Pseudo-euclidean Jordan algebras

A Jordan algebra J is said to be pseudo-euclidean if J is endowed with an associative non-degenerate symmetric bilinear form B. B is said an associative scalar product on J. First, we provide a description of the pseudo-euclidean Jordan K-algebras in terms of double extensions and generalized double extensions. In particular, we shall use this description to construct all pseudo-euclidean Jordan algebras of dimension less than or equal to 5. And then, from one of these algebras, we shall construct a twelve dimension Lie algebra by the "TKK" construction. Second, a description of symplectic pseudo-euclidean Jordan algebras is provided and finally we describe a particular class of these algebras namely the class of symplectic Jordan-Manin Algebras. In addition to these descriptions, this paper demonstrates that these last two classes are identical and provides several information on the structure of pseudo-euclidean Jordan algebras.