The factorization problem for Jordan algebras. Applications

TL;DR

This paper explores the factorization problem in Jordan algebras, introduces matched pairs and bicrossed products, and proposes a new deformation method to classify complements, advancing the structural understanding of Jordan algebras.

Contribution

It introduces the concepts of matched pairs, bicrossed products, and a new deformation approach to solve the factorization and classifying complements problems in Jordan algebras.

Findings

Any Jordan algebra factorizing through two given algebras is isomorphic to a bicrossed product.

Introduces matched pairs and bicrossed products for Jordan algebras.

Proposes a new deformation method for classifying complements.

Abstract

We investigate the factorization problem as well as the classifying complements problem in the setting of Jordan algebras. Matched pairs of Jordan algebras and the corresponding bicrossed products are introduced. It is shown that any Jordan algebra which factorizes through two given Jordan algebras is isomorphic to a bicrossed product associated to a certain matched pair between the same two Jordan algebras. Furthermore, a new type of deformation of a Jordan algebra is proposed as the main step towards solving the classifying complements problem.