A note about algebras obtained by the Cayley-Dickson process

TL;DR

This paper extends the concepts of level and sublevels in composition algebras to those created via the Cayley-Dickson process, providing new insights into their structure and division algebra constructions.

Contribution

It generalizes level and sublevel concepts to Cayley-Dickson algebras and constructs division algebras with specified dimensions and levels.

Findings

Generalized level and sublevel definitions for Cayley-Dickson algebras.

Constructed division algebras of dimension 2^t with prescribed levels.

Provided a framework for creating division algebras with specific properties.

Abstract

In this paper, we generalize the concepts of level and sublevels of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every $t\in \Bbb{N},$ a division algebra $A_{t}$ of dimension $2^{t}$ over the power-series field $K\{X_{1},X_{2},...,X_{t}\}.$ This gives us the possibility to construct a division algebra of dimension 2$^{t}$ and prescribed level 2$^{k}$ $ k, t\in \Bbb{N}^{*}.$