Jordan algebra

Let R be a commutative ring with 10. An R-algebraMathworldPlanetmathPlanetmathPlanetmath A with multiplication not assumed to be associative is called a (commutativePlanetmathPlanetmathPlanetmath) Jordan algebraMathworldPlanetmathPlanetmath if

  1. 1.

    A is commutative: ab=ba, and

  2. 2.

    A satisfies the Jordan identity: (a2b)a=a2(ba),

for any a,bA.

The above can be restated as

  1. 1.

    [A,A]=0, where [,] is the commutator bracket, and

  2. 2.

    for any aA, [a2,A,a]=0, where [,,] is the associatorMathworldPlanetmath bracket.

If A is a Jordan algebra, a subset BA is called a Jordan subalgebra if BBB. Let A and B be two Jordan algebras. A Jordan algebra homomorphism, or simply Jordan homomorphism, from A to B is an algebra homomorphism that respects the above two laws. A Jordan algebra isomorphism is just a bijective Jordan algebra homomorphism.


  • If A is a Jordan algebra such that char(A)2, then A is power-associative (

  • If in addition 2=1+1char(A), then by replacing a with a+1 in the Jordan identity and simplifying, A is flexible (

  • Given any associative algebra A, we can define a Jordan algebra A+. To see this, let A be an associative algebra with associative multiplication and suppose 2=1+1 is invertiblePlanetmathPlanetmathPlanetmath in R. Define a new multiplication given by

    ab=12(ab+ba). (1)

    It is readily checked that this new multiplication satisifies both the commutative law and the Jordan identity. Thus A with the new multiplication is a Jordan algebra and we denote it by A+. However, unlike Lie algebrasMathworldPlanetmath, not every Jordan algebra is embeddable in an associative algebra. Any Jordan algebra that is isomorphicPlanetmathPlanetmathPlanetmath to a Jordan subalgebra of A+ for some associative algebra A is called a special Jordan algebra. Otherwise, it is called an exceptional Jordan algebra. As a side note, the right hand side of Equation (1) is called the Jordan product.

  • An example of an exceptional Jordan algebra is H3(𝕆), the algebra of 3×3 Hermitian matricesMathworldPlanetmath over the octonions.

Title Jordan algebra
Canonical name JordanAlgebra
Date of creation 2013-03-22 14:52:15
Last modified on 2013-03-22 14:52:15
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 17C05
Synonym Jordan homomorphism
Synonym Jordan isomorphism
Defines Jordan identity
Defines special Jordan algebra
Defines exceptional Jordan algebra
Defines Jordan algebra homomorphism
Defines Jordan subalgebra
Defines Jordan algebra isomorphism