Coordinatization of Jordan Algebras over Locally Ringed Spaces

Abstract

The classical theory of reduced Jordan algebras is rephrased in the setting of locally ringed spaces. Coordinatization of such algebras leads to the concept of composition triples, i.e. a generalization of composition algebras, over locally ringed spaces. Over a fixed locally ringed space composition triples define a category as well as reduced Jordan algebras do. A natural equivalence between these categories is shown. A generalization of the classical Cayley-Dickson-Doubling Process allows the classification of composition triples over the projective line. By taking global sections composition triples of rank 8 induce Jordan algebras over the base field with pretty big radicals.