Jordanalgebras

Jordanalgebras, also known as Jordan algebras, are a type of non-associative algebra named after the mathematician Pascual Jordan. They were introduced in the 1930s as a generalization of quadratic forms and have since found applications in various areas of mathematics and physics. Jordan algebras are defined over a field of characteristic not equal to 2, and they are characterized by the identity x^2 * (x * y) = x * (x^2 * y), where x and y are elements of the algebra. This identity is known as the Jordan identity.

One of the most well-known examples of a Jordan algebra is the algebra of symmetric matrices over a field. Another important example is the exceptional Jordan algebra, which is a 27-dimensional algebra that plays a crucial role in the classification of simple Lie algebras. Jordan algebras have also been studied in the context of quantum mechanics, where they arise naturally in the study of spin systems and other quantum mechanical systems.

In recent years, there has been renewed interest in Jordan algebras due to their connections to other areas of mathematics, such as representation theory and algebraic geometry. Jordan algebras have also been studied in the context of non-commutative geometry, where they play a role in the study of spectral triples and other geometric structures. Overall, Jordan algebras are a rich and interesting area of study that continues to be an active area of research in mathematics and physics.