Maximal dimensional subalgebras of general Cartan type Lie algebras
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910718386765824 |
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| author | Bell, Jason Buzaglo, Lucas |
| author_facet | Bell, Jason Buzaglo, Lucas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_06001 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Maximal dimensional subalgebras of general Cartan type Lie algebras Bell, Jason Buzaglo, Lucas Rings and Algebras Primary: 17B66, 17B35, 16P40. Secondary: 17B65, 17B68 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. |
| title | Maximal dimensional subalgebras of general Cartan type Lie algebras |
| topic | Rings and Algebras Primary: 17B66, 17B35, 16P40. Secondary: 17B65, 17B68 |
| url | https://arxiv.org/abs/2311.06001 |