{"title":"坦波利-利布塔和魏尔代数","authors":"Matthew Harper, Peter Samuelson","doi":"10.1112/jlms.70174","DOIUrl":null,"url":null,"abstract":"<p>We define a monoidal category <span></span><math>\n <semantics>\n <mi>W</mi>\n <annotation>${\\mathbf {W}}$</annotation>\n </semantics></math> and a closely related 2-category <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mi>Weyl</mi>\n </mrow>\n <annotation>${\\mathbf {2Weyl}}$</annotation>\n </semantics></math> using diagrammatic methods. We show that <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mi>Weyl</mi>\n </mrow>\n <annotation>${\\mathbf {2Weyl}}$</annotation>\n </semantics></math> acts on the category <span></span><math>\n <semantics>\n <mrow>\n <mi>TL</mi>\n <mo>:</mo>\n <mo>=</mo>\n <msub>\n <mo>⨁</mo>\n <mi>n</mi>\n </msub>\n <msub>\n <mo>TL</mo>\n <mi>n</mi>\n </msub>\n <mo>−</mo>\n <mi>mod</mi>\n </mrow>\n <annotation>$\\mathbf {TL}:=\\bigoplus _n \\operatorname{TL}_n\\mathrm{-mod}$</annotation>\n </semantics></math> of modules over Temperley–Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of <span></span><math>\n <semantics>\n <mi>W</mi>\n <annotation>${\\mathbf {W}}$</annotation>\n </semantics></math> and a third category we define <span></span><math>\n <semantics>\n <msup>\n <mi>W</mi>\n <mi>∞</mi>\n </msup>\n <annotation>${\\mathbf {W}}^\\infty$</annotation>\n </semantics></math> are closely related to the Weyl algebra. We formulate a sense in which <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>W</mi>\n <mi>∞</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$K_0({\\mathbf {W}}^\\infty)$</annotation>\n </semantics></math> acts asymptotically on <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>K</mi>\n <mn>0</mn>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>TL</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$K_0(\\mathbf {TL})$</annotation>\n </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"111 5","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70174","citationCount":"0","resultStr":"{\"title\":\"The Temperley–Lieb tower and the Weyl algebra\",\"authors\":\"Matthew Harper, Peter Samuelson\",\"doi\":\"10.1112/jlms.70174\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We define a monoidal category <span></span><math>\\n <semantics>\\n <mi>W</mi>\\n <annotation>${\\\\mathbf {W}}$</annotation>\\n </semantics></math> and a closely related 2-category <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>2</mn>\\n <mi>Weyl</mi>\\n </mrow>\\n <annotation>${\\\\mathbf {2Weyl}}$</annotation>\\n </semantics></math> using diagrammatic methods. We show that <span></span><math>\\n <semantics>\\n <mrow>\\n <mn>2</mn>\\n <mi>Weyl</mi>\\n </mrow>\\n <annotation>${\\\\mathbf {2Weyl}}$</annotation>\\n </semantics></math> acts on the category <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>TL</mi>\\n <mo>:</mo>\\n <mo>=</mo>\\n <msub>\\n <mo>⨁</mo>\\n <mi>n</mi>\\n </msub>\\n <msub>\\n <mo>TL</mo>\\n <mi>n</mi>\\n </msub>\\n <mo>−</mo>\\n <mi>mod</mi>\\n </mrow>\\n <annotation>$\\\\mathbf {TL}:=\\\\bigoplus _n \\\\operatorname{TL}_n\\\\mathrm{-mod}$</annotation>\\n </semantics></math> of modules over Temperley–Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of <span></span><math>\\n <semantics>\\n <mi>W</mi>\\n <annotation>${\\\\mathbf {W}}$</annotation>\\n </semantics></math> and a third category we define <span></span><math>\\n <semantics>\\n <msup>\\n <mi>W</mi>\\n <mi>∞</mi>\\n </msup>\\n <annotation>${\\\\mathbf {W}}^\\\\infty$</annotation>\\n </semantics></math> are closely related to the Weyl algebra. We formulate a sense in which <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n <mn>0</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <msup>\\n <mi>W</mi>\\n <mi>∞</mi>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$K_0({\\\\mathbf {W}}^\\\\infty)$</annotation>\\n </semantics></math> acts asymptotically on <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>K</mi>\\n <mn>0</mn>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>TL</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$K_0(\\\\mathbf {TL})$</annotation>\\n </semantics></math>.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":\"111 5\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-05-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70174\",\"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.70174\",\"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.70174","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
我们用图解的方法定义了一个单类W ${\mathbf {W}}$和一个密切相关的2类2 Weyl ${\mathbf {2Weyl}}$。我们证明2 Weyl ${\mathbf {2Weyl}}$作用于类别TL：templeley - lieb代数上的模的= n TL n - mod $\mathbf {TL}:=\bigoplus _n \operatorname{TL}_n\mathrm{-mod}$，其生成的1-态由归纳和限制作用。W ${\mathbf {W}}$的Grothendieck群和我们定义的第三类W∞${\mathbf {W}}^\infty$与Weyl代数密切相关。我们给出了K 0 (W∞)$K_0({\mathbf {W}}^\infty)$作用于渐近的意义k0 (TL) $K_0(\mathbf {TL})$。
We define a monoidal category and a closely related 2-category using diagrammatic methods. We show that acts on the category of modules over Temperley–Lieb algebras, with its generating 1-morphisms acting by induction and restriction. The Grothendieck groups of and a third category we define are closely related to the Weyl algebra. We formulate a sense in which acts asymptotically on .
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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