Pseudo-euclidean Jordan algebras

Amir Baklouti (1) , Said Benayadi (1)

1 LMAM - Laboratoire de Mathématiques et Applications de Metz

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.

Mots clés: double extensions; Jordan Algebras; Tits Kantor Koecher construction; Pseudo-euclidean symplectic Jordan algebras; symplectic quadratic Lie algebras; representations of Jordan algebras; Jordan bi-algebras; Jordan-Manin algebras; Jordan Yang Baxter equation; r-matrices; T*-extension; double extensions.

Domaines: Anneaux et algèbres [math.RA]

Amir Baklouti, Said Benayadi. Pseudo-euclidean Jordan algebras. 2008. ⟨hal-00340666⟩