Pseudo-Euclidean Jordan Algebras

Abstract

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.

Key Words:

• Double extensions
• Jordan Algebras
• Jordan bialgebras
• Jordan–Manin algebras
• Jordan Yang Baxter equation
• r-matrices
• Representations of Jordan algebras
• Symplectic quadratic Lie algebras
• T*–extension
• Tits–Kantor–Koecher construction

2010 Mathematics Subject Classification:

• 17C10
• 17C20
• 17C50
• 17C55
• 17B06
• 17B62

ACKNOWLEDGMENTS

We thank W. Bertram, M. Bordemann, A. Elduque, and A. Pasquale for very interesting discussions. Moreover, we are very grateful to A. Elduque and A. Pasquale for its remarks, which improved the readability of this paper.

Notes

Communicated by A. Elduque.