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 𝕆0, n are exceptional.