{"title":"On Octonionic Submodules Generated by One Element","authors":"Qinghai Huo, Guangbin Ren","doi":"10.1007/s00006-024-01355-6","DOIUrl":null,"url":null,"abstract":"<div><p>The aim of this article is to characterize the octonionic submodules generated by one element, which is very complicated compared with other normed division algebras. To this end, we introduce a novel identity that elucidates the relationship between the commutator and associator within an octonionic bimodule. Remarkably, the commutator can be expressed in terms of the linear combination of associators. This phenomenon starkly contrasts with the quaternionic case, which leads to a unique right octonionic scalar multiplication compatible with the original left octonionic module structure in the sense of forming an octonionic bimodule. With the help of this identity, we get a new expression of the real part and imaginary part of an element in an octonionic bimodule. Ultimately, we obtain that the submodule generated by one element <i>x</i> is <span>\\({\\mathbb {O}}^5x\\)</span> instead of <span>\\({\\mathbb {O}}x\\)</span>.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00006-024-01355-6","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
The aim of this article is to characterize the octonionic submodules generated by one element, which is very complicated compared with other normed division algebras. To this end, we introduce a novel identity that elucidates the relationship between the commutator and associator within an octonionic bimodule. Remarkably, the commutator can be expressed in terms of the linear combination of associators. This phenomenon starkly contrasts with the quaternionic case, which leads to a unique right octonionic scalar multiplication compatible with the original left octonionic module structure in the sense of forming an octonionic bimodule. With the help of this identity, we get a new expression of the real part and imaginary part of an element in an octonionic bimodule. Ultimately, we obtain that the submodule generated by one element x is \({\mathbb {O}}^5x\) instead of \({\mathbb {O}}x\).
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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