Maximal dimensional subalgebras of general Cartan-type Lie algebras

Let k$\mathbb {k}$ be a field of characteristic zero and let Wn=Der(k[x1,& mldr;,xn])$\mathbb {W}_n = \operatorname{Der}(\mathbb {k}[x_1,\ldots,x_n])$ be the nth$n{\text{th}}$ general Cartan-type Lie algebra. In this paper, we study Lie subalgebras L$L$ of Wn$\mathbb {W}_n$ of maximal Gelfand-Kirillov (GK) dimension, that is, with GKdim(L)=n$\operatorname{GKdim}(L) = n$.For n=1$n = 1$, we completely classify such L$L$, proving a conjecture of the second author. As a corollary, we obtain a new proof that W1$\mathbb {W}_1$ satisfies the Dixmier conjecture, in other words, End(W1)\{0}=Aut(W1)$\operatorname{End}(\mathbb {W}_1) \setminus \lbrace 0\rbrace = \operatorname{Aut}(\mathbb {W}_1)$, a result first shown by Du.For arbitrary n$n$, we show that if L$L$ is a GK-dimension n$n$ subalgebra of Wn$\mathbb {W}_n$, then U(L)$U(L)$ is not (left or right) noetherian.