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Quasialgebra structure of the octonions. (English) Zbl 0999.17006
Summary: We show that the octonions are a twisting of the group algebra of \(\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2\) 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 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.

MSC:
17A35 Nonassociative division algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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