Jordan Algebra

In abstract algebra, a Jordan algebra is an (not necessarily associative) algebra over a field whose multiplication satisfies the following axioms:

  1. (commutative law)
  2. (Jordan identity).

The product of two elements x and y in a Jordan algebra is also denoted xy, particularly to avoid confusion with the product of a related associative algebra. The axioms imply that a Jordan algebra is power-associative and satisfies the following generalization of the Jordan identity: for all positive integers m and n.

Jordan algebras were first introduced by Pascual Jordan (1933) to formalize the notion of an algebra of observables in quantum mechanics. They were originally called "r-number systems", but were renamed "Jordan algebras" by Albert (1946), who began the systematic study of general Jordan algebras.

Read more about Jordan AlgebraSpecial Jordan Algebras, Examples, Derivations and Structure Algebra, Formally Real Jordan Algebras, Peirce Decomposition

Other articles related to "jordan algebra":

Jordan Algebra - Peirce Decomposition
... If e is an idempotent in a Jordan algebra A (e2=e) and R is the operation of multiplication by e, then R(2R−1)(R−1) = 0 so the only eigenvalues of R are 0, 1/2, 1 ... If the Jordan algebra A is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces A = A0(e) ⊕ A1/2 ...

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