论文标题
Cayley-Dickson代数的扭曲组代数结构
The twisted group algebra structure of the Cayley-Dickson algebra
论文作者
论文摘要
由于缺乏明确的乘法表，Cayley-Dickson代数长期以来一直是一个挑战。尽管通过归纳构建可以进行构造，但其明确的结构一直难以捉摸。在本文中，我们通过揭示Cayley-Dickson代数为具有显式扭曲函数$σ（a，b）$的扭曲组代数，提出了解决这个长期问题的解决方案。我们表明，此函数满足等式$$ e_ae_b =（-1）^{σ（a，b）} e_ {a \ oplus b} $$，并为Cayley-Dickson代数和Cayley-Dickson代数之间的关系提供了一个公式，从而使Cayley Allbra septly Allbrawarbra的表达式表达了Cayley Allbra的表达。我们的方法不仅解决了Cayley-Dickson代数和分裂Cayley-Dickson代数缺乏明确的结构，而且还阐明了这种基本数学对象的基础结构。
The Cayley-Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we propose a solution to this long-standing problem by revealing the Cayley-Dickson algebra as a twisted group algebra with an explicit twist function $σ(A,B)$. We show that this function satisfies the equation $$e_Ae_B=(-1)^{σ(A,B)}e_{A\oplus B}$$ and provide a formula for the relationship between the Cayley-Dickson algebra and split Cayley-Dickson algebra, thereby giving an explicit expression for the twist function of the split Cayley-Dickson algebra. Our approach not only resolves the lack of explicit structure for the Cayley-Dickson algebra and split Cayley-Dickson algebra but also sheds light on the algebraic structure underlying this fundamental mathematical object.
