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arXiv · 2311.06001

Maximal dimensional subalgebras of general Cartan type Lie algebras

Jason Bell·Lucas Buzaglo
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Abstract

Let $\Bbbk$ be a field of characteristic zero and let $\mathbb{W}_n = \operatorname{Der}(\Bbbk[x_1,\cdots,x_n])$ 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 $\operatorname{GKdim}(L) = n$. 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, $\operatorname{End}(\mathbb{W}_1) \setminus \{0\} = \operatorname{Aut}(\mathbb{W}_1)$, 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.

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Jason Bell, Lucas Buzaglo. 2023-11-10. Maximal dimensional subalgebras of general Cartan type Lie algebras. https://arxiv.org/abs/2311.06001

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