Let $L$ be a Lie Algebra such that $\dim Z(L)=1$ and $L/Z(L)$ is abelian. Prove that $L\cong H_{2n+1}$ (Heisenberg algebra) for some $n\in\mathbb{N}$.
I was able to show that if $\dim L$ is odd, then the statement is true. I'm stuck however on showing that $L$ must be off odd dimension. My professor gave a clue that this has something to do with a non-degenerate skew-symmetric bilinear form (on $L$), but I'm not sure how to follow.
Any help would be appreciated.
Since $L/Z(L)$ is abelian, this shows that $[L,L]\subset Z(L)$. Since $\text{dim}Z(L)=1$, we have two cases: either $[L,L]=\{0\}$, in which case $L$ is abelian (which is not included in your exercise), or $[L,L]=Z(L)$, in which case $L$ is a symplectic extension of $\mathbb{R}^{2n}\cong L/Z(L)$(which is exactly what you denote by $H_{2n+1}$). to see this, let $Z(L)=\langle z\rangle $, then consider the map $$w:L/Z(L)\times L/Z(L) \to Z(L), (\overline{x},\overline{y}) \to [x,y]$$ This is a well defined bilinear antisymmetric nondegenerate (for any noncentral element $x$, there is a non central lement $y$ such that $w(x,y)=[x,y]\neq 0$) form (that is why it is called a symplectic extension). Now, it is well-known that such forms exist only on even dimension, so $\text{dim} L=2n+1$. Hence, $L/Z(L)$ has a Symplectic Basis, pick representatives $e_{i},f_{i}\cdots i=1..n$ satisfying $[e_{i},f_{j}]=\delta_{i,j}z$.