Quasialgebra Structure of the Octonions

✍ Scribed by Helena Albuquerque; Shahn Majid

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 228 KB
• Volume: 220
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1998.7850

✦ Synopsis

We show that the octonions are a twisting of the group algebra of

in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. In particular, we show that they are quasialgebras associative up to a 3-cocycle isomorphism. We show that one may make general constructions for quasialgebras exactly along the lines of the associative theory, including quasilinear algebra, representation theory, and an automorphism quasi-Hopf algebra. We study the algebraic properties of quasialgebras of the type which includes the octonions. Further examples include the higher 2 n -onion Cayley algebras and examples associated to Hadamard matrices.