Jordan-Algebra

In mathematics, ie, a commutative algebra A is a Jordan algebra if for all x, y in A, the so-called Jordan identity is satisfied.

An alternative definition is (x, y of A, x is invertible ).

That is, A is not associative, usually, but it is a weak form of the associative law.

It is named after the German physicist Pascual Jordan, who wanted to use them for the axiomatization of quantum physics.

Under a noncommutative Jordan algebra is meant an algebra, which still meets the Flexibility Act in addition to the Jordan identity.

Special and exceptional Jordan algebras

For an associative algebra of characteristic not equal to 2, a Jordan algebra can be constructed by defining a new multiplication with unchanged Addition:

Jordan algebras which are isomorphic to so formed are called special Jordan algebras, the other exceptional Jordan algebras.

The exceptional Jordan algebra M (3.8 ) (also termed ) is defined by matrices of the following type

Given. Here, a, b, c are real numbers, and X, Y, Z octonions, the multiplication is as defined above, but it is not a special Jordan algebra, as the multiplication of the octonions is not associative.

Over the complex numbers is M (3,8 ) is the only exceptional Jordan algebra, while there are over the real numbers three isomorphism classes of exceptional Jordan algebras.

Formal real Jordan algebras

A Jordan algebra is called formally real if they can not be represented as a nontrivial sum of squares. Formal real Jordan algebras were classified in 1934 by Jordan, von Neumann and Wigner.