Repeated Cayley–Dickson Processes and Subalgebras of Dimension 8

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY Accounts of Chemical Research Pub Date : 2023-08-09 DOI:10.1007/s00006-023-01289-5

Jacques Helmstetter

引用次数: 0

Abstract

Let K be a field of characteristic other than 2, and let \(\mathcal {A}_n\) be the algebra deduced from \(\mathcal {A}_1=K\) by n successive Cayley–Dickson processes. Thus \(\mathcal {A}_n\) is provided with a natural basis \((f_E)\) indexed by the subsets E of \(\{1,2,\ldots ,n\}\). Two questions have motivated this paper. If a subalgebra of dimension 4 in \(\mathcal {A}_n\) is spanned by 4 elements of this basis, is it a quaternion algebra? The answer is always “yes”. If a subalgebra of dimension 8 in \(\mathcal {A}_n\) is spanned by 8 elements of this basis, is it an octonion algebra? The answer is more often “no” than “yes”. The present article establishes the properties and the formulas that justify these two answers, and describes the fake octonion algebras.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

重复Cayley-Dickson过程和维数为8的子代数

设K是除2以外的特征域，并且设\（\mathcal{A}_n\)是从\（\mathcal）推导出的代数{A}_1=K\）通过n个连续的Cayley-Dickson过程。因此\（\mathcal{A}_n\)提供了由\（\｛1,2，\ldots，n \｝\）的子集E索引的自然基\（（f_E）\）。两个问题激发了本文的写作动机。如果\（\mathcal）中维数为4的子代数{A}_n\)由这个基的4个元素跨越，它是四元数代数吗？答案总是“是”。如果\（\mathcal）中维数为8的子代数{A}_n\)由这个基的8个元素跨越，它是一个八元代数吗？答案往往是“不”而不是“是”。本文建立了证明这两个答案的性质和公式，并描述了伪八元代数。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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.

期刊最新文献

Management of Cholesteatoma: Hearing Rehabilitation. Congenital Cholesteatoma. Evaluation of Cholesteatoma. Management of Cholesteatoma: Extension Beyond Middle Ear/Mastoid. Recidivism and Recurrence.