Pseudo-Euclidean Jordan Algebras

A pseudo-euclidean Jordan algebra is a Jordan algebra 𝔍 with an associative non-degenerate symmetric bilinear form B. We study the structure of the pseudo-euclidean Jordan algebras over a field 𝕂 of characteristic not two, and we obtain an inductive description of these algebras in terms of double extensions and generalized double extensions. Next, we study the symplectic pseudo-euclidean Jordan 𝕂-algebras, and we give some informations on a particular class of these algebras, namely the class of symplectic Jordan–Manin Algebras.