Abstract
Let \({(X,\mathcal{O}_X)}\) be a locally ringed space. We investigate the structure of symmetric composition algebras over X obtained from cubic alternative algebras \({\mathcal{A}}\) over X generalizing a method first presented by J. R. Faulkner. We find examples of Okubo algebras over elliptic curves which do not have any isotopes which are octonion algebras and of an octonion algebra which is a Cayley-Dickson doubling of a quaternion algebra but does not contain any quadratic étale algebras.
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Pumplün, S. Symmetric composition algebras over algebraic varieties. manuscripta math. 132, 307–333 (2010). https://doi.org/10.1007/s00229-010-0348-2
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DOI: https://doi.org/10.1007/s00229-010-0348-2