Alternative Clifford-like algebras [Articol]

Abstract

The real, complex, and quaternion division algebras may be constructed as Clifford algebras. However, while the octonions and split octonions share many features with Clifford algebras, such as an involutary anti-automorphism, their nonassociativity prevents their realization as Clifford algebras. We now introduce Kingdon algebras: alternative Clifford-like algebras over vector spaces which are equipped with a symmetric bilinear form, such that the octonions and split octonions arise as Kingdon algebras. In the process, they acquire 2-graded (superalgebra) structure. A comparable construction yields 3-graded algebras over spaces equipped with an alternating trilinear form.