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

We investigate the construction and properties of Clifford algebras by a similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z n 2 by a cocycle. Our approach is more general than the usual one based on generators and relations. We obtain in particular the periodicity properties and a new construction of spinors in terms of left and right multiplication in the Clifford algebra.

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The structure of G-graded quasialgebras, introduced in [1] , is studied for the simplest nontrivial group: G ¼ Z 2 . The resulting notion is a class of Z 2 -graded algebras A ¼ A 0 È A 1 which are either associative or satisfy the ''antiassociative condition'' ðxyÞz ¼ ðÀ1Þx y z xðyzÞ for homogeneous elements. A full description is given in case A 0 is semisimple and A 1 is a unital A 0 -bimodule. 2161