Quasialgebra structure of the octonions

We show that the octonions are a twisting of the group algebra of Z(2) x Z(2) x Z(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 far 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. (C) 1999 Academic Press.