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关于局部复代数与低维Cayley-Dickson代数

On locally complex algebras and low-dimensional Cayley-Dickson algebras

Matej Bresar, Peter Semrl, Spela Spenko

摘要:

本文从Frobenius和(分别)的经典定理的简短证明开始。Zorn on associative and(resp.)交错实除代数这些定理表征了前三个(分别为)。4)Cayley-Dickson代数。然后引入并研究了由任意非纯量元生成的子代数同构于C的实单位元非结合代数类。我们称之为局部复代数。特别是,我们描述所有这样的代数的维数最多为4。我们的主要动机,但是,引进局部复代数是,这个概念使我们有可能扩大弗罗贝纽斯'和佐恩定理的方式,它也涉及第五凯莱-迪克森代数,sedenions。

英文摘要:

The paper begins with short proofs of classical theorems by Frobenius and (resp.) Zorn on associative and (resp.) alternative real division algebras. These theorems characterize the first three (resp. four) Cayley-Dickson algebras. Then we introduce and study the class of real unital nonassociative algebras in which the subalgebra generated by any nonscalar element is isomorphic to C. We call them locally complex algebras. In particular, we describe all such algebras that have dimension at most 4. Our main motivation, however, for introducing locally complex algebras is that this concept makes it possible for us to extend Frobenius' and Zorn's theorems in a way that it also involves the fifth Cayley-Dickson algebra, the sedenions.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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