A note about levels and sublevels of algebras obtained by the Cayley-Dickson process

TL;DR

This paper extends the concepts of level and sublevel to algebras formed via the Cayley-Dickson process, enabling the construction of division algebras with specific dimensions, levels, and sublevels.

Contribution

It generalizes the notions of level and sublevel to Cayley-Dickson algebras and constructs division algebras with prescribed properties using Brown's method.

Findings

Constructed division algebras of dimension 2^t with specific levels and sublevels.

Established the existence of division algebras of dimension 2^t+1 with prescribed levels and sublevels.

Extended the theory of levels and sublevels to a broader class of algebras.

Abstract

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