Pseudo-euclidean Jordan algebras

TL;DR

This paper characterizes pseudo-euclidean Jordan algebras using double extensions, constructs low-dimensional examples, and explores their symplectic and Jordan-Manin subclasses, revealing their structural relationships.

Contribution

It provides a comprehensive description of pseudo-euclidean Jordan algebras via double extensions and introduces the equivalence of symplectic and Jordan-Manin subclasses.

Findings

Constructed all pseudo-euclidean Jordan algebras up to dimension 5.

Built a 12-dimensional Lie algebra from a Jordan algebra using TKK construction.

Showed the equivalence of symplectic and Jordan-Manin classes.

Abstract

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.