{"title":"非交换Laurent多项式环上线性群、相关$K_1$-群和$K_2$-群的泛中心扩张","authors":"Ryusuke Sugawara","doi":"10.21099/tkbjm/20214501013","DOIUrl":null,"url":null,"abstract":"We prove that linear groups over rings of non-commutative Laurent polynomials $D_{\\tau}$ have Tits systems with the corresponding affine Weyl groups and have universal central extensions if $|Z(D)|\\geq 5$ and $|Z(D)|\\neq 9$. We also determine structures of $K_1$-groups and identify generators of $K_2$-groups.","PeriodicalId":44321,"journal":{"name":"Tsukuba Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Universal central extensions of linear groups over rings of non-commutative Laurent polynomials, associated $K_1$-groups and $K_2$-groups\",\"authors\":\"Ryusuke Sugawara\",\"doi\":\"10.21099/tkbjm/20214501013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove that linear groups over rings of non-commutative Laurent polynomials $D_{\\\\tau}$ have Tits systems with the corresponding affine Weyl groups and have universal central extensions if $|Z(D)|\\\\geq 5$ and $|Z(D)|\\\\neq 9$. We also determine structures of $K_1$-groups and identify generators of $K_2$-groups.\",\"PeriodicalId\":44321,\"journal\":{\"name\":\"Tsukuba Journal of Mathematics\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2020-05-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Tsukuba Journal of Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.21099/tkbjm/20214501013\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tsukuba Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.21099/tkbjm/20214501013","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
Universal central extensions of linear groups over rings of non-commutative Laurent polynomials, associated $K_1$-groups and $K_2$-groups
We prove that linear groups over rings of non-commutative Laurent polynomials $D_{\tau}$ have Tits systems with the corresponding affine Weyl groups and have universal central extensions if $|Z(D)|\geq 5$ and $|Z(D)|\neq 9$. We also determine structures of $K_1$-groups and identify generators of $K_2$-groups.
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