1
$\begingroup$

What is an example of a nilpotent Lie algebra $\cal g$ with dimension four and:

  • $dim\ \mathcal g'=1$,
  • $dim \ \mathcal g'=2$,
  • $dim \ \mathcal g'=3$?

Is there a way to construct them?

$\endgroup$
  • 1
    $\begingroup$ Yes, we can construct all possible $4$-dimensional Lie algebras. The nilpotent ones are the abelian ones and $\mathfrak{h}_1\oplus K$, $\mathfrak{f}_4$. However, $codim (g')=1$ is impossible, as you know. $\endgroup$ – Dietrich Burde Feb 17 '16 at 10:51
2
$\begingroup$

Take the $3$-dimensional Heisenberg Lie algebra $\mathfrak{h}_1$ with basis $(x,y,z)$ and Lie bracket $[x,y]=z$ and put $L=\mathfrak{h}\oplus K$ with the $1$-dimensional abelian Lie algebra $K$. Then $\dim L=4$ and $\dim L'=\dim [L,L]=1$. For $\dim L'=2$ take the standard graded filiform Lie algebra $\mathfrak{f}_4$ of dimension $4$, with basis $(e_1,\ldots e_4)$ and Lie brackets $$ [e_1,e_i]=e_{i+1}, \; i=2,3. $$ Then $L'=\langle e_3,e_4\rangle$.
We cannot have $\dim L'=3$ for a nilpotent Lie algebra $L$ of dimension $4$, because $\dim L/L'\ge 2$, see your question here.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.