Jordanalgebran

Jordanalgebra is a mathematical structure that generalizes the concept of Jordan algebras. Unlike standard Jordan algebras which are based on a commutative product, Jordanalgebras incorporate an additional non-commutative product, often referred to as the "Jordan product." This structure allows for the study of both commutative and non-commutative aspects of algebraic systems in a unified framework.

The formal definition of a Jordanalgebra involves a vector space over a field, equipped with two bilinear operations. One is the Jordan product, which must satisfy certain identities analogous to those of a Jordan algebra (specifically, the flexible law and a modified associator identity). The second operation, the non-commutative product, does not have these specific constraints but is crucial for the overall structure.

Jordanalgebras find applications in various areas of mathematics and physics. They are used in the study of non-associative algebras, quantum mechanics, and even in certain areas of operator theory. The interplay between the commutative Jordan product and the non-commutative product allows for the analysis of systems that exhibit both commutative and non-commutative behaviors. Research in this field focuses on classifying different types of Jordanalgebras, understanding their properties, and exploring their connections to other algebraic structures and physical phenomena.