Jordan algebra

In mathematics , a commutative algebra A is called a Jordan algebra if the so-called Jordan identity is fulfilled for all x, y in A. It is named after the German physicist Pascual Jordan . ${\ displaystyle x (x ^ {2} y) = x ^ {2} (xy)}$

An alternative definition is (x, y from A, x invertible). ${\ displaystyle x ^ {- 1} (xy) = x (x ^ {- 1} y)}$

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

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

A non-commutative Jordan algebra is an algebra which, in addition to the Jordan identity, also fulfills the law of flexibility .

Special and exceptional Jordan algebras

A Jordan algebra can be constructed from an associative algebra with a characteristic other than 2 by defining a new multiplication with unchanged addition : ${\ displaystyle A}$ ${\ displaystyle A ^ {+}}$ ${\ displaystyle \ cdot _ {J}}$

{\ displaystyle x \ \ cdot _ {J} \ y = {xy + yx \ over 2}.}

Jordan algebras that are isomorphic to those formed in this way are called special Jordan algebras , the other exceptional Jordan algebras .

The exceptional Jordan algebra M (3,8) (also referred to as ) is represented by matrices of the following type ${\ displaystyle E_ {3}}$

${\ displaystyle {\ begin {pmatrix} a & X & Y \\ {\ bar {X}} & b & Z \\ {\ bar {Y}} & {\ bar {Z}} & c \ end {pmatrix}}}$ given. Here a, b, c are real numbers and X, Y, Z are octonions , the multiplication is as given above, but it is not a special Jordan algebra, since the multiplication of the octonions is not associative.

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

Formally real Jordan algebras

A Jordan algebra is formally called real if it can not be represented as a nontrivial sum of squares. Formally real Jordan algebras were classified by Jordan, von Neumann, and Wigner in 1934. ${\ displaystyle A}$ ${\ displaystyle 0 \ in A}$

literature

Hel Braun , Max Koecher : Jordan-Algebren , Springer Berlin 1998 ISBN 3540035222
Tonny A. Springer : Jordan Algebras and Algebraic Groups , Springer-Verlag Heidelberg 1998
Pascual Jordan, John von Neumann , Eugene Wigner (1934), "On an Algebraic Generalization of the Quantum Mechanical Formalism", Annals of Mathematics (Princeton) 35 (1): 29-64

Web links

Walter Feit: On the work of Efim Zelmanov (with an outline of the history of the theory of Jordan algebras)