{"title":"具有几乎多项式增长的导数的列半简单代数和 PI 代数品种","authors":"Sebastiano Argenti","doi":"10.1090/proc/16896","DOIUrl":null,"url":null,"abstract":"<p>We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U upper T 2 left-parenthesis upper W Subscript lamda Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">UT_2(W_\\lambda )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E n d left-parenthesis upper W Subscript mu Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">End(W_\\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some integral dominant weight <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda comma mu\"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\lambda ,\\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"mu not-equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mu \\neq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the special case <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L equals German s German l Subscript 2\"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"fraktur\">s</mml:mi> <mml:mi mathvariant=\"fraktur\">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">L=\\mathfrak {sl}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that this is a sufficient condition too.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"214 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth\",\"authors\":\"Sebastiano Argenti\",\"doi\":\"10.1090/proc/16896\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper U upper T 2 left-parenthesis upper W Subscript lamda Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">UT_2(W_\\\\lambda )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper E n d left-parenthesis upper W Subscript mu Baseline right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:mi>d</mml:mi> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>μ</mml:mi> </mml:msub> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">End(W_\\\\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some integral dominant weight <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"lamda comma mu\\\"> <mml:semantics> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\lambda ,\\\\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"mu not-equals 0\\\"> <mml:semantics> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mu \\\\neq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the special case <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper L equals German s German l Subscript 2\\\"> <mml:semantics> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant=\\\"fraktur\\\">s</mml:mi> <mml:mi mathvariant=\\\"fraktur\\\">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">L=\\\\mathfrak {sl}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove that this is a sufficient condition too.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"214 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-05-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/proc/16896\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/proc/16896","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
摘要
我们考虑的是具有某种有限维半简单李代数派生作用的关联代数。我们证明，如果一个微分方程具有几乎多项式的增长，那么它是由 U T 2 ( W λ ) UT_2(W_\lambda)或 E n d ( W μ ) End(W_\mu ) 中的一个代数生成的，对于某个积分主重 λ , μ \lambda ,\mu μ ≠ 0 \mu \neq 0。在 L = s l 2 L=\mathfrak {sl}_2 的特殊情况下，我们证明这也是一个充分条件。
Lie semisimple algebras of derivations and varieties of PI-algebras with almost polynomial growth
We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras UT2(Wλ)UT_2(W_\lambda ) or End(Wμ)End(W_\mu ) for some integral dominant weight λ,μ\lambda ,\mu with μ≠0\mu \neq 0. In the special case L=sl2L=\mathfrak {sl}_2 we prove that this is a sufficient condition too.
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