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 nonlinear 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 ν;

3. We make a precise connection with the Cayley–Dickson process for structurable algebras.

Keywords:

• Cayley–Dickson process
• cubic norm structures
• isotopies
• structurable algebras

2010 Mathematical Subject Classification:

• 17C50
• 17D99
• 17A60
• 16W10
• 17B60

Acknowledgments

I am grateful to Bruce Allison for some very inspiring and enlightening conversations based on an earlier version of this paper and, not in the least, for encouraging and stimulating me to make this work accessible in a published form. I also thank Bernhard Mühlherr and Richard Weiss for making their work [14] available to me prior to publication; their work has been the main source of inspiration for the current paper, even though this might not be directly visible in the results.

Notes

1 Recall that a map $T:J\times J\to K$ is called hermitian if $T(sa,tb)=s{t}^{\sigma }T(a,b)$ and $T{(a,b)}^{\sigma }=T(b,a)$ for all $s,t\in K$ and all $a,b\in J$.