A catalog of Cayley-Dickson-like products

2011/07/07 by John W. Bales, Bales, John W.
Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA

paper · pdf · doi:10.48550/arxiv.1107.1301

openalex publication_date 2011/07/07 · arxiv created 2015/04/14 · arxiv updated 2015/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A catalog of all 32 Cayley-Dickson-like doubling products on ordered pairs (a,b),(c,d) for which (1,0) is the left and right identity and for which xx* and x*x equal the square of the norm of x given the conjugate (a,b)*=(a*,-b). Only eight of these are true Cayley-Dickson doubling products, since 24 of them do not satisfy the quaternion properties. Each of the eight Cayley-Dickson products has a distinctive representation in the Fano Plane.

Citations

• Properly twisted groups and their algebras
• On the Algebras Formed by the Cayley-Dickson Process

Related

• An alternate Cayley-Dickson product
• The Cayley-Dickson process for dialgebras
• About some properties of algebras obtained by the Cayley-Dickson process
• Automorphism groups of Cayley-Dickson loops
• Structure of the Cayley-Dickson algebras