Quasialgebra Structure of the Octonions

Abstract

We show that the octonions are a twisting of the group algebra of 2 × 2 × 2 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 2n-onion Cayley algebras and examples associated to Hadamard matrices.