The twisted group algebra structure of the Cayley-Dickson algebra

2022/05/16 by Ren, Guangbin, Zhao, Xin
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2205.07986

Abstract

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⊕ 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.

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