Levels and sublevels of division algebras obtained by the Cayley-Dickson process

Abstract

{\small \ We generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to division algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \mathbb{N},${\small \ a nonassociative 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}{\small and prescribed \thinspace level and sublevel}{\small ,\thinspace \thinspace}$k,t\in \mathbb{N}-\{0\}$ and dimension $2^t${\small and prescribed \thinspace level}$2^k+1${\small ,\thinspace \thinspace}$k\in \mathbb{N}$

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• Values for the levels and sublevels of algebras obtained by the Cayley-Dickson process