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 al gebra 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 theo ry, and an automorphism quasi-Hopf algebra. We study the algebraic properti es of quasialgebras of the type which includes the octonions. Further examp les include the higher 2(n)-onion Cayley algebras and examples associated t o Hadamard matrices. (C) 1999 Academic Press.