Maximal dimensional subalgebras of general Cartan-type Lie algebras

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-12-16 DOI:10.1112/blms.13216

Jason Bell, Lucas Buzaglo

{"title":"Maximal dimensional subalgebras of general Cartan-type Lie algebras","authors":"Jason Bell, Lucas Buzaglo","doi":"10.1112/blms.13216","DOIUrl":null,"url":null,"abstract":"Let <math>\n <semantics>\n <mi>k</mi>\n <annotation>$\\mathbb {k}$</annotation>\n </semantics></math> be a field of characteristic zero and let <math>\n <semantics>\n <mrow>\n <msub>\n <mi>W</mi>\n <mi>n</mi>\n </msub>\n <mo>=</mo>\n <mo>Der</mo>\n <mrow>\n <mo>(</mo>\n <mi>k</mi>\n <mrow>\n <mo>[</mo>\n <msub>\n <mi>x</mi>\n <mn>1</mn>\n </msub>\n <mo>,</mo>\n <mtext>…</mtext>\n <mo>,</mo>\n <msub>\n <mi>x</mi>\n <mi>n</mi>\n </msub>\n <mo>]</mo>\n </mrow>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathbb {W}_n = \\operatorname{Der}(\\mathbb {k}[x_1,\\ldots,x_n])$</annotation>\n </semantics></math> be the <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mtext>th</mtext>\n </mrow>\n <annotation>$n{\\text{th}}$</annotation>\n </semantics></math> general Cartan-type Lie algebra. In this paper, we study Lie subalgebras <math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math> of <math>\n <semantics>\n <msub>\n <mi>W</mi>\n <mi>n</mi>\n </msub>\n <annotation>$\\mathbb {W}_n$</annotation>\n </semantics></math> of maximal Gelfand–Kirillov (GK) dimension, that is, with <math>\n <semantics>\n <mrow>\n <mo>GKdim</mo>\n <mo>(</mo>\n <mi>L</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>n</mi>\n </mrow>\n <annotation>$\\operatorname{GKdim}(L) = n$</annotation>\n </semantics></math>.For <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$n = 1$</annotation>\n </semantics></math>, we completely classify such <math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math>, proving a conjecture of the second author. As a corollary, we obtain a new proof that <math>\n <semantics>\n <msub>\n <mi>W</mi>\n <mn>1</mn>\n </msub>\n <annotation>$\\mathbb {W}_1$</annotation>\n </semantics></math> satisfies the Dixmier conjecture, in other words, <math>\n <semantics>\n <mrow>\n <mo>End</mo>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>W</mi>\n <mn>1</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <mo>∖</mo>\n <mrow>\n <mo>{</mo>\n <mn>0</mn>\n <mo>}</mo>\n </mrow>\n <mo>=</mo>\n <mo>Aut</mo>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>W</mi>\n <mn>1</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\operatorname{End}(\\mathbb {W}_1) \\setminus \\lbrace 0\\rbrace = \\operatorname{Aut}(\\mathbb {W}_1)$</annotation>\n </semantics></math>, a result first shown by Du.For arbitrary <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math>, we show that if <math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math> is a GK-dimension <math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> subalgebra of <math>\n <semantics>\n <msub>\n <mi>W</mi>\n <mi>n</mi>\n </msub>\n <annotation>$\\mathbb {W}_n$</annotation>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>L</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$U(L)$</annotation>\n </semantics></math> is not (left or right) noetherian.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 2","pages":"605-624"},"PeriodicalIF":0.8000,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13216","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.13216","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

Let $k$ be a field of characteristic zero and let $W_{n} = Der (k [x_{1}, …, x_{n}])$ be the $n th$ general Cartan-type Lie algebra. In this paper, we study Lie subalgebras $L$ of $W_{n}$ of maximal Gelfand–Kirillov (GK) dimension, that is, with $GKdim (L) = n$ .

For $n = 1$ , we completely classify such $L$ , proving a conjecture of the second author. As a corollary, we obtain a new proof that $W_{1}$ satisfies the Dixmier conjecture, in other words, $End (W_{1}) ∖ {0} = Aut (W_{1})$ , a result first shown by Du.