{"title":"关于复数李群 $ (F_{4,R})^C, (E_{6,R})^C, (E_{7,R})^C ,(E_{8,R})^C$ 和那些紧凑实数形式 $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$ 的实数化","authors":"Toshikazu Miyashita","doi":"arxiv-2409.07760","DOIUrl":null,"url":null,"abstract":"In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C,\n{E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually\nuse the Cayley algebra $ \\mathfrak{C} $. In the present article, we consider\nreplacing the Cayley algebra $ \\mathfrak{C} $ with the field of real numbers\n$\\mathbb R$ in the definition of the groups above, and these groups are denoted\nas in title above. Our aim is to determine the structure of these groups. We\ncall realization to determine the structure of the groups.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"283 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On realizations of the complex Lie groups $ (F_{4,R})^C, (E_{6,R})^C, (E_{7,R})^C ,(E_{8,R})^C$ and those compact real forms $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$\",\"authors\":\"Toshikazu Miyashita\",\"doi\":\"arxiv-2409.07760\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C,\\n{E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually\\nuse the Cayley algebra $ \\\\mathfrak{C} $. In the present article, we consider\\nreplacing the Cayley algebra $ \\\\mathfrak{C} $ with the field of real numbers\\n$\\\\mathbb R$ in the definition of the groups above, and these groups are denoted\\nas in title above. Our aim is to determine the structure of these groups. We\\ncall realization to determine the structure of the groups.\",\"PeriodicalId\":501312,\"journal\":{\"name\":\"arXiv - MATH - Mathematical Physics\",\"volume\":\"283 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.07760\",\"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 - Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.07760","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On realizations of the complex Lie groups $ (F_{4,R})^C, (E_{6,R})^C, (E_{7,R})^C ,(E_{8,R})^C$ and those compact real forms $F_{4,R},E_{6,R},E_{7,R},E_{8,R}$
In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C,
{E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually
use the Cayley algebra $ \mathfrak{C} $. In the present article, we consider
replacing the Cayley algebra $ \mathfrak{C} $ with the field of real numbers
$\mathbb R$ in the definition of the groups above, and these groups are denoted
as in title above. Our aim is to determine the structure of these groups. We
call realization to determine the structure of the groups.