Maximal dimensional subalgebras of general Cartan type Lie algebras

Let $\mathbb{k}$ be a field of characteristic zero and let ${\mathbb{W}}_n=\mathrm{Der}(\mathbb{k}[x_1,\cdots ,x_n])$ be the $n^{\text{th}}$ general Cartan type Lie algebra. In this paper, we study Lie subalgebras $L$ of ${\mathbb{W}}_n$ of maximal Gelfand-Kirillov (GK) dimension, that is, with $\mathrm{GKdim}(L)=n$. For $n=1$, we completely classify such $L$, proving a conjecture of the second author. As a corollary, we obtain a new proof that ${\mathbb{W}}_1$ satisfies the Dixmier conjecture, in other words, $\mathrm{End}({\mathbb{W}}_1)\setminus \{0\}=\mathrm{Aut}({\mathbb{W}}_1)$, a result first shown by Du. For arbitrary $n$, we show that if $L$ is a GK-dimension $n$ subalgebra of ${\mathbb{W}}_n$, then $U(L)$ is not (left or right) noetherian.