begin{bmatrix} J\_{1}\\\ & \ddots \\\ && J\_{r}\end{bmatrix}$$


where $J\_{i}$ is an $n\_{i}\times n\_{i}$ $0$-Jordan block and $n\_{1}\geq \ldots \geq n\_{r} >0$. Then, $n\_{1} +\ldots +n\_{r}=n$ is a partition of $n$, which we will denote $\pi(\xi): 1^{s\_{1}}2^{s\_{2}} \cdots $, so that $$1s\_{1}+2s\_{2}+\ldots =n$$ is the partition of $n$ determined by $J$.


Let $\xi \in \mathfrak{g}$ and let $\pi(\xi) : 1^{s\_{1}}2^{s\_{2}} \cdots $ be the partition associated to $ad\_{\xi}$ and suppose that $(x\_{1},\ldots,x\_{n})$ is a Jordan basis of $ad\_{\xi}$, so that the matrix of $ad\_{\xi}$ is in Jordan form. Thus, $x\_{1},x\_{n\_{1}+1},\ldots, x\_{n\_{r-1}+1}$ is a basis of $$Z\_{\mathfrak{g}}(\xi)=\ker ad\_{\xi}.$$ 


As $\mathfrak{g}$ is nilpotent we must have $Z(\mathfrak{g}) \neq \{0\}$. Let $w \in Z(\mathfrak{g})$ be nonzero. Then, $w \in Z\_{\mathfrak{g}}(\xi)$ and so $w= c\_{1}x\_{1}+\ldots + c\_{r} x\_{n\_{r-1}+1}$.


Assume that $s\_{1}=0$ in the partition $\pi$ (so that $\pi$ contains no parts equal to $1$). Thus, $x\_{2},\ldots, x\_{n\_{r-1}+2} \notin Z\_{\mathfrak{g}}(\xi)$ and $x=c\_{1}x\_{2}+\ldots +c\_{r}x\_{n\_{r-1}+2}$ is such that $[\xi,x] =w\in Z(\mathfrak{g})$.


While this condition is sufficient it is not necessary: for example, consider the nilpotent Lie algebra $\mathfrak{n}\_{3}$ of strictly upper triangular $3\times 3$ complex matrices. Then,


$$\xi = \begin{bmatrix} 0&1&0\\\0&0&1\\\0&0&0\end{bmatrix}$$


is such that $im \; ad\_{\xi}=Z(\mathfrak{n}\_{3})$, while $\pi(\xi): 12$.


