Classification of the Algebras 𝕆_p,q

Abstract

We study a series of real nonassociative algebras 𝕆 _{p, q} introduced in [5]. These algebras have a natural -grading, where n = p + q, and they are characterized by a cubic form over the field ℤ₂. We establish all the possible isomorphisms between the algebras 𝕆 _{p, q} preserving the structure of -graded algebra. The classification table of 𝕆 _{p, q} is quite similar to that of the real Clifford algebras Cl _{p, q} , the main difference is that the algebras 𝕆 _{n, 0} and 𝕆_{0, n} are exceptional.

Key Words:

• Graded algebras
• Nonassociative algebras
• Octonions
• Twisted group algebras

2010 Mathematics Subject Classification:

• 17A70
• 16W55

ACKNOWLEDGMENTS

The authors are grateful to Valentin Ovsienko for enlightening discussions.

Notes

Communicated by A. Elduque.

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