{"title":"关于雅各布森-莫罗佐夫定理向偶数特征的扩展","authors":"David I. Stewart, Adam R. Thomas","doi":"10.1112/jlms.70007","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mi>G</mi>\n <annotation>$G$</annotation>\n </semantics></math> be a simple algebraic group over an algebraically closed field <span></span><math>\n <semantics>\n <mi>k</mi>\n <annotation>$\\mathbb {k}$</annotation>\n </semantics></math> of characteristic 2. We consider analogues of the Jacobson–Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in <span></span><math>\n <semantics>\n <mrow>\n <mi>g</mi>\n <mo>:</mo>\n <mo>=</mo>\n <mo>Lie</mo>\n <mo>(</mo>\n <mi>G</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathfrak {g}:=\\operatorname{Lie}(G)$</annotation>\n </semantics></math> and also those with overalgebras isomorphic to the algebras <span></span><math>\n <semantics>\n <mrow>\n <mo>Lie</mo>\n <mo>(</mo>\n <msub>\n <mi>SL</mi>\n <mn>2</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{Lie}(\\mathrm{SL}_2)$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mo>Lie</mo>\n <mo>(</mo>\n <msub>\n <mi>PGL</mi>\n <mn>2</mn>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$\\operatorname{Lie}(\\mathrm{PGL}_2)$</annotation>\n </semantics></math>. This leads us to calculate the dimension of the Lie automiser <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>n</mi>\n <mi>g</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>k</mi>\n <mo>·</mo>\n <mi>e</mi>\n <mo>)</mo>\n </mrow>\n <mo>/</mo>\n <msub>\n <mi>c</mi>\n <mi>g</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>e</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathfrak {n}_\\mathfrak {g}(\\mathbb {k}\\cdot e)/\\mathfrak {c}_\\mathfrak {g}(e)$</annotation>\n </semantics></math> for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70007","citationCount":"0","resultStr":"{\"title\":\"On extensions of the Jacobson–Morozov theorem to even characteristic\",\"authors\":\"David I. Stewart, Adam R. Thomas\",\"doi\":\"10.1112/jlms.70007\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span></span><math>\\n <semantics>\\n <mi>G</mi>\\n <annotation>$G$</annotation>\\n </semantics></math> be a simple algebraic group over an algebraically closed field <span></span><math>\\n <semantics>\\n <mi>k</mi>\\n <annotation>$\\\\mathbb {k}$</annotation>\\n </semantics></math> of characteristic 2. We consider analogues of the Jacobson–Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>g</mi>\\n <mo>:</mo>\\n <mo>=</mo>\\n <mo>Lie</mo>\\n <mo>(</mo>\\n <mi>G</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathfrak {g}:=\\\\operatorname{Lie}(G)$</annotation>\\n </semantics></math> and also those with overalgebras isomorphic to the algebras <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>Lie</mo>\\n <mo>(</mo>\\n <msub>\\n <mi>SL</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{Lie}(\\\\mathrm{SL}_2)$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>Lie</mo>\\n <mo>(</mo>\\n <msub>\\n <mi>PGL</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\operatorname{Lie}(\\\\mathrm{PGL}_2)$</annotation>\\n </semantics></math>. This leads us to calculate the dimension of the Lie automiser <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>n</mi>\\n <mi>g</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>k</mi>\\n <mo>·</mo>\\n <mi>e</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>/</mo>\\n <msub>\\n <mi>c</mi>\\n <mi>g</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>e</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\mathfrak {n}_\\\\mathfrak {g}(\\\\mathbb {k}\\\\cdot e)/\\\\mathfrak {c}_\\\\mathfrak {g}(e)$</annotation>\\n </semantics></math> for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70007\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70007\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70007","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
让 G $G$ 是特征为 2 的代数闭域 k $\mathbb {k}$ 上的一个简单代数群。我们考虑雅各布森-莫罗佐夫定理在这种情况下的相似性。更准确地说，我们将那些在 g : = Lie ( G ) $\mathfrak {g}:=\operatorname{Lie}(G)$中具有简单三维 Lie 上代数的无幂元素以及那些具有与 Lie ( SL 2 ) $\operatorname{Lie}(\mathrm{SL}_2)$ 和 Lie ( PGL 2 ) $\operatorname{Lie}(\mathrm{PGL}_2)$ 同构的上代数的无幂元素进行了分类。这样我们就可以计算 Lie 自动机的维度 n g ( k - e ) / c g ( e ) $\mathfrak {n}_\mathfrak {g}(\mathbb {k}\cdot e)/\mathfrak {c}_\mathfrak {g}(e)$ 适用于所有零势轨道；在偶数特征中，这个量对同源性非常敏感。
On extensions of the Jacobson–Morozov theorem to even characteristic
Let be a simple algebraic group over an algebraically closed field of characteristic 2. We consider analogues of the Jacobson–Morozov theorem in this setting. More precisely, we classify those nilpotent elements with a simple 3-dimensional Lie overalgebra in and also those with overalgebras isomorphic to the algebras and . This leads us to calculate the dimension of the Lie automiser for all nilpotent orbits; in even characteristic this quantity is very sensitive to isogeny.
期刊介绍：
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.
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