Structurable algebras of skew-dimension one and hermitian cubic norm structures

Abstract

We study structurable algebras of skew-dimension one. We present two different equivalent constructions for such algebras: one in terms of non-linear isotopies of cubic norm structures, and one in terms of hermitian cubic norm structures. After this work was essentially finished, we became aware of the fact that both descriptions already occur in (somewhat hidden places in) the literature. Nevertheless, we prove some facts that had not been noticed before: (1) We show that every form of a matrix structurable algebra can be described by our constructions; (2) We give explicit formulas for the norm $\nu$; (3) We make a precise connection with the Cayley-Dickson process for structurable algebras.