{"title":"作为扭曲环群的紧凑异常李代数𝔤^{𝔠}₂","authors":"Cristina Draper","doi":"10.1090/proc/16821","DOIUrl":null,"url":null,"abstract":"<p>A new highly symmetrical model of the compact Lie algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German g 2 Superscript c\"> <mml:semantics> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\"fraktur\">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {g}^c_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is provided as a twisted ring group for the group <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z 2 cubed\"> <mml:semantics> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\"application/x-tex\">\\mathbb {Z}_2^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R circled-plus double-struck upper R\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {R}\\oplus \\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German s German u left-parenthesis 2 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">s</mml:mi> <mml:mi mathvariant=\"fraktur\">u</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {su}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of the Gell-Mann matrices in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German s German u left-parenthesis 3 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"fraktur\">s</mml:mi> <mml:mi mathvariant=\"fraktur\">u</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {su}(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a bonus, the split Lie algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German g 2\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\"fraktur\">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">\\mathfrak {g}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also seen as a twisted ring group.</p>","PeriodicalId":20696,"journal":{"name":"Proceedings of the American Mathematical Society","volume":"31 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The compact exceptional Lie algebra 𝔤^{𝔠}₂ as a twisted ring group\",\"authors\":\"Cristina Draper\",\"doi\":\"10.1090/proc/16821\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>A new highly symmetrical model of the compact Lie algebra <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"German g 2 Superscript c\\\"> <mml:semantics> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\\\"fraktur\\\">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathfrak {g}^c_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is provided as a twisted ring group for the group <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper Z 2 cubed\\\"> <mml:semantics> <mml:msubsup> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {Z}_2^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the ring <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper R circled-plus double-struck upper R\\\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {R}\\\\oplus \\\\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"German s German u left-parenthesis 2 right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"fraktur\\\">s</mml:mi> <mml:mi mathvariant=\\\"fraktur\\\">u</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathfrak {su}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of the Gell-Mann matrices in <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"German s German u left-parenthesis 3 right-parenthesis\\\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"fraktur\\\">s</mml:mi> <mml:mi mathvariant=\\\"fraktur\\\">u</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathfrak {su}(3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As a bonus, the split Lie algebra <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"German g 2\\\"> <mml:semantics> <mml:msub> <mml:mrow> <mml:mi mathvariant=\\\"fraktur\\\">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathfrak {g}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is also seen as a twisted ring group.</p>\",\"PeriodicalId\":20696,\"journal\":{\"name\":\"Proceedings of the American Mathematical Society\",\"volume\":\"31 1\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-03-09\",\"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/16821\",\"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/16821","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
本文提供了紧凑李代数 g 2 c \mathfrak {g}^c_2 的一个新的高度对称模型,作为 Z 2 3 \mathbb {Z}_2^3 群和环 R ⊕ R \mathbb {R}\mathbb {R}oplus \mathbb {R} 的一个扭曲环群。这个模型是自足的,可以在没有关于根、八元数的推导或交叉积的知识的情况下使用。特别是,它提供了一个具有整数结构常量的正交基础,完全由半简单元素组成,是 s u ( 2 ) \mathfrak {su}(2) 中的保利矩阵和 s u ( 3 ) \mathfrak {su}(3) 中的盖尔-曼矩阵的广义化。作为奖励,分裂的李代数 g 2 (mathfrak {g}_2 )也被视为一个扭曲的环群。
The compact exceptional Lie algebra 𝔤^{𝔠}₂ as a twisted ring group
A new highly symmetrical model of the compact Lie algebra g2c\mathfrak {g}^c_2 is provided as a twisted ring group for the group Z23\mathbb {Z}_2^3 and the ring R⊕R\mathbb {R}\oplus \mathbb {R}. The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in su(2)\mathfrak {su}(2) and of the Gell-Mann matrices in su(3)\mathfrak {su}(3). As a bonus, the split Lie algebra g2\mathfrak {g}_2 is also seen as a twisted ring group.
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