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

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

2011-01-27

arxiv.org/abs/1101.5237v2

Mathematics

Rings and Algebras

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Scientific paper

{\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}$% {\small \,2}^{t}\,${\small and prescribed \thinspace level and sublevel}$% {\small \,\,2}^{k}${\small ,\thinspace \thinspace}$k,\,t\in \mathbb{N}-\{0\} $ and dimension ${\small \,2}^{t}\,${\small and prescribed \thinspace level}${\small \,\,2}^{k}+1${\small ,\thinspace \thinspace}$k\in \mathbb{N}% -\{0\},t\in \mathbb{N},t\geq 2.$

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