{"title":"简单排列的 Kac-Moody 算法的最大紧凑子代数的高自旋表征","authors":"Robin Lautenbacher, Ralf Köhl","doi":"arxiv-2409.07247","DOIUrl":null,"url":null,"abstract":"Given the maximal compact subalgebra $\\mathfrak{k}(A)$ of a split-real\nKac-Moody algebra $\\mathfrak{g}(A)$ of type $A$, we study certain\nfinite-dimensional representations of $\\mathfrak{k}(A)$, that do not lift to\nthe maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$\nassociated to $\\mathfrak{g}(A)$ but only to its spin cover $Spin(A)$.\nCurrently, four elementary of these so-called spin representations are known.\nWe study their (ir-)reducibility, semi-simplicity, and lift to the group level.\nThe interaction of these representations with the spin-extended Weyl-group is\nused to derive a partial parametrization result of the representation matrices\nby the real roots of $\\mathfrak{g}(A)$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras\",\"authors\":\"Robin Lautenbacher, Ralf Köhl\",\"doi\":\"arxiv-2409.07247\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given the maximal compact subalgebra $\\\\mathfrak{k}(A)$ of a split-real\\nKac-Moody algebra $\\\\mathfrak{g}(A)$ of type $A$, we study certain\\nfinite-dimensional representations of $\\\\mathfrak{k}(A)$, that do not lift to\\nthe maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$\\nassociated to $\\\\mathfrak{g}(A)$ but only to its spin cover $Spin(A)$.\\nCurrently, four elementary of these so-called spin representations are known.\\nWe study their (ir-)reducibility, semi-simplicity, and lift to the group level.\\nThe interaction of these representations with the spin-extended Weyl-group is\\nused to derive a partial parametrization result of the representation matrices\\nby the real roots of $\\\\mathfrak{g}(A)$.\",\"PeriodicalId\":501038,\"journal\":{\"name\":\"arXiv - MATH - Representation Theory\",\"volume\":\"26 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Representation Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.07247\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Representation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07247","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras
Given the maximal compact subalgebra $\mathfrak{k}(A)$ of a split-real
Kac-Moody algebra $\mathfrak{g}(A)$ of type $A$, we study certain
finite-dimensional representations of $\mathfrak{k}(A)$, that do not lift to
the maximal compact subgroup $K(A)$ of the minimal Kac-Moody group $G(A)$
associated to $\mathfrak{g}(A)$ but only to its spin cover $Spin(A)$.
Currently, four elementary of these so-called spin representations are known.
We study their (ir-)reducibility, semi-simplicity, and lift to the group level.
The interaction of these representations with the spin-extended Weyl-group is
used to derive a partial parametrization result of the representation matrices
by the real roots of $\mathfrak{g}(A)$.