# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Drinfeld, V.G., Quasihopf algebras, Leningrad math. J., 1, 1419-1457, (1990) · Zbl 0718.16033 [2] Mac Lane, S., Categories for the working Mathematician, Gtm, 5, (1974), Springer-Verlag Berlin/New York · Zbl 0906.18001 [3] Gurevich, D.I.; Majid, S., Braided groups of Hopf algebras obtained by twisting, Pac. J. math., 162, 27-44, (1994) · Zbl 0787.17010 [4] Majid, S., q-Euclidean space and quantum Wick rotation by twisting, J. math. phys., 35, 5025-5033, (1994) · Zbl 0833.17015 [5] Majid, S., Foundations of quantum group theory, (1995), Cambridge Univ. Press Cambridge · Zbl 0857.17009 [6] Majid, S., Tannaka-Krein theorem for quasihopf algebras and other results, Contemp. math., 134, 219-232, (1992) · Zbl 0788.17012 [7] Sweedler, M.E., Hopf algebras, (1969), Benjamin Elmsford [8] Joyal, A.; Street, R., Braided monoidal categories, Mathematics reports, 86008, (1986), Macquarie University [9] Majid, S., Quantum double for quasi-Hopf algebras, Lett. math. phys., 45, 1-9, (1998) · Zbl 0940.16018 [10] Zhevlakov, K.A.; Slin’ko, A.M.; Shestakov, I.P.; Shirshov, A.I., Rings that are nearly associative, (1982), Academic Press San Diego · Zbl 0487.17001 [11] Dixon, G., Division algebras: octonions, quaternions, complex numbers and the algebraic design of physics, (1994), Kluwer Academic Dordrecht/Norwell · Zbl 0807.15024 [12] Majid, S., Quantum and braided diffeomorphism groups, J. geom. phys., 28, 94-128, (1998) · Zbl 1050.58005 [13] Majid, S., Algebras and Hopf algebras in braided categories, Lecture notes in pure and applied mathematics, 158, (1994), Dekker New York, p. 55-105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.